Simultaneous Correlation of Liquid−Liquid, Liquid−Solid, and Liquid

Sep 14, 2007 - Mathematical model of liquid–liquid equilibrium for a ternary system using the GMDH-type neural network and genetic algorithm. H. Gha...
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Ind. Eng. Chem. Res. 2007, 46, 7030-7037

CORRELATIONS Simultaneous Correlation of Liquid-Liquid, Liquid-Solid, and Liquid-Liquid-Solid Equilibrium Data for Water + Organic Solvent + Salt Ternary Systems. Anhydrous Solid Phase M. M. Olaya, A. Marcilla, M. D. Serrano, A. Botella, and J. A. Reyes-Labarta* Dpto. Ingenierı´a Quı´mica, UniVersidad de Alicante, Apdo. 99, Alicante 03080, Spain

The simultaneous correlation of liquid-liquid (LL), liquid-solid (LS), and LLS equilibrium data for four water + organic solvent (alcohol or acetone) + inorganic salt (NaCl) ternary systems at 25 °C has been carried out. First, the NRTL equation was used to formulate the excess Gibbs energy (GE), assuming nondissociation of the dissolved salt in the liquid phase. Afterward, a modification of the NRTL equation taking into account the ionic behavior of the solid (electrolyte-NRTL) was introduced into the simultaneous correlation of the equilibrium regions for the same systems. The binary parameters and the comparison of the experimental and calculated equilibrium data obtained with both the molecular and electrolyte models are presented in this work. The stability analysis based on the Gibbs energy function of mixing, together with the common tangent criteria have been used to validate the parameters calculated. Introduction The addition of a nonvolatile solute, for example an electrolyte, to a solvent mixture modifies the interaction among the various solvents and solute molecules resulting in a phase equilibrium modification. If the liquid-liquid equilibrium (LLE) of an aqueous solute-organic solvent mixture is considered, the addition of an electrolyte generally enriches the organic phase with the solute component improving the extraction conditions (salting-out effect). An aqueous liquid-liquid equilibrium is the result of the action of intermolecular forces, mainly of the hydrogen-bonding type, and therefore, the addition of salt to such systems introduces ionic forces that affect the phase equilibria. Many are the industrial applications where the salting-out effect on LLE is used. In conventional solvent extraction, the addition of salt generally increases the distribution coefficients of the solute and the selectivity of the solvent for the extraction of the solute. Separation by solvent extraction becomes increasingly more difficult when tieline slopes decrease, thus becoming parallel to the solvent axis (solutropic solutions). The addition of an adequate salt significantly changes the mutual solubility, eliminating the solutrope. The salting-out effect on LLE also finds application in extractive fermentation,1-4 extractive crystallization,5,6 and solvent dehydration.7 A vast number of papers have been published on experimentally obtained LLE data for ternary and quaternary systems containing an inorganic salt. As example, some of these recently published papers are cited.8,9 Correlation of the equilibrium data facilitates equipment design. The first attempt to empirically correlate the LLE data of mixed solvents with the presence of a salt was the Setschenow equation.10 This was followed by the Hand equation.11 Eisen and Joffe12 proposed a new equation that was basically a * To whom correspondence should be addressed. Tel.: (34) 965 903789. Fax: (34) 965 903826. E-mail: [email protected].

