Fluid Phase Equilibria, 25 (1986) 147- 160 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

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EVALUATION OF THE METHODS OF CORRELATION AND INTERPOLATION OF QUATERNARY LIQUID-LIQUID EQUILIBRIUM DATA. APPLICATION TO THE SYSTEM WATER-ETHANOL-CHLOROFORM-TOLUENE AT 25°C

FRANCISCO RUIZ*, DANIEL PRATS and VINCENE GOMIS

Departamento de Quimica TPcnica, Universidad de Alicante, Aptdo. 99 Alicante (Spain)

(Received April 14, 1985; accepted in final form August 6, 1985)

ABSTRACT

Ruiz, F., Prats, D. and Gomis, V., 1986. Evaluation of the methods of correlation and interpolation of quaternary liquid-liquid equilibrium data. Application to the system water-ethanol-chloroform-toluene at 25°C. Fluid Phase Equilibria, 25: 147-160.

The advantages and disadvantages of three ways to correlate and interpolate quaternary liquid-liquid equilibrium data: (1) the graphical methods, (2) the analytical methods using models describing the mixture Gibbs energy and (3) the analytical methods using models for the equilibrium ratios, are studied. To carry out this objective, firstly, data published previously for the system water-ethanol-chloroform-toluene at 25’C are correlated using the three different methods; then, a new set of tie line data for the same system are determined experimentally; finally, these experimental tie lines are compared with the interpolated ones from the three correlations, and the accuracy of each one is checked.

INTRODUCTION

Calculations in the design of liquid-liquid extraction equipment become complex when the feed is a mixture of three components or when a mixture of two solvents (either miscible (mixed solvent) or immiscible (double solvent)) is used because the corresponding quaternary liquid-liquid equi- libria (QLLE) have to be considered. It is necessary to represent graphically or analytically experimental liquid-liquid phase equilibrium data so that the data can be interpolated. There are different methods for doing this, but studies comparing the advantages and accuracy of each one have not been done.

The objective of this work is to study the advantages and disadvantages of the three main ways to correlate and interpolate QLLE data: (1) the graphical methods such as the Ruiz and Prats method (1983), (2) the

037%3812/86/$03.50 0 1986 Elsevier Science Publishers B.V.

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analytical method that use models describing the mixture Gibbs energy such as the UNIQUAC model, and (3) the analytical methods that use models for the equilibrium ratios.

To carry out this objective, this paper is organized into three parts. In the first one, we review the three main methods of correlation and interpolation of QLLE data and apply them to a set of QLLE data of the system water-ethanol-chloroform-toluene at 25°C (Ruiz et al., 1985). In the sec- ond part, we determine a new set of tie lines for the system water-ethanol-chloroform-toluene at 25°C. In the third part, these experi- mental tie lines are calculated again using the correlations obtained in the first part. In this way, the experimental equilibrium data are compared with the interpolated ones using the three different methods, and the accuracy of each one is checked.

METHODS OF CORRELATION AND INTERPOLATION OF QLLE DATA. APPLICA- TION TO THE QUATERNARY SYSTEM: WATER-ETHANOL-CHLOROFORM- TOLUENE AT 25°C

Graphical methods

In the literature various methods have been used to represent graphically the compositions of equilibrium phases for quaternary systems; for example, Hunter (1942) Smith (1944), Chang and Moulton (1953), Powers (1954), Treybal (1963), Frolov (1965) and Ruiz and Prats (1983). Of all these methods the Ruiz and Pratz method is the most appropriate because it allows the direct use of experimental equilibrium data whereas other meth- ods require calculation or interpolation to draw the distribution curves which correlate the compositions between the conjugate phases and, moreover, the iterations needed to obtain nonmeasured equilibrium data are simpler using the Ruiz and Prats method than those using the other methods.

Let us consider the system water (W)-ethanol (E)-chloroform (C)-toluene (T). In a previous paper (Ruiz et al., 1985) 12 ternary and 17 quaternary tie lines were determined experimentally for that system. The initial mixtures for determining these 29 tie lines were selected on the conditions that Xw = Xc + X, and that the proportions of chloroform and toluene were taken such that the ratio Xc/( Xc + X,) = M and X, = L vary systemati- cally (M = 0.0, 0.2, 0.4, 0.6, 0.8 and 1.0 and L increased until the homoge- neous region was reached). Xi represents the weight percentage of compo- nent i. These initial mixtures are shown in Fig. 1. The tie lines obtained with them allow the graphical representation of the quaternary system using the method of Ruiz and Prats.

