correlation of ternary liquid-liquid equilibria in system isobutyl acetate-acetic acid-water

Pergamon Chemical En#ineering Science, Vol. 51, No. 6, pp. 893-903, 1996 Copyright © 1996 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0009-2509/96 $15.00 + 0.00

0009-2509(95)00345-2

CORRELATION OF TERNARY LIQUID-LIQUID EQUILIBRIA IN SYSTEM ISOBUTYL ACETATE-ACETIC ACID-WATER

JAROSLAV PROCH.AZKA* and ALES HEYBERGER Institute of Chemical Process Fundamentals, Academy of Sciences of Czech Republic,

165 02 Prague 6-Suchdol, Czech Republic

(Received 27 March 1995)

Abstract--A method of correlating ternary liquid-liquid equilibrium data based on the model presented by Rod (1976, Chem. Engng J. 11, 105-110) is discussed and compared with other methods available in the literature. Several variants of the method are defined and corresponding computation algorithms are described. Their properties are tested on equilibrium data for the system isobutyl acetate-acetic acid-water at 25°C presented by Heyberger et al. (1977, Collection Czech. Chem. Commun. 42, 3355-3362). The accuracy of the individual variants and the stability of the respective algorithms are demonstrated. A procedure for stabilizing the optimization and finding a unique solution is developed. The present correlation procedure compares favourably with other published methods.

INTRODUCTION

To design a liquid-liquid extraction process, accurate data are needed since usually high yields and high purities of solute are desired, The simplest case of liquid-liquid extraction is the physical extraction in a ternary or pseudo-ternary system. Various methods have been proposed for correlating the equilibrium data in such systems. For correlating binodal data of type I and II ternary systems, empirical functions of varying complexity have been developed (Spalding, 1970; Bulatov and Yachmenev, 1971; Hlavat~,, 1972). For distribution data, Hand (1930) and Othmer and Tobias (1942) presented power function type relations. Heyberger et al. (1977) used orthogonal polynomials of 15th degree for correlating binodal data and of third degree for distribution data in the system isobutyl acetate-acetic acid-water. Their method compared favourably with those of Hlavat~, Hand and Othmer and Tobias.

In contrast to these purely empirical procedures, Rod (1976) presented a method for correlating both distribution and binodal data based on thermodyna- mical grounds. He introduced the distribution coefficients of all three components, mi, as functions of the solute concentration in the x-phase:

exp(~a i j x~) , i=1 ,2 ,3 , j = 0 , 1 . . . . . n. (1) m i x

Here x~ is the mass fraction of the component 1 (solute) in the x-phase, aij are empirical constants and n is the degree of the polynomial. Renon and Prausnitz (1968) developed the NRTL equation describing the variation of activities of components in the coexisting

phases with composition as a result of molecular interactions. This method is better suited for predic- ting ternary or quaternary equilibria from binary v-1 equilibrium data than for correlating sets of ternary 1-1 equilibrium data. Recently, Brandani (1994) presented a regression algorithm for ternary 1-1 equilibria based on the NRTL model.

The present work is a part of a broader study of equilibria in multicomponent liquid-liquid systems. For correlating and interpolating experimental data in pseudo-ternary systems, a reliable procedure was sought. It should allow to reproduce the data with low systematic error in the whole concentration range. The model variant for simulating equilibria should be easy to integrate into complex systems for process design. With respect to these requirements, the Rod model seems most promising and therefore it has been chosen for further study. In the present work several model variants of different complexity have been defined and the respective algorithms are reported. The properties of these variants are tested and mutually compared using a set of equilibrium data for the system isobutyl acetate-aceticacid-water at 25°C published by Heyberger et aL (1977). It is a type I system with rather unsymmetrical mutual solubilities of solvents in coexisting phases. Therefore, it poses high demands on the model flexibility. For correlating their data the authors employed the Hlavat~, (1972), Hand (1930) and Othmer and Tobias (1942) correla- tions, as well as their own method using orthogonal polynomials. Accordingly, their results can also be utilized for comparing the above methods with the extended Rod method developed in the present work.

*Corresponding author. Fax: 342073, E-mail: icecas@icpf. cas.cz.

