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On the use of the NRTL method for calculation of liquid-liquid equilibria in ternary systems

Alvarez Juliá, J.¹; Barrero, C.R.²; Grande, M. del C.²; Marschoff, C.M.³

Universidad de Buenos Aires, Facultad de Ingeniería, Paseo Colón 850 (1063) Buenos Aires, Argentina
¹Departamento de Matemáticas
²Departamento de Química
³Departamento de Ingeniería Química
Fax: +54 11 4576 3240, e-mail:cmarschoff@gmail.com

Received August 4, 2008.
In final form October 15, 2008

Abstract
Application of the NRTL method to experimental data obtained on several systems at different temperatures shows that the energy parameter set that minimizes the objective function at a given temperature, T₁, can be used for estimating phase compositions at other T values without serious error. It is concluded that this approach should be preferred to the use of interpolated or extrapolated g_ij values.

Keywords: Thermodynamics ; Liquid-liquid equilibria; NRTL method

Resumen
La aplicación del método NRTL para ajustar curvas de equilibrio líquido – líquido en varios sistemas a distintas temperaturas muestra que, para estimar las composiciones de equilibrio a una temperatura arbitraria T, es preferible emplear el conjunto de parámetros de energía obtenidos al ajustar los datos obtenidos a una temperatura dada antes que utilizar valores de los g_ij interpolados o extrapolados de las series obtenidas.

Palabras clave: Termodinámica; Equikibrio líquido-líquido; Método NRTL

Introduction
Prediction of thermodynamic properties in liquid mixtures is of great importance for the chemical engineer. Among the different approaches that have been proposed in order to achieve this goal, the Non-Ramdom Two Liquid method (NRTL) [1] is still one of the most frequently employed options. However, and as we have already pointed out previously [2], the original predictive goal of this method has been replaced by ex-post calculation of the fitting parameters that best correlate experimental data.
In Reference 2 particular attention was devoted to the fact that, when a system is experimentally studied at different temperatures, the fitting parameters that result when the usual calculation techniques are applied do not show a cogent behaviour and, in fact, it is not possible to obtain fitting parameters at an arbitrary temperature value by interpolating or extrapolating from the calculated values.
In this work, we further explore the behaviour of the NRTL equations in the case of three component mixtures showing liquid – liquid phase equilibria.

The NRTL method
The basic equation of the method stems out from an essentially heuristic approach that combines a modification of the quasi – chemical theory of Guggenheim [3] with the model of Scott [4]. In this approach a binary solution, 1-2, is considered as a mixture of two types of cell: one centered in a molecule of species 1 and the other in a molecule of species 2. A non-random distribution results that is described in terms of local mole fractions:

x₂₁/x₁₁ = (x₂/x₁) exp[-a₁₂(g₂₁ – g₁₁)/RT]          (1)

where x₂₁ and x₁₁ are, respectively, the mole fractions of components 2 and 1 in the neighborhood of molecule 1, g₂₁ and g₁₁ are the Gibbs energies of interaction for the 2-1 and 1-1 pairs and a₁₂ is a characteristic constant of the mixture. Calculation of Gibbs excess energies yields:

ln g_j = x_i² {[t_ijG_ij²/(x_j + x_iG_ij)²] + [t_jiG_j1/(x_i + x_jG_ji)²]            (2)

where

t_ij = (g_ij – g_jj)/RT          (3)

and

G_ij = exp (-a_ij t_ij)           (4)

with these equations and employing experimental data from liquid – vapour and liquid – liquid equilibria of binary systems the a_ij and (g_ij – g_jj) values can be obtained.