The method consists of representing the two conjugate phases on the same graph, in such a way that the tie lines are defined by pairs of points which

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w

Fig. 1. Representation of the solubility surface and initial mixtures for the quaternary system water (W)-ethanol (E)-chloroform (C)-toluene (TJ at 25°C. The value of the ratio Xw/( Xc

+ X,) for the points F and G is 1.

have the same values of the two parameters M and L for each one of the phases. To carry out the representation a tetrahedral projection is made onto a plane parallel to two nonintersecting edges (Cruickshank et al., 1950).

In Fig. 2, the equilibrium of the quaternary system water (W)-ethanol (E)-chloroform (C)-toluene (T) has been represented using this method. Figure 2a shows the tetrahedral projection onto a plane parallel to the edges water-toluene and ethanol-chloroform. The origin of coordinates is situated on the chloroform apex, and the coordinates are

z = x, + x,

y=x,+x, (1) (4

Figure 2b shows the tetrahedral projection onto a plane parallel to the edges water-chloroform and ethanol-toluene. The origin of coordinates is situated on the toluene apex, and the coordinates are

z’=XE+XC (3)

y’ = xw -I- X, (4)

The problem of determining nonmeasured equilibrium data can be re- solved very quickly: knowing the composition of one phase, that of the conjugate phase is obtained immediately from the values of M and L deduced from the known phase. To obtain the tie line corresponding to a quaternary mixture of any composition, the point that defines its composi-

Fig. 2. Graphical representation of the equilibrium data for the quaternary system water (W)-ethanol (E)-chloroform (C)-toluene (T) at 25’C: (a) projection onto a plane parallel to the W-T and E-C edges (z = X, -t X, and y = X, + X,); (b) projection onto a plane parallel to the W-C and E-T edges (z’ = X, + X, and y’= X, + X,).

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tion is located on the two projections and the tie line that passes through that point and which defines the same values of A4 and L for the two phases and the two projections is obtained by trial and error.

Analytical methods using models describing the mixture Gibbs energy

To apply these analytical methods, one must have available a model giving the mixture Gibbs energy or, equivalently, the activity coefficients as functions of composition and temperature and, moreover, a method for calculating liquid-liquid equilibrium compositions using the above model.

Sorensen et al. (1979) described the three different types of model which have been used to correlate liquid-liquid equilibrium data using mixture Gibbs energy functions:

(1) models for activity coefficients or the excess Gibbs function, e.g., Margules, Van Laar, Black (Black, 1958), equations of local composition model such as the modifications of the Wilson equation (Hiranuma, 1974; Novak et al., 1974; Nagata et al., 1975), NRTL (Renon and Prausnitz, 1968) and UNIQUAC (Abrams and Prausnitz, 1975);

(2) equations of state, e.g., Redlich-Kwong (Redlich and Kwong, 1949) and Peng-Robinson (Peng and Robinson, 1976) equations; and

(3) group contribution methods, e.g., ASOG (Derr and Deal, 1969) and UNIFAC (Fredenslund et al., 1975).

These models have a set of adjustable parameters which must be obtained using the experimental liquid-liquid equilibrium data.

Among the many models, those most often used to correlate liquid-liquid equilibrium data are the local composition ones, especially NRTL and UNIQUAC.

Sorensen et al. (1979) also described the two different main strategies for obtaining parameters from liquid-liquid equilibrium data at constant tem- perature and pressure. Expressed in terms of the least-squares principle, they are :

(1) minimization of activity differences; and (2) minimization of the distances between experimental mole fractions

xijk and calculated mole fractions Zijk

Q = i i i (xijk - iijk)2

where i denotes component, j denotes phase and k the tie line. Objective functions stated in terms of concentrations are more com-

plicated, computationally, than objective functions in terms of activities but express directly our desired goal: to represent as accurately as possible the experimental tie lines. Several investigators have used objective functions of

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concentrations to correlate liquid-liquid. equilibrium data: for example, Renon et al. (1971), Guffey and Wehe (1972), Anderson and Prausnitz (1978) Sorensen and Arlt (1979-1980), Sorensen et al. (1979), Magnussen et al. (1980) and Ruiz et al. (1984,1985).