THEORETICAL

According to the phase rule, a ternary two-phase isothermal isobaric system at equilibrium has only

893

894 J. PROCHAZKA and A. HEYBERGER

one degree of freedom. Therefore, the compositions of coexisting phases can be expressed as functions of the concentration of one component in one phase (e.g. the mass fraction of solute in the x-phase, x~). With respect to the requirement of equality of component activities in coexisting phases, the solution to the problem is to find suitable functions ~xl (x~), 7y~ (x~), or F~(x~) in the relations

YiYri(xl) = x f f , , i ( x l ) or mi = y i / x l = y , , i ( x l ) / y r i (x l )

= Fi(xl), i = 1,2,3. (2)

For the functions Fi(x~), Rod proposed the form (1). To secure that these functions are single-valued in the vicinity of the plait point, the choice of the x-phase must satisfy the condition

( d y ~ / d x ~ ) < 1 (3)

in this region. The distribution relations

Yi = m i ( x l ) x i (4)

and the conditions of consistency

F, x~ = F, y~ = 1 (5) i i

yield formulae for the remaining mass fractions in the x-phase:

X 2 = [1 - - m 3 + ( m 3 - - m l ) x l ] / ( m 2 - m 3 ) ,

x3 = [1 -- mE + (m2 -- r n l ) x l ] / ( m 3 -- mE). (6)

Thus, given the parameters a~ i and xl all other equilibrium concentrations can be determined.

At the plait point holds

xl = x~p, m; = 1. (7)

At the limit x~ --, 0, eqs (1) reduce to

lim In mi = aio. (8) X l ~ 0

Taking into account the constraints (7) and (8), eqs (1) transform to

lnmi = ~ , b i j ( x x - X lp) J, i = 1,2,3, j = I . . . . . n. J

(9)

Another relation to be taken into account is the Gibbs-Duham equation

~, x i ( d In r n J d x l ) = 0. (10) i

Its limit at the plait point is

x l = x l p , ~ x i ( d m i / d x O = O. i

Equations (9) and (11) yield the relation

(11)

X l = Xl .a , Z x i b i l = 0 . ( 1 2 ) i

Accordingly, the parameters b , must differ in signs. From eqs (5) and (12) one obtains for the mass fractions of the remaining two components at the plait

point

x2p : [-(b31 - b l l ) X l p -- b31]/(b21 - b31),

x3p = [(b21 - b l l ) x l p - b21]/(b31 - b21). (13)

These expressions, identical with the limits of relations (6) at the plait point, represent constraints assur- ing smooth transition between the x- and y-branches of the binodal curve.

The basic form of the mathematical model for type I ternary systems consists of eqs (4), (6), and (9) containing 3n + 1 parameters. In the present work, polynomials of third degree in formulae (9) have been found sufficient. Thus, 10 parameters, bt ~ . . . . , b 3 3 , Xlp,

are needed to describe the equilibrium. Given the values of these parameters and a set of values of the mass fractions of solute in the x-phase, Xlk , the calculation of the compositions of coexisting phases is straightforward.

The development of the model equations presented above follows, in principle, the lines of that given by Rod, but a different way of deriving eqs (8)-(13) has been chosen. Also a different meaning is attributed to the relations (13) which are conceived as a constraint, rather than as formulae for determining coordinates of the plait point. This important point will be discussed later.

P A R A M E T E R E S T I M A T I O N

The model parameters have to be evaluated by means of an optimization procedure. The Marquardt (1963) algorithm has been used and the corresponding FORTRAN subroutine BSOLVE (Kuester and Mize, 1973) has been incorporated in the program.

The appropriate form of the objective function in the Marquardt method is a sum of squares of deviations of calculated and measured values of quantities to be correlated. Modifications of this general form may be considered to meet various requirements im- posed on the correlation procedure: (i) type of data available; (ii) types of errors expected and strategy of their minimization; and (iii) structure of mathematical model.

A particular data set may contain either the distribution data only, or both the distribution and binodal data. In the former case, the distribution data must be complete, i.e. in each of the conjugate phases the concentrations of at least two components must be measured. In the latter case the subset of distribution data may contain only the concentrations of solute, since the mathematical model interconnects both cat- egories of data and thus allows to calculate complete compositions of conjugate phases for each value of x~. Weighted sum of squares may be used if the reliability of some parts of the data set or the requirements on the precision of their correlation differ. Another modi- fication uses relative instead of absolute deviations, which tends to reduce the absolute correlation error in the low concentration range.

In the present case, the main aim has been to prove whether the investigated model is flexible enough to

Correlation of ternary

be able to correlate the data with a correlation error comparable with the r a n d o m error of individual measurements. Therefore, all data have been taken with the weight 1. This is in contrast to Heyberger et al. (1977) who supposed systematic error of their binodal data in the low concentration range and applied to them the weight 0.1.