combination of the Setschenow and Hand equations. Marcilla and co-workers13 proposed a modification of the Eisen-Joffe equation that resulted in a better fit of the experimental equilibrium data. On the other hand, molecular interaction based models for phase equilibria, such as NRTL or UNIQUAC, and their modified forms, to account for the presence of electrolytes, are increasingly used for data correlation of these systems. However, it is very important to point out that, in general, the papers dealing with any type of correlation involving inorganic salts only consider the liquid-liquid equilibrium region.14-18 Nevertheless, the parameters of a molecular thermodynamic model should be capable of representing all the equilibrium data of a system, at least at constant temperature and pressure, regardless of the aggregation state and the number of phases present. Some important efforts have recently been made to carry out the simultaneous correlation of solid-liquid-vapor (SLV) equilibrium data, but poor or inconsistent results are obtained for many systems. For example, Iliuta et al.19 and Thomsen et al.20 have used the extended UNIQUAC model for electrolyte solutions to predict the SLV equilibrium behavior of many systems, yielding poor results in some cases. Salts like NH4NO3, Na2CO3, and K2CO3 cause a liquid-phase split when added to a homogeneous solvent mixture, such as, for example, aqueous ethanol. However, the electrolyte-UNIQUAC model does not reproduce the LLE region in aqueous ethanol solutions with NH4NO3 or Na2CO3 and produces a three liquid-phase region (LLLE) in the water-rich part that does not actually exist in the system containing K2CO3.20 In previous papers,21,22 we presented and discussed the problematic nature of the calculation of phase equilibria involving solid components, illustrating some topological properties of the LLS equilibrium and looking mainly for a systematic procedure that would allow for the simultaneous correlation of all the different equilibrium regions LL, LS, and LLS present in water + organic solvent + salt systems. As a first approximation to the problem, the model used to formulate

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this is not the case for the equilibria involving different aggregation state phases such as solid-liquid equilibria. In these cases, different expressions for the Gibbs energy function must M S be applied for each phase, gM solid (g for a pure solid) and gliquid and the Gibbs stability criterium must be satisfied taking into account the possible presence of a solid phase in the equilibrium state of the system. LS and LLS equilibrium regions are defined by the one or two liquid composition points on the gM liquid surface, respectively, where the tangent plane to that surface also contains the Gibbs energy of the solid (gs) and no other point should exist below that tangent plane (minor tangent plane equilibrium condition). Data correlation procedure

Figure 1. Triangular prism employed for the qualitative representation of the phase diagram (top face) and the Gibbs energy surface for a water + organic solvent + salt ternary system.

the liquid-phase activity coefficients and the Gibbs energy was the original NRTL model, without taking into account the ionic behavior of the dissolved salt. The goal of this paper is to simultaneously correlate all the equilibrium regions present in four water + solvent + salt ternary systems at constant temperature and pressure, improving the previous correlation results obtained with the NRTL model22 and using the electrolyte-NRTL model suggested by Chen et al.23,24 with the Born contribution.25 The effectiveness of both types of correlations are evaluated and compared. In addition, the parameters obtained have been validated using phase equilibria stability criteria based on a topological analysis of the Gibbs energy surface together with the minor common tangent plane condition (tangent plane stability test). The sets of experimental data used in this work have been previously published by the corresponding authors.13,26,27 Stability Condition for LLSE in Water + Organic Solvent + Salt Ternary Systems In Figure 1, a triangular prism has been used as a qualitative representation of the phase equilibrium diagram (top face) for the LLSE of a ternary system with two partially miscible components (1 and 2) and an inorganic salt as the third component, where the height of the prism represents the dimensionless Gibbs energy of mixing values (gM ) GM/RT). Figure 1 is explained with great detail in a previous paper.22 The Gibbs energy of mixing surface for liquid mixtures shown in Figure 1 is topologically coherent or compatible with the phase equilibria diagram of this type of system, considering that the Gibbs energy for the pure solid salt is represented by gS in this figure.22 Gibbs proved that a necessary and sufficient condition for absolute stability of a mixture at a fixed temperature, pressure, and overall composition is that the Gibbs energy of mixing (gM) surface at no point be below the plane tangent to the surface at a given overall composition. This stability condition has been widely used and discussed in many references and is more frequently applied to liquid-liquid equilibrium,28 where the analytical expression for the Gibbs energy of mixing surface is the same for both phases (gM liquid). Obviously,

