Ind. Eng. Chem. Res. 1991, 30, 2018-2019
2018
Alternative Method for the Evaluation of the Wilson Binary Constants An alternative method to calculate the constants of the Wilson equation for the correlation of activity coefficients (ri)of binary liquid solutions is proposed. The method is based on the integration of both the analytical expression and the experimental data of the excess Gibbs energy (GE).The experimental GE values were first correlated by an orthogonal discrete polynomial expansion. The Wilson constants obtained by this method were satisfactorily compared with that calculated through other usual techniques.
Introduction Analyzing the difficultiesrelated to the evaluation of the constants Aij of the Wilson equation, Apelblat and Wisniak (1989) proposed a simple method for their calculation. It is based on the resolution of the equation system that results from the expression derived by Wilson for the excess Gibbs energy and the corresponding expression for its first derivative with respect to concentration. Therefore, by knowing both magnitudes at a given composition, the A, values can be calculated. The extreme of the experimental dependence GE vs x i was chosen for such a calculation. The location of this value was obtained through the expansion as a polynomial of the function CE in the central concentration region. Nevertheless,in many cases this region is not well-defined, particularly when only a few data values are available. This fact produces uncertainties in the determination of the maximum and therefore in the calculation of the A i . values. The present work deals with the derivation of an alternative method for the evaluation of the Wilson constants, starting from the complete experimental GE vs x, data set.
The Wilson equation for the excess Gibbs energy can be written as follows:
(GE/RIT?WiLon= Gf2(~1,A12)+ GFi(x29A21)
(2)
GF(xi,Aij) = -xi In [Aij + (1 - Aij)xi]
(3)
where
The left-hand side of eq 1 results:
L"(GE/R7')WiL"n dx 1
-
-
where x2 = 1 - x l . This integration can be done analytically:
Therefore, the complete expression for eq 5 is
Theory The proposed method consisted in the derivation, in a manner similar to that of Apelblat and Wisniak, of expressions of the type f = f(Aij,Aji,xi),from which Wilson constants can be calculated. In this case, the evaluation of the following integral is proposed:
~ x l ( C E / R T ) W i Ldx " n = L'1(GE/R7')exp dx, = f(A12+421~1)(1)
With the values x1 = 0.5 and x1 = 1 chosen, the pair of equations is generated. To evaluate the integral of the experimental excess Gibbs energy data, it is proposed to adjust the (GE/ values by the following polynomial: = CCkPk(x1)
lGE/[RTxl(l - X , ) ] f "
where different limits of integration can be used.
(7)
k
Table I system (reference) n-hexane (1)-l,l,l-trichloroethane(2);60 O C (Hanson and Van Winkle, 1967) n-hexane (1)-2-butanone (2);60 "C (Hanson and Van Winkle, 1967) n-hexane (lk2-butanol (2): 60 O
C
(Hanson and Van Winkle, 1967)
diisopropylamine (1)-water (2); 10 "C (Davison, 1968) benzene (1)-ethanol (2); 25
O C
(Smith and Robinson, 1970)
2-propanol (1)-n-heptane (2);60 "C Van Ness et al., 1967) n-octane (1)-dioxane (2);80 "C (Tassios and Van Winkle, 1967) n-nonane (1)-dioxane (2);80 O C (Taasios and Van Winkle, 1967)
0888-5885/91/2630-2018$02.50/0
method present work Apelblat and Wisniak Hirata present work ADelblat and Wisniak Hjrata Dresent work Apelblat and Wisniak Hirata present work Apelblat and Wisniak Hirata present work Apelblat and Wisniak Hirata present work Apelblat and Wisniak Hirata present work Apelblat and Wisniak Hirata present work Apelblat and Wisniak Hirata
A12
'421
0.71630 0.81558 0.61607 0.53257 0.63235 0.64871 0.57764 0.51966 0.47709 0.10684 0.11618 0.10306 0.51217 0.50089 0.46589 0.16901 0.16999 0.16554 0.30702 0.32378 0.29298 0.17876 0.25867 0.29778
0.99823 0.88886 1.09504 0.44423 0.37073 0.36797 0.17842 0.23342 0.20058 0.43606 0.46697 0.42809 0.10100 0.10246 0.11025 0.35057 0.37169 0.26335 0.76939 0.76179 0.78399 0.94723 0.67974 0.72565
0 1991 American Chemical Society
1@E, 6.51 10.23 24.79 17.17 25.73 30.80 31.92 57.81 70.74 82.92 78.34 86.09 22.91 17.62 19.32 76.50 86.52 118.3 21.68 26.52 28.66 91.49 208.7 113.4
2019
I n d . Eng. Chem. Res. 1991,30,2019-2020
where the Pk(xl)are defined as follows: Po(x1) = 1
Pl(xl) = (xl
- al)Po(xl)
p/t+i(xi)= (xi - ak+iPk(xi) - bd’k-i(x1) k>l (8) The parameters ah and bk are evaluated so as to generate the set of orthogonalpolynomials Pk(xl) from the complete experimental data set. After the evaluation of these coefficients, the ck are calculated for a least-squares method. The polynomial expansion obtained in this way allow the integration of the function (GE/RT)W according to eq 2 or the application of the method proposed by Apelblat and Wisniak, using the entire composition range. The constants A, are obtained from the resolution of the following system of equations:
PUlon(A12,A21,~1=O.5) = PF’(A12,A2+1=0.5)
(9)
Results and Discussion The proposed method was applied to different isothermic systems. The expansion equation 8 allows also the application of Apelblat and Wisniak method. The results obtained by computer calculations are shown in Table I. The correlation ability of the