With respect to process and equipment design it is often desirable to keep the systematic error of the model in the low concentration range at a minimum. Accordingly the sum of squares of re la t i ve deviations has been chosen as the objective function

S = Sa + Sx + Sy = rain! (14)

where S, Sd, S , Sy are the total sum and the sums related to the distribution data and the data for the x- and y-branches of the binodal curve, respectively. If the numbers of measurements in the individual sums are K, L, and M, then

= s =Esx,, s , - - E s , . , k = l . . . . . K, k l ra

l = 1 . . . . . L, m = 1, ... ,M (15)

and

Sdk -~- E ( X i k / X i k - - 1) 2 -~ (Yik/Yik -- 1) 2, i

sxl = ~ ( X i , / x , , - 1) 2, s m = ~ ( Y i , / Y i m - 1) 2,

i = 1,2,3. (16)

Here x, y are the measured and X, Y the calculated mass fractions, respectively.

Rod (1976) considered also the objective function in the form of a sum of squares of relative deviations of measured and calculated values of the d i s t r ibu t ion

coe f f i c i en t s

S o = ~ S D k , k = l . . . . . K (17) k

and

SDk -~- E ( Y i k X i k / X i k Y i k - - 1) 2, i = 1,2,3. (18) i

liquid-liquid equilibria 895

In this case, of course, complete distribution data are needed. Also a less satisfactory correlation of the individual concentrations for the same data set may be expected. The data set namely consists now of 2K or 3K independent values of distribution coefficients, instead of the 4K or 6K independent values of mass fractions in the former case.

Until now it has been tacitly assumed that the measured values of the independent variable, xl, are free of error, i.e. that for the distribution data, as well as for the x-phase binodal data, holds xl = XI. How- ever, the general case of all data being subject to error can also be considered, as has been mentioned by Rod. A procedure minimizing the sums (16) by varying X~,

sek = min!, s~ = min!, srm = rain! (19)

has been incorporated in the respective variants.

E X P E R I M E N T A L D A T A

The set of equilibrium data for the system isobutyl acetate-aceticacid-water at 25°C has been used (Heyberger et al., 1977). It consists of three subsets, the two subsets of binodal data and that of distribution data. The binodal data have been obtained by the turbidity method. The distribution data, containing only the mass fractions of solute, have been determined by potentiometric titration. The paper presents 62 binodal points and 16 pairs of equilibrium solute concentrations. The mutual solubilities of water and isobutyl acetate at 25°C are also included. As the binodal data consist of pairs of repetitions, 30 aver- aged points have been used instead. For purposes of comparison of model variants, the composition of the coexisting phases has been completed by graphical interpolation of the binodal data with an average error not exceeding the random error of the experiments. The resulting data set is summarized in Tables 1 and 2. Various partial sets of this general data set have been used in the present work to simulate data types commonly found in the literature.

Table 1. Distribution data at 25°C

k x l x2 x3 Yl Y2 Y3 talk m2k mak

1" 0 0.0136 0.986 0 0.9960 0.0040 73.2 0.0041 2 0.0067 0.0138t 0.980t 0 . 0 1 8 2 0 . 9 7 9 t 0.OO50t 2.72 70.7t 0.OO51t 3 0.0107 0.0143t 0.975t 0 . 0 2 8 3 0.967t 0.0052t 2.64 67.5t 0.OO53t 4 0.0147 0.0153t 0.970t 0 . 0 3 8 1 0.956t 0.OO60t 2.59 62.5 t 0.OO62t 5 0.0190 0.0180t 0.963t 0 . 0 4 7 9 0 . 9 4 5 t 0.0070t 2.52 52.5t 0.OO73t 6 0.0290 0.0190t 0.952t 0 . 0 6 7 0 0 . 9 2 6 t 0.OO75t 2.31 48.7t 0.OO79t 7 0.0370 0.0205t 0.943t 0 . 0 8 4 0 0.908t 0.0077t 2.27 44.3 t 0.OO82t 8 0.0490 0.0240t 0.927t 0 . 1 0 5 0 0 . 8 8 7 t 0.0080t 2.14 37.0 t 0.0086t 9 0.0690 0.0314t 0.900t 0 . 1 4 0 0 0 . 8 5 0 t 0.0100t 2.03 27.1t 0.0111t

10 0.0790 0.0310t 0.890t 0.1570 0.832 t 0.0110t 1.99 26.8t 0.0124t 11 0.1070 0.0380t 0.855t 0 . 2 0 1 0 0.786t 0.0130t 1.88 20.7t 0.0152t 12 0.1470 0.0480¢ 0.805t 0 . 2 5 4 0 0 . 7 2 8 t 0.0180~ 1.73 15.2"~ 0.0223t 13 0.1750 0.0575 t 0.768t 0:2910 0.686 t 0.0230t 1.66 11.97 0.0300? 14 0.2450 0.1040~f 0.651t 0 . 3 5 6 0 0.596t 0.0480t 1.45 5.73t 0.0737t 15 0.2680 0.11205" 0.620t 0 . 3 7 8 0 0.5571" 0.0850t 1.41 4.97~" 0.1371t 16 0.2800 0.12OOt 0.6OO~f 0 . 3 8 6 0 0.542t 0.0725t 1.38 4.51t 0.1208t 17 0.3140 0.1510t 0.535t 0 . 4 0 8 0 0.482t 0.1100t 1.30 3.19t 0.2056t

* Mutual solubility of solvents. tObtained by graphical interpolation of binodal and distribution data.


