Ind. Eng. Chem. Res. 1992,31,2783-2794 McGraw-Hik New York, 1984; Section 16, pp 5-10. Warshawsky, A. Extraction with Solvent-Impregnated Resins. In Zon Exchange and Solvent Extraction; Marinsky, J. A., Marcus, Y., Eds.; Marcel-Dekker: New York, 1981; Vol. 8, Chapter 3, pp 229-310. Yoshizuka, K.; Sakomoto, Y.; Baba, Y.; Inoue, K. Distribution
2783
Equilibria in the Adsorption of Cobalt(I1) and Nickel(I1) on Levextrel Win Containing Cyanex 272. Hydrometallurgy 1990,23, 3W318. Received for review June 8, 1992 Accepted September 9, 1992
Multiphase Equilibrium Analysis via a Generalized Equation of State for Associating Mixtures S. Jayaraman Suresht and J. Richard Elliott, Jr.* Chemical Engineering Department, University of Akron, Akron, Ohio 44325-3906
A detailed analysis of thermodynamic models for multiphase equilibria is presented with special
+
emphasis on hydrocarbon water systems. Evaluation of representing association sites in water by four-site, three-site, and two-site models by Wertheim's thermodynamic perturbation theory shows that any of the models may be accurately applied, but the four-site model is difficult to apply for generalized analysis of multiphase equilibria. The two-site model is recommended because ita accuracy is a t least equivalent to that of the three-site model or four-site model, but it is more convenient to apply in general. A predictive correlation is proposed for hydrocarbon water phase equilibria which offers improvements in accuracy and general applicability over the best available models. An analytical solution for the thermodynamics of mixtures containing two associating species is derived. The proposed model is shown to provide accurate representation of liquid-liquid, vapor-liquid, and vapodiquid-liquid equilibria of binary mixtures that include water, hydrocarbons, alcohols, ketones and H2S. A single binary interaction parameter was adjusted to provide the best fit of vapor-liquid and liquid-liquid equilibria for all binary mixtures.
+
Introduction Associating mixtures comprise a large number of hydrogen-bonding and complexing solutions ranging from aqueous amino acids to nylon blends. The thermodynamics and phase equilibria of these systems are of practical interest because they are central to many applications, and they are of theoretical interest because of the complex anomalies exhibited. One method to treat associating systems is to assume a potential model for association and develop expressions based on statistical mechanics. The underlying assumption in such approaches is that the contribution to the potential which gives rise to association is strong and short-ranged, and that longer-ranged contributions can be handled better by considering them separately. Assuming a well-defined potential model which is consistent with this perspective, it is possible to derive tractable expressions for the thermodynamic quantities of interest (Andersen, 1974; Wertheim, 1984a). While the validity of splitting the contributions to the potential in this way is still something of an open question, an accurate theory for the thermodynamics of associating mixtures would provide substantial impetus for considering it. Thus we consider here the level of accuracy that can be obtained by incorporating just the short-range association effect along with the usual attractive and repulsive dispersion terms, and neglecting specific electrostatic terms. As one might expect, the expressions resulting from the statistical mechanical approach are very similar to expressions that can be derived by assuming chemical equilibria between the various oligomers (cf. Heidemann and Prausnitz, 1976; Economou and Donohue, 1991). The key difference between the two approaches is that the statistical mechanical approach encompasses many com'Present address: Department of Chemical and Petroleum Engineering, University of Pittsburgh, Pittsburgh, PA 15260.
plex issues like branched or cyclic networks in a way that is very general and self-consistent. The chemical perspective, on the other hand, requires explicit specification of many inaccessible quantities like the change in heat capacity due to association and the entropy change of branching, but it can be more straightforwardly tailored to some specific anomalous systems like hydrofluoric acid solutions. In our previous work on associating mixtures (Elliott et al., 1990; Suresh and Elliott, 1991),we focused primarily on the chemical perspective and highlighted many of the similarities between the two approaches. In related work, Chapman et d (1990) have shown that the thermodynamic perturbation theory (TPT) developed by Wertheim (1984a,b, 19-b) combined with attractive dispersion and repulsive interactions can accurately represent many aspeds of the complex phase behavior in associating systems. Furthermore, Nezbeda and Kolafa (1990) demonstrated the accuracy of TPT relative to association in simulated water and methanol, and Nezbeda and Iglesias-Silva (1990) demonstrated accurate representation of some of the anomalous behavior for the water molecule using TPT. Considering the importance of water in many systems of practical interest and its clear tendency to form highly branched networks, we have undertaken this study of applying the various conceivable TPT models of water within the framework of a generalized cubic equation of State. In the discussion below, we consider three potential models of water which can be treated by TPT. These models differ according to the number of possible association sites allowed per water molecule: two, three, or four. Throughout our analysis, we compare the theories for accuracy in treating multiphase equilibria because we find that representation of simple vapor-liquid equilibrium is relatively insensitive to detailed assumptions of the models. We show that all three models provide accurate representations of phase equilibria if only a single system
0 1992 American Chemical Society O~SS-5SS5/92/2631-27S3~03.00/0
2784 Ind. Eng. Chem. Res., Vol. 31, No. 12, 1992
is considered, but the two-site and three-site models are more accurate when applied in general to a wide range of mixture types. We then consider generalized application of the TPT for alcohols, ketones, and H2S and find that an analytical solution can be derived for mixtures of two associating species, which greatly facilitates practical applications for such mixtures. In concluding, we present arguments which support a recommendation of the twosite TPT model for generalized engineering applications and suggest that this model may provide greater accuracy and adaptability than any available models like the Kabadi and Danner (1985) model for hydrocarbon water systems.
