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Ind. Eng. Chem. Res. 2008, 47, 2100-2108
CORRELATIONS Simultaneous Correlation of Liquid-Liquid, Liquid-Solid, and Liquid-Liquid-Solid Equilibrium Data for Water + Organic Solvent + Salt Ternary Systems: Hydrated Solid Phase Formation Antonio Marcilla,* Juan Antonio Reyes-Labarta, Marı´a del Mar Olaya, and Marı´a Dolores Serrano Departamento de Ingenierı´a Quı´mica, UniVersidad de Alicante, Apdo. 99, Alicante 03080, Spain
The condensed phase equilibrium behavior of water + organic solvent + salt systems, when different hydrates are formed in the solid phase, increases the number of equilibrium zones, substantially complicating the simultaneous correlation of the equilibrium data in all the existing regions. In this paper, a procedure to perform the correlation of this type of system is presented in great detail, using, as an example, the water + 1-butanol + LiCl ternary system, where the salt appears in two forms: anhydrous and monohydrated (LiCl‚ H2O). To this end, the non-random two-liquid (NRTL) model for the excess Gibbs energy (GE) has been implemented, because the assumption of an electrolytic character for the solution in the liquid phase, despite being potentially more correct physically, does not give more flexibility to the GE function, as was discussed in a previous work. The common tangent plane criterion is used as an equilibrium condition to determine the equilibrium compositions in all the regions. The results obtained show the limitations of the NRTL equation to model this type of system, as a consequence of its lack of flexibility, in terms of the topological concepts that are related to the Gibbs energy of mixing (GM) function and the tangent plane criterion. 1. Introduction The industrial applications in which the salting-out effect is used on liquid-liquid equilibria (LLE) are many: extractive fermentation,1-4 extractive crystallization,5,6 solvent dehydration,7 etc. The addition of a salt to a solvent mixture modifies the interaction among the various solvent and solute molecules, resulting in a phase equilibrium modification that is usually favorable to the desired separation. For example, for an aqueous-organic solvent mixture, the addition of an electrolyte generally causes an enrichment of the organic phase with the solute component, improving the extraction conditions. Consequently, LLE for water + organic solvent + salt ternary systems are of interest for many unit operations. Molecular-interaction-based models for phase equilibria, such as non-random two-liquid (NRTL) or UNIQUAC, and their modified forms (to account for the presence of electrolytes), are generally used for data correlation of solvent mixture + salt systems where solid phases are present. The ability of a given model to represent all the equilibrium regions using a unique set of parameters is a necessary condition for the model to be considered to be thermodynamically consistent. In addition, this condition is very advantageous for design calculations. Despite this situation, many papers that involve equilibrium data far from critical conditions, where solid and liquid phases are both present, only consider LLE when trying to fit experimental data with a model.8-12 Sometimes, when the solid phase is included in the phase equilibria regression, the liquid-liquid (LL) and solid-liquid (SL) data sets are fitted separately, adducing the difficulty of classical models to describe different types of equilibria with a single set of parameters.13 When simultaneous correlation of all the * To whom correspondence should be addressed. Tel.: (34) 965 903789. Fax (34) 965 903826. E-mail address:
[email protected].