Table2. Binodaldata at 25°C

l , m x I x 2 x 3 Yl Y2 Y3

1 0.0103 0.0148 0.9750 0 0 0 2 0.0359 0.0197 0.9444 0 0 0 3 0.0715 0.0294 0.8992 0 0 0 4 0.0804 0.0337 0.8859 0 0 0 5 0.1239 0.0416 0.8345 0 0 0 6 0.1649 0.0564 0.7787 0 0 0 7 0.2034 0.0685 0.7281 0 0 0 8 0.2378 0.0917 0.6705 0 0 0 9 0.2743 0.1147 0.6111 0 0 0

10 0.3010 0.1392 0.5598 0 0 0 11 0.3204 0.1606 0.5190 0 0 0 12 0.3409 0.1953 0.4638 0 0 0 13 0.3773 0.2559 0.3668 0 0 0 14 0.3911 0.2559 0.3668 0 0 0 15 0.4012 0.3250 0.2738 0 0 0 1 0 0 0 0.0217 0.9727 0.0057 2 0 0 0 0.0424 0.9514 0.0063 3 0 0 0 0.1034 0.8869 0.0098 4 0 0 0 0.1454 0.8430 0.0117 5 0 0 0 0.1836 0.8036 0.0128 6 0 0 0 0.2195 0.7664 0.0142 7 0 0 0 0.2985 0.6770 0.0245 8 0 0 0 0.3110 0.6526 0.0364 9 0 0 0 0.3291 0.6300 0.0409

10 0 0 0 0.3623 0.5850 0.0529 l l 0 0 0 0.3928 0.5374 0.0699 12 0 0 0 0.3960 0.5183 0.0858 13 0 0 0 0.4133 0.4759 0.1108 14 0 0 0 0.4148 0.4170 0.1682 15 0 0 0 0.4077 0.3516 0.2407

As the solute is more soluble in the aqueous phase, the organic phase has been chosen as the x-phase to fulfil condit ion (3). There is, however, some ambiguity about the position of the plait point, since the range of the distribution data does not reach as far as that of the binodal data. This point will be discussed later.

MODEL VARIANTS

The individual variants compared comprise combi- nations of the principal model [eqs (4), (6) and (9)] with various types of objective functions and various modifications of the principal data set. The variants with fixed and optimized solute concentrations in the x-phase are also included. The variants are listed in Table 3. Variant I uses complete distribution data and binodal data, keeps the x-phase solute concentrations at the measured values and uses the objective function in the form of eqs (14)-(16). Variant II optimizes the x-phase solute concentrations. In the variant III only complete distribution data are used with the above form of objective function and with fixed x-phase solute concentrations. The variant IV differs from III in optimizing the x-phase concentrations and the variant V in using the objective function based on eq. (17). Variants VI and VII differ from the variants I and II, respectively, in using distribution data of solute only.

As a measure of goodness of fit of the individual model variants the relative standard deviation of the

Table 3. Model variants

Variant no. Characteristics*

I DDc; BD; S; Xle II DDc, Bo, S, XIo III DDc; S; Xlr IV DDc; S; Xlo V DDc; SD; XIr VI DDs; Bo; S; X1F VII DDs; BD; S; Xlo

*DDc - - complete distribution data; DDs--distribution data of solute only; BD--binodal data; S--objective function according to eqs (14)-(16); SD--objective function according to eq. (17); Xlr - X1 fixed; X l o - X1 optimized.

measured and calculated values of mass fractions, a, has been employed:

a = ( S / N ) 1/2, N = n k K + n t L + n ~ M - rip. (20)

Here N is the total number of degrees of freedom, nk n~, nm, the numbers of measured concentrations at each point of the respective set and np the number of model parameters to be determined. For mutual comparison of the fit of the model variants, the F-test has been used.

ALGORITHMS

A F O R T R A N program T E R E Q U I was developed for correlating ternary l iquid-l iquid equilibrium data and simulating these equilibria. The flowsheet of T E R E Q U I is depicted in Fig. 1.