Model Description. The systems considered in this paper contain electrolytes, and therefore, ionic species are present in the liquid mixtures. However, as a first approximation to the problem, the NRTL model29 was used, in which dissociation in the liquid phase is considered to not occur. After that, the electrolyte-NRTL model23,24 with the Born contribution25 was used to formulate the activity coefficients and the Gibbs energy function. These two models include the corresponding binary parameters (Aij) to be calculated by the simultaneous correlation of the equilibrium data for all the regions. The electrolyte-NRTL model was originally proposed for aqueous electrolyte systems by Chen et al.23 It was later extended to mixed solvent electrolyte systems.24 Chen proposed an excess Gibbs energy expression which contains two contributions: one contribution for the local interactions that exist in the immediate neighborhood of any central species and the other for the long-range ion-ion interactions that exist beyond the immediate neighborhood of a central ionic species. The NRTL model is used for the local interactions and is developed as a symmetric model, based on pure solvent and pure completely dissociated liquid electrolyte reference states. The PitzerDebye-Hu¨ckel (PDH) model and the Born equation are used to take into account the long-range interactions. The PitzerDebye-Hu¨ckel model is formulated using unsymmetric reference states for solvent and ions species (* denotes an unsymmetric reference state). The Born equation is used to account for the Gibbs energy of transfer of ionic species from an infinite dilution state in a mixed solvent to an infinite dilution state in the aqueous phase. All these contributions are added to give the following equation:

G*E G*E,lc G*E,PDH G*E,Born ) + + RT RT RT RT

(1)

which leads to /PDH ln γ/i ) ln γ/lc + ln γ/Born i + ln γi i

(2)

The local composition contribution NRTL model has been used without modifications as it was first proposed by Chen et al.24 but must be normalized by the infinite dilution activity coefficients for the cation and anion (γ∞c and γ∞a ) in order to obtain the unsymmetric model:

G*E,lc GE,lc ) - xc ln γ∞c - xa ln γ∞a RT RT

(3)

The PDH equation, generalized to include mixed solvents, is given as

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G*E,PDH RT

ln

γ/PDH i

) -(

[

∑i xi)(4AφIx/F) ln(1 + FIx1/2)

zi2xIx - 2xIx3 2zi2 ln(1 + FxIx) + ) -Aφ F 1+F I

xx

(4)

]

(5)

where zi is the charge number of specie i; F is the “closest approach” parameter fixed at 14.9 as is used by Chen et al.;23,24 Ix the ionic strength Ix ) 1/2Σixizi2 (mole fraction scale); and Aφ is the Debye-Hu¨ckel parameter calculated as follows:

Aφ )

x( )

x

1 3

1000‚2πNAds Ms

e2 kT

3

(6)

where e is the electron charge,  is the dielectric constant for the solvent mixture, k is Boltzmann’s constant, ds is the solvent mixture density, NA is Avogadro’s number, and Ms is the solvent mixture molecular weight. The Born contribution is

G*E,Born RT

)

( )(∑ )

e2 1

-

2kT 

) ln γ/Born i

xizi2

1

w

(

i

(7)

ri

)

2 e2 1 1 zi 2kT  w ri

(8)

where  is the dielectric constant for the solvent mixture, w is the dielectric constant for water, and ri is the Born radius for Na+ and Cl-.30 Calculation of the Gibbs Energy for the Solid (gS). When it is possible for the system to consist of solid phases in equilibrium with one or more liquid phases, then at the temperature and pressure of the system, the Gibbs energy of the solid phase must be calculated and taken into consideration for the equilibrium conditions. To carry out that calculation, different procedures have been used depending on the model for the Gibbs energy function utilized. If the NRTL model is used to formulate the Gibbs energy, and therefore the dissociation of the salt in the liquid phase is not taken into account, the standard state for the salt component is a pure liquid at the system T and P and the gs value can be calculated as the chemical potential change from the pure liquid to the pure solid salt at those T and P conditions, using a thermodynamic cycle:31

gS )

( )

Figure 2. Graphical representation of the Gibbs energy calculation for the pure solid NaCl at 25 °C (gS,E) using the electrolyte-NRTL model.