phase equilibrium regions of systems that involve solids under moderate temperature and pressure conditions has been conducted, poor results or inconsistencies that are due to the behavior of the models have frequently been found. For example, Iliuta et al.14 and Thomsen et al.15 have used the extended UNIQUAC model for electrolyte solutions to predict the solidliquid-liquid-vapor (SLLV) equilibrium behavior of many systems, but poor results are obtained for some of them. Salts such as 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 that contain NH4NO3 or Na2CO3, and produces a three-liquid-phase region (LLLE) in the water-rich part that actually does not exist in the system that contains K2CO3.15 Moreover, no article has been found that includes a liquidliquid-solid equilibrium (LLSE) data correlation in which the solid phase corresponds to different hydrates of a salt. In a previous paper,16 we presented a procedure to simultaneously correlate the equilibrium data of all the equilibrium regions for a ternary system: water + organic solvent + inorganic salt. Afterward, some correlations were improved and the results that were obtained using the molecular NRTL model17 and the electrolyte-NRTL model18-20 were compared.21 No improvement was found in the regression results by assuming dissociation of the salt into ions: the restrictions imposed on the equation for the activity coefficient, taking the electrolyte into account, resulted in a loss of flexibility of the model that still kept the same number of variables. The K-value method, which imposes equality of chemical potentials as equilibrium condition,15 is frequently used to formulate phase equilibrium conditions. Nevertheless, it has been proven22 that, because of the high nonlinearity of the Gibbs energy models, this method is slowly convergent and can result
10.1021/ie071290w CCC: $40.75 © 2008 American Chemical Society Published on Web 02/27/2008
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Figure 1. Qualitative representation of the phase diagram for a ternary system water + organic solvent + salt when the solid phase can be present in both anhydrous and monohydrated forms.
in multiple solutions that seem to satisfy the isofugacity condition, being local Gibbs energy minima. Therefore, all equilibrium calculations should be carefully validated using a stability analysis: the necessary and sufficient condition for the absolute stability of a mixture at a certain pressure, temperature, and overall composition is that at no point should the Gibbs energy of mixing (GM) surface be below the tangent plane to the surface at the overall composition. In a previous paper,21 we performed the fitting of equilibrium data sets for some water-organic solvent-salt systems, using the equality of activities for the LLE and global Gibbs energy minimization for the liquid-solid equilibria (LSE). Results were validated geometrically, so that the tangent plane criterion was met. When different hydrates are present in the solid phase of a system, the number of equilibrium regions increases and this complicates the simultaneous correlation of the equilibrium data in all the existing regions. In this paper, a procedure for the simultaneous correlation of LLSE data at constant temperature and pressure for water + solvent + salt systems, where hydrates are formed in the solid phase, is presented. To illustrate the method, the water + 1-butanol + LiCl23 ternary system at 298 K, where the salt appears in both anhydrous and monohydrated form, is used as example. Gibbs energy minimization, which improves convergence and avoids false equilibrium solutions, is used as the equilibrium condition. This equilibrium requirement is based on the equations proposed by Iglesias-Silva et al.,24 which guarantees fulfillment of the common tangent plane condition. The ideas presented in this paper can be extended to similar systems with different numbers of hydrates and, therefore, to systems with different numbers of equilibrium regions and of varying complexity. The excess Gibbs energy (GE) has been formulated using the NRTL equation, but different models could also be used. 2. Phase Diagram for LLSShE in Water + Organic Solvent + Salt Ternary Systems and the Gibbs Stability Criterion For water + organic solvent + salt ternary systems, with two possible forms for the solid phasesanhydrous (S) and monohydrated (Sh)ssix different heterogeneous equilibrium regions are found. These regions (Figure 1) are liquid-liquid (LL), liquid-anhydrous solid (LS), liquid-anhydrous solid-monohydrated solid (LSSh), liquid-liquid-monohydrated solid (LLSh), and two separate liquid-monohydrated solid (LSh) regions. Figure 2a shows a triangular prism used for the qualitative representation of the phase equilibrium diagram (top face), for a ternary system with two partially miscible components (1 and 2) and one inorganic salt as the third component (3), in which