4, [ s ~ , , , o . I

of

Input: vadant type 1-~1 approx, of bij Xlp, X3p, r ~,~J

Vadants I, II, VI, VII I

Calculation of I expetin'~ntal or est mated mij

~ Con'ection of b31~l ¢ E;(22) l "

variation of ~1 ~ l Calculation of I bii,•Xlp ~1 I I rnij, Xij, Vii, ~k,

-1_ [ [~,symorsDk [" r I ] Eqs (4), (6), (9),

S = mint ? I [ (16) °r (18) I I

[ E q . (t4) Of (17) VII~ I Yes~F~ : ~Van 'ant t IV,

sigma, Eq.(20) [

Correlation of ternary liquid-liquid equilibria 897

graphical representation of the data and narrow limits have been set for optimizing the model parameter Xlp. The relations (13) have been transformed to

b31 -= (b l lx lp + b21x2p)/(xlp + x2p - 1) (21)

which ensures smoothness of the binodal curve at the Variants III, IV, V I ~, plait point. Various ways of incorporating condition

Calculation of (21) into the computation algorithm may be con- exp~m~t~ rail sidered, e.g. including X2p in the set of model para-

I meters to be optimized, instead of b31. In such case, _ _ eq. (21) would become a part of the main model. This

approach, however, leads to an appreciable reduction of stability of convergence, since the resulting algorithm is extremely sensitive to the variations of X~p. Therefore, an alternative procedure has been chosen: in each iteration cycle of the subroutine BSOLVE the value of b31 obtained is corrected using the relation

b31 = rb31 + (1 - - r){bl lXlp q- b21X2p)/

(Xlp + Xzp - 1) (22)

where r e (0; 1) is an adjustable constant. In the course of optimization by the subroutine

BSOLVE, a procedure FUNC is repeatedly used, which solves the system of eqs (4), (6) and (9) and evaluates the sums (16) or (18). In the variants III, IV and V this procedure is straightforward, since here the x-phase solute concentrations are measured. In the variants I, II, VI and VII for the y-phase binodal data experimental xl-values are not available. In this case for starting the solution of eqs (4), (6) and (9) the first estimates of ml, mentioned above, are used and the first estimates of X1 are calculated according to eq. (4a):

Xx = Yl/mx. (4a)

In the variants II, IV and VII, the Xa values have to be optimized to fulfil the conditions (19). An iterative procedure based on the principle of halving the steps has been developed, which is a part of the subroutine FUNC.

Another source of convergence difficulties is the lack of solvent distribution data (variants VI and VII). As can be seen from Table 1, the distribution coefficients of solvents (graphically estimated) vary more rapidly with Xa than those of the solute, especially at higher solute concentrations. Therefore, the calculated solvent concentrations are very sensitive to errors in the solute content. Since the respective measured values are missing, however, the resulting deviations cannot be evaluated and included into the objective function. This effect is still stronger when x~ is assumed to be subject to error, as in the variant VII.

According to this reasoning a successive solution of the individual model variants has been adopted using the set of parameter values obtained by solving the preceding variant as the first estimate for the next variant. In general, the variant I has been found the easiest to begin with.

Fig. 1. Flowsheet of program TEREQUI.

In the simulation mode, for a value of x, and a given set of values of model parameters, {bu, xlp}, it computes complete composition of coexisting phases. This procedure comprises a straightforward solution of the system of eqs (4), (6) and (9). In the correlation modes the calculation procedure is iterative; an optimum set of values of model parameters {bu, X~p} being sought to minimize the objective functions S or SD.

The program discriminates between the individual model variants. In the case of variants III, IV and V complete experimental distribution data are available. Hence, for all data points experimental values of distribution coefficients, mu, can be determined. In the case of variants I, II, VI and VII the data set contains also binodal data and, in the last two cases, the distribution data are incomplete. Accordingly, an incomplete set of measured m u values is obtained. For the y- phase binodai data first estimates of ml are obtained by interpolating the subset of distribution data.

For optimizing the model parameters b u, Xlp the subroutine BSOLVE is used. The first estimates of these parameters are a part of input data. With the system and data set used in the present work selecting an appropriate set of initial estimates of {b u, Xip} values has been found to be a difficult task. The main reasons are the strong asymmetry of binodal curve, the lack of distribution data near the plait point and the lack of measurements of solvents distribution. These features of the data to be correlated promote the model's tendency of converging to various local optima. A powerful remedy against this tendency has been found in including into the algorithm the conditions at the plait point. An estimate of the plait point coordinates, xtp, xz~, has been obtained from a

898

RESULTS AND DISCUSSION

The results of calculat ions performed according to the individual model var iants are summarized in Table 4. In its upper par t the opt imized values of the respective model parameters , including the plai t point coordinates , are shown. The lower par t conta ins the statistics character iz ing the corre la t ion by the individual model variants. These are the relative s tandard deviat ion of measured and calculated concent ra t ions [eq. (20)] and the pa ramete r SK, the deviat ion of the sum ~ x i p b i l in eq. (12) from zero. The F-test has been used for compar ing the goodness of fit of model variants. As the var ian t VII exhibits the best fit, its variance has been chosen as the basis of comparison. The values of the 5% confidence limit, F0.o 5, are also included.