Objective functions in terms of concentrations contain the mole fractions which, for a given current parameter set, must be predicted for each tie line. The methods of doing that are based on one of the formulations of the equilibrium criterion. Some authors use a method based on the minimization of the molar Gibbs energy of mixing, but most of them use the isoactivity method based on the equation

a;, = ai2 (6)

where LZ,~ represents the activity of component i in phase j and must be obtained using the excess Gibbs function which is given by the model used.

Equation (6) and the stoichiometric equations

cxil = 1 and cxjZ = 1 (7)

form a set of equations which may be solved for the 2N unknowns (N is the number of components) if the pressure, the temperature and either N - 2 concentrations in one of the phases or the feed molar composition x,? is specified

where @ is the molar fraction of the feed that splits into phase 1. Several methods of solving the system of equations (6-8) have been

devised in the literature so far: Renon et al. (1971) used the Newton-Raph- son method, Guffey and Wehe (1972) and Marina and Tassios (1973) took advantage of the minimization method while Fredenslund et al. (1977) advocated successive substitution with analytic criteria and Null (1970), Ragaini et al. (1974) and Mukhopadhyay and Sahasranaman (1982) advoc- ated successive substitution with the Newton method.

In the present work, the 29 experimental tie lines of the quaternary system water-ethanol-chloroform-toluene at 25°C have been correlated using the UNIQUAC model as slightly modified by Anderson and Prausnitz (1978) and the pure component molecular structure constants indicated in Table 1. The interaction parameters obtained are shown in Table 2.

The parameter estimation procedure was as reported in a previous paper (Ruiz and Gomis, 1985). The estimation was carried out by minimizing,using the Nelder-Mead procedure (Nelder and Mead, 1965) the objective func- tion in terms of concentrations (eqn. 5) where the compositions have been predicted by setting the concentration in the global initial mixture equal to that in the middle of the corresponding experimental tie line. This objective

1.53

TABLE 1

Pure component molecular structure UNIQUAC constants a

Component r 4 4’

Water 0.92 1.40 1.00

Ethanol 2.11 1.97 0.92 Chloroform 2.70 2.34 2.34 Toluene 3.92 2.97 2.97

a Prausnitz et al., 1980.

function can be stated as

(9)

(xilk + xiZk)/2 belonging to the tie lie A&r2k. The absolute mean deviation between the correlated experimental com-

positions and the ones calculated using the UNIQUAC parameters shown in Tables 1 and 2 is 1.9%.

Analytical methods using models for the equilibrium ratios

Chimowitz et al. (1983, 1984) developed local models for ternary systems. For a ternary system l-2-3 containing the immiscible pair l-2 the equa- tions of the model are

In K; = Ajlx$ + AiSxz2 + A;, 00)

where xij represents the mole fraction of component i in the phase where component j is dominant and Ki the distribution ratio of component i

(defined as xi2/xj1). The Ai; parameters are adjustable and they can be calculated by standard linear regression using the experimental liquid-liquid equilibrium data.

TABLE 2

UNIQUAC interaction parameters (K) for the QLLE data a

W E C T

W 0.0 - 102.6 413.7 308.2

E - 316.8 0.0 104.4 -41.8 C 1375.9 - 432.5 0.0 414.7

T 1326.9 e 321.1 - 202.1 0.0

a Ruiz et al., 1985. Water (W)-ethanol (E)-chloroform (C)-toluene (T) at 2YC.

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The method for calculating new liquid-liquid equilibrium compositions consists of the solution of a system of equations which includes the equa- tions of the local model (lo), the stoichiometric equations (7) and the material balance (8). This system of equations can easily be solved using, for example, the Newton-Raphson method, because the analytical derivates are obtained easily.

For the quaternary system water (W)-ethanol (E)-chloroform (C)-toluene (T) we have developed a similar model. Now, the water is the dominant component in the aqueous phase, but there are two dominant components in the organic phase: chloroform and toluene. Therefore, the corresponding equations are

In K, =Ail(xw(aq))’ +AiZ(xc(or))* +Ai3(xT(or))* +Ai4 (11) where the Aii are the adjustable parameters and xw(aq), x&or) and xT(or) represent the mole fraction of water in aqueous phase, chloroform in the organic phase and toluene in the organic phase, respectively.

These equations are reduced to the corresponding ones for the ternary systems water-ethanol-chloroform and water-ethanol-toluene when either X ,=Oorx,=O.