+
Theory We perform this evaluation using the equation of state developed by Elliott, Suresh, and Donohue (1990),referred to as the ESD-EOS in the rest of this paper. The ESDEOS is designed to incorporate findings from molecular simulations of associating molecules as well as experimental data for anomalous systems. The emphasis of our effort has been to derive an equation of state which is as easy and general to apply as conventional equations of state like the Soave (1972) equation, while including the key insights afforded by the fundamental analysis. In our analysis of vaporliquid equilibrium (VLE), we found that the association model was always as accurate as the Soave equation or more accurate, with substantial enhancements in accuracy for solutions containing a nonassociating diluent mixed with associating molecules (Elliott et al., 1990, Suresh and Elliott, 1991). These previous analyses of associating systems were performed using the chemical theory formulated by Heidemann and Prausnitz (1976). The chemical theory provides exact expressions for mixtures containing one associating species and any number of diluents. However, the extension of the chemical theory to treat mixtures comprised of more than one associating species is not straightforward (Suresh and Elliott, 1991). Approximate expressions provided by Ikonomou and Donohue (1988) and Anderko (1989) have been applied, and both of these approximations have been shown to provide reasonable accuracy in representing the VLE of multiple associating species. Occasional problems of inaccuracies and thermodynamic inconsistencies do arise, however (Economou and Donohue, 1992), and extensions to multiphase equilibria tend to amplify the effects of inaccuracies. For mixtures involving associating species, an important assumption in the chemical theory is regarding the value of the change in specific heat on association (ACp). VLE calculations are generally quite insensitive to the value of ACp For example, Ikonomou and Donohue (1988) assume that there is no change in the specific heat on association while Elliott et al. (1990) assigned a value of ACp = -R. In each of the above casea, it was shown that accurate VLE correlations could be obtained. In this paper, we show that liquid-liquid equilibrium (LLE) representation, especially of water + hydrocarbon systems, is quite sensitive to the value of ACp. In our analysis, alcohols and ketones are treated as molecules with two sites (one positive and one negative). Even though the two-site model may not be very appealing at all conditions when compared with NMR studies (Karachewski et al., 19891, this kind of a description is found to be adequate for the representation of phase equilibria (Suresh and Elliott, 1991). As for water, on the other hand, it has long been known that the molecule possesses multiple sites. The most striking evidence of this is the tetrahedral coordination in solid. Recent calculations by
Buckingham and Fowler (1988) indicate that the two negative sites may in effect serve as a single site, thus leading to a picture of the water molecule effectively possessing three sites in the fluid phase. Wei et al. (1991) showed that only three out of four possible sites in water molecules are hydrogen-bonded at any given instant (two negative sites and a positive site or one negative site and two positive sites). To elucidate all of the trade-offs, we have evaluated the two-site, thresite, and four-sitemodels independently. In addition to alcohols, ketones, and water, a compound of substantial interest in the petrochemical industry is H a . Recent spectroscopic results (Sennikov et al., 1990) have shown that H2S self-associates with a hydrogen bond enthalpy of -1.4 kcal/mol and cross-associates with water with a hydrogen bond enthalpy of -3.1 kcal/mol. While these associations are weak, we also find some evidence for H2S self-association in the curvature of the vapor pressure at low temperatures. Hence, our association model taka into account both self-associationof H a and cross-association between H2S and water. This analysis was performed with the intent of providing a direct comparison between our association model and the Soave equation while maintaining an equal level of complexity in terms of the number of adjustable parameters. Hence we invoke a single adjustable binary interaction parameter (kij) at all conditions. By maintaining this basis for comparison, we obtain a clear indication of the merita of explicitly recognizing association in the equation of state. Pure Associating Species. A brief comparison between TPT and the chemical theory is provided here to highlight the role of ACp in the derivations. For a detailed comparison, readers are referred to the work of Ekonomou and Donohue (1991). TPT provides a simple expression for the excess Helmholtz energy due to association (Chapman et al., 1988) as follows:
where M is the number of association sites on each molecule, XAis the mole fraction of sites not bonded at site A, and the summation is over all the association sites on a molecule. These terms are then added to the usual terms of the equation of state for the repulsive and attractive contributions
where and ZOatt are evaluated on the basis of the properties of the monomer (as if no association had occurred). An expression for ZB"" may be derived from TPT by applying the ESD-EOS terms for ZorePand 2,". The key term is ZOrePfrom which the value of the pair correlation function at contact, g'"), may be estimated. Following Boublik (19741, g'") can be estimated from Zorepby (4)
and are related to g(")through XAin eq 1. XA is given by the following equation:
2""
Ind. Eng. Chem. Res., Vol. 31, No. 12, 1992 2786
where Applying eq 6 and lumping the parameter c with K m since they are both molecular parameters, one obtains (7)
where u* is the volume parameter, 7 is the reduced density, CHB is the hydrogen-bonding square-well depth, and Km is the measure of bonding volume available to the molecule. It should be noted that the precise shape dependency of eq 4 is a subject of current research, but the nature of the lumping clearly shows how this issue m a y be approximated at present. For a two-site model, the extent of association is given as follows (Jackson et al., 1988):
and
where N T is the true number of moles and Nois the superficial number in the case of the two-site model. To see how TPT relates to the chemical theory and the equation of state presented by Elliott et al. (1990), we must recall the result from the chemical theory,
It should be noted that the exact equivalence through eq 17 is a fortunate result of the simple form for Zom adopted by the ESD-EOS. Using the Camahan-Starling equation for ZomP would give a complex relationship for the quantity referred to as eg by Economou and Donohue (1991). Phase equilibria in all the nonaqueous mixtures that were studied were found to be insensitive to the value of AC,. For example, Elliott et al. (1990) chose a value of ACp = -R and were able to accurately represent VLE in nonaqueous systems. However, the estimation of phase equilibria in aqueous systems, especially LLE, is quite sensitive to the value of ACp. Turning to the four-site model of pure water, the TPT provides exact expressions for this model. However, it is not yet clear as to how the chemical theory can address a four-site model. The additional problem of chemical theory for a multisite model is the extra complexity of accounting for branching. While there should be some equivalent chemical relations that account for the branching, it is not clear what they should be or how any alternatives might be more justifiible than the TPT result. Applying TFT, then, eqs 1-8 with the four-site model yield 2 XA = (18) 1 + (1 + 8poAm)1/2
For the threesite model of pure water, the expressions lead to the following form. 2 (20) NT/No= 1 POAAB (1 + 6poAm+ ( ~ o A m ) ~ ) ' / ~
+
XA = XB = (Note: The last term in eq 11 was mistakenly omitted from the previous papers, but was included in all calculations.) Comparing to eq 3,
For the extent of association, the chemical theory provides
where a =
70
EK*
1 - 1.970
+
2 1 - POAAB+ (1 + 6poAm
+ (POAAB)~)''~ (21)
xc =
2 1 + p0A.m
(22)
+ (1 + 6poAm + ( P O A ~ ) ~ ) " '
where superscript C corresponds to the proton-acceptor site and A and B refer to the proton-donor sites. All of these representations of the thermodynamica of association can be applied with equal computational effort when considering the pure species. It should be noted that NT/No for the four-site and three-site models derived in this way (from the Helmholtz free energy) no longer has the physical significance of reciprocal mean cluster size. Mixtures. For mixtures, TPT gives the following expressions for
(14)
K* = 'K TcT,AC,/R+l
where
u*
(16)
where AH is hydrogen-bonding enthalpy, K , is the equilibrium constant at the critical point, and AC, is the change in specific heat on association. It is clear that expressions 7-9 are familiar in form to expression 13-16. By relating eqs 7 and 14, it can be seen that TPT assigns
