Extended Flexibility for GE Models and Simultaneous Description of

A method to increase the flexibility of the composition dependence of GE models or equations of state is proposed. The formalism is based on a nonline...
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Ind. Eng. Chem. Res. 2005, 44, 7600-7608

CORRELATIONS Extended Flexibility for GE Models and Simultaneous Description of Vapor-Liquid Equilibrium and Liquid-Liquid Equilibrium Using a Nonlinear Transformation of the Concentration Dependence Ju 1 rgen Rarey School of Chemical Engineering, University of Kwa-Zulu Natal, Durban 4041, RSA, and Industrial Chemistry, Institute for Pure and Applied Chemistry, University of Oldenburg, 26111 Oldenburg, Germany

A method to increase the flexibility of the composition dependence of GE models or equations of state is proposed. The formalism is based on a nonlinear transformation of the concentration space and can be applied to any mixture model without re-deriving the model equations. Any number of additional binary parameters can be introduced, which show nearly no intercorrelation with the original model parameters. The reason is that the additional parameters only effect the form but not the size and symmetry of, for example, GE. The mathematical formalism for the introduction into GE models was derived and applied to the UNIQUAC model. The modified model (FlexQUAC) was then used to regress a large number of vapor-liquid equilibrium (VLE) data sets (approximately 4000), whereby significant improvements were observed for systems with medium to large positive deviation from Raoult’s law. As a result of the very small intercorrelation with the original model parameters it was assumed that the additional parameters have no negative effect on the prediction of multicomponent behavior from binary data. This was verified for 13 carefully selected systems. The motivation for this development was not primarily the improvement of the regression capability but the simultaneous description of VLE and liquid-liquid equilibrium (LLE) data. Typically models such as NRTL or UNIQUAC predict immiscible regions that are way too large from VLE data or VLE separation factors that are too low from LLE. This shortcoming of the original models is corrected by the concentration space transformation. Examples for the simultaneous description of VLE and LLE for binary and ternary systems are presented and compared to the corresponding results of the UNIQUAC equation. Introduction For the prediction of different multicomponent mixture properties from binary data various thermodynamic models are used, mainly GE models and equations of state (EOS). These models contain one (e.g., EOS with quadratic mixing rule), two (e.g., Wilson, UNIQUAC), or three (e.g., NRTL) binary parameters for each pair of components, which have to be derived from experimental data. This limited number of parameters is often not sufficient to reproduce binary data within, or sometimes even close to, the experimental uncertainty. Increasing the number of parameters for a better description of mixture properties very often leads to numerical instabilities and uncertain prediction of multicomponent behavior. As a result of a strong intercorrelation, increasing the number of model parameters tends to deprive them of their physical significance. This paper will present a simple mathematical procedure to enhance the flexibility of any model for the calculation of mixture properties using one or more additional binary parameters. As the intercorrelation of these parameters with the original model parameters * To whom correspondence should be addressed. E-mail: [email protected].

is usually neglectible, they do not worsen the prediction of multicomponent behavior. As an example, the procedure is used for the improvement of the widely used UNIQUAC model by one additional binary parameter. Enhanced fitting capability and reliable prediction for multicomponent behavior is shown for numerous examples. In addition, the modified UNIQUAC equation (FlexQUAC) is able to describe vapor-liquid equilibrium (VLE) and liquid-liquid equilibrium (LLE) data simultaneously using only one set of parameters for both properties. Besides the FlexQUAC-equations, the equation for a modified NRTL (FlexNRTL) is also given. The same procedure may be used with an in principle unlimited number of binary, ternary, and higher parameters, thus, improving model flexibility even further without any mathematical or numerical complications. Deficiencies of Current Two- or Three-Parameter Models Various deficiencies of the currently most widely used GE models are well-known: (a) The simultaneous description of VLE and LLE data is not possible. Using binary interaction parameters derived from LLE data in nearly all cases leads

10.1021/ie050431w CCC: $30.25 © 2005 American Chemical Society Published on Web 08/19/2005

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Figure 1. Prediction of VLE using UNIQUAC GE model parameters from LLE1 and regression of VLE data for the system water (1)1-pentanol (2) at 101.3 kPa together with experimental xyP data.2