species, but the aqueous salt infinitely diluted. In this sense, no information in literature has been found to solve this reference state change. For this reason, an alternative calculation procedure is suggested that requires the experimental data corresponding to the solubility of the salt in water and the Gibbs stability condition. First of all, the gM liquid curve is plotted against the water mole fraction along the salt-water binary using the electrolyte-NRTL model and the specific binary parameters for it. Next, the experimental solubility value of the salt in water at the T of the system is located, and then, the tangent line to S,E the gM liquid curve at this point is calculated. The required g value must correspond to the y-axis value obtained as the intersection of this tangent line, at the gM liquid curve in the experimental salt solubility composition, with the y-axis. In Figure 2, this procedure is schematically illustrated for the calculation of the Gibbs energy for the pure solid NaCl at 25 °C using the electrolyte-NRTL model with the binary parameters for water-NaCl from van Bochove et al.25 (A13 ) -3875.95 K; A31 ) -587.65 K). The result is

gS,E ) -1.53 Equilibrium Conditions. In order to simultaneously correlate all the equilibrium regions, the use of the isoactivity criteria for the liquid-liquid region (LLE) and the minimization of the overall Gibbs energy of mixing for the liquid-solid equilibrium regions (LSE and LLSE) is combined. The isoactivity criteria, defined as follows, must be satisfied by all the components of the system

Tf GS ∆µS ∆hf ) ) 1+ RT RT RTf T 1 RT

∫TT ∆Cp dT - R1 ∫TT f

aLi 1 ) aLi 2 or xLi 1‚γLi 1 ) xLi 2‚γLi 2 ∆Cp dT (9) T

f

where ∆hf is the enthalpy of fusion at the normal melting temperature, ∆Cp is the difference between heat capacity of the liquid and that of the solid phase, and Tf is the normal melting temperature. The value obtained for the Gibbs energy of pure solid NaCl at 25 °C using eq 9 was

gS ) -10.94 When the electrolyte-NRTL model is used, the Gibbs energy value for the pure solid phase (gS,E) cannot be calculated using a thermodynamic cycle as with the NRTL model due to the fact that the reference state for the salt is not the pure liquid

(10)

where aLi and γLi denote the activity and the activity coefficient, respectively, of the component i in liquid phases L1 and L2. On the other hand, the LSE calculations by minimization of the overall Gibbs energy require that the contribution of the liquid mixture and the solid phase to the Gibbs energy be taken into account. The overall Gibbs energy of mixing for 1 mol of overall composition mixture is given by

gM overall

)

GM overall RT

)

GS RT

+

GM liquid RT

∆µS

) s‚

+

RT ∆µLi

c

(1 - s)

∑ i)1

xLi ‚

RT

(11)

Ind. Eng. Chem. Res., Vol. 46, No. 21, 2007 7033 Table 1. Correlated Ternary Water + Alcohol (or Acetone) + NaCl Systems at 25 °C system

components

A

water (1) + 1-butanol (2) + NaCl (3) water (1) + 3-pentanol (2) + NaCl (3) water (1) + acetone (2) + NaCl (3) water (1) + ethanol (2) + NaCl (3)

B C D

heterogeneous regions

ref

LL, LLS, and LS

13

LL, LLS, and LS

26

LL, LLS, and LS

27

LS

13

Table 2. Correlation Results Using Both the NRTL and Electrolyte-NRTL Methods for the Water (1) + 1-Butanol (2) + NaCl (3) System at 25 °C, Binary Parameters Aij (K), Objective Function (OF), and the Standard Deviation (σ) NRTL

electrolyte-NRTL

j 1 i

1 2 3

j

2

3

1530.09

1016.14 -447.61

-320.17 -1878.67 9465.99 OF(LL) ) 5.1765 × 10-3 OF(LS) ) 1.0730 × 10-3 σ ) 0.659

1

2

3

1507.23

-345.86 20.59

-314.33 -715.43 8919.39 OF(LL) ) 3.4051 × 10-2 OF(LS) ) 2.1374 × 10-2 σ ) 1.962

Table 3. Correlation Results Using Both the NRTL and Electrolyte-NRTL Methods for the Water (1) + 3-Pentanol (2) + NaCl (3) System at 25 °C, Binary Parameters Aij (K), Objective Function (OF), and the Standard Deviation (σ) NRTL