The summation on j is over all molecules and the summation on Bj is over all sites in molecule j . In mixtures containing one associating species and any number of diluents, TPT and the chemical theory give similar expression (Economou and Donohue, 1991). In mixtures containing two associating species and any number of
2786 Ind. Eng. Chem. Res., Vol. 31, No. 12, 1992
diluents, the cross-associating terms, AAaj and AB~A~, need to be evaluated. Here, we can make an important assumption: AAiq = A B ~ A = ~Ai;* = (AA~B~AA,B)'/~ (25)
Where Ai;* is value of A for the molecular pair ij. To clarify the nature of the above assumption, consider hydrogen bonding between water and H2S. Equation 25 implies that the cross-association parameter is the same for S-H.-0 bonds and for 0-H-.S bonds between H2S and water. In other words, AMi is the average cro88-&98ociBtion parameter between a water (i) and a H2S 0') molecule [=Aij*]. With this assumption, eq 25 gives the following cubic equation for the two-site two-site model which can be solved analytically: (XiA)3[po2Ai;*Aii*~i- po2Aii*2~i] (XiA)2[poAi;* - poAii* po2Aii*Aij*] + (XiA)[poAii* - 2poAi;*] poAij* = 0 (26)
+
+
+
Quation 26 generates three roota for X t . One of the roots was found to never occur in all the systems studied. The remaining two roots are
XiA= 2(-Q)l/' cos(cp/3) - a1/3 XiA= 2(-Q)1/2 cos(cp/3 + 240') - a1/3 where cp =
S=
COS-'(S/(-Q~)'/~) gala2 - 27a3 - 2az 54
The appropriate root is the one occurring between 0 and 1, and only one root is valid for any given binary system. Without the assumption of eq 25, it is still possible to solve for the equilibrium properties, but the equation's solution is not analytical. It is possible to consider developing an analytical solution for the mixture of a three-site with a two-site model, but the assumption of eq 25 makes this endeavor of questionable value. The energies associated with the *+- sites of the three-site models will be very different from those of the two-site model, because there are more of them. This means that equating S-H-0 to 0-He-S bonds as in eq 25 is highly questionable. Considering the importance of obtaining analytical solutions for engineering applications, we are led to a strong engineering motivation for general application of the two-site model, even though the chemical and physical justification of such a model may be subject to debate. Procedure To Evaluate Phase Equilibria. The LLE calculations were performed using a flash routine. Experimental values for the two liquid-phase compositions were used to initiate the guess for the K values. Liquid roots and fugacity coefficients were then generated for the respective liquid phases and the iterations proceeded.
Table I. Values of cm,/k (K), v* (cms/molecule), c , tm/RT,, Km/v*, and the Number of Associating Sites for some Asrociating ComDounds no. of component sites aD,,/k u*(lOZ3) c rm/RT, KABIV' methanol 326.06 3.382 1.1202 5.17 0.0226 269.72 3.909 1.5655 4.86 ethanol 0.0283 1-propanol 242.51 4.172 2.7681 2.50 0.1Ooo 2-propanol 236.54 4.600 2.3148 3.75 0.0500 phenol 354.33 4.981 2.0972 2.14 0.1220 water 427.25 1.563 1.0053 4.00 0.1Ooo water 442.17 1.720 1.0401 3.35 0.0500 333.84 1.939 1.0416 2.00 0.0442 H2S acetone 247.70 5.027 2.1001 0.51 0.1Ooo
The three-phase calculations were performed as follows: 1. Experimental values of pressure and the liquid-phase compositions were used to initiate the guess for the K values. 2. LLE calculation was performed at a particular temperature based on the above guesses. 3. Bubble point pressure calculation was performed with one of the liquid-phase compositions (from step 2). 4. If the pressure was equal to the guessed pressure, the iteration was stopped. If not, another value of pressure was guessed and steps 1-4 were repeated.
Results and Discussion In previous calculations based on VLE (Elliott et al., 1990), it was found that providing explicit representation of the hydrogen-bonding effect was overridingly important and that the accuracy of the model was relatively insensitive to the other specifications such as c, tDisp/k, or u*. Hence, to develop a consistent method to obtam association parameters, we used the fact that the 2, of an associating compound is depressed relative to ita nonassociating homomorph. However, the situation is quite different in the case of LLE. The size parameter, u*, has a large impact on the entropy of mixing, which in turn plays a major role in liquid-liquid immiscibilities. Therefore, representation of LLE and VLE by the same thermodynamic model requires careful attention to both the strength of hydrogen bonding and the size parameters. In this context we have found it necessary to develop new values for the parameters that represent the hydrogen-bonding effect of associating components (Table I). The revised values of ZCm do not match any prespecified conditions, however, and this tends to have a detrimental effect on the accuracy of density estimates. We assigned a higher value to being able to predict the VLE and LLE simultaneously for these complex mixtures because density estimates from simple equations of state tend to be inaccurate in general and methods of correcting the density estimates are available (Mathias et al., 1989). To determine values of pure component parameters which permit simultaneous representation of LLE and VLE, we found it necessary to consider the LLE of one binary mixture for each determination. Thus, for example, we determined pure component parameters of water which provided the best fit of water + benzene LLE data while matching T,,P,, and w and maintaining the error in vapor pressure of water to less than 3 % . The focus of this determination was basically to adjust u* such that the accuracy of the LLE could be improved. These same values of pure component parameters (Table I) for water were then used in further calculations with paraffins, naphthenes, aromatica, alcohols, and HzS. Similar determinations were made for methanol by studying the LLE of methanol
Ind. Eng. Chem. Res., Vol. 31, No. 12, 1992 2787 C
-1
-1
9
-3
-4
-5
-6 350
0.45
0.55
0.75
0.65
0.85
do0
500
450
T(N
0.95
Tr
Figure 1. Coefficient of compressibility of water vs reduced temperature for four-site model. (-) PR = 0.503. (---) PR = 2.52. (--) P R = 10.1.
+
cyclohexane and for ethanol by studying ethanol + hexadecane. By adjusting u* in this way, it could be argued that a second parameter has been introduced for LLE, but application of the same value of u* to LLE and VLE of many systems should indicate whether u* has been "tuned" to a specific data set. The number of adjustable parameters for both the association model and the Soave equation was kept to a minimum to improve the predictive capabilities. Hence, a single binary interaction parameter ( I z i j ) was adjusted to provide the best fit for LLE and VLE for all binary systems. Multiphase Equilibria of Aqueous Systems. Water has long been recognized as a complex molecule with many anomalies related to its small size, strong hydrogen bonding, and multipole moments. Even the relatively simple models of water considered here require some elaboration to understand the trade-offs between the various alternatives. The discussion below shows that the four-site model has the advantage relative to some anomalies of pure water but the three-site or the two-site model represents LLE behavior in water + hydrocarbon systems best. One of anomalies that water exhibits is that the coefficient of compressibility (29)
first decreases at low prwures, paases through a minimum, and then increases with increasing temperature. Nezbeda and Iglesias-Silva (1990) showed that a four-site TPT model with no disperse attraction term was capable of qualitatively reproducing this anomaly. We looked for this anomaly using the two-site, the three-site, and the four-site models. It was found that the two-site and the threesite models do not exhibit the anomaly. The four-site model without the dispersion term is able to exhibit the anomaly similar to the findings of Nezbeda and Iglesias-Silva (1990), but the inclusion of the disperse attraction term, which leads to more accurate representation of the vast majority of water's properties, tends to decrease the four-site association effect and the anomaly disappears. Figure 1 shows a plot of the coefficient of compressibility for the equation of state accounting for the repulsive and association effects (represented by the four-site model). The
Figure 2. Three-phase diagram for water (1) + hexane (2). (0) Experimental (Tsonoupoulos and Wilson, 1983). (- -) ESD using AC, = -1. (-) ESD using ACp from eq 17. The lower curve corresponds to the mole fraction of hexane in the water-rich phase. The upper curve corresponds to the mole fraction of water in the hexane-rich phase.