The extension of the NRTL approach to multicomponent systems is based on the usual assumption that only pair interactions can be considered and, hence, activity coefficients for the different species in a given phase can be written as:

ln g_j = (S_j t_jiG_jix_j/S_kG_kix_k) + S_j{(x_jG_ij/S_kx_kG_kj) [t_ij – (S_l x_lt_ljG_lj/S_kx_kG_kj)]}            (5)

Thus, the method allows to make predictions on phase compositions for liquid – liquid equilibria in ternary systems employing energy parameters obtained experimentally from phase equilibria in binary systems.
However, almost all published papers dealing with experimental determination of liquid – liquid equilibria in ternary systems do not apply the NRTL approach in order to verify its predictive capability but, instead, the method is employed for calculating the t_ij values that minimize an objective function like:

[S_k=1 S_j=1,2 S_j=1-3 (x_ijk – x^*_ijk)²]^1/2. N ^-1            (6)

where x_ijk is the experimental mole fraction of component i in the j phase for the kth tie line, x^*_ijk is the corrresponding calculated value and N is the number of experimental tie lines.
As we have shown in previous work [2], the interaction parameter obtained for a given pair of species is neither independent of the nature of the third component nor of the temperature at which the experimental data set was obtained. Moreover, the observed change in the interaction parameters does not show a cogent behaviour with, for instance, changes in aliphatic chain length of a component or temperature variation. Hence, an estimation of energy parameters for a given mixture made by interpolation or extrapolation of previously obtained values is a very risky choice.
In this respect it is not surprising that the nature of molecule 3 affect the interaction between molecules 1 and 2. However, the apparently chaotic variation of energy parameters with temperature is somewhat baffling and, consequently, we decided to carry out further work in studying this particular behaviour.

Calculations and discussion
In order to further examine the output of the NRTL method when applied to liquid – liquid equilibria in ternary systems at varying temperature literature data were chosen for systems that widely differ in their chemical characteristics. Thus, data from the following ternary mixtures were considered:

a) heptadecane + propylbenzene + N-methyl-2-pyrrolidone (NMP) at 298.15 K, 308.15 K, 318.15 K and 328.15 K [5].
b) limonene + ethanol + water at 293.15 K, 303.15 K, 313.15 K and 323.15 K [6].
c) methyl tert-butyl ether (MTBE) + 1-hexanol + water at 288.15 K, 298.15 K and 308.15 K [7].
d) methyl propyl ketone (MPK) + propanoic acid + water at 298.15 K, 308.15 K, 318.15 K and 328.15 K [8].
e) benzonitrile + propanone + water at 298.15 K, 303.15 K, 308.15 K, 313.15 K and 318.15 K [9].

In Tables 1 – 5 the g_ij values obtained for the five systems when the objective function of Eqn. (6) is minimized for each temperature are displayed together with the gij obtained when data at all temperatures are employed in a single minimization process. In all cases the steepest descent method was employed with a = 0.3 and, since the relevant parameters in the calculations are the t_ij values, all values are referred to g₁₁ = 0. It is clear from these results that the variation of the g_ijvalues with temperature is not regular and, hence, it is impossible to obtain the fitting parameters at an intermediate temperature by interpolation. Also, it is readily seen that the results obtained when data at all temperatures are simultaneously employed in the minimization process fit very poorly with experimental data.

Table 1: Fitting parameters for the NRTL approach for the system heptadecane (1) + propylbenzene (2) + NMP (3) calculated from data in Ref. [5].

Table 2: Fitting parameters for the NRTL approach for the system limonene (1) + ethanol (2) + water (3) calculated from data in Ref. [6].

Table 3: Fitting parameters for the NRTL approach for the system MTBE (1) + 1-hexanol (2) + water (3) calculated from data in Ref. [7].

Table 4: Fitting parameters for the NRTL approach for the system water (1) + propanoic acid (2) + MPK (3) calculated from data in Ref. [8].

Table 5: Fitting parameters for the NRTL approach for the system water (1) + benzonitrile (2) + propanone (3) calculated from data in Ref. [9].

