Development and Evaluation of a More Precise Excess Gibbs Energy

School of Chemical Engineering and Specialty Separations Center, Georgia ... the empirical Gibbs energy of mixing expressions used to represent such d...
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Ind. Eng. Chem. Res. 1998, 37, 4512-4519

Development and Evaluation of a More Precise Excess Gibbs Energy Expression for Vapor-Liquid and Liquid-Liquid Phase Behavior Leon Scott E. I. du Pont de Nemours and Company, Wilmington, Delaware 19898-0302

David Bush, Noel H. Brantley, and Charles A. Eckert* School of Chemical Engineering and Specialty Separations Center, Georgia Institute of Technology, Atlanta, Georgia 30332-0100

The rational design of separations in multicomponent systems depends on the quality of vaporliquid equilibrium (VLE) data and the precision to which these data can be represented. In recent years, new experimental techniques have begun to yield better quality VLE data. Often these data warrant the use of more than two parameters per binary pair, which is common in the empirical Gibbs energy of mixing expressions used to represent such data. In this work, we propose two new expressions for representing the Gibbs free energy of mixing: the four parameter Scott-Wilson and the five parameter Scott-Regular Solution-Wilson. A method for evaluating such expressions is presented, including our proposed technique of predicting γ∞ in a mixed solvent from binary data only. For this, we measured γ∞ in a mixed solvent system using differential ebulliometry. Introduction The rational design of separations in multicomponent systems requires multicomponent phase equilibrium data, which are very expensive to obtain and of limited general use. Over the years, methods have been developed for representing such data with various empirical expressions for the Gibbs energy of mixing. Concomitantly, techniques have become available for taking accurate binary vapor-liquid equilibrium (VLE) data and scaling them to multicomponent systems, leading to less expensive data acquisition and more generality. Current methods now work well for VLE, but only marginally for liquid-liquid equilibria (LLE), which are much more sensitive to the quality of the Gibbs energy expression.1,2 The best currently available expressions, Wilson,3 NRTL,4 and UNIQUAC,5 predict multicomponent behavior using only parameters obtained from binary data, almost always with only two parameters per binary pair. In fact, one can often use limiting activity coefficients to get these parameters rapidly and accurately.6 Although the Wilson equation correlates VLE data better than any other two-parameter model,7 it cannot predict liquid phase separation, which is a severe limitation. Moreover, the other equations, which can predict LLE, cannot readily be used in conjunction with the Wilson equation for a multicomponent system. Although the current multicomponent VLE methods are adequate for most current designs, they do not handle well very close separations, especially in extractive distillation, nor are they good for enthalpy calculations. For LLE, only Type II systems (two immiscible pairs) can be predicted reliably, which is useless for liquid extraction. The Type I system (one immiscible * Corresponding author. Phone: 404-894-7070. Fax: 404894-9085. E-mail: [email protected].

pair) predictions are most unreliable, especially anywhere in the vicinity of a plait point. The real limitation is that although the current Gibbs energy expressions are all valid solutions to the GibbsDuhem (differential) equation, they are not correct. They are approximations useful under limited conditions, much as we use the ideal gas law or cubic equations correctly for limited ranges of variables. We do not know the “true” functionality or even whether there exists any single true functionality. VLE are relatively easy to handle because most of the separation comes from differences in vapor pressure, with activity coefficients being just a correction factor. However, in LLE we must take second derivatives even to determine if phase-splitting occurs, and the equilibrium equation depends solely on the activity coefficients. Therefore, LLE data for Type I systems are incredibly sensitive to the quality of the Gibbs energy expression. In recent years, new experimental techniques have begun to yield a small number of better quality VLE data that are really too good to represent with just two parameters per binary pair. For example, even if the two limiting activity coefficients are known, it is possible to have different behavior at finite concentrations. Shown in Figure 1 is the prediction of the P-x diagram for a realistic system based on γ1∞ ) 5 and γ2∞ ) 1.5. The azeotrope composition ranges from 0.16 to 0.25, and the total pressure varies by 10 Torr. Therefore, the purpose of this work is to demonstrate a type of (still empirical) equation that permits two important advantages: the use of more than two parameters for those binary pairs for which the data warrant such treatment, and the mixing of Gibbs energy expressions for better and more flexible representation of multicomponent systems. Moreover, we suggest a new and perhaps more sensitive method of testing the quality of the data and equations by using limiting activity coefficients in a mixed solvent.