-
anomaly is quite distinct at low pressures and weakens at higher pressures. Another anomaly is that the density of pure water increases upon melting at 1atm and continues to rise up to a temperature of 4 OC. This anomalous behavior in density has not been observed for the two-site, the three-site, or the four-site model. On the basis of this analysis, it can be expected that the four-site model is most appropriate for representing the density of pure water, but the extent of the advantage is small since accurate representation of the compressibility anomaly necessitates reduced accuracy for important properties like the vapor pressure. It would appear that reproducing details of such anomalous behavior would probably require including the polar momenta and that the coupling between the preferred long-range orientations and short-range associations would likely be capable of the kind of redundant enhancement that would reproduce the anomalies. However, the practical benefit of accurately representing the anomalies would need to be balanced against the cost of the added complexities in the model. We now demonstrate the effect of the value of AC in representing threephase equilibria in water + hydrocarkn systems. We chose the water + hexane system (Figure 2) to demonstrate this effect. The experimental data for this system were taken from Tsonopoulos and Wilson (1983). The curves show that the solubility trends with respect to temperature are inaccurate for AC, = -R when compared to the value of AC, obtained from eq. 17. It should be noted that the logarithmic scale in Figure 2 tends to mask the magnitude of the difference in errors for these two models. Hence, in the further comparisons below, the association model using the revised value of AC, will only be analyzed and compared with the Soave equation and its modified forms. In all previous studies, we had compared our results with the Soave equation. However, for waterhydrocarbon systems, the Soave equation modified by Kabadi and Danner (1985) provides one of the best available representations. Hence, we have compared the association model with the Kabadi-Danner equations for the water + hydrocarbon systems. Figure 3 shows a three-phase diagram for the system water + ethylcyclohexane. The experimental data for this system were taken from GPA Research Report-62 (Brady et al., 1982). It is clear from
2788 Ind. Eng. Chem. Res., Vol. 31, No. 12, 1992 Table 11. Mole Fractions of Benzene in the Water-Rich Phase and Water in the Benzene-Rich Phase Compared for the Two-Site Model (Eq 30), the Three-Site Model (Eq31), and the Kabadi-Danner Equation with Experimental Data (Tsonopoulos and Wilson, 1983)" % error in composition Xbnzene(water-rich phase) - ~T (K) exp 2-site 3-site KD 2-site 3-site KD 313.15 0.4433-3* 0.3603-3 0.2635-3 0.5073-3 18.8 40.5 14.3 _. 373.15 0.0953-2 0.1093-2 0.0963-2 0.0383-2 14.8 0.7 59.7 0.2763-2 0.0 5.9 13.9 0.2433-2 0.2293-2 0.2423-2 423.15 0.511E-2 0.5093-2 0.4983-2 5.4 5.7 7.8 0.5403-2 473.15 % error in composition Xwster(benzene-rich phase)
T (K) 313.15 373.15 423.15 473.15
exP 0.5013-2 0.2083-1 0.7133-1 1.8503-1
2-site 0.2503-2 0.1913-1 0.691E-1 1.971E-1
KD
3-site 0.2733-2 0.2003-1 0.7204-1 2.0483-1
2-site 50.0 8.2 3.1 6.5
0.4243-2 0.2543-1 0.8933-1 2.8483-1
3-site 45.4 3.8 0.9 10.7
KD 15.4 22.1 25.2 53.9
?& error in composition = I[X - X(exp)]/X(exp)l X 100. *0.443E-2 represents 0.433 X lo-*, etc.
two-site and the three-site models are equally competitive in phase equilibria correlations in systems containing water. Since one of our main goals is to develop a simple equation of state that can be generally applied to associating mixtures, we have analyzed only the two-site and the three-site models in the rest of this paper. From our analysis of water + hydrocarbon systems, we were able to develop predictive correlations for the binary interaction parameter in t e r n of temperature (kelvin) and the difference in the values of tDisp/u* between hydrocarbon and water for the two-site model as follows. k,,(water-alkanes) =
kJwater-aromatics) = -0.0523 350
500
Figure 3. Three-phase diagram for water (1) + ethylcyclohexane Experimental (Brady et al., 1982). (-) ESD using two-site (2). (0) model. (---) ESD using three-site model. (--) ESD using four-site model. The lower curve corresponds to the mole fraction of ethylcyclohexane in the water-rich phase. The upper curve corresponds to the mole fraction of water in the ethylcyclohexane-rich phase.