The calculat ions star ted with the model var iant I. For a b road range of initial pa ramete r values the procedure converged to a single solution. As the calculated plait poin t coordinates were close to the values found graphically from dis t r ibut ion and binodal da ta ( x l p = 0.398; x z p = 0.310; x3p = 0.292), the above solut ion was accepted as global op t imum for var iant I and the respective pa ramete r values were used as f i r s t e s t i m a t e s for solving var iant II.

In the da ta set used (Table 1) mutua l solubilities of pure solvents are included. As relative deviat ions (16) are used, very small bu t finite values of measured solute concent ra t ions have to be ascribed to these points. The fact tha t this first tie line does not exhibit excessively large deviat ion in compar i son with the ne ighbour ing binodal points, raises doubts abou t the assumpt ion made by Heyberger et al. (1977) of a systematic error of the b inodal da ta in the low concentra- t ion range.

As can be seen from Table 4, a significant improve- men t of fit has been ob ta ined using the var iant II instead of var iant I. A closer analysis of the Sak , Sxl , Sy m

values for the individual exper imental points has shown, tha t the improvement is mainly due to a bet ter

J. PROCH/~ZKA and A. HEYBERGER

fit of the Y3 coordinates at higher solute concentra- t ions (Fig. 2). The figure also shows a selection of calculated tie lines covering the range of experimental d is t r ibut ion data. A systematic shift of the tie lines received by var iant II can be observed. Nevertheless, the optimized values of the plait poin t coordinates for bo th var iants differ only little from each other and from the graphically est imated values. Apparently, including the opt imizat ion of x 1 values into the model significantly contr ibutes to its flexibility.

As the next step the var ian t III using only complete d is t r ibut ion data and fixed x 1 coordinates was investigated. First, the plait point coordinates were kept at the op t imum values for var iant II. As can be seen from Table 4, this var iant Correlates the dis t r ibut ion da ta a lmost as well as var iant II does the set of dis t r ibut ion and binodal data. This is consis tent with the finding that inadequate fitting of b inodal da ta near the plait point is the main cause of the deter iora t ion of fit of var iant I in compar i son with var iant II. Next, var iant III was optimized with variable plait point coordinates (variant I l i a in Table 4). In this case the fit is significantly better. However, the opt imized values of plait poin t coordinates differ from those for var iant II. This is an example of uncer ta in ty of solut ion due to the lack of d is t r ibut ion data near the plait point, as ment ioned above. Figure 3 shows the experimental d is t r ibut ion data (completed by graphical interpola- t ion of the solvent concentrat ions) , as well as the results of s imulat ion using the solutions III and I l ia .

In cont ras t to var iant III in the var iant IV the xl coordinates are also subject to opt imizat ion. The re- suits for var iant IV in Table 4 have been ob ta ined keeping the plait poin t coordinates at the values for var iant III. The significantly bet ter fit of var iant IV testifies, similarly as in compar ing var iants I and II, the increased flexibility of models with opt imized solute concent ra t ions in the x-phase.

In the next step var iant V was examined, i.e. the same distr ibution data were treated using the objective

Table 4. Optimized parameter values of model variants

Variant

Parameter I II III lIIa IV V VI VII

bll - 2.872 - 2.834 - 2.955 - 2.136 - 2.971 - 3,387 - 3.837 - 2.992 bl2 - 1.240 -- 1.539 - 1.735 0.1324 - 1.893 -- 2.997 -- 4.164 -- 1.792 b13 0.0536 0.0390 0.0370 0.0379 0.0324 - 0.0167 0.0199 - 0.0008 b21 - 11.08 - 10.91 - 11.02 - 10.03 - 10.95 - 10.97 - 10.89 - 10.92 b22 - 0.2065 - 0.2622 - 0.2783 - 0.1779 - 0.3091 - 1.078 - 0.5264 - 1.054 b23 2.429 1.588 1.896 0.0422 1.705 0,4431 0.6434 0.1175 b3 x 16.09 15.70 16.90 14.91 16.20 16.95 16.02 14.66 b32 3.004 2.498 2.442 1.808 2.177 1.350 0.3474 0.2269 b33 - 11.16 - 10.01 - 18.40 - 9.993 - 14.05 - 21.76 - 18.59 - 10.04 Xxp 0.3980 0.3981 0.3981 0.4224 0.3981 0.3981 0,3981 0.3981 X2p 0.3144 0.3114 0.3114 0.3114 0.3114 0.3114 0.3114 0.3114 Xa~ 0.2876 0.2905 0.2905 0.2905 0.2905 0.2905 0.2905 0.2905 a 0.235 0.113 0.116 0.088 0.058 0.079 0.113 0.038 S K - 0.042 - 0.035 0.302 - 0.009 0.113 0.159 - 0.265 - 0.333 F 38.24 8.84 9.31 5.36 2.33 4.32 8.84 1.00 Fo.os 1.42 1.42 1.49 1.50 1.49 1.49 1.46 1.46