Now, a quick inspection of Eqns. (5) and (6), which depend on the t_ij and the a_ij, indicates that the objective function must have a number of local minima. The immediate question that arises is how good will be the correlation at a given temperature, T₁, if the parameters obtained by looking for the absolute minimum in Eqn. (6) at another temperature, T₂, are employed. We have performed these calculations and the results are summarized in Tables 6 – 10. In Figures 1 – 5 some of the resulting ternary diagrams are shown.

Table 6: Objective functions defined by Eqn. (6) at different temperatures obtained with the g_ij sets calculated at T(g_ij) for the system heptadecane + propylbenzene + NMP.

Table 7: Objective functions defined by Eqn. (6) at different temperatures obtained with the g_ij sets calculated at T( g_ij ) for the system limone + ethanol + water.

Table 8: Objective functions defined by Eqn. (6) at different temperatures obtained with the g_ij sets calculated at T( g_ij ) for the system MTBE + 1-hexanol + water.

Table 9: Objective functions defined by Eqn. (6) at different temperatures obtained with the g_ij sets calculated at T( g_ij ) for the system MPK + propanoic acid + water.

Table 10: Objective functions defined by Eqn. (6) at different temperatures obtained with the g_ij sets calculated at T( g_ij ) for the system benzonitrile + propanone + water.

The obtained results indicate that the objective function values that result for the different systems studied remain within the same order of magnitude when the energy parameter set calculated at a given temperature is changed by the parameter set obtained at other temperature. Moreover, in some cases, and as a consequence of experimental uncertainties, a better objective function value results when the parameter set obtained at a different temperature is employed.

Figure 1: Experimental (O and solid curve) and calculated phase compositions for the {heptadecane + propylbenzene + NMP} system [5] at 328.15 K, employing the g_ij values that minimize the objective function of Eqn. (6) at the indicated temperature: ‚, 298.15 K; x, 308.15 K;D, 318.15 K; ◊, 328.15 K.

Figure 2: Experimental (O and solid curve) and calculated phase compositions for the {limonene + ethanol + water} system [6] at 293.15 K, employing the g_ij values that minimize the objective function of Eqn. (6) at the indicated temperature: ‚, 293.15 K; +, 303.15 K; *, 313.15 K; ◊, 323.15 K.

Figure 3: Experimental (o and solid curve) and calculated phase compositions for the {MTBE + 1-hexanol + water} system [7] at 288.15 K, employing the g_ij values that minimize the objective function of Eqn. (6) at the indicated temperature: ?, 288.15 K;‚ , 298.15 K; x , 308.15 K

Figure 4: Experimental (O and solid curve) and calculated phase compositions for the {MPK + propanoic acid + water} system [8] at 298.15 K, employing the g_ij values that minimize the objective function of Eqn. (6) at the indicated temperature: ‚, 298.15 K; x, 308.15 K; D, 318.15 K; ◊, 328.15 K.

Figure 5: Experimental (o and solid curve) and calculated phase compositions for the {benzonitrile + propanone + water} system [9] at 318.15 K, employing the g_ij values that minimize the objective function of Eqn. (6) at the indicated temperature: ‚, 298.15 K; +, 303.15 K; x, 308.15 K; *, 313.15 K; D, 318.15 K.

By the other hand, and looking at Figures 1 – 5 it is clear that the shape of the curves is maintained when energy parameters obtained at temperatures different from the experimental one are employed and that larger quantitative differences are observed near the plait point zone but, nevertheless, the results are acceptable for prediction purposes.
The final conclusions are:
a) if data are available at a given temperature, T₁, for the liquid – liquid equilibrium of a ternary system, compositions at T₂ can be safely estimated employing the energy parameter set obtained from the experimental values.
b) if data are available at several temperatures and estimations are needed at a different thermal condition any of the energy parameter sets should be used rather than employing extrapolated or interpolated g_ij values.

Acknowledgement: The authors gratefully recognize financial support from the Agencia Nacional de Promoción Científica y Tecnológica (FONCYT), through project PICT38467.

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