10.1021/ie9800546 CCC: $15.00 © 1998 American Chemical Society Published on Web 09/24/1998

Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998 4513

Activity coefficients can be calculated from these equations by use of the following relationship

ln γi ) GE/RT +

( ) ∂GE/RT ∂xi

-

T,P,xj

∑k

xk

( ) ∂GE/RT ∂xk

T,P,xj

(7) The composition dependencies are

( ) ∂GE/RT ∂xk

T,P,xj

∑i

Figure 1. Prediction of P-x diagram from infinite dilution activity coefficients for four gE expressions.

gE Expression Development

where

The Wilson model describes the Gibbs energy of a mixture with the following expression:

Gmix/RT )

∑i xi ln(xi/∑j xjΛij)

(1)

where Λij is the binary parameter between component i and j and Λii ) 1. The term in parentheses is exactly the same as the function for calculating the vapor mole fraction of components from an ideal liquid mixture

yi ) xi/

Pj

∑j P

xj

(2)

∂gij ∂xk

)

(

-

-

∑j

(

( ∑( )

bjmxm

Λik

Λij

gik

xj

bimxm ∑ m

-

Pjγj

xj

(3)

j i

If we let γij ) γi/γj and assume that this ratio can be derived from the expression

GE/RT )

∑i xi ln(∑j xjbij)

(4)

where bij and bji are two additional binary parameters and bii ) 1. Equation 4 is similar to the Wilson equation except for the sign. Adding our correction into eq 1 will yield the following expression for the excess Gibbs energy

GE

)-

RT

[

Λij

∑i xi ln ∑j γ

ij

]

xj

(5)

where

γij )

bimxm ∑ m bjmxm ∑ m

[

exp

∑l

xl blmxm ∑ m

]

(bli - blj)

(6)

with bii ) 1. The parameters bij and bji increase the number of parameters to four for each binary pair. We call this the Scott-Wilson equation.

(gij)2

xj

))

∂gij ∂xk

)

(8)

(

+

bjmxm) ∑ m

2

(

bki - bkj

∑ m

∑l

-

exp

))

xlblk (bli - blj)

[

bkmxm

where Pj/Pi ) Λij. For a nonideal mixture, the mole fraction is given by eq 3

∑j P γ

(

Λij

bjk

i

yi ) xi/

xj

xi

γij

j

bjmxm ∑ m

bimxm ∑ m

gkj

j

bik

∑ m

[∑ ( )] Λkj

) -ln

∑l

(

∑ m

blmxm)2

×

]

xl

∑ m

(bli - blj) (9) blmxm

This equation will describe phase separation and fits VLE data better that any of the previous excess Gibbs energy expressions. By setting the parameters bij and bji to 1, the equation reduces to the Wilson expression. It has one problem in that it cannot yield values of the Gibbs energy of mixing >0. Of course, such mixtures never exist in reality, but this does preclude calculation of the unstable portion of the gE curve for highly immiscible systems such as hydrocarbons and water. Nova´k, Matous, and Pick8 show that including regular solution theory in the Wilson equation will allow for Gmixing > 0. The additional parameter gives the following expression for the Regular Solution-Wilson (RW)

GE/RT ) -

∑i xi

(

)

1

∑j xjΛij) - 2 ∑j xjCij

ln(

(10)

where Cij ) Cji and Cii ) 0. This improves the ability of the Wilson equation to fit VLE data and also allows the equation to represent phase splitting. However, the third parameter is still insufficient to represent both VLE and LLE with a single set of parameters. To solve this problem, regular solution theory is added to the Scott-Wilson model in the same way Novak et al.8 modified the Wilson equation.