the aforementioned different combinations of the liquid-liquidanhydrous solid-monohydrated solid (LLSSh) equilibrium are present. The height of the prism represents the dimensionless Gibbs energy of mixing values (gM ) GM/(RT)), assuming that all the components are in their standard states as pure liquids at the same temperature and pressure of the system. This figure complements the one presented with great detail in a previous paper16 for a mixed solvent-anhydrous salt ternary system. The Gibbs energy of mixing surface for the liquid mixtures shown in Figure 2a is topologically coherent or compatible with the phase equilibrium diagram for this type of system. In this figure, the Gibbs energy for the pure solid salt is represented by gS and the Gibbs energy for the monohydrated solid salt is represented by gSh. Gibbs proved that a necessary and sufficient condition for absolute stability of a mixture at a fixed temperature, pressure, and overall composition is that at no point should the Gibbs energy of mixing (gM) surface be below the plane tangent to the surface at a given overall composition. This stability condition has been widely used and explained in many references and, more frequently, applied to LLE,25 where the analytical expression of the Gibbs energy of mixing surface is the same for both equilibrium phases (gM). Obviously, this is not the case for equilibria that involve different aggregation state phases such as a solid-liquid equilibrium (SLE). In these cases, different expressions for the Gibbs energy function must be applied for each phase: gS for a pure solid, gSh for the monohydrated solid, and gM for the liquid. According to the equilibrium conditions based on the tangent plane criterion, the following are the topological conditions required for the Gibbs energy surface, defined by any model, to reproduce simultaneously the equilibrium behavior of the system in all the regions: The LL region: There must be a zone on the gM surface with conjugated pairs of points (R,β) where common tangent planes to the surface can be found (see Figure 2b). These conjugated points define the LL tielines of the system. The LS region: The Gibbs liquid surface must have a curve that connects all the points (such as R in Figure 2c), where the tangent planes also contain the Gibbs energy value for the anhydrous solid (gS). The LSh regions: There must be two separate Gibbs surface regions for which two curves can be found that connect points (such as R in Figure 2d), where all tangent planes to the surface contain the Gibbs energy for the monohydrated solid (gSh). The LLSh region: A plane containing the Gibbs energy for the monohydrated solid should be tangent to the liquid surface at two points that correspond to two liquid phases in equilibrium with the monohydrate Sh (see Figure 2e). The LSSh region: Another tangent plane passing through the Gibbs energy for the anhydrous and monohydrated solid should only have a point on the liquid Gibbs energy surface (see Figure 2f). The 1L homogeneous regions. In the 1L homogeneous regions, the curvature changes on the liquid-gM surface do not permit the existence of common tangent planes at different points on the surface or between the surface and any of both of the solid points. In other words, the planes that are tangent to the surface in this region only touch it a single point and never contain the Gibbs energy of any of the solids. Therefore, the phase equilibrium behavior for this type of system is very complex. Many are the requirements that must satisfy the Gibbs energy of mixing surface for the liquid, together with the gM values for the two existing solid phases (gS and gSh), to fulfill the phase equilibrium conditions that are based on the tangent plane criterion.
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Figure 2. Phase equilibria representation for a water + organic solvent + anhydrous/monohydrated salt in a triangular prism: (a) ternary phase diagram and gM function; (b) liquid-liquid (LL) diagram; (c) liquid-anhydrous solid (LS) diagram; (d) liquid-monohydrated solid (LSh); (e) liquid-liquidmonohydrated solid (LLSh) diagram; and (f) liquid-anhydrous solid-monohydrated solid (LSSh).
3. Data Correlation Procedure For the correlation of the equilibrium data, the procedure suggested by Iglesias24 has been used for the equilibrium calculations. Such a procedure solves orthogonal derivatives, the tangent plane equations, and the mass balances simulta-
neously. These authors used this procedure for binary and ternary systems with a different number of equilibrium phases, but all of them were liquid phases. In the present paper, we have adapted this method to the simultaneous correlation of several equilibrium regions, combining liquid and solid phases.