Figure 3 that the two-site and the three-site models are quite accurate while the four-site model overestimates the composition of water in the ethylcyclohexane-rich phase. Similar results were obtained for other water + alkane and water + naphthene systems. This result is contrary to our expectation that the four-site model would be most appropriate for the water molecule. It is possible to obtain accurate results for the four-site model by adjusting the u* parameter of hydrocarbons. However, since we insist that our equation of state be generalized with respect to all hydrocarbons for convenient engineering applications, the simultaneous representation of the anomalous behavior of pure water and phase equilibrium data for water + hydrocarbon systems does not seem possible with the four-site model of hydrogen bonding where cross-association between water and hydrocarbon is neglected. It is possible to build cross-association into the model, and recent spectroscopic results suggest significant association between benzene and water (Suzuki et al., 1992). But the cross-association coefficient would introduce an additional adjustable parameter and preclude an analytical solution, both of which we were trying to avoid at this juncture. Overall, the four-site model seems physically reasonable but it does not provide the accurate, general, and efficient representation of phase equilibria which we sought. Hence, it appears that the four-site model in its present form cannot be applied through our equation of state. The
'
0.406 x
p i ( ; )
-
water
I+(*:
-
0.3482 - 0.427 X
)water
(+) I + aromatic
kij(water-naphthenes) = 0.09103
-
(s
)alkanj
0.16 x 10-377
+ 0.2535 X
(30)
For the three-site model, we were able to develop correlations for the binary interaction parameter as follows. kii(water-alkanes) =
kij(water-aromatics) = 0.00531
);
3.960 x 10-41(
water
-
+
(s) I +
2.60 x 10-577
aromatic
ki;(water-naphthenes) =
where t/u* has the units of (J/cm3). The temperature dependencies in the above correlations represent a small penalty since the same dependency is applied to each family of compounds. It should be noted that the Kabadi-Danner relation also uses temperature-dependent parameters and further assumes different values of the parameters depending on the phase. Tables II-VI11 show the percent deviations in composition for various water + hydrocarbon systems for the Kabadi-Danner equation and the associating model using the above correlations for kip Table M shows the kij values for various mixtures. Figure 4 shows a three-phase diagram for the water + benzene system as represented by the various models. Figure 5 shows the phase diagram for the water + propane system. It should be noted that the k , values for this system for
Ind. Eng. Chem. Res., Vol. 31, No. 12, 1992 2789 Table 111. Mole Fractions of Hexane in the Water-Rich Phase and Water in the Hexane-Rich Phase Compared for the Two-Site Model (Eq 30), the Three-Site Model (Eq 31), and the Kabadi-Danner Equation with Experimental Data (TsonoDoulos and Wilson. 1983)" Xhelane(water-rich phase) % error in composition T (K) exu 2-site 3-site KD 2-site 3-site KD 0.6983-5 0.7903-5 373.15 0.620E-5b 1.0283-5 12.5 27.4 65.3 0.3773-4 0.3533-4 423.15 0.3393-4 0.2423-4 11.1 4.1 28.4 0.1603-3 0.1343-3 473.15 0.1853-3 0.5353-4 13.5 27.6 71.1 Xwatar(hexane-rich phase) % error in composition T (K) exP 2-site 3-site KD 2-site 3-site KD 1.0803-2 1.3203-2 0.9193-2 52.3 86.2 29.6 373.15 0.7093-2 0.5063-1 0.5593-1 423.15 0.3113-1 0.4403-1 62.6 79.7 41.5 1.8923-1 1.9603-1 473.15 1.1103-1 2.2123-1 70.4 76.1 99.3 ~~
% error in composition = ([X - X(exp)]/X(exp)JX 100. '0.6203-5 represents 0.620 X
etc.
Table IV. Mole Fractions of Ethylcyclohexane (Etch) in the Water-Rich Phase and Water in the Ethylcyclohexane-Rich Phase Compared for the Two-Site Model (Eq 30), the Three-Site Model (Eq 31), and the Kabadi-Danner Equation with Exwrimental Data (Brady et al.. 1982)" XEtch(water-rich phase) % error in composition T (K) exP 2-site 3-site KD 2-site 3-site KD 0.2303-5 0.2403-5' 0.2793-5 0.0763-5 16.2 4.1 68.2 367.59 0.1643-4 0.1903-4 0.0443-4 18.0 5.0 78.0 423.43 0.2003-4 0.0833-3 0.1193-3 479.48 0.12 13-3 0.020E-3 31.4 1.6 83.4 X,,, (Etch-rich phase) % error in composition T (K) exP 2-site 3-site KD 2-site 3-site KD' 0.1213-1 0.1363-1 0.0923-1 86.1 109.2 41.5 367.59 0.0653-1 0.5993-1 0.3803-1 0.5183-1 0.4403-1 36.3 57.6 15.8 423.43 1.6543-1 1.9473-1 1.9853-1 54.5 81.9 85.5 479.48 1.0703-1
" % error in composition = I[X - X(exp)]/X(exp)( X
100. '0.2403-5 represents 0.240 X LO-5, etc.
Table V. Mole Fractions of Ethylbenzene in the Water-Rich Phase and Water in the Ethylbenzene-Rich Phase Compared for the Two-Site Model (Eq 30), the Three-Site Model (Eq 31), and the Kabadi-Danner Equation with Experimental Data (Brady et al.. 1982)" % error in composition XEtbn(water-rich phase) T (K) exu 2-site 3-site KD 2-site 3-site KD 0.6483-4 1.6523-4 6.9 24.6 92.1 0.8003-4 367.59 0.860E-4b 0.2703-3 0.3813-3 35.2 23.3 73.9 0.2193-3 0.2963-3 423.43 0.7923-3 63.2 64.2 32.0 0.9793-3 0.9853-3 479.48 0.6003-3 0.3133-2 0.3473-2 0.1483-2 3.7 6.8 54.5 536.09 0.3253-2 XWater (Etbn-rich phase) % error in composition T (K) exP 2-site 3-site KD 2-site 3-site KD 0.1743-1 0.2173-1 13.4 6.8 0.1863-1 0.161E-1 16.7 367.59 0.716E-1 0.9353-1 0.5963-1 0.6733-1 12.9 20.1 56.9 423.43 2.2003-1 3.6613-1 1.6303-1 2.0553-1 26.1 34.9 124.6 479.48 5.4203-1 6.5483-1 4.0803-1 5.1933-1 27.3 32.8 60.5 536.09 ~
a
% error in composition = I[X - X(exp)]/X(exp) X 100. '0.8603-4 represents 0.860 X
etc.
Table VI. Mole Fractions of Xylene in the Water-Rich Phase and Water in the Xylene-Rich Phase Compared for the Two-Site Model (Eq 30), the Three-Site Model (Eq 31), and the Kabadi-Danner Equation with Experimental Data (Anderson and Prausnitz, 1986)" Xxylene (water-rich phase) % error in composition T (K) exP 2-site 3-site KD 2-site 3-site KD 0.1273-3 0.0933-3 27.9 37.7 54.5 0.204E-3b 0.1473-3 398.15 0.1563-3 12.7 20.6 47.6 0.2603-3 0.2373-3 423.15 0.2983-3 0.2543-3 12.4 15.1 50.8 0.4533-3 0.4393-3 448.15 0.517E-3 0.3953-3 19.6 20.5 59.0 0.7753-3 0.7663-3 473.15 0.9643-3 % error in composition Xaratcr(xylene-rich phase) T (K) exp 2-site 3-site KD 2-site 3-site KD 0.481E-1 21.2 39.4 73.0 0.3373-1 0.3883-1 398.15 0.2783-1 18.3 37.4 75.2 0.705E-1 0.8993-1 0.6063-1 423.15 0.513E-1 17.1 39.0 89.2 1.2293-1 1.6733-1 0.8843-1 1.0353-1 448.15 2.991E-1 10.8 34.2 98.1 1.6733-1 2.0263-1 473.15 1.5103-1 % error in composition = ([X - X(exp)]/X(exp)X 100. *0.204E-3 represents 0.204 X
the two-site and the three-site models were predicted using eqs 30 and 31. In other words, the water + propane system was not included in the regression of parameters for eqs
etc.