GE RT

)-

∑i xi

(( ln

Λij

∑j g

ij

)

xj -

1 2

)

∑j Cijxj

(11)

4514 Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998 Table 1. Fit of Binary VLE Data at 50 °Ca chloroform/ethanol chloroform/heptane ethanol/heptane acetone/chloroform acetone/methanol chloroform/methanol acetonitrile/ethanol acetonitrile/water ethanol/water a

error

Wilson

NRTL

Scott-Wilson

RS-Wilson

Scott-RS-Wilsonb

mean max mean max mean max mean max mean max mean max mean max mean max mean max

3.97 10.32 0.31 0.61 1.42 6.86 1.39 3.16 0.37 1.32 4.13 9.56 0.76 2.05 2.31 6.03 1.49 2.89

2.24 6.28 0.43 0.73 2.32 9.19 1.23 2.81 0.31 1.15 0.94 1.51 0.59 1.96 0.39 1.14 0.26 0.92

0.25 0.60 0.17 0.54 0.83 2.10 0.30 0.72 0.25 1.10 0.23 0.75 0.58 1.91 0.41 0.87 0.22 0.63

3.95 10.17 0.17 0.51 1.12 3.42 1.39 3.16 0.32 1.20 3.97 9.34 0.59 2.16 0.65 2.54 1.42 2.67

0.20 0.52 0.27 0.57 0.34 1.38 0.65 1.79 0.35 1.26 0.34 0.73 0.37 1.00 0.18 0.55 0.28 0.98

Mean/maximum error in regressed pressure, Torr. b Regressed using eq 12a.

Table 2. Prediction of Ternary VLE Data at 50 °C from Binary VLE Data Regressionsa system chloroform/ethanol/heptane acetone/chloroform/methanol acetonitrile/ethanol/water a

error

Wilson

NRTL

Scott-Wilson

RS-Wilson

Scott-RS-Wilsonb

mean max mean max mean max

4.98 9.84 5.05 13.32 10.39 19.80

8.35 17.57 8.93 23.70 2.17 5.67

3.63 11.91 3.25 14.30 3.62 7.60

5.38 11.85 4.11 12.68 11.35 23.56

2.53 9.61 1.05 12.68 0.59 1.53

Mean/maximum error in regressed pressure, Torr. b Regressed using eq 12a.

The Scott-Regular Solution-Wilson (SRW) equation has five adjustable parameters per binary pair, and we will show that satisfactory results can be obtained for regressing VLE and LLE in highly immiscible systems. Correlation and Prediction of VLE data The data sets of Abbott and Van Ness21 were chosen because of their experimental accuracy and because the sets include binary and ternary data. The P-x data were reduced using Barker’s method9 for the following binary pairs: chloroform/ethanol, chloroform/heptane, ethanol/heptane, acetone/chloroform, acetone/methanol, chloroform/methanol, acetonitrile/ethanol, acetonitrile/ water, and ethanol/water. Second virial coefficients were taken from the compilation of Dymond and Smith10 or were estimated using the Hayden and O’Connell method.11 Molar volumes at 25 °C were extrapolated to other temperatures using the modified Rackett technique.12 Parameters were determined using the downhill Simplex algorithm13 minimizing the objective function (OF) in eq 12 n

OF )

∑i (Pi - Pˆ i)2

(12)

where the carat denotes a predicted value. When fitting only binary data using the Wilson, NRTL, Regular Solution-Wilson, Scott-Wilson, and Scott-Regular Solution-Wilson models, the five-parameter Scott-Regular Solution-Wilson model fits the data better than any of the other models for all systems tested. However, when predicting ternary data from parameters taken from binary data, the ternary predictions in many cases are inferior to those made by the two-parameter Wilson equation because when five binary parameters per pair are used in the regression,

overfitting may often result and the parameters become highly correlated. Adding ternary data into the regressions reduces the correlation between parameters and produces parameters that are superior for fitting both binary and ternary data together. Because ternary (or higher order) data are seldom available, a method is needed to reduce this overfitting while using only binary data. When no ternary data are available, overfitting can be reduced by modifying the objective function in the binary regression n

OF )

∑i (Pi - Pˆ i)2 + Wf((b12 - 1)2 + (b21 - 1)2 + C122) (12a)

where Wf is a weighting factor, arbitrarily chosen to reduce the overfitting effect of the bij and Cij parameters. If Wf is chosen so that the OF shown in eq 12 is doubled over that obtained when Wf ) 0, the overfitting is significantly reduced. Even with this reduced fit of the binary data, the equation still fits the binary data as well or better than any of the other equations and predicts ternary data significantly better for the three systems evaluated. Table 1 gives the fit of the binary data, and Table 2 shows how the parameters obtained from fitting the binary data (using eq 12) predict ternary data using the various models. Also shown is the fit using the ScottRegular Solution-Wilson model, while applying eq 12a with Wf chosen so that the OF is increased by a factor of 2 over that when Wf ) 0. The five-parameter Scott-Regular Solution-Wilson model, with weighting by eq 12a, performs best. The fit of the binary data is superior in every case but two, where the fits are only slightly poorer than that of the

Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998 4515 Table 3. Regression of Both Binary and Ternary VLE Data at 50 °Ca system chloroform/ethanol/heptane acetone/chloroform/methanol acetonitrile/ethanol/water a

error

Wilson

NRTL

Scott-Wilson

RS-Wilson

Scott-RS-Wilson

mean max mean max mean max

2.46 10.18 3.30 10.15 3.23 10.26

1.98 11.33 1.26 7.13 0.78 6.81

0.77 4.66 0.60 3.17 0.89 4.70

2.42 10.17 2.07 12.48 0.70 3.10

0.51 1.93 0.57 8.08 0.25 1.65

Mean/maximum error in regressed pressure, Torr.

Figure 2. Activity coefficients and gE/x1x2RT for chloroform (1)/ ethanol (2) at 323.15 using Wilson and Scott-Wilson models. Parameters regressed from binary P-x data.

NRTL model, and the predictions of ternary data are significantly better in all three cases. When both binary and ternary data are available, the higher parameter models will fit better. Because the Regular Solution-Wilson and the Scott-Wilson models have the same form as the Wilson model, they must regress data the same or better than the Wilson model. Table 3 shows how well the models fit both the binary and ternary data using all the data in the regression and using eq 12 for all regressions. The flexibility of the model is demonstrated in Figure 2 for the shape of gE/x1x2RT for the chloroform/ethanol system. The Scott-Wilson model fits the P-x data well because it can represent the S-shape of the function gE/ x1x2RT, which gives a maximum in the activity coefficient at finite concentrations. Prediction of Infinite Dilution Activity Coefficients Another test of an excess Gibbs energy expression has been the prediction of infinite dilution activity coefficients from finite P-x data. We propose an even more stringent test of gE expressions in the prediction of γ∞ in a mixed solvent from binary data only. In this work we evaluate the Scott-Wilson model by comparing predicted and experimentally determined limiting activity coefficients of chloroform in pure heptane and ethanol and in a mixture of heptane and ethanol. The limiting activity coefficients were determined experimentally from boiling point depression data, (∂T/ ∂x1)P∞, using the ebulliometric method. The ebulliometric method has been developed and used extensively for determining binary limiting activity.14,15 Extending this method to ternary systems required the development of a new equation to calculate limiting activity coefficients for ternary systems from (∂T/∂x1)P∞ data. Described in this paper is the development of this equation and its application to determine limiting activity coef-

Figure 3. System used to measure limiting activity coefficients.

ficients. In addition to the Scott-Wilson model, the Wilson model and the combination of the two are evaluated based on their prediction of limiting activity coefficients of the ternary system. Apparatus. The apparatus for the boiling point depression measurements was that used by Trampe with a modified pressure control system.15 Figure 3 is a diagram of the experimental apparatus. This system consists of boilers of the modified Scott design. In these experiments, constant pressure is maintained using a Moore model 43-20 sub-atmospheric pressure regulator connected to a N2 supply and a Fischer Scientific Maxima D8A vacuum pump. This regulator uses the pneumatic null-balance principle of operation. The N2 is passed through drierite before entering the constantpressure manifold. The system pressure is displayed on a MKS Baratron system, consisting of types 170M6B and 170M-44A electronics units and a type 310 BHS1000 sensor head. This digital pressure gauge has a resolution of 0.01 Torr. The system pressure is controlled to 0.04-0.14 Torr, depending on the total pressure and boiling characteristics of the solvent. Effects of pressure fluctuations in this system are greatly reduced by measuring a temperature difference. Materials. The following materials were used in this experiment: chloroform, Aldrich, 99+% anhydrous, 0.5% ethanol used as stabilizer, dried over 4 Å molecular sieves; ethanol, Quantum Chemical Corporation, 200 proof, dried over 4 Å molecular sieves; n-heptane, Aldrich, 99% anhydrous, used as purchased. All of the materials were stored and handled in a dry N2 environment. Experimental Procedure. The experimental procedure is similar to that used by Trampe.15 A brief description will be presented here. Solvent mixtures are made gravimetrically under a dry N2 environment. Then, 400 mL of solvent or solvent mixture is loaded into each boiler gravimetrically. The pressure is set so that the system boils at 50 °C by adjusting the Moore sub-atmospheric pressure regulator. The boilers are heated to a steady-state refluxing condition. The initial