electrolyte-NRTL

j 1 i

1 2 3

j 2

3

1303.83

1560.10 3033.98

-116.18 -1883.29 2701.56 OF(LL) ) 3.760 × 10-3 OF(LS) ) 8.0684 × 10-6 σ ) 0.670

1

2

3

1425.24

-2801.65 7248.74

-114.98 -169.10 34326.39 OF(LL) ) 2.4426 × 10-2 OF(LS) ) 1.8187 × 10-2 σ ) 2.252

Table 4. Correlation Results Using NRTL for the Water (1) + 1-Acetone (2) + NaCl (3) System at 25 °C, Binary Parameters Aij (K), Objective Function (OF), and the Standard Deviation (σ) NRTL j 1 i

1 2 3

2

3

557.74 1460.40 -170.73 651.89 -1923.51 9413.90 OF(LL) ) 3.5644 × 10-3 OF(LS) ) 1.3229 × 10-3 σ ) 0.595

where s is the ratio (moles of solid phase)/(moles of mixture) and (1 - s) the ratio (moles of liquid phase)/(moles of mixture). The liquid contribution in eq 11 has been calculated using both the NRTL and electrolyte-NRTL models. The solid contribution to the Gibbs energy of the overall system is calculated as it was described in the previous section. Equation 11 solves the LS equilibrium problem, i.e., for any overall composition which separates into a solid and a liquid phase, a minimum in the overall gM with respect to “s” must exist. The composition of such a minimum, if it exists, does not depend on the composition of the initial mixture, although the actual value of gM overall, logically, varies with the overall composition. The application of this Gibbs stability criterion, that implies the minimization of eq 11 with respect to s, has been discussed elsewhere.21

Figure 3. Graphical comparison of the experimental and calculated (NRTL and eletrolyte-NRTL) equilibrium data (molar fractions) for the ternary system water + 1-butanol + NaCl at 25 °C.

The simultaneous correlation of the LL, LS, and LLS equilibrium data requires solving eq 10 together with the minimization of eq 11, employing the same set of parameters of the model used to formulate the activity coefficients in the LLE region and the Gibbs energy of the liquid mixtures in the LSE regions. Model Parameter Estimation. The optimization of the binary parameters (Aij) of the model has been done using the simplex flexible method with the objective function defined as in eq 12. This objective function is the result of two contributions: one corresponding to the two liquids zone, OF(LL), and the other corresponding to the zone where a solid is present in equilibrium, OF(LS): nLL 3

OF ) OF(LL) + OF(LS) )

∑∑[[((xi)n,or)exp -

n)1 i)1

((xi)n,or)cal]2 + [((xi)n,ac)exp - ((xi)n,ac)cal]2] + nLS 3

[[((xi)n,L)exp - ((xi)n,L)cal]2] ∑ ∑ n)1 i)1

(12)

Where, (xi)n is the molar fraction of component i on tieline n; nLL and nLS denote the number of tielines in the LL and LS regions, respectively; or, aq and L represent the organic, aqueous, and liquid phases; and exp and cal are the experimental and calculated equilibrium data. In the LL region, the Newton Raphson method is applied to solve eq 10 together with the

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Table 5. Correlation Results Using Both the NRTL and Electrolyte-NRTL Methods for the Water (1) + Ethanol (2) + NaCl (3) System at 25 °C, Binary Parameters Aij (K), Objective Function (OF), and the Standard Deviation (σ)

1 i

1 2 3

NRTL

electrolyte-NRTL

j

j

2

3

10.386

-125.51 -450.79

-389.32 -849.12 713.21 OF(LS) ) 3.1347 × 10-6 σ ) 0.0218

1

2

3

1009.76

-896.21 5809.27

-66.30 -989.27 10358.24 OF(LS) ) 3.6353 × 10-4 σ ) 0.234

mass balances required, in order to obtain calculated compositions for all the components of the system. The Newton Raphson method is strongly initialization dependent and can lead to trivial solutions; special attention to these difficulties has been paid while solving equilibrium calculations. The calculated compositions in the LS zone are obtained by minimizing the overall Gibbs energy of mixing given by eq 11. For the correlation of the experimental tietriangle (LLSE data), we divided it into one LL and two LS conjugated equilibrium compositions, which are included in the corresponding contribution to the objective function represented by eq 12. The parameter Rij of the NRTL model is, initially, a fitting parameter. Nevertheless, a good correlation is obtained with a