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To perform simultaneous phase equilibrium calculations for all the equilibrium regions of systems that contain liquid and solid phases, the Gibbs energy of mixing for the liquid mixtures and for all the solid phases that could be present must be calculated. 3.1. Gibbs Energy for the Liquid Mixtures. For liquid mixtures, the dimensionless Gibbs energy of mixing (gM) can be formulated as
gM )
GE
+
RT
∑i
xi ln xi
(1)
where, in the present paper, the NRTL model17 has been used for the excess Gibbs energy (GE). The model includes the corresponding binary parameters (Aij) to be calculated via simultaneous correlation of the equilibrium data for all the regions of the system. 3.2. Gibbs Energy for the Anhydrous Solid (gS). Taking into account the use of a unique reference state (pure liquid at the same temperature and pressure of the system), for all the components of the system,26 the Gibbs energy for the anhydrous solid salt can be calculated as the chemical potential change from the pure liquid to the pure solid salt at the system temperature (T) and pressure (P) conditions, using a thermodynamic cycle:27
gS )
( )
Tf ∆µS ∆hf 1+ ) RT RTf T 1 RT
where A and B are tabulated parameters:30 A ) 3.084, B ) -2905.6. The values of G0salt and GH0 2O are set to zero at all temperatures. The Gibbs energy for the monohydrated salt LiCl‚H2O has been calculated using eq 3: 0 GLiCl‚H 2O
RT
3.4. Equilibrium Conditions. Assuming that the molar Gibbs energy is a function of the mole fraction,
g ) g(x1,x2,x3)
f
f
∆Cp dT (2) T
where µS is the chemical potential for the solid (expressed in units of J/mol), ∆hf the enthalpy of fusion at the normal melting temperature28 (also expressed in units of J/mol), ∆Cp the difference between the liquid and solid heat capacities29 (expressed in units of J mol-1 K-1), and Tf the normal melting temperature28 (in Kelvin). The dimensionless Gibbs energy for pure LiCl solid at 298 K and 1 atm, with respect to the reference state previously defined, has been calculated using eq 2:
∆µS ) gS ) -5.443 RT 3.3. Gibbs Energy for the Hydrated Solid. The Gibbs energy of formation (G0) of the hydrated solid phase, with respect to reference states (hypothetical subcooled molten salt and liquid water), must be calculated. According to the resemblance of salt hydration to gas adsorption on solid surfaces, the Brunauer-Emmett-Teller (BET) multilayer gas adsorption model has been applied to describe the relationship between water activity and salt concentration in extremely concentrated solutions.30 The calculation can be outlined as follows:
salt‚mH2O(s) T salt(l) + mH2O(l) 0 RTgSh ) Gsalt‚mH ) Gsalt + mGH2O 2O
) G0salt + RT ln(asalt(l)) + mGH0 2O + mRT ln(a H2O(l))
g ) g(x1,x2)
(
) RT ln(ksalt‚mH2O(l)) ) RT A +
B T
)
(3)
(5)
The corresponding nonlinear equations that must be solved simultaneously for all the equilibrium regions are formulated as follows.24 The LL region: A ternary mixture of two liquid phases R and β in equilibrium is defined geometrically by a tie-line as defined by the following conditions:
F2 ≡
( ) ( ) ∂g ∂x1 ∂g ∂x2
R
( ) ( ) ( ) ∂g ∂x1
-
T,P,x2 R
∂g ∂x2
-
T,P,x1
F3 ≡ (gβ - gR) - (xβ1 - xR1 )
∂g ∂x1
R
β
)0
(6)
)0
(7)
T,P,x2 β T,P,x1
-
T,P,x2
( )
(xβ2 - xR2 )
∂g ∂x2
R
) 0 (8)
T,P,x1
The LS region (anhydrous or hydrated solid): A ternary mixture of one liquid phase (R phase) in equilibrium with a solid is defined graphically by a plane tangent to the Gibbs energy surface at the liquid equilibrium composition xRi that also includes the Gibbs energy for the solid (g*).
F4 ≡ (g* - gR) - (x*1 - xR1 )
( ) ∂g ∂x1
R
-
T,P,x2 R (x* 2 - x2 )
( ) ∂g ∂x2
R
) 0 (9)
T,P,x1
The asterisk symbol (*) refers to an anhydrous (S) or hydrated (Sh) solid, so eq 9 can be used to model both types of equilibrium (LS and LSh). The LLSh region: In this region, the phase equilibrium is defined geometrically by a plane tangent to the Gibbs energy surface at two points for two liquid phases (e.g., R and β), and this plane must also contain the point corresponding to the Gibbs energy for the hydrated solid (gSh).