Figure 2. Correlation of VLE using the UNIQUAC model for the system ethanol (1)-heptane (2) at 50 °C together with experimental xP data.3

to a very poor description of VLE data and activity coefficients at infinite dilution. Typical examples are given in Figures 1 and 2. (b) The models show only a limited flexibility with respect to the concentration dependence of GE. While the behavior of nearly ideal systems, which is mainly governed by the differences in pure-component vapor pressures, can easily be described, the models fail in cases of medium or strong deviations from Raoult’s law. A not even severe example is shown in Figure 1. In the xy diagram (Figure 1a), the curve calculated from LLE is not steep enough in the homogeneous region, while the regression of VLE data predicts a way too large immiscible region. Figure 1b shows the separation factor R12, the key property for distillation. It is obvious that both sets of parameters fail to reproduce the experimental results within the required accuracy. Systems with complete miscibility but rather strong positive deviation from Raoult’s law pose another problem for data regression, which will fail either in the dilute region or falsely predict an immiscible region. A

typical example is shown in Figure 2. The upper regression curve represents the best fit to the data. During the second regression (lower curve) all parameter values which led to a negative value of the second derivative of G(x) with respect to x in the whole concentration and a small temperature range around 50 °C were discarded. The lower curve represents the closest fit which still shows no heterogeneous region. While the NRTL model performs a little better because of the additional adjustable parameter, it is still far from adequate for these types of data. (c) All GE models contain an implicit temperaturedependence of GE, which is often at least quantitatively right but fails to predict the correct heat of mixing in most cases. While in the last case it is relatively simple to correct the erroneous results by introducing temperaturedependent interaction parameters, a modification of the concentration dependence is much more difficult and likely to affect the ability of the model to predict the properties of multicomponent mixtures. If the model would lose this predictive capability, little or no advantages would be left over flexible mathematical expressions such as the n-suffix Margules equation or a Legendre- or Redlich-Kister polynomial. Different Approaches for a Modification of the Concentration Dependence of GE A number of different methods were used to improve the concentration dependence of GE models in the past. A simple way is, for example, to construct GE(x) by adding the contributions of two different models:

GE(x) ) GaE(x) + GbE(x)

(1)

If the models are sufficiently different with respect to the concentration dependence of GE, this method is able to solve the problem of simultaneous description of VLE, LLE, and γ∞. Typically a simple two-suffix Margules expression is used to extend the flexibility of a local-

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Figure 3. Binary GE curves of different sizes, symmetries, and shapes.

composition GE model. If applied to the Wilson model, this modification is then able to describe LLE. Another possibility is to multiply the GE calculated from a given model by a function of concentration. This method was, for example, employed by Cha and Prausnitz4 for an improved description of ternary LLE:

GE(x) ) GaE(x)RT exp(βx1λ1x2λ2x3λ3)

(2)

A third approach is to change the model equations directly, for example, by defining a concentration dependence of the interaction parameters in the Boltzmann term of a local composition model:

(gij - gjj) + τ′ij )

∑k xk∆gijk

RT

(3)

This approach was, for example, tried by Fredenslund et al.,5 but the work was discontinued because of a variety of problems. All these modifications are in principle empirical. Other ways to improve models such as UNIQUAC or NRTL are by improving the theoretical basis (e.g., by taking into account association). This is normally not possible without the introduction of parameters, which should be derived from independent methods such as neutron scattering, Monte Carlo simulations, and so forth. Because of the limited knowledge about most liquid mixtures, these methods are seldomly applicable for typical industrial applications. Currently it is still difficult to predict even the size and symmetry of the GE curve for mixtures of practical importance from molecular parameters with sufficient accuracy. There is even far less chance to predict the slight differences in shape, which are responsible for a correct and simultaneous description of VLE, LLE, and γ∞. Simple Mathematical Procedure for the Modification of the Concentration Dependence of GE The GE models most often used for the calculation of phase equilibria in nonelectrolyte multicomponent mixtures are Wilson, NRTL, and UNIQUAC. While the Wilson model is not able to describe LLE, it is often useful for the description of homogeneous mixtures with