Figure 5. Three-dimensional representation of the surface, the gs value, and their common tangent lines and triangles using (a) the NRTL and (b) the electrolyte-NRTL models with the parameters obtained in the correlation of the water + 1-butanol + NaCl system at 25 °C.

constant value of Rij; 0.2 is frequently used in the literature.31 For the correlations carried out using the electrolyte-NRTL model, the values for Rij suggested by van Bochove et al.25 have been used: R1,2 ) 0.2, R1,3 ) 0.1, R2,3 ) 0.3. Data Correlation Results and Discussion Experimental equilibrium data at 25 °C for four ternary systems involving liquid and solid phases have been correlated using the suggested procedure (Table 1). Tables 2-5 show the parameters Aij obtained in the correlations carried out and the accuracy of the data fit given by the objective function values (OF) calculated using eq 12 and the standard deviations given by the following:

σ ) 100 × Figure 4. Graphical comparison of the experimental and calculated (NRTL and eletrolyte-NRTL) equilibrium data (molar fractions) for the ternary system water + 3-pentanol + NaCl at 25 °C.

x

n

c

cal 2 exp cal 2 {(xexp ∑ ∑ ji - xji )I + (xji - xji )II } j)1 i)1

2cn

(13)

Experimental and calculated results have also been graphically represented in Figures 3-6. Some magnifications of a part of

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Figure 7. Representation of the two NRTL binary parameters for both totally miscible (1 L) and partially miscible (LL) binary systems with Rij ) 0.2 (data obtained from the DECHEMA Chemistry Data Series32).

Figure 6. Graphical comparison of the experimental and calculated (NRTL) equilibrium data (molar fractions) for the ternary system water + acetone + NaCl at 25 °C.

the triangular diagram in these figures have been included for a better comparison of the results. It can be observed that fitting all of the regions using the same set of Aij Values is possible and yields acceptable results when the NRTL model is used. For both systems, water + 1-butanol (or 3-pentanol) + NaCl at 25 °C (see Figures 3 and 4, respectively), the NRTL model can successfully describe all the equilibrium regions of the systems within a good approximation. However, the electrolyte-NRTL model has not been able to provide a good approximation to the LLS s equilibrium region (tietriangle). The gM liquid surface and the g (gS,E) value for NaCl as a pure solid are quite different depending on the model used. For instance, in Figure 5a and b, the gM liquid surface, the gs value, and the corresponding tangent lines and triangles have been plotted using the NRTL and the electrolyteNRTL models, respectively, with the parameters obtained from the correlation of the water + 1-butanol + NaCl system. For both systems, water + 1-butanol (or 3-pentanol) + NaCl at 25 °C, the electrolyte-NRTL function seems to be less flexible in simultaneously reproducing, using the same set of parameters, the LL and LLS equilibrium regions. The water + acetone + NaCl system (Figure 6) presents an additional difficulty, the presence of the LL region because the water + acetone binary system is completely miscible. In our opinion, a set of parameters must be consistent with all the phase equilibrium behavior of the system, at least at constant T and P. Therefore, in correlating this system, the restriction of total miscibility must be used for the two parameters corresponding

Figure 8. Three-dimensional representation of the surface, the gs value, and their common tangent lines and triangles using the NRTL with the parameters obtained in the correlation of the water + acetone + NaCl system at 25 °C.