The parameters Aij have been fitted by standard linear regression using the 29 experimental tie lines. The calculated parameters are shown in Table 3.

DETERMINATION OF A NEW SET OF TIE LINES FOR THE QUATERNARY SYS- TEM WATER-ETHANOL-CHLOROFORM-TOLUENE AT 25°C

Experimental method

The analytical method to determine the new set of tie lines is the same as that used in a previous work (Ruiz et al., 1985). Basically, it consists of chromatographic analysis of each one of the phases in which a synthetic heterogeneous initial mixture splits.

TABLE 3

Equations for the equilibrium ratios for the QLLE data a

In K, = -4.68(x,(aq))‘-1.68(~c(or))~ -2.94(x,(or))2 +1.03 In K, = 0.30(x,(aq))2 -0.83(x,(or))2 -2.08(xr(or))2 +0.39 In Kc = 7.46(xw(aq))2 +l.ll(xc(or))2 +2.06(xr(or))‘-1.13 In K, = 9.18(xw(aq))2 +4.74(xc(or))2 + 1.99(x,(or))’ - 1.40

a Ruiz et al., 1985. Water (W)-ethanol (E)-chloroform (C)-toluene (T) at 25°C. (x,(j) = mole fraction of component i in the phase j; (or) = organic phase; (as) = aqueous phase; K = xi(or)/xi(aq)).

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(cl) cb) Fig. 3. Representation of the solubility surface and the new set of initial mixtures for the quatemary system water (W)-ethanol (E)-chloroform (C)-toluene (T) at 25°C. The value of the ratio X,/(X, + X,) for the points F and G is: (a) l/3; (b) 3.

The initial mixtures for these tie lines were selected with the following conditions: (1) M= X,-/(X, + X,) = 0.2, 0.4, 0.6 and 0.8, (2) X,/( Xc + X,) = 3 or l/3 and (3) the ethanol levels (L) were increased stepwise until the homogeneous region was reached. The prepared initial mixtures have been represented schematically in Fig. 3a and b. The initial mixtures used in the correlations in the first part of this paper belong to the plane X,/( Xc + X,) = 1 (Fig. 1)

Experimental results

Table 4 shows the compositions (wt.%) for the new quatemary tie lines determined experimentally. The values of M, L and X,/( Xc + X,) for the initial mixtures are also included.

INTERPOLATED TIE LINES OBTAINED USING THE GRAPHICAL AND ANALYTI- CAL METHODS

The graphical representation using the method of Ruiz and Prats, the UNIQUAC parameters and the parameters of the model for the equilibrium ratios shown in the first part of this paper have been used to calculate the same set of tie lines determined experimentally.

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TABLE 4

Quaternary tie line data (wt.%) obtained in this work for the quaternary system water (W)-ethanol (E)-chloroform (C)-toluene (T) at 25°C

Tie line M L number

Organic phase Aqueous phase

Xw/(& + Xr) =I/3 1 0.2 10 2 20 3 30 4 40 5 0.4 10 6 20 7 30 8 0.6 10 9 20

10 30 11 0.8 10 12 20 13 30

X,/(X, + X-r) = 3

0.18 2.78 19.2 77.8 72.9 26.7 0.22 0.21 0.61 7.27 18.0 74.1 54.4 42.8 1.09 1.76 1.27 11.8 16.3 70.6 38.2 51.7 3.15 6.91 3.02 19.8 13.5 63.7 22.1 51.7 6.26 20.0 0.22 3.00 38.6 58.2 73.5 25.9 0.44 0.19 0.86 9.01 35.4 54.7 56.0 40.4 2.42 1.21 2.26 16.4 31.3 50.1 39.5 48.7 6.50 5.33 0.28 3.67 57.6 38.5 74.8 24.5 0.59 0.07 1.38 12.0 51.7 35.0 59.6 36.7 3.07 0.59 4.27 22.5 43.1 30.1 44.2 44.8 8.07 2.90 0.46 4.71 75.5 19.3 76.2 23.0 0.83 0.035 2.14 15.0 66.0 16.9 65.0 31.9 2.84 0.19 6.75 27.4 52.5 13.4 53.6 39.2 6.30 0.84