F5 ≡
) RT ln(asalta H2O(l)m)
(4)
if the material balance ∑3i xi ) 1 is used to substitute x3 for x1 and x2, the aforementioned equation becomes
F1 ≡
∫TT ∆Cp dT - R1 ∫TT
) gSh ) -6.661
F6 ≡
( ) ( ) ∂g ∂x1
∂g ∂x2
R
-
T,P,x2
R
T,P,x1
-
( ) ( ) ∂g ∂x1
∂g ∂x2
β
)0
(10)
)0
(11)
T,P,x2 β T,P,x1
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Figure 3. Flux diagram for simultaneous phase equilibria calculations of water + organic solvent + salt (anhydrous/monohydrated) systems.
F7 ≡ (gβ - gR) - (xβ1 - xR1 )
( ) ∂g ∂x1
R T,P,x2
(xβ2 - xR2 )
( )
R F8 ≡ (gSh - gR) - (xSh 1 - x1 )
∂g ∂x1
F9 ≡ (gS - gR) - (x1S - x1R)
-
R
( ) ∂g ∂x2
R
R (xSh 2 - x2 )
T,P,x1 R F10 ≡ (gSh - gR) - (xSh 1 - x1 )
( ) ∂g ∂x2
R
) 0 (13)
T,P,x1
The LSSh region: The plane that contains the anhydrous and hydrated solid Gibbs energy (gS, gSh) is tangent to the Gibbs energy surface at the composition of the liquid phase (xRi ).
∂g ∂x1
R
-
T,P,x2
(x2S - x2R)
) 0 (12)
-
T,P,x2
( )
( ) ∂g ∂x1
R
( ) ∂g ∂x2
R
) 0 (14)
T,P,x1
-
T,P,x2
R (xSh 2 - x2 )
( ) ∂g ∂x2
R
) 0 (15)
T,P,x1
3.5. Model Parameter Estimation. The optimization of the binary parameters (Aij) of the model has been done using the Simplex Flexible method,31 with the objective function defined as given below in eq 16. This objective function is the result of
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Figure 4. Dimensionless Gibbs energy of mixing (gM) and its derivatives for a binary system.
two contributions: one corresponds to the LLE zone (O.F.(LL)) and the other corresponds to the zones where a solid (anhydrous or hydrated) is present in the equilibrium (O.F.(LS)).
binary interaction parameter, Aij (K)
O.F. ) O.F.(LL) + O.F.(LS) nLL
)
i
3
∑ ∑ {[((xi)n,or)exp - ((xi)n,or)cal]2 + [((xi)n,aq)exp -
n)1 i)1
nLS
((xi)n,aq)cal] } + 2
3
∑ ∑ [((xi)n,L)exp - ((xi)n,L)cal]
Table 1. NRTL Binary Interaction Parameters Aij Obtained from the Correlation of the Equilibrium Data for the System Water (1) + 1-Butanol (2) + LiCl (3) at T ) 298 K (r ) 0.2)
2
(16)
n)1 i)1
where (xi)n is the molar fraction of component i on tie-line n; nLL and nLS denote the number of tie-lines in the LL and LS regions, respectively. The subscripts or, aq, and L denote the organic, aqueous, and liquid phases, respectively, and the subscripts exp and cal respectively denote the experimental and calculated equilibrium data. For the correlation of the LLSh tie-triangle data, we consider it to be divided into one LL and two LSh pairs of conjugated compositions. For the correlation of the LSSh data, we consider it to be divided into two LS and LSh pairs of conjugated compositions. All these equilibrium data are included in the corresponding contribution to the objective function, as represented by eq 16. All the aforementioned equations (eqs 6-15) must be solved simultaneously to ascertain the parameters of the model that give a better approximation to the equilibrium behavior of a system that is composed of water + organic solvent + salt (anhydrous/monohydrated). The procedure used to perform these calculations has been outlined in Figure 3. The set of nonlinear equations is solved using the Newton-Raphson method. This method is strongly dependent on initial guesses and can lead to trivial solutions; special attention must be been given to these difficulties when equilibrium calculations are performed. The parameter Rij of the NRTL model is, initially, a fitting parameter. Nevertheless, a good correlation is obtained with a constant value of Rij ) 0.2, which is frequently used in the literature.27
1 2 3
j)1 -409.706 -1558.824
j)2
j)3
2715.194
-1413.224 -1526.429
4909.617
O.F. (LL) ) 8.46 × 10-2 O.F. (LS) )3.40 × 10-3 O.F. (LSh) ) 1.06 × 10-1 O.F. global ) 1.94 × 10-1 σ ) 5.99
3.6. Limited Composition Space for the Liquid-Liquid Equilibrium Root Determination. To find the conjugated compositions from the LLE calculations for the LL tie-lines and the LLSh tie-triangle, a method to constrain the composition zones under consideration is used to eliminate most of the problems arising from multiple solution roots and the need to make smart initial guesses. The procedure takes advantage of the topological information about the surface curvature changes given by the second derivative of the Gibbs energy of mixing function. Figure 4 shows the regions to which equilibrium points are confined. For a binary system, the equilibrium compositions xI1 and xII1 