30 and 31. The two-site model is more accurate than the Kabadi-Danner equation but is only shghtly more accurate than the three-site model. Figure 6 shows a three-phase
2790 Ind. Eng. Chem. Res., Vol. 31, No. 12, 1992 Table VII. Mole Fractions of Octane in the Water-Rich Phase and Water in the Octane-Rich Phase Compared for the Two-Site Model (Eq30), the Three-Site Model (Eq 31), and the Kabadi-Danner Equation with Experimental Data (Brady et al.. 1982)" Xoetane(water-rich phase) % error in composition T (K) exo 2-site 3-site KD 2-site 3-site KD 0.39634 0.550E-6 0.178E-6 20.7 10.0 64.4 0.500E-6* 366.48 0.4883-5 0.3803-5 0.4733-5 0.080E-5 24.5 28.4 78.9 422.04 0.3403-4 0.400E-4 0.3913-4 0.0323-4 2.2 15.0 92.0 477.59 X,,,, (octane-rich phase) % error in composition T (K) e=P 2-site 3-site KD 2-site 3-site KD 0.6203-2 0.9423-2 1.210E-2 0.7153-2 51.9 95.2 15.3 366.48 0.4983-1 0.5693-1 0.3923-1 26.4 44.4 0.5 422.04 0.3943-1 1.8433-1 1.9633-1 477.59 1.260E-1 2.006E-1 46.3 55.8 59.2 ~~~
a
~
% error in composition = I[X - X(exp)]/X(exp)l X 100. *0.500E-6 represents 0.500 X lo*, etc.
Table VIII. Summary of Percent Deviations for the Two-Site Model (Eq30), the Three-Site Model (Eq 31), and the Water Kabadi-Danner Equation in Representing the Phase Equilibria (LLE or Three Phase) for Various Hydrocarbon Systemsa r a hyd ter-rich Xkb-p %AADP 2-site 3-site KD 2-site 3-site KD 2-site 3-site KD propane 20.5 9.7 59.6 28.1 38.8 29.8 LLE LLE LLE hexane 12.4 19.7 54.9 61.7 80.7 56.8 1.5 1.8 3.5 octane 15.8 17.8 78.4 41.5 65.1 24.8 2.2 1.8 2.8 ethylcyclohexane 21.9 3.6 76.5 59.0 82.9 54.7 3.8 1.9 1.6 benzene 9.7 13.2 23.9 16.9 15.2 29.1 3.9 5.2 1.6 xylene 18.1 23.5 52.9 16.8 37.5 83.9 LLE LLE LLE ethylbenzene 27.2 29.7 63.1 19.9 23.6 64.7 1.6 2.2 1.2 H2S 13.9 12.7 3.1
+
average 17.4 16.7 58.5 32.1 49.1 49.1 2.7 2.6 2.1 = % deviation for the hydrocarbon in the water-rich phase. Xkbk:'"h = % deviation for the water in the hydrocarbon-rich phase. %AADP = % deviation for the pressure in three-phase calculations. D
rater-rich hyd
Table IX. Values of ki, for Binary Systems by Two-Site Model and Not Covered by Eq 30 mixture association model" Soave" methanol + 0.033 0.127 cyclohexane methanol + 0.047 0.112 hexadecane ethanol + 0.021 -0.018 hexadecane phenol pentane 0.096 - 0.300 X 10-3T -0.081 + 0.226 X 10-3T 0.212 - 1.500 x 1 0 - 4 ~-0.01 H2S + water
+
"I -1'
8
-2.
ONote that T is in kelvin.
diagram for the water + propane system at various temperatures. Figure 7 shows the phase diagram for the water + ethylbenzene system. Again, the two-site and the three-site models show a definite improvement over the Kabadi-Danner equation. The explanation for the accurate representation by the two-site model probably centers on a cancellation of errors. The two-site model tends to underestimate the self-association of water relative to the three-site and four-site models, but a larger value for the disperse attraction energy results in virtually the same trends for water's fugacity. This trade-off is similar to that of setting the disperse attraction energy to zero to reproduce the compressibility anomaly via the four-site model. The optimal resolution of this trade-off from an engineering perspective depends on improvements in predictive accuracy relative to penalties in computational efficiency. At this time, we find no improvement in accuracy for the more complex model, so we opt for computational efficiency. Nonaqueous Systems. Figure 8 shows a plot of ethanol + hexadecane. The experimental data for this system were taken from the compilation by Macedo and Rasmussen
-4
300
350
400
450
500
T(@
Figure 4. Three-phase diagram for water (1) + benzene (2). (0) Experimental (Tsonoupoulos and Wilson, 1983). (-1 ESD using two-site model. (- - -1 ESD using three-site model. (--) KabadiDanner equation. The lower c w e corresponds to the mole fraction of benzene in the water-rich phase. The upper curve corresponds to the mole fraction of water in the benzene-rich phase.
(1987). The ESD-EOS shows a clear improvement in the representation of LLE. Figure 9 shows a plot of VLE and LLE for the methanol + cyclohexane system. The experimental data for this system were taken from the compilation by Soerensen and Arlt (1979). I t is clear that the association model accurately represents VLE and LLE simultaneously for this system while the Soave equation shows large deviations. Figure 10 shows similar trends in the representation of LLE for methanol + hexadecane. Figure 11 shows LLE for phenol + pentane systems. For this system, a signif-
Ind. Eng. Chem. Res., Vol. 31, No. 12,1992 2791
0.00
0.W
0.04
'0.9994
0.12
0.9997
xd)oYd)
Figure 5. Phase diagram for water (1)+ propane (2)at 366.48 K. (0) Experimental data (Kobayashi and Katz, 1953). (-) ESD using twesite model. (--) ESD using three-site model. (-- -) Kabadi-Danner equation. C
-1
-21
-2
8 -3
-4 -4
-c
375
395
415
435
T(Q
Figure 6. Three-phase diagram for water (1)+ propane (2). (0) Experimental data (Kobayashi and Katz, 1953). (-) ESD using two-site model. (--) ESD using three-site model. (---) KabadiDanner equation.
icant improvement in correlation was possible through the application of a temperature-dependent kij parameter. Figure 11 shows a comparison of LLE representation for three models: (1)ESD with temperature-dependent kij; (2) ESD with temperature-independent kij; (3) Soave equation. It is clear that the ESD equation of state with temperature-independent kij correlates the LLE quite well at temperatures away from the critical point. However, if the liquid-liquid critical point needs to be matched as well, a temperature-dependent kij is required for this system. Figures 8-10 show that a temperature-independent k , is sufficiently accurate for those systems, even in the critical region. Table X shows the error in compositions for some nonaqueous systems. Mixtures Involving Two Associating Components. Now we extend our treatment to systems involving more than one associating species using eqs 23-27. Our initial consideration demonstrates the superiority of the TPT solution over the available approximations relevant to analysis of VLE. Further considerationsdemonstrates that the superiority also applies to multiphase equilibria.