14 0.2 15 16 17 18 19 0.4 20 21 22 23 24 0.6 25 26 27 28 0.8 29 30 31

10 0.10 0.73 20 0.14 2.17 30 0.25 4.45 40 0.64 7.63 50 1.06 10.1 10 0.12 0.94 20 0.20 2.61 30 0.59 5.88 40 0.97 9.94 50 3.49 13.0 10 0.14 1.11 20 0.26 3.42 30 0.82 8.96 40 2.21 15.4 10 0.15 1.32 20 0.43 5.18 30 2.17 15.6 40 6.35 27.2

19.7 79.5 87.0 12.8 0.10 0.086 19.2 78.5 75.3 24.4 0.19 0.19 18.0 71.2 63.6 35.3 0.51 0.67 15.4 76.3 51.4 45.1 1.18 2.37 11.5 77.3 39.6 51.9 2.04 6.45 38.9 60.0 86.9 12.8 0.21 0.047 37.8 59.4 75.0 24.4 0.43 0.14 34.8 58.7 63.4 34.9 1.08 0.54 29.9 59.2 50.8 44.3 2.80 2.14 15.9 67.6 38.4 50.7 4.50 6.39 58.5 40.2 86.8 12.7 0.42 0.026 56.7 39.6 75.1 24.1 0.73 0.083 51.5 38.7 63.1 34.6 1.87 0.41 42.8 39.6 50.0 43.2 4.91 1.86 78.3 20.2 86.9 12.5 0.61 0.025 74.7 19.7 75.0 23.8 1.14 0.050 63.1 19.2 63.6 33.3 2.94 0.24 47.8 18.7 49.2 41.4 7.92 1.52

The absolute mean deviations between experimental and calculated QLLE data using the graphical method, the UNIQUAC model and the model for the equilibrium ratios are 2.0, 2.2 and 5.1 mol%, respectively.

The greatest deviation corresponds to the results obtained using the model for the equilibrium ratios, while the deviations obtained are smaller (and similar) when the graphical or UNIQUAC methods are used. However, a

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great part of the deviation obtained using the model for the equilibrium ratios is due to the contribution of the tie lines whose initial global mixture is very near the homogeneous region and therefore very near the plait point curve. Actually, some of these tie lines have not been interpolated but extrapolated. If we eliminate the five tie lines nearest the plait point curve, the absolute mean deviation obtained with the 26 remaining tie lines is reduced to 2.2 mol% when the model for the equilibrium ratios is used. Now, the absolute mean deviations obtained using the graphical method and the UNIQUAC model are 1.7 and 2.1 mol%, which are also smaller than those obtained with the 31 tie lines, but the results obtained using the three methods are now more similar.

CONCLUSIONS

Using the graphical method of Ruiz and Prats, we have obtained the best results for the interpolated quaternary tie lines of the system water-ethanol-chloroform-toluene when they are compared with the ex- perimental ones. For that, this graphical method represents in many cases a viable alternative to computer methods requiring analytical correlation of equilibrium data. However, the method requires that adequate equilibrium data (with determined values of 44, L and X,/( Xc + X,) in this case) be available and that extensive graphical construction be done. Furthermore, graphical iterations are necessary to interpolate, and the method thus proves rather tedious in practice.

Using the UNIQUAC model we have calculated similar results for the quaternary system water-ethanol-chloroform-toluene to those obtained using the graphical method. However, complicated and extensive programs consuming a lot of computational time are needed to correlate the experi- mental QLLE data because of the great number of tie lines which, in general, must be correlated to obtain parameters representing the entire system. Furthermore, the calculation of each quaternary tie line involves more computational time than a binary or ternary tie line because the number of unknowns in the system of equations which we must solve is higher.

The use of models for the equilibrium ratios has led to good results which are similar to those obtained with the other methods, as long as the interpolated tie lines are not very near the plait point curve. On the other hand, the method can be implemented easily on a microcomputer because it is very simple since the parameters can be fitted by standard linear regres- sion and the analytical derivatives of the obtained equations can be calcu- lated easily.

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LIST OF SYMBOLS

A adjustable parameter C chloroform E ethanol K equilibrium ratio L X, in the initial mixture M Xc/( Xc + Xr) in the initial mixture N number of components

Q objective function T toluene w water x weight percentage x mole fraction

y, z, Y’, z’ coordinates

Greek letter

@ molar fraction of feed that splits into phase I

Superscripts

,.

F calculated value feed

Subscripts

i

j k

component phase tie line

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