rather large limiting activity coefficients. In these cases, NRTL and UNIQUAC most often falsely predict an immiscible region. As the proposed modification aims at the simultaneous description of LLE and VLE, the Wilson model is not further considered here. Both the NRTL and the UNIQUAC model contain two binary interaction parameters, which have to be regressed to experimental or estimated (via UNIFAC, mod. UNIFAC, ASOG, etc.) data. An additional adjustable binary parameter in NRTL (nonrandomness factor R) gives the model a little more flexibility but should only be varied within a certain numerical range. It is usually not possible to obtain a simultaneous description of VLE and LLE via variation of R. In addition, UNIQUAC uses two pure-component parameters, which should not be adjusted during the regression of binary data to avoid inconsistencies when predicting multicomponent behavior. With two binary parameters the models possess two degrees of freedom, and it is, therefore, possible to exactly reproduce two experimental data points [a complete binary VLE data point (x, y, P, T) represents two activity coefficients and is, therefore, equivalent to two data points]. These could be, for example, the two limiting activity coefficients defining the limiting separation factors for distillation or the concentrations of two liquid phases in equilibrium required for extraction or the decanter. Both sets of parameters are significantly different except for very large miscibility gaps. For the simultaneous correlation of the different experimental data (VLE, LLE, γ∞, etc.) an additional adjustable parameter is required. The two parameters available are able to describe the size (e.g., max value) and the symmetry (e.g., location of the maximum) of the binary GE curve. Once these parameters are fixed, the shape of the curve is only governed by the model equation. Figure 3 illustrates size (large, small), symmetry (symmetric, unsymmetric), and shape (sharp or round) as used in this context. When the adjusted binary parameters are used, the models mostly allow a sufficiently precise prediction of multicomponent VLE and an approximate prediction of multicomponent LLE. It is of primary importance not to deprive these parameters of their physical significance to retain the ability of the model to predict the multicomponent behavior. The shape of the curve is defined by the concentration dependence of GE as given by the model equations. How

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is it possible to introduce a shape parameter with little or no effect on the two binary interaction parameters? The most simple approach is to replace the mole fraction vector x by a nonlinear vector function f(x) with the following properties: (1) f(x) obeyes the boundary conditions fi ) 0 if xi ) 0 and fi ) 1 if xi ) 1. (2) f(x) obeyes the summation condition ∑ifi ) 1. (3) f(x) is continuous and differentiable. (4) f(x) is symmetrical with regard to x in binary systems to avoid affecting the symmetry. (5) f(x) is defined for systems containing any number of components. For dfi(x)/dxi larger than unity near xi ) 0 and xi ) 1 (in a binary system) the shape of the GE curve is more round, and in the opposite case it is more sharp. Size and symmetry of GE[f(x)] are not affected. An additional advantage of this approach is that it does not require re-derivation of the expression for the activity coefficient γ by deriving nTGE with respect to the mole numbers. The modification can easily be applied to any model via the chain rule: E

A simple expression satifying the different conditions is

xi(1 +

∑k

∑k xkdik)

xk(1 +

∑l

(4)

xldkl)

with dii ) 0 and dij ) dji. There is no reason not to construct an even more powerful flexibilization by expanding the parameter dij as function of xj - xi [whereby the expansion should only use powers of 1, 3, 5, ... to avoid affecting the symmetry of GE(x)]:

xi(1 + fi(x) )

∑k

∑k xk∑l (xi - xk)

xk(1 +

l-1

∑l ∑ m xl

dikl)

∑k ∑l xk

[

(5)

]

∂(nTGE) ∂ni

(xi - xk)l-1dikl +

xmfiklm + ...)}/ ∑k xk∑l xleikl + ∑k xk∑l xl∑ m {∑xk[1 + ∑xl∑(xk - xl)m-1dklm + ∑xl∑xmeklm + k l m l m x x x f ∑l l∑ m∑ n klmn + ...]} (6) m n whereby only ejkl with j * k * l and fjklm with j * k * l * m should be nonzero. To ensure symmetric flexibilization, set ejkl ) ekjl ) elkj ) eljk ) ejlk ) eklj and similar for fklmn. Application of the Chain Rule for the Calculation of Activity Coefficients for the Modified Models GE, HE, and CPE for the flexibilized model can easily be calculated by first calculating f(x) and then using this

(7)

T,P,nj*i

or

ln γi )

[ ] ∂(nTQ) ∂ni

with Q )

T,P,nj*i

GE RT

Simple differentiation leads to

ln γi ) Q + nT

( ) ∂Q ∂ni

(8)

T,P,nj*i

Usually Q is given as a function of the mole fractions. Applying the chain rule leads to

( ) ( )( ) ∂Q ∂nj

T

)

∂Q ∂xi

T

∂xi ∂nj

(9)

where (∂Q/∂nj) and (∂Q/∂xi) are vectors and (∂xi/∂nj) is the gradient of x (Jacobi matrix ∇x, see appendix). Equation 9 can be more conveniently written as