4516 Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998

equilibrium temperature offset between the reference boiler and each sample boiler is measured. This initial offset is used to correct all subsequent ∆T readings. Solute injections are made gravimetrically via a septum into each of the boilers. The volume of the solute injections is between 0.4 and 1.2 mL. A total of five or six solute injections are made into each boiler during an experimental run. After each injection, the system is allowed to reequilibrate (∼10-30 min depending on size of injection and boiling characteristics of system), and the temperature difference between the sample boiler and the reference boiler is recorded. For the ternary system the value of the pressure derivative with respect to temperature, (∂P/∂T)x1f0, of the solvent mixture is necessary. This value is measured experimentally by measuring vapor pressures of the solvent mixture in the reference boiler as a function of temperature in the region of 50 °C. These data are fit to a polynomial and the slope at 50 °C is determined. Data Reduction. Binary ebulliometric data are analyzed following the development of Gatreaux and Coates.16 The necessary expressions can be found elsewhere.17,14 To analyze the ternary ebulliometric data, a new expression had to be developed. The results of this development, neglecting vapor phase nonideality, is

γ1∞ ) P - x2

( ) ∂γ2 ∂x1

T

Psat 2

- x3

( ) ∂γ3 ∂x1

T

Psat 3

( )( )

∂T ∂x1

P

∂P x ∂T 1f0

Psat 1 (13) The results of this development taking into account vapor phase nonideality as well as a complete description of these derivations is given in Appendix A. The expression for the activity coefficient at infinite dilution in the ternary system requires the solvent mixture mole fractions, the activity coefficients at finite dilution of each solvent, and the derivative of the finite activity coefficient of each solvent with respect to the infinitely dilute solute. The finite activity coefficients can be estimated with an appropriate model. Fugacity coefficients are obtained from second virial coefficients. Experimental virial coefficients are used where available, and otherwise are estimated using the method of Hayden and O’Connell.11 The limiting composition derivative of the temperature, (∂T/∂x1)P∞, is obtained directly from the ebulliometric data. This value is found by fitting the data to various functionalities as described by Trampe.15 Saturation vapor pressures are measured experimentally or are calculated by a corresponding states correlation.12 Molar liquid volumes are calculated by the modified Rackett equation.12 Results. The results of the experimental determination of γ∞ of chloroform in the binary solvent mixture ethanol/n-heptane are presented in Table 4. These values were calculated taking into account vapor phase corrections using eq A4 in the Appendix. The uncertainties reported were determined by a progression of error procedure in which the uncertainties with each term in the calculation are summed to give an overall uncertainty. A large portion of the reported uncertainty for each limiting activity coefficient is attributed to the uncertainty associated with the experimental (∂T/∂x1)P∞ value. Uncertainty for this

Figure 4. Prediction of limiting activity coefficients for the system chloroform (1)/ethanol (2)/heptane (3) using parameters determined from finite binary VLE data only. Table 4. Experimental Limiting Activity Coefficients for the System Chloroform (1)/Ethanol (2)/n-Heptane (3) literature20

experimental x2

x3

T, K

γ1∞

T, K

γ1∞

1 1 0.799 0.609 0.603 0.305 0 0

0 0 0.201 0.391 0.397 0.695 1 1

322.9 322.9 322.9 322.8 322.9 323.0 323.0 323.0

1.65 ( 0.08 1.53 ( 0.08 1.29 ( 0.10 1.17 ( 0.09 1.12 ( 0.09 1.12 ( 0.08 1.33 ( 0.06 1.33 ( 0.06