to the water + acetone binary (A12 and A21). We have found a very useful relation between the values of the NRTL binary parameters and the phase equilibrium behavior of a binary system (Figure 7). First, we plotted Aij against Aji for many totally and partially miscible binary systems, using the NRTL model with the parameter Rij ) 0.2. The DECHEMA Chemistry Data Series Collection32 was used as a source for this data. In that representation, the points are divided into two zones: one for the liquid-phase split and the other for total miscibility of the binary system. After that, we looked for the boundary between these two zones, assigning values to the two Aij and Aji parameters and checking the equilibrium behavior of a binary system calculated using the NRTL model. We found that there is a line that separates the zone where the binary systems are completely miscible (1 L) and the zone where splitting into two liquid phases occurs (LL). This line can be approximated by the following polynomial function

Aij ) aAji3 + bAji2 + cAji + d with the following values for the constants

a ) -4.46564 × 10-8 b ) 2.95745 × 10-4 c ) 1.20662 d ) 766.908

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Figure 9. Dimensionless Gibbs energy of mixing (NRTL) versus molar fraction of the acetone (2) in a cross section that contains one tieline in the middle of the LL region of the water + acetone + NaCl system at 25 °C.

When the electrolyte-NRTL model has been used for the correlation of the water + acetone + NaCl system at 25 °C, it has not been possible to obtain a set of parameters that roughly fit the ternary experimental data, and which would also be coherent with the total miscibility behavior required for the water + acetone binary system at 25 °C. For the water + ethanol + NaCl ternary system at 25 °C, a good data fit is obtained using both the NRTL and electrolyteNRTL models, as is shown in the results presented in Table 5 and Figure 10. We found no improvement by assuming the dissociation of the salt into ions. The restrictions included in the equation for the activity coefficient due to the electrolyte consideration result in a loss of flexibility of the corresponding equation, that still keeps two parameters per binary. Consequently, the equations to model the electrolytic behavior of the salt should be developed on a physically sound basis, but should also be capable of providing improved correlations as compared to those corresponding to the molecular approximations that only take into account the short-range interactions. Conclusions

Figure 10. Graphical comparison of the experimental and calculated (NRTL and eletrolyte-NRTL) equilibrium data (molar fractions) for the ternary system: water + ethanol + NaCl at 25 °C.

The parameters given in Table 4, that correspond to the calculated equilibrium represented in Figure 6, satisfy the condition of total miscibility for the water + acetone pair at 25 °C. The experimental behavior of the system requires that a small liquid-liquid ternary region appears close to the completely miscible water + acetone binary system. The consequence of this fact in the topology of the gM surface versus composition is the slight curvature of that surface in the 2 L s region. Figure 8 represents the gM liquid surface, the g value, and the tangent lines and triangle for the parameters given in Table 4. Figure 9 shows, as an example, the cross section of the gM surface versus composition on one of the calculated tielines in the middle of the LL region for the water + acetone + NaCl system at 25 °C, obtained using the NRTL model with the parameters given in Table 4. We can observe that the gM line between the calculated equilibrium compositions is very straight, which may result in difficulties in the calculation, and even false tielines can be obtained with a very low isoactivity objective function. Therefore, the minor tangent plane criteria or the minimization of the global Gibbs energy of mixing have been used for this system to validate the given solution. Many are the examples that we have found in correlating equilibrium data using the NRTL model, where the topology of the gM function gives poorly defined equilibrium regions because of the flat curve, surface, or hyper-surface obtained between the equilibrium compositions.

The analysis of the shape of the Gibbs energy function and application of the proposed procedure allows for the simultaneous correlation of the LL, LS, and LLS equilibrium regions for ternary systems containing an inorganic salt. Although the electrolyte-NRTL model is formally more correct for the systems considered in this work, the correlations obtained were not improved, and even poorer results were obtained when the ionic interactions were taken into account. This fact seems to be related to the topology of the gM liquid surface obtained with the electrolyte-NRTL model and the Gibbs energy for the pure solid NaCl (gS,E), that makes the simultaneous fitting of the LL and LLS regions difficult. The main problems arise for the correlation of the water + acetone + NaCl system at 25 °C, because of the existence of one ternary liquid-liquid region (LL) because the water + acetone binary system is completely miscible at this temperature. A rough correlation of the experimental equilibrium data for this system is obtained using the NRTL model, but the parameters obtained give a very flat gM surface which indicates a poor definition of the equilibrium regions. The best results obtained in the simultaneous correlation of this system using the electrolyte-NRTL model have not been acceptable. Also, a polynomial equation has been deduced that relates the two NRTL binary parameters, Aij and Aji, which generates either completely miscible or partially miscible behavior for the binary system. This equation can be used in the ternary or higher equilibrium data correlations using the NRTL model, to obtain parameters that are coherent with the behavior of the binary systems. Nomenclature ai ) activity of component i γi ) activity coefficient of component i Aij ) binary interaction parameters Rij ) NRTL nonrandomness factor c ) number of components lc ) local composition Mi ) molecular weight of component i d ) density (kg/m3) k ) Boltzmann constant (J/K)