( )

( )

∂Q ∂Q ) ∇xijT ∂nj ∂xi

(10)

If GE is considered a function of the transformation f(x) instead of the mole fraction x, the gradient ∇x only needs to be replaced by ∇f in the above equation. Activity coefficients can then be calculated from

(xk - xl)m-1dklm)

Ternary and higher parameters can easily be introduced as required:

fi(x) ) {xi(1 +

RT ln γi )

E

dG [f(x)] dG (f) df(x) ) dx df dx

fi(x) )

vector instead of the mole fraction vector in the respective model equations. Any property requiring the derivation of GE with respect to concentration needs to take into account the derivation of f(x) with respect to concentration. The activity coefficient of component i is defined via the partial molar GE at constant T, P, and nj*i

ln γi ) Q(f) + ∇fT

[ ] ∂Q(f) ∂f

T,P,nj*i

(11)

The elements of ∇f and the derivation of Q ) GE/RT with respect to the mole fraction x for the UNIQUAC and NRTL equations are given in the appendix. Test of the FlexQUAC Model To verify the applicability of the new model, several aspects have to be considered: (a) Does the model give improved correlation of experimental binary VLE data? (b) Is the flexibilized model able to predict multicomponent VLE from binary data with a quality similar to (or even better than) the original model? (c) Can the model be used for the simultaneous regression of VLE and LLE data? (d) Is the model able to predict the ternary LLE behavior from binary data alone? As both NRTL and UNIQUAC suffer very similar deficiencies, the tests will be applied to the FlexQUAC

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Figure 4. Mean relative squared deviations in pressure as a function of the geometric mean of the activity coefficients at infinite dilution calculated from the UNIQUAC regression.

model only. Similar results can then be expected for the modified NRTL (FlexNRTL).

Figure 5. Regression results for the system tetrahydrofuranwater at T ) 298.15 K using the UNIQUAC and FlexQUAC models together with exp. Px data.7

Application of FlexQUAC to the Correlation of Binary VLE Data As a first test the FlexQUAC model was applied to the correlation of binary VLE data stored in the Dortmund Data Bank (DDB1) using only one additional parameter. The DDB was developed by J. Gmehling and co-workers for the development or further development of group contribution methods such as UNIFAC, mod. UNIFAC, ASOG, PSRK, or LIFAC and is listed as ref 1. Different techniques are employed for the measurement of VLE data. Very high precision xPT data are available using the static method as described by Van Ness et al.6 In this method the pressure in an equilibrium cell with known amounts of the components is measured at fixed temperature. The concentrations in the liquid and vapor phase are iteratively calculated using an nVT-flash calculation. In the DDB more than 4000 binary data sets can be found from this type of measurement. All these data sets were regressed individually with the mean relative squared deviation in pressure as the objective function F:

F)

1

n

∑

ni)1

(

)

Pexp,i - Pcalc,i Pexp,i

Figure 6. Graphical comparison of the predictive capabilities of the UNIQUAC and FlexQUAC models for selected ternary mixtures.

2

(12)

The relative reductions of the objective function compared to that of the UNIQUAC regression obtained for 4106 isothermal xP data sets are shown in Figure 4 as a function of the geometric mean of the activity coefficients at infinite dilution calculated from the UNIQUAC regression. Only data sets with 10 or more data points were used to ensure sufficient information on the dependence of the system pressure on composition. For mean limiting activity coefficients between 1 (ideal systems) and 5, the ratio of the objective functions for FlexQUAC and UNIQUAC decreases linearly to about 0.4. As shown in Figure 4, above a mean limiting activity coefficient of 15, the improvement is less reproducible as these systems show limited miscibility and the data in the two-phase region are often of questionable quality. Nevertheless, for the data sets with a mean limiting activity coefficients greater 5 (in total 930), the objective function was reduced by a factor of 0.38. For 235 of these data sets the mean improvement was more than 1 order

Figure 7. Simultaneous correlation of VLE and LLE using FlexQUAC GE model parameters from LLE and VLE data for the system water (1)-1-pentanol (2) (9 , VLE data;2 [, LLE data;1 2, azeotropic data;1 solid line, VLE calculation; dotted line, LLE calculation; dashed line, azeotropic composition calculation).

of magnitude. This can be considered as proof for a very significant and physically meaningful increase in flexibility of GE with respect to x. A typical example is given in Figure 5 for the system tetrahydrofuran-water. In this case the UNIQUAC model falsely predicts an immiscible region. FlexQUAC is able to reproduce these very precise data within their experimental uncertainty.