323

1.62

293

1.47

value was assigned based on the standard error of the fit of the mole fraction versus temperature reduction data and the residuals of this fit. Multiple runs would reduce this uncertainty. Table 4 also contains a comparison of the γ∞chloroform values obtained in this study with those presented in the literature for the pure single component solvents. The values for the binary system reported in this work compare well, within the experimental uncertainty, with the literature values. Discussion. Shown in Figure 4 is the infinite dilution activity coefficient of chloroform (1) in a mixture of ethanol (2) and heptane (3). The solid lines are the Wilson and Scott-Wilson gE models with parameters regressed to the binary finite concentration P-x data of Van Ness and Abbott. The Wilson equation predicts accurately γ∞ for chloroform in pure heptane, but is 30% higher than experimental data for γ∞ in pure ethanol. This result is expected because the Wilson equation describes adequately the P-x data for chloroform heptane but fails for the chloroform/ethanol system. The Scott-Wilson model predicts the γ∞ in the pure solvents for each regression but does not predict the mixed solvent data accurately. If parameters regressed from binary and ternary P-x data are used, the predictions improve, which are shown in Figure 5. The Wilson equation still cannot reproduce the experimental γ∞ data, which the Scott-Wilson equation can accomplish within the experimental uncertainty. The Wilson equation, with only two parameters, is not flexible enough to describe this system. Even though the Scott-Wilson model can predict the γ∞ with parameters from binary and ternary data, we would prefer to use only binary data. In examining the fit of the finite concentration data, the Scott-Wilson model was only substantially better than the Wilson model for the systems ethanol/ heptane and chloroform/ethanol.

Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998 4517 Table 5. Binary Parameters Obtained from LLE and VLE for Benzene, Water, Acetone, Ethanol, and Tetrahydrofuran Λ12

Λ21

b12

0.204 0.0790 0.481 2.757 0.394 1.497 0.0269

2.51 0.473 0.337 0.197 0.118 1.413 0.00359

3.119 0.747 2.676 0.970 2.828 0.971 0.760

a

b21

C12

0.100 1.28 1.620 -0.0596 0.121 0.403 1.096 0.689 0.098 0.392 1.021 0.584 1.25 1.43

compound 1 compound 2 water ethanol water acetone water THF water

ethanol benzene acetone benzene THFa benzene benzene

Tetrahydrofuran.

Figure 5. Prediction of limiting activity coefficients for the system chloroform (1)/ethanol (2)/heptane (3) using parameters determined from finite binary and ternary VLE data.

Figure 7. Ethanol/benzene/water LLE regressed with the ScottRegular Solution-Wilson equation.

Figure 6. Scott-Wilson model as in Figure 4. Mixed model using Wilson equation for chloroform (1)/heptane (3) binary VLE and Scott-Wilson for chloroform (1)/ethanol (2) and ethanol (2)/heptane (3) binary VLE data. No ternary data were used for the mixed model.

In Figure 6 we show a prediction based on a combination of the Wilson and Scott-Wilson equations. The parameters are from only binary data, with the Wilson parameters used for the chloroform/heptane system and the Scott-Wilson used for chloroform/ethanol and ethanol/heptane systems. The explanation is that if we regress four binary parameters for the chloroform/ ethanol system, they will be highly correlated. Adding ternary data into the regressions reduces the correlation between parameters. When ternary data are not available, fewer parameters should be used, which yielded, in this example, a satisfactory result. For systems that have near-ideal solution behavior, the Scott-Wilson model always has highly correlated parameters. Our recommendation is to set the bij parameters to 1 (to give the Wilson equation) when only binary data are available.

Figure 8. Acetone/benzene/water LLE regressed with the ScottRegular Solution-Wilson equation.

for ethanol/benzene were measured by differential ebulliometry by the technique of Scott.19 The data can be found in the Supporting Information. The objective function used to fit the parameters is

∑i ((xi - xˆ i)2 + (x′i - xˆ ′i)2) + ∑i (Pi - Pˆ i)

w Fit of LLE Data The NRTL and UNIQUAC models cannot correlate LLE and VLE with the same set of parameters. We tested the Scott-Regular Solution-Wilson (SRW) equation for fitting a Type I LLE set and at the same time fit the VLE data for that system (the Type II LLE system is not as demanding on the gE model). For the immiscible pair, we chose benzene and water. The third components were ethanol, acetone, and tetrahydrofuran. The LLE data sets were taken from the compilation of Sorensen and Arlt.18 All the VLE binary pairs except

(14)

where w is a weighting factor used to give more emphasis on tie line data. The three ternary LLE data and the six binary VLE data were regressed together to give a unique set of parameters which are given in Table 5. The fit of the LLE data are shown in Figures 7-9. Conclusion With better VLE data becoming available, twoparameter gE models may not be adequate to fit all

4518 Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998

Consider first simplifications that can be made at low pressures. The ratios of fugacity coefficients and the exponent in the Poynting correction are approximately equal to unity.