350
450 T(Q
400
500
550
Figure 7. Three-phase diagram for water (1)+ ethylbenzene (2). (0) Experimental (Brady et al., 1982). (-) ESD using two-site model. (---) ESD using three-site model. (--) Kabadi-Danner equation. The lower curve corresponds to the mole fraction of ethylbenzene in the water-rich phase. The upper curve corresponds to the mole fraction of water in the ethylbenzene-rich phase. Table X. Percent Deviations for the ESD (Two Site) and the Soave Equations of State in Representing LLE for Various Alcohol Hydrocarbon Systems"
+
Xfig
Xtypol-rich
system ESD-EOS Soave ESD-EOS Soave methanol + cyclohexane 7.6 20.4 1.1 97.0 methanol + hexadecane 7.0 35.6 25.2 100.0 ethanol + hexadecane 7.9 18.9 5.8 99.0 phenol + pentane 1.6 6.2 3.4 183.3 average
6.0
20.3
8.9
120.0
h d-rich I % deviation for the alcohol in the hydrocarbon-rich XJwho~mhol-rich xhyd I% deviation for the hydrocarbon in the alcophase. II
hol-rich phase.
Figure 12 shows a VLE plot for the acetone + methanol system for the ESD-EOS.TPT and Anderko's mixing d e were evaluated for this system. While Anderko's mixing rules works well for most cross-associatingmixtures, it was found to perform poorly for the acetone + methanol sys-
2792 Ind. Eng. Chem. Res., Vol. 31, No. 12, 1992 335
33c
330
32C
325
E
310
320
300 315
310
290 0.00
0.25
0.50
0.75
1.oo
000
0.25
0.50
1.oo
0.75
X(1)
X(1)
Figure 8. Solubility plot for ethanol (1) + hexadecane (2). (0) Experimental (Soerensen and Arlt, 1979). (--) ESD. (--) Soave equation.
Figure 11. Solubility plot for phenol (1) + pentane (2). (0) Experimental (Soerensen and Arlt, 1979). (- - -) ESD with temperature-dependent kij. (-) ESD with temperature-independent &+ (--) Soave equation.
1
0.0721
II
i
280 0 00
0.25
0.50
0.75
1.oo
X(l),Y(V
Figure 9. LLE and VLE plot for methanol (1) + cyclohexane (2) at 760 mmHg. (0) Experimental (Soerensen and Arlt, 1979). (-) ESD. (--) Soave equation. 1^^
4 w .
370
E + 350
330 310
00
02
08
06
04
I/,
0
,’
XU)
0.00
0.25
0.50
0.75
1.00
X(1)
Figure 10. Solubility plot for methanol (1) + hexadecane (2). (0) Experimental (Soerensen and Arlt, 1979). (-) ESD. (--) Soave equation.
tem as shown in Figure 12. Economou and Donohue (1992) have discussed related shortcomings for Anderko’s mixing rule. The simple mixing rule of Ikonomou and Donohue (1988) is accurate for this system but is generally less
+
accurate than the TPT result, especially for the H2S water system. For example, in the H2S water system, the mixing rule proposed by Ikonomou and Donohue (1988)does not indicate a liquid-liquid phase split at 344.26 K while the experimental data clearly does. This deficiency is related to its failure to match the limit for weakly associating species. The ESD-EOS in conjunction with TPT provides a quite accurate representation of VLE and LLE. Since an analytical solution is available for the two-site + two-site model, there is a very small loss in computational efficiency. This computational efficiency represents a significant engineering advantage for the two-site model relative to the models of water presently available. The H2S water system is a mixture containing two associating compounds. For this analysis, water is represented by a two-site model. Figure 13 shows the phase diagram for H2S + water system at 344 and 444 K. The original Soeve equation of state is used for comparison with the ESD equation of state since the Kabadi-Danner equation is only applicable for water + hydrocarbon systems. It can be seen that the two-site model is able to accurately represent VLE and LLE simultaneously at 344 K while the LLE representation is poor for the Soave
+
290
I O
Figure 12. VLE for acetone + water at 45 O C . (0) Experimental (Marinichev and Susarev, 1965). (-) ESD using Wertheim’ssolution. (---) ESD using Anderko’s solution.
+
410
390
j
00401
Ind. Eng. Chem. Res., Vol. 31,No. 12, 1992 2793 3c
2c
E
E 1C
c 0.0
0.1
0.2
0.3
0.4
0.5
0.8
0.7
0.8
0.9
1.0
X(l)tY(l)
ExperiFigure 13. Phase diagram for H2S (1) + water (2). (0) mental data at 344.26 K (Selleck et al., 1952). (+) Experimentaldata at 444.26 K (Selleck et al., 1952). (-) ESD. (--) Soave Equation. 1.0
0.8
0.6
2 X
0.4
0.2
equation of state approach like our association model. 4. The two-site, three-site, and four-site models have been compared as models for the water molecule. The anomalous behavior in the coefficient of compressibility was found in the four-site model but not in the two-site and the three-site models. However, the accurate representation of phase equilibria in water + hydrocarbon systems could be well represented by the four-site model only if more adjustable parameters were introduced into the model. The two-site and the three-site models were found to accurately represent the phase equilibria in water + hydrocarbon systems. While aqueous systems still represent a significant challenge, a reasonably accurate and computationally efficient engineering model is obtained by assuming a two-site model for water. Future improvement will likely require considering the polar moments of water and cross-association between aromatics and water, and all three models of water should be reevaluated at that time. Since phase equilibrium calculation using a generalized equation of state was our primary emphasis, we recommend the two-site model for the present. 5. For water + hydrocarbon systems, a generalized correlation was obtained for the k,j parameter. Hence, phase equilibria in water + hydrocarbon systems can be predicted. A comparison with the Kabadi-Danner equation of state shows that the association model is more accurate. The accuracy in representing these systems is very encouraging because they have been the stumbling block for many previous theories. 6. The representation of phase equilibria for the H2S water system is a substantial improvement over the Soave equation. In summary, the success of the theory in representing multiphase equilibria as well aa VLE using an association model opens the door to a broad range of new and promising applications.
- ._._._._._._._._._._._._._._._._._.. Acknowledgment 0
0.0
340
350
360 T(K)
370
380
Figure 14. Three-phase diagram for H2S (1) + water (2). (0) Experimental (Selleck et al., 1952). (-) ESD. (---) Soave equation.
equation. Also, the vapor-phase composition of H,S is accurately represented by the ESD equation at 444 K while the Soave equation performs poorly. Figure 14 shows that the ESD equation is again superior to the Soave equation in representing the three-phase diagram for H2S + water a t various temperatures.