are confined to the shaded regions from 0 to xA1 and from xB1 to 1, respectively. These compositions (xA1 and xB1 ) are the intersections of the second derivatives of gM with the x-axis, and, therefore, they are the two inflection points of the gM curve. This procedure can be extended to ternary systems, where the inflection points are used to define the boundaries of a closed region where all mixtures spontaneously split. The conjugated compositions for LLE must be sought between the inflection points and the diagram limits. No trivial solution would be found, because the constrained composition zone for both phases is separated. This procedure is less time-consuming, because it avoids the need to search for equilibrium solutions where no inflections are present.
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Figure 5. Experimental and calculated data for the ternary system water (1) + 1-butanol (2) + LiCl (3) at 298 K.
4. Data Correlation Results and Discussion Selected experimental data for the water + 1-butanol + LiCl system at 298 K,23 i.e., those that define the shape of the phase diagram in all the equilibrium regions (the three binary data, one ternary tie-line that describes the “S-shaped” 2L region, and both LLS and LSSh triangles (see Figure 5)) have been simultaneously correlated using the suggested procedure. With the calculated values of the dimensionless Gibbs energy for the pure LiCl solid (gS), the monohydrated salt LiCl‚H2O (gSh) (recall eqs 2 and 3) and the NRTL model to formulate the gM function for the liquid mixtures (recall eqs 6-15) are solved to obtain the set of binary parameters that minimizes the objective function (eq 16). Table 1 shows the parameters Aij obtained in the correlations that have been performed, the accuracy of the data fit given by the objective function (O.F.) values calculated using eq 16, and the standard deviation using eq 17, which is defined as given in Sørensen and Artl:32
σ ) 100 ×
x
n
c
cal 2 exp cal 2 {(xexp ∑ ∑ ji - xji )I + (xji - xji )II } j)1 i)1
2cn
(17)
The high value of the mean deviation between the experimental and calculated equilibrium data, σ ) 5.99, illustrates the difficulty of obtaining a good fit for the experimental data. Experimental and calculated results have also been graphically represented in Figure 5. Fitting all the regions with NRTL, using the same set of Aij, is qualitatively possible; however, a large deviation exists between the experimental and calculated LLSh tie-triangle. In the correlation process, the most important feature that stood out was the difficulty encountered to fit both the LL and LLSh equilibrium regions simultaneously. To corroborate this
Table 2. NRTL Binary Interaction Parameters Aij Obtained from the Correlation of the Equilibrium Data for the System Water (1) + 1-Butanol (2) + LiCl (3) at T ) 298 K (r ) 0.2), Using the Gibbs Energy for the Monohydrated Salt (gSh) as a Fitting Parameter binary interaction parameter, Aij (K) i 1 2 3
j)1 -210.171 -823.116
j)2
j)3
1216.745
-1286.518 -1727.670
2971.737
O.F. (LL) ) 1.39 × 10-2 O.F. (LS) ) 6.07 × 10-3 O.F. (LSh) )1.32 × 10-2 O.F. global ) 3.32 × 10-2 σ ) 2.48
idea, a test was performed: the Gibbs energy for the monohydrated salt (gSh) was treated as a new fitting parameter, in an attempt to give more flexibility to the equilibrium calculation, and it showed that the accuracy of the fit improved remarkably. Table 2 gives the values for the NRTL binary parameters obtained using this last procedure, and Figure 6 shows a better approximation between the experimental and calculated equilibrium data, with a mean deviation of σ ) 2.48. The problems encountered in regard to representing both the LL and LLSh regions simultaneously, using the NRTL model, were avoided, to a great extent. In the calculations, the dimensionless Gibbs energy of mixing for the LiCl‚H2O (gSh) was restricted to be lower than the liquid Gibbs energy surface at the same composition, to avoid sets of parameters that could not reproduce the LLSh equilibrium region. The Gibbs energy of mixing for the LiCl‚H2O calculated by this procedure was gSh ) -3.744. For both parameter sets, it was verified that the calculated tie-lines and tie-triangles were contained in planes tangent to the gM surface that did not intersect the Gibbs energy surface at any other point. This guaranteed that all were stable equilibrium solutions.