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n ) number of experimental data points xi ) molar fraction of component i GE ) excess Gibbs energy (J/mol) GM, gM ) Gibbs energy of mixing (J/mol and dimensionless, respectively) gS, gS,E ) Gibbs energy of the solid (dimensionless) with NRTL and electrolyte-NRTL respectively µi ) chemical potential of component i (J/mol) S ) (moles of solid phase)/(moles of mixture) ∆hf ) heat of fusion (J/mol) ∆Cp ) difference between heat capacity of the liquid and that of the solid phase at the melting point (J/(mol K)) T ) temperature (K) Tf ) melting point (K) R ) gas constant (J/(mol K))  ) dielectric constant (C2/(N m2)) e ) electron charge (C) F ) “closest approach” parameter ri ) Born radius for Na+ or Cl- (m) zi ) charge number of species i Ix ) ionic strength Aφ ) Debye-Hu¨ckel parameter σ ) average deviation Subscripts i, j ) component i,j n ) tieline n c ) cation a ) anion w ) water exp ) experimental value cal ) calculated value or ) organic phase ac ) aqueous phase Superscripts L1, L2 ) liquid phases S ) solid phase ∞ ) infinite dilution Literature Cited (1) Malinowski, J. J.; Daugulis, A. J. Liquid-liquid and vapour-liquid behavior of oleyl alcohol applied to extractive fermentation processing. Can. J. Chem. Eng. 1993, 71, 431. (2) Malinowski, J. J.; Daugulis, A. J. Salt effects in extraction of ethanol, 1-butanol and acetone from aqueous solutions. AIChE J. 1994, 40, 1459. (3) Demirci, A.; Pometto, A. L.; Hok-L, G. Ethanol production by Saccharomyces cerevisiae in biofilm reactors. J. Ind. Microrbiol. Biot. 1997, 19, 299-304. (4) Veljkovic, V. B.; Lazic, M. L.; Stankovic, M. Z. Repeated use of yeast biomass in ethanol fermentation of juniper berry sugars. World J. Microb. Biot. 2006, 22, 519. (5) Lynn, S.; Schiozer, A. L.; Jaecksch, W. L.; Cos, R.; Prausnitz, J. M. Recovery of anhydrous Na2SO4 from SO2-scrubbing liquor by extractive crystallization: liquid-liquid equilibria for aqueous solutions of sodium carbonate, sulfate, and/or sulfite plus acetone, 2-propanol, or tert-butyl alcohol. Ind. Eng. Chem. Res. 1996, 35, 4236. (6) Dionysiou, D.; Tsianou, M.; Botsaris, G. Extractive crystallization for the production of calcium acetate and magnesium acetate from carbonate sources. Ind. Eng. Chem. Res. 2000, 39, 4192. (7) Meissner, H. P.; Stokes, C. A. Solvent dehydratation by salting out. Ind. Eng. Chem. 1944, 36, 816. (8) Santos, F. S.; d’Avila, S. G.; Aznar, M. Salt effect on liquid-liquid equilibrium of water plus 1-butanol plus acetone system: experimental determination and thermodynamic modelling. Fluid Phase Equilib. 2001, 187, 265.

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ReceiVed for reView April 22, 2007 ReVised manuscript receiVed July 17, 2007 Accepted July 26, 2007 IE0705610