Ind. Eng. Chem. Res., Vol. 44, No. 19, 2005 7605 Table 1. Test of the Prediction of Ternary from Binary VLE Data mean relative deviation in P [%] methanol (1)-n-hexane (2)9 methanol (1)-cyclohexane (2)10 n-hexane (1)-cyclohexane (2)9 methanol (1)-n-hexane (2)-cyclohexane (3)9 methanol (1)-cyclohexane (2)10 methanol (1)-n-hexane (2)11 n-hexane (1)-cyclohexane (2)11 methanol (1)-n-hexane (2)-cyclohexane (3)11 methanol (1)-cyclohexane (2)12 methanol (1)-n-hexane (2)13 n-hexane (1)-cyclohexane (2)14 methanol (1)-n-hexane (2)-cyclohexane (3)15 acetone (1)-chloroform (2)16 acetone (1)-methanol (2)16 chloroform (1)-methanol (2)16 acetone (1)-methanol (2)-chloroform (3)16 acetone (1)-chloroform (2)16 acetone (1)-methanol (2)16 chloroform (1)-methanol (2)16 acetone (1)-methanol (2)-chloroform (3)16 acetone (1)-chloroform (2)17 acetone (1)-methanol (2)17 chloroform (1)-methanol (2)17 acetone (1)-methanol (2)-chloroform (3)17 ethanol (1)-acetonitrile (2)18 acetonitrile (1)-water (2)18 ethanol (1)-water (2)18 ethanol (1)-acetonitrile (2)-water (3)18 acetone (1)-ethanol (2)19 acetone (1)-water (2)19 ethanol (1)-water (2)19 acetone (1)-ethanol (2)-water (3)19 ethanol (1)-1,4-dioxane (2)20 water (1)-1,4-dioxane (2)20 ethanol (1)-water (2)20 ethanol (1)-water (2)-1,4-dioxane (3)20 acetone (1)-acetonitrile (2)21 acetone (1)-methyl acetate (2)21 methyl acetate (1)-acetonitrile (2)21 acetone (1)-methyl acetate (2)-acetonitrile (3)21 benzene (1)-cyclohexane (2)22 cyclohexane (1)-aniline (2)23 benzene (1)-aniline (2)23 benzene (1)-cyclohexane (2)-aniline (3)24 benzene (1)-cyclohexane (2)25 n-hexane (1)-cyclohexane (2)25 n-hexane (1)-benzene (2)25 n-hexane (1)-benzene (2)-cyclohexane (3)25 1-heptene (1)-n-heptane (2)26 1-heptene (1)-n-octane (2)26 n-heptane (1)-n-octane (2)26 1-heptene (1)-n-heptane (2)-n-octane (3)27

γ∞ (FlexQUAC)

T [K]

UNIQUAC

FlexQUAC

1 in 2

2 in 1

293.15 293.15 293.15 293.15 303.15 303.15 303.15 303.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 328.15 328.15 328.15 328.15

3.25 7.49 0.06 7.01 7.01 4.36 0.07 6.12 4.17 3.30 0.02 4.46 0.34 0.06 0.43 1.68 0.34 0.06 0.33 1.33 0.36 0.19 0.33 0.56 0.26 1.23 0.39 1.41 0.13 0.33 0.23 1.62 0.19 1.64 0.21 1.08 0.17 0.03 0.06 0.42 0.10 0.97 0.22 0.65 0.26 0.25 0.17 0.65 0.03 0.38 0.11 1.64

1.52 4.68 0.02 3.61 3.14 1.91 0.06 3.82 2.62 0.77 0.03 2.47 0.35 0.02 0.43 1.71 0.35 0.02 0.33 1.36 0.36 0.08 0.33 0.51 0.19 0.16 0.39 1.16 0.02 0.04 0.23 1.32 0.12 1.64 0.21 1.29 0.17 0.03 0.04 0.37 0.07 0.33 0.22 0.83 0.26 0.25 0.17 0.63 0.03 0.11 0.11 1.64