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Figure 6. Experimental and calculated data for the ternary system water (1) + 1-butanol (2) + LiCl (3) at 298 K, with the Gibbs energy for the monohydrated salt (gSh) as a fitting parameter.
5. Conclusions A procedure to conduct a simultaneous correlation of all liquid-liquid (LL), liquid-anhydrous solid (LS), liquidliquid-anhydrous solid (LLS), and liquid-anhydrous solidmonohydrated solid (LSSh) equilibrium regions in a water + organic solvent + anhydrous/hydrated salt system is presented. This procedure is based on the tangent plane criterion as the equilibrium condition and searches for roots in a composition space where the second derivative of the Gibbs energy is positive. This method allows for faster convergence than the isoactivity-criterion-based method and avoids false and trivial equilibrium solutions. From the results that have been obtained, it can be observed that, using the non-random two-liquid (NRTL) model, it is qualitatively possible to find binary parameters that generate all the equilibrium regions for these types of ternary systems, where multiple regions that combine liquid and solid phases exist simultaneously. Nevertheless, deviations obtained between the experimental and calculated equilibrium data are high, so the difficulty of the NRTL equation to model these types of system is illustrated. In these systems, many topological conditions must occur on the Gibbs energy surface to reproduce the system behaviors in all the equilibrium regions, according to the tangent plane criterion, simultaneously. The most important difficulty was the simultaneous fitting of both the LL and LLSh equilibrium regions. The correlation was repeated with the Gibbs energy for the monohydrated salt (gSh) used as a parameter to be fitted, which improved the correlation results considerably. This result may be interpreted as follows: (1) The models for the liquid Gibbs energy of mixing (gM) function must be very flexible to represent the condensed phase equilibria of complex systems simultaneously (for example,
when different solid phases are in equilibrium with liquid phases). (2) To correlate equilibrium regions that combine liquid and solid phases simultaneously, the Gibbs energies for all the solid phases and those for the liquid mixtures must be compatible. Acknowledgment For financial support, the authors thank the University of Alicante (project GRJ05-13) and the Generalitat Valenciana (projects GV05/191 and GV2007/125). Nomenclature ai ) activity of component i Aij ) binary interaction parameters Rij ) NRTL nonrandomness factor c ) number of components n ) tie-line number m ) moles of H2O per mole of hydrated salt xi ) molar fraction of component i GE ) excess Gibbs energy (J/mol) G0 ) Gibbs energy of formation (J/mol) GM ) Gibbs energy of mixing (J/mol) gM ) dimensionless Gibbs energy of mixing µi ) chemical potential of component i (J/mol) gS ) dimensionless Gibbs energy of the anhydrous solid gSh ) dimensionless Gibbs energy of the monohydrated solid ∆hf ) melting heat (J/mol) ∆Cp ) difference between heat capacity of the liquid and that of the solid phase at the normal melting temperature (J mol-1 K-1) T ) temperature (K) P ) pressure (atm)
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ReceiVed for reView September 26, 2007 ReVised manuscript receiVed January 7, 2008 Accepted January 11, 2008 IE071290W