¢0

X

1.00

0.80

0.60

0.40

0.20

0.00

0.00 0.10 0.20 0.30 0.40

X1,Y1

Fig. 2. Comparison of measured distribution data (complemented by graphical interpolation) and binodal data with results of calculations using model variants I and II: (+) measured; (x, - - ) variant I;

(a, ----) variant I1.

function (17) and the parameter values obtained with variant III as first estimates. Table 4 shows a significant improvement of fit. This result is unexpected, since optimizing the ratios of equilibrium concentrations, instead of the concentrations themselves, im- poses less severe conditions on the system. A plausible explanation of this finding is that, on the one hand, the objective function (17) gives the system more flexibility and, on the other hand, fixing of the plait point and using the parameter values for model III as the first estimate helps to find a global optimum near to that for the variants III or IV. Indeed, with various different first estimates of the parameter vector b = {b o, Xi~} the system converges to different optima with higher variance (multiplicity of solutions).

Finally, the variants VI and VII were investigated. These variants do not use the distribution data for components 2 or 3. The respective data set consists only of the distribution data for solute, the binodal data and the plait point coordinates. Table 4 shows

that variant VI correlates the data much better than variant I and variant VII significantly better than variant IL Hence, omitting the graphically interpo- lated distribution data for components 2 and 3 leads to an improvement of correlation. The possible error of interpolation can hardly explain this striking differ- ence. Apparently, it should be attributed to greater flexibility of the variants VI and VII. Figure 4 compares the distributions simulated by variants I and VI, Fig. 5 those using variants II and VII. However, the variants VI and VII exhibit very low stability of solution. This problem has been overcome using the solutions of variants I and II as first estimates and fixing the plait point coordinates on the values obtained by the variant II (method of successive solution).

The results in Table 4 show that generally the present model allows a better correlation of binodal data than the model by Hlavat~, (1972) (see Heyberger et al., 1977). With the variant VII equal fit of the distribution data has been obtained as with the orthogonal polynomials used by Heyberger et al.

900

X

J. PROCHAZKA and A. HEYBERGER

1.00 ~

0.80

0.60

0.40

0.20

0.00 0.10

×

+ A

x

o

o

~ x A x

~. ~ n ~< | |

0.20

X1,Y1

0.30 0.40

Fig. 3. Comparison of measured distribution data, complemented by graphical interpolation, with results of calculations using model variants III and Ilia: ( + ) measured; (x) variant III; (~) variant Illa; ([~) fixed

plait point; (©) optimized plait point.

These results obtained by correlating one particular ternary system with the Rod method can be generaliz- ed as follows: At the beginning a first estimate of the plait point coordinates has to be obtained by interpolating the data. If the data set to be correlated contains incomplete distribution data, it should be completed by a suitable interpolation. According to the structure of the data set the correlation should begin with either the variant I or III with fixed x-phase solute concentrations. An optimization of the plait point coordinates can be tried. To obtain a better fit the solution thus found can be used as the first estimate of the parameter values for solution of variants VI and VII or IV and V, respectively. In each step of this procedure variation of the parameter r in eq. (22) can help to reduce the value of SK and thus to improve the smoothness of the binodal curve at the plait point.

As was pointed out earlier, the main aim of this work was to find a flexible and reasonably stable correlation for such highly nonlinear and asymmetric

ternary systems like that investigated. The parameter values of the empirical expressions (9) for the activity function (2) in Table 4 cannot be expected to reflect fundamental properties of the system, Nevertheless, for the successively solved variants the corresponding parameters mostly agree in sign and order of magni- tude, which manifests the stability of the method of successive solutions.