( )

φsati =1 φˆ i

[

]

Vli(P - Psat i ) =1 exp RT Substituting the above approximation into eq A2, taking the derivative, and noting the following as x1 f 0

( )

∂x1 )1 ∂x1

Figure 9. Tetrahydrofuran/benzene/water LLE regressed with the Scott-Regular Solution-Wilson equation.

systems. We present examples of more flexible gE models that have the advantage of being “mixable” for multicomponent systems, permitting different forms to be used for each binary. These empirical expressions can be useful and may form the basis for testing other gE concepts on with both VLE and LLE as better experimental data become available. The Scott-Wilson model correlates VLE data better than current models with the addition of two or three parameters. For predicting multicomponent behavior without ternary data, the four-parameter and fiveparameter models should only be used for those binaries deviating far from ideal solution behavior and in those cases, the regression should be conducted with eq 12a for the objective function as described earlier. The Scott-Wilson model can handle immiscible systems, but appears to require an additional parameter for hydrocarbon/water systems. The measurement of limiting activity coefficients in a solvent mixture has been demonstrated and suggested as an additional test of the quality of gE expressions.

( )

∂x2 1 )) -x2 ∂x1 (1 + x3/x2)

( )

∂x3 x3/x2 ) -x3 )∂x1 (1 + x3/x2) γ1 f γ1∞

( ) ( ) ( ) ∂P ∂x1

3

P)

∑i

xiγiPsat i

φsat i φˆ i

(

exp

)

νli(P - Psat i ) RT

∂xi

)

( )

∂ ∂xi

{

3

∑i

xiγiPsati

φsat i φˆ i

(

exp

RT

)}

where xi, γi, φi, and P are functions of x1.

T

Psat 2 - x3

( ) ∂γ3 ∂x1

T

Psat 3 -

( )( ) ∂T ∂x1

P

∂P ∂T

x1f0

(A3) Considering now a derivation that takes into account vapor phase nonidealities, the derivation proceeds noting the following additional derivatives

( ) ( ) ( )( )

∂φˆ j ∂ (φˆ -1) ) -(φˆ -2 j ) ∂x1 j ∂x1

(A1)

νli(P - Psat i )

∂γ2 ∂x1

Psat 1

∂φˆ j ∂x1

The derivative of the total system pressure with respect to solute mole fraction is taken as follows

∂P

P

results in an expression for γ1∞

Psystem - x2

Derivation of Ternary Equation. The derivation of the equation necessary for calculating limiting activity coefficients for a ternary system closely follows the derivation for a binary system. The total pressure of a ternary system is given by

∂P ∂T x ∂T 1 ∂x1

sat x2γ2Psat 2 + x3γ3P3 ) Psystem

γ1∞ ) Appendix

)

T

)

T

∂φˆ j ∂P ∂P ∂x1

j ) 2, 3 j ) 2, 3

T

Substituting and solving for γ1∞ results in the following expression

(A2) γ1∞ )

A - B + C - D + (1 + E - F + G - H)I J

(A4)

Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998 4519

where the terms A through J are

A)

(

) ( ( ) ( ( ) ( ( ) (

B ) x2

C)

) ) ) )

{ { { {

} } } }

φsat VL2(P - Psat 1 2 2 ) γ2Psat exp 2 1 + x3/x2 φˆ 2 RT ∂γ2 ∂x1

Psat 2

T,x1f0

φsat VL2(P - Psat 2 2 ) exp φˆ 2 RT

x3/x2 φsat VL3(P - Psat 3 3 ) γ3Psat exp 3 1 + x3/x2 φˆ 3 RT

D ) x3

∂γ3 ∂x1

Psat 3

T,x1f0

E ) x2γ2Psat 2

( )( ) { φsat 2

(φˆ 2)2

F ) x2γ2Psat 2 G ) x3γ3Psat 3

φsat VL3(P - Psat 2 3 ) exp φˆ 3 RT

( ) {

}( )

φsat VL2(P - Psat VL2 2 2 ) exp φˆ 2 RT RT

( )( ) {

H ) x3γ3Psat 3

}

∂φˆ 2 VL2(P - Psat 2 ) exp ∂P RT

φsat 3

(φˆ 3)2

( ) {

}( )