52.81 75.41 1.19

27.56 33.26 1.14

67.11 53.46 1.16

25.08 26.17 1.10

59.34 39.77 1.12

32.90 24.19 1.10

0.35 2.01 2.53

0.47 1.98 9.21

0.35 2.01 2.62

0.47 1.98 8.09

0.33 2.01 2.58

0.53 2.07 8.56

3.20 12.97 5.57

3.68 7.67 2.63

2.14 9.49 5.42

2.00 5.63 2.66

2.49 9.34 5.67

2.92 10.84 2.61

1.01 1.14 1.29

1.00 1.12 1.34

1.46 7.21 1.83

1.53 9.36 2.21

1.32 1.05 1.54

1.47 1.04 1.56

1.08 1.22 1.07

1.10 1.64 1.22

Prediction of Ternary VLE Using Binary Parameters In the next step, the performance of FlexQUAC for the prediction of the VLE in ternary homogeneous systems from binary data has to be verified. In case the ability of UNIQUAC for the prediction of multicomponent systems is preserved, a better prediction is to be expected because of the improved correlation of the binary subsystems. For this test a set of binary and ternary data as proposed by Gierycz8 was used. Table 1 contains a list of these data sets as well as the mean relative deviation in pressure in case of regression (binary systems) and prediction from binary data (ternary systems). As additional information, the γ∞ values calculated from the parameters of the FlexQUAC regression are given to judge the mixture behavior. The test set consists of 13 ternary and 39 binary sets, whereby the ternary and binary sets of a system were usually determined in the

same well-known laboratory and the systems show different behaviors ranging from strong negative to strong positive deviations from Raoult’s law. For nearly all cases of medium to large positive deviation from Raoult’s law the regression and prediction results of FlexQUAC are significantly better than those of UNIQUAC. This is especially true for the mixture methanol (1)-n-hexane (2)-cyclohexane (3), which shows the highest activity coefficients of all the test systems. In case of activity coefficients lower than or close to unity, no significent influence of the model modification can be observed as could be expected. Figure 6 shows a graphical comparison of the results for both models. A test of the predictive capability of the NRTL model revealed results significantly better than those of UNIQUAC but still worse than those of FlexQUAC. As the model modification proposed here is not aimed at improving the predictive capability for the multi-

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Figure 8. Separation factor as a function of composition using FlexQUAC GE model parameters from the simultaneous regression of VLE and LLE data for the system water (1)-1-pentanol (2) at 101.3 kPa together with experimental xyP data.2

Figure 9. Pxy diagram for the system 2-butanol (1)-water (2) at 45.04 °C together with experimental xyP data28 and curves calculated from FlexQUAC using different values of d12 (a, 0.0; b, 0.05; c, 0.1; d, 0.15; e, 0.2; f, 0.22; g, 0.25). Interaction parameters were regressed to LLE composition29 (dashed lines).

component mixture behavior, the goal of these tests was to verify that the predictive capability is not worsened. An improvement of the predictive capability is caused only by the improved representation of binary data. As the results show, the predictive capability is retained. Simultaneous Description of Binary VLE and LLE Data As already shown above (see Figures 1 and 2) and discussed in many articles and textbooks, the simultaneous description of VLE and LLE is usually not possible using the UNIQUAC or NRTL model. Figure 7 shows the data from Figure 1 together with the results

from the simultaneous correlation of VLE and LLE using FlexQUAC. LLE data and azeotropic data measured by various authors were taken from the DDB and are also shown in Figure 7. It can be seen that a nearly perfect description of all these properties is achieved. The separation factor as a function of composition is shown in Figure 8. FlexQUAC is able to reproduce the experimental values with sufficient accuracy contrary to the UNIQUAC equation (Figure 1). The slight maximum at low concentrations of water is due to the temperature change along the isobaric curve. The temperature dependence of the separation factor is influenced by both the difference in the heats of vaporization of the components and the temperature dependence of the activity coefficients. Instead of presenting similar results for several other systems, the effect of the additional parameter d12 is demonstrated in Figure 9 in case of the system 2-butanol-water. The interaction parameters were regressed to experimental LLE compositions for different fixed values of d12. The lowest curve (a, d12 ) 0) represents the results for the unmodified UNIQUAC equation. In this case neither the azeotropic composition nor the equilibrium pressure is described well. The slope dP/dx at low concentrations of the components is especially very poorly reproduced. This leads to a poor description of the activity coefficients and separation factors at infinite dilution. In addition, the calculation results in a heterogeneous instead of a homogeneous azeotrope. With increasing values of the parameter d12 the calculated curve comes closer to the experimental results with an optimum at d12 ≈ 0.23. Prediction of Ternary LLE Behavior from Binary Data When calculating the ternary LLE from binary data it is mostly observed that a too large immiscible region is predicted. A typical example is shown in Figure 10 for the system methanol (1)-1,4-dioxane (2)-cyclohexane (3) at 25 °C. Binary interaction parameters for the binary systems 1-2 and 2-3 were regressed to VLE data stored in the DDB, and the parameters for the binary 1-3 were regressed to binary LLE data. While in case of the UNIQUAC equation the immiscible region extends to a 1,4-dioxane concentration of 0.2, a more realistic value of about 0.1 is predicted using FlexQUAC with d13 ) 0.1. Conclusion The nonlinear transformation in composition space allows the selective modification of the concentration dependence of GE in mixture models. As was shown in