CONCLUSIONS

In the present work the Rod model for correlating ternary liquid-liquid equilibrium data has been re- defined and expanded into seven variants. The individual variants differ in complexity, type of data re- quired and type of objective function used. Suitable optimization algorithms based on the Marquardt method have been developed. This method has been successfully used for correlating the equilibrium data on the system isobutyl acetate-aceticacid-water, published by Heyberger et al. (1977). This system


1.00

0.80

0.60

0.111

0.20

0.00

0.00 0.t0 0.20 0.30

X1,Y1

Fig. 4. Distribution calculated using model variants I and VI: (x,

0.40

) variant I; (A, - . . . . ) variant VI.

exhibits a highly asymmetric equilibrium and has been chosen for testing the properties of the correlation method.

In general, in the arrays of the model variants I, II, VI and VII, or III and IV or V with increasing flexibility (fitting of data) their stability (convergence to a unique solution) decreases. To overcome the insta- bility of higher members of the array, a procedure of successive solution has been applied in which the solution of the preceding member is used as first estimate of the solution for the subsequent member.

For a variant to converge to a unique solution it is essential that either the data set comprises distribution data covering the whole range of solute concentrations, or a reliable estimate of the plait point coordinates can be made using in addition to the distribution data also binodal data covering this region.

In the present work the data set containing only binodal data and solute distribution data could be successfuUy correlated combining graphical estimation of missing distribution data and plait point coot-

dinates with the method of successive solution of model variants.

The present method has been found superior to other correlation methods investigated in the work of Heyberger et al. (1977), either for its higher goodness of fit, or, in the case of the method of orthogonal polynomials, for its lower parametricity.

a b F FO.O5 K

L

m

M

n

NOTATION

constant in eq. (1) constant in eq. (9) ratio of variances 5% confidence interval of F number of distribution experiments number of x-phase binodal experiments distribution coefficient number of y-phase binodal experiments degree of polynomial (1); number of measured concentrations in an experimental point

C~S 51-6-[


1.00

0.80

0.60

>- (6 X

0.40

0.20

0.00

0.00 0.t0 0.20 0.30

XI,Y1

Fig. 5. Distribution calculated using model variants II and VII: ( ×,

0.40

) variant II; (A, - . . . . ) variant VII.

N r s

S

x X

Y Y

total number of degrees of freedom i adjustable constant in eq. (22) sum of squares of relative deviations in j one experiment k total sum of squares of relative deviations l measured mass fraction in x-phase calculated mass fraction in x-phase m measured mass fraction in y-phase calculated mass fraction in y-phase p

x

Y Greek letters

activity coefficient F function in eq. (2) a relative standard deviation of measured

and calculated mass fractions

Subscripts d D

distribution concerning differences of distribution coefficients

ith component (1--acetic acid; 2--water; 3-- isobutyl acetate) j th terms of polynomials (1) and (9) kth member of set of distribution measurements /th member of set of binodal measurements in x-phase ruth member of set of binodal measurements in y-phase plait point; number of model parameters x-phase (organic phase) y-phase ( aqueous phase)

REFERENCES

Brandani, S., 1994, An algorithm for ternary liquid-liquid equilibria calculation. Chem. Biochem En#ng Quart. 8, 163-166.

Bulatov, S. N. and Yachmenev, L. T., 1971, Approximation of binodal curves of heterogeneous ternary systems. Teor. Osn. Khim. Technol. 5, 644-650.

Hand, D. B., 1930, Dimeric distribution, I. The distribution of a consolute liquid between two immiscible liquids. J. phys. Chem. 34, 1961-2000.

Correlation of ternary liquid liquid equilibria 903

Heyberger, A., Horh~ek, J., Buli6ka, J. and Prochfizka, J., 1977, Equilibrium of isobutyl acetate-acetic acid water. Collection Czech. Chem. Commun. 42, 3355-3362.

Hlavat~,, K., 1972, Correlation of the binodal curve in a ternary liquid mixture with one pair of immiscible liquids. Collection Czech. Chem. Commun. 37, 4005 4007.

Kuester, J. L. and Mize, J. H., 1973, Optimization Techniques with FORTRAN. McGraw Hill, New York.

Marquardt, D. W., 1963, An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Ind. Appl. Math. 11, 431-441.

Othmer, D. F. and Tobias, P. E., 1942, Liquid-liquid extraction data. tie line correlation. Ind. En#ng Chem. 34, 693-696.

Renon, H. and Prausnitz, J. M., 1968, Local compositions in thermodynamic excess functions for liquid mixtures. A.1.Ch.E.J. 14, 135-144.

Rod, V., 1976, Correlation of equilibrium data in ternary liquid-liquid systems. Chem. Engng J. 11, 105-110.

Spalding, W. M., 1970, Determination of tie lines in liquid systems. Br. Chem. Engn 9 15, 62.

correlation of ternary liquid-liquid equilibria in system isobutyl acetate-acetic acid-water

Documents