Conclusions In summary, we have presented a number of significant findings with regard to multiphaee equilibria in associating systems. 1. The accurate representation of LLE in aqueous mixtures was found to be sensitive to the value of ACT 2. Simultaneous representation of LLE and VLE by the same thermodynamic model is an extremely demanding undertaking for a simple equation of state. While we have used only one set of pure component association parameters, a comparable study by Magnussen et al. (1981) concluded that separate parameters should be used to represent LLE and VLE using UNIFAC. A similar study by Wenzel and Krop (1990)showed that their equation of state required two or more adjustable parameters to represent VLE and LLE simultaneously. 3. Methods to incorporate supercritical components like H2S add further complications to activity coefficient models like the UNIFAC while it is straightforward for an
S.J.S. would like to acknowledge the Financial support provided by BP America. This material is based on work supported in part by the National Science Foundation under Grant No. CTS 9110285. The Government has certain rights in this material. Nomenclature A = Helmholtz energy c = shape factor for the repulsive term AC, = specific heat change on association E = exp(H(1- l/TJ) FAB = exp(eHB/RT) - 1 g'"' = pair correlation function at contact H = AH/RTc - AC,/R AH = enthalpy on association k-value = molar distribution coefficient K* = K$T,/v* K , = equilibrium constant at the critical temperature KAB= measure of bonding volume kl = 1.7745 kz = 1.0617 k,, = binary interaction parameter M i= total number of sites in molecule i NAv = Avogadro number NT = true number of moles No = superficial number of moles = 1 + 1.90476(~- 1) R = gas constant T = temperature u* = characteristic size parameter = .n/6u3 XA= fraction of sites not bonded at A
2794 Ind. Eng. Chem. Res., Vol. 31, No. 12, 1992
xi = mole fraction of species i y = exp(fifDisp! :.k2
2 = compressibility factor Z M = 9.49 a = [ q / ( l - 1.9q)]EK* AAB = d ' ) F A B K A B Aij* = value of A for the molecular pair ij = potential energy well depth
= reduced density = p (u* ) 9 = fugacity coefficient p = molar density c = effective molecular diameter 7
Subscripts c = critical property Disp = dispersion t e r m HB = hydrogen bonding i = ith associating species 0 = quantity is independent of association r = reduced property T = quantity is calculated after accounting for association Superscripts assoc = association property att = disperse attraction contribution A = site A rep = disperse repulsion contribution Registry No. H2S, 7783-06-4; methanol, 67-56-1; ethanol, 64-17-5; 1-propanol, 71-23-8; 2-propanol, 67-63-0; phenol, 108952; water, 7732-18-5; acetone, 67-64-1.
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ented at the AIChE Winter National Meeting, Atlanta, GA, March 11-14,1984; paper no. 34b. Heidemann, R. A.; Prausnitz, J. M. A van der Waals-type equation of state for fluids with associating molecules. Proc. Natl. Acad. Sci. USA 1976, 73, 1773. Ikonomou, G. D.; Donohue, M. D. Extension of the Associating Perturbed Chain Theory to Mixtures with more than One Associating Component. Fluid Phase Equilib. 1988, 39, 129. Jackson, G.; Chapman, W. G.; Gubbins, K. E. Phase Equilibria of Associating Fluids: Spherical Molecules with Multiple Bonding Sites. Mol. Phys. 1988, 65, 1. Kabadi, V. N.; Danner, R. P. Modified Soave-Redlich-Kwong Equation of State for Water-Hydrocarbon Phase Equilibrium Calculations. Znd. Eng. Chem. Process Des. Dev. 1985,24, 537. Karachewski, A. M.; McNiel, M. M.; Eckert, C. A. A Study Of Hydrogen Bonding In Alcohol Solutions Using NMR Spectroscopy. Znd. Eng. Chem. Res. 1989,28, 315. Kobayashi, R.; Katz, D. L. Vapor-liquid Equilibria for Binary Mixtures of Hydrocarbon-Water Systems. Znd. Eng. Chem. 1953,45, 440. Macedo, E. A.; Rasmussen, P. Liquid-Liquid Equilibrium Data Collection; DECHEMA: Frankfurt/Main, 1987; Vol. V, Part 4. Magnuwen, T.; Rasmuesen, P.; Fredunslund, A. UNIFAC Parameter Table for Prediction of Liquid-Liquid Equilibria. Znd. Eng. Chem. Process Des. Dev. 1981,20, 331. Marinichev, A. N.; Susareve, M. P. (1965) In Vapor-Liquid Equilibria Data Collection. Gmehling, J., Onken, U., Eds.; DECHEMA: Frankfurt, Germany, 1977; Vol. 1/2a, p 79. Mathias, P. M.; Naheiri, T.; Oh, E. M. A Density Correction for the Peng-Robinson Equation of State. Fluid Phuse Equilib. 1989,47, 77. Nezbeda, I.; Iglesias-Silva, G. A. Primitive Model of Water. 111. Analytic Theoretical Results with Anomalies for the Thermodynamic Properties. Mol. Phys. 1990, 69, 767. Nezbeda, I.; Kolafa, I. Primitive model of water. 11. Czech. J . Phys. 1990, 40, 138. Selleck, F. T.; Carmichael, L. T.; Sage, B. H. Phase Behavior in the Hydrogen Sulfide-Water. Znd. Eng. Chem. 1952,44,2219-2226. Sennikov, P. G.; Raldugin, D. A.; Shkrunin, V. E.; Tokhadze, K. G. J . Mol. Struct. 1990, 219, 203. Soave, G. Equilibrium Constants from a Modified Redlich-Kwong Equation of State. Chem. Eng. Sci. 1972,27, 1197. Soerensen, J. M.; Arlt, W. Liquid-Liquid Equilibrium Data Collection; DECHEMA: Frankfurt/Main, 1979 Vol. V, Part 1. Suresh, S. J.; Elliott, J. R., Jr. Applications of a Generalized Equation of State for Associating Mixtures. Znd. Eng. Chem. Res. 1991, 30,524-532. Suzuki, S.; Green, P. G.; Bumgarner, R. E.; Dasgupta, S.; Goddard, W. A., 111; Blake, G. A. Benzene Forms Hydrogen Bonds with Water. Science 1992,257,942. Tsonopoulos, C.; Wilson, G. M. High-Temperature Mutual Solubilities of Hydrocarbon and Water. AIChE J . 1983,29,990-999. Wei, S.; Shi, Z.;Castleman, A. W., Jr. Mixed Cluster Ions as a Structure Probe: Experimental evidence for Clathrate Structure of (H20)&+ and (H20)21H+.J. Chem. Phys. 1991, 94, 3267. Wenzel, H.; Krop, E. Phase Equilibrium by Equation of State: A Short Cut Method Mowing for Association. Fluid Phase Equilib. 1990,59, 147-169. Wertheim, M. S. Fluids with Highly Directional Attractive Forces. I. Statistical Thermodynamics; J. Stat. Phys. 1984a, 35, 19. Wertheim, M. S. Fluids with Highly Directional Attractive Forces. II. ThermodynamicPerturbation Theory and Integral Equations. J . Stat. Phys. 1984b, 35, 35. Wertheim, M. S. Fluids with Highly Directional Attractive Forces. 111. Multiple Attraction Sites. J. Stat. Phys. 1986a, 42, 459. Wertheim, M. S. Fluids with Highly Directional Attractive Forces. IV. Equilibrium Polymerization. J. Stat. Phys. 198613, 42, 477. Receiued for review July 6, 1992 Accepted September 29, 1992