Figure 10. Binodal curves in the ternary system methanol (1)-1,4-dioxane (2)-cyclohexane (3) at 25 °C from UNIQUAC and FlexQUAC calculations together with mutual solubility data.30

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this work, this leads to improved correlation of experimental VLE data and enables the simultaneous regression of VLE and LLE data as well as an improved prediction of ternary LLE from binary information. The procedure can be applied in a simple way to any mixture model without generating inconsistencies or additional intercorrelations of parameters. Besides the use for modifying the UNIQUAC model (FlexQUAC), it may also be a promising possibility to solve similar problems arising from other models such as EOS or group contribution GE models such as UNIFAC or modified UNIFAC. In the case of strongly asymmetric systems it might be advantageous to apply the transformation of the UNIQUAC model to the surface fraction instead of the mole fraction. FORTAN and VBA code for FlexQUAC as well as further detailed information like parameters used for the calculations in this work can be supplied by the author upon request. Appendix Analytical Derivation of the Jacobi Matrix. Case a: Mole Fraction x(n). The elements of the Jacobi matrix are defined as

∇x(n)i,j )

∂xi ∂(ni/nT) ) ∂nj ∂nj

[

∂ ∂f(xi)

∇x(n)i,j )

∂nj

)

∑k xkdik)

]

∑k xk(1 + ∑l xldkl) ∂nj

∂

(

for i * j

)

∑k nkdik

∑k nk∑l nldkl ∂nj

[

)

] ]

with

ST ) 1 +

φi

]}

- Qi ln(Si)

i ) 1, ..., n

with

φi ) r)

Si )

ri r

Θi )

∑i xiri

q)

qi q

∑i xiqi

( )

∑j xjΘjτji

τij ) exp

-∆uij RT

i, j ) 1, ..., n

where ∆uij is the interaction parameter between component i and component j (J/mol), ri is the relative van der Waals volume of component i, qi is the relative van der Waals surface of component i, and the derivation of GE/RT with respect to xi is given by

) qi[5 ln(Θi) - ln(Si)] + (1 - 5qi)

(

T,xj*i

)

τij

∑j xjqjS

j

j ) 1, ..., n

Analytical Derivation of GE/RT with Respect to the Mole Fraction x (NRTL). In the case of NRTL, it holds that

T,xj*i

) ln(γi)

Literature Cited

(1 + Si)(1 + Sj) ∂f(xi) 1 ) x (1 + dij) - 2xi ∂nj ST i ST

∑k xkdik

() Θi

ln(φi) + (5q - 1)φi - Θi 4q +

∂f(xi) (1 + Si)2 1 ) 1 + xi + Si - 2xi ∂nj ST ST

Si )

{[

GE ∂ /∂xi RT

ninT + ni

[

∑i

[( ) ]

nT2 +

for i ) j

RT

xi ln(φi) + 5Qi ln

∂ /∂xi RT

Case b: Simple One-Parameter Transformation Equation (Only Binary Parameters). The elements of the Jacobi matrix are defined as

xi(1 +

)

[( ) )

∂xi ∂(ni/nT) ) ) -xj ∂nj ∂nj

for i * j

GE

GE

∂xi ∂(ni/nT) ) ) 1 - xi ∂nj ∂nj

for i ) j

Analytical Derivation of GE/RT with Respect to the Mole Fraction x (UNIQUAC). The UNIQUAC equation for GE/RT is

∑k xkSk

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Received for review April 8, 2005 Revised manuscript received June 30, 2005 Accepted July 18, 2005 IE050431W