φsat VL3(P - Psat VL3 3 3 ) exp φˆ 3 RT RT

( ) ( )

I) -

J ) Psat 1

}

∂φˆ 3 VL3(P - Psat 3 ) exp ∂P RT

∂P ∂T

x1f0

( ) {

∂T ∂x1

P,x1f0

}

φsat VL1(P - Psat 1 1 ) exp φˆ 1 RT

Acknowledgment The authors are most grateful for the financial support of the Du Pont Company in this research effort. They also thank Sam Edge, Steven Sherman, and Mark Cain for their assistance. Supporting Information Available: Tables of isothermal vapor-liquid equilibrium data (2 pages). Ordering information is given on any current masthead page. Literature Cited (1) Nicolaides, G. L.; Eckert, C. A. Optimal Representation of Binary Liquid Mixture Nonidealities. Ind. Eng. Chem. Fundam. 1978, 17, 331-340.

(2) Lafyatis, D. S.; Scott, L. S.; Trampe, D. M.; Eckert, C. A. Test of the functional dependence of gE(x) liquid-liquid equilibria using limiting activity coefficients. Ind. Eng. Chem. Res. 1989, 28, 585-590. (3) Wilson, G. M. Vapor-Liquid Equilibrium. XI: A New Expression for the Excess Free Energy of Mixing. J. Am. Chem. Soc. 1964, 86, 127-130. (4) Renon, H.; Prausnitz, J. M. Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures. AIChE J. 1968, 14, 135-144. (5) Abrams, D. S.; Prausnitz, J. M. Statistical Thermodynamics of Liquid Mixtures: A New Expression for the Excess Gibbs Energy of Partly or Completely Miscible Systems. AIChE J. 1975, 21, 116-128. (6) Eckert, C. A.; Sherman, S. R. Measurement and Prediction of Limiting Activity Coefficients. Fluid Phase Equil. 1996, 116, 333-342. (7) Walas, S. M. Phase Equilibrium in Chemical Engineering; Butterworth: Boston, 1985. (8) Nova´k, J. P.; Matous, J.; Pick, J. Liquid-Liquid Equilibria; Elsevier: New York, 1987. (9) Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed.; Prentice Hall: Englewood Cliffs, NJ, 1986. (10) Dymond, J. H.; Smith, E. B. The Virial Coefficients of Pure Gases and Mixtures. A Critical Compilation; Clarendon: Oxford, 1980. (11) Hayden, J. G.; O’Connell, J. P. A Generalized Method for Predicting Second Virial Coefficients. Ind. Eng. Chem., Process Des. Dev. 1975, 14, 209-216. (12) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids; McGraw-Hill: New York, 1987. (13) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.; Cambridge University: New York, 1992. (14) Thomas, E. R.; Newman, B. A.; Long, T. C.; Wood, D. A.; Eckert, C. A. Limiting Activity Coefficients from Differential Ebulliometry. J. Chem. Eng. Data 1982, 27, 233-240. (15) Trampe, M.; Eckert, C. A. Limiting activity coefficients from an improved differential boiling point technique. J. Chem. Eng. Data 1990, 35, 156-162. (16) Gatreaux, M. F., Jr.; Coates, J. Activity Coefficients at Infinite Dilution. AIChE J. 1955, 1, 496-500. (17) Eckert, C. A.; Newman, B. A.; Nicolaides, G. L.; Long, T. C. Measurement and Application of Limiting Activity Coefficients. AIChE J. 1981, 27, 33-40. (18) Sørensen, J. M.; Arlt, W. Liquid-Liquid Equilibrium Data Collection, DECHEMA: Frankfort, 1979. (19) Scott, L. S. Determination of Activity Coefficients by Accurate Measurement of Boiling Point Diagram. Fluid Phase Equil. 1986, 26, 149-163. (20) Yang, Y.; Wu, H.; Xie, S. Chengdu Keji Daxue Xuebao 1983, 1, 27. (21) Gmehling, J.; Onken, V.; Arlt, W. Vapor-Liquid Equilibrium Data Collection; Dechema: Frankfurt, Germany, 1977.

Received for review January 30, 1998 Revised manuscript received July 6, 1998 Accepted July 9, 1998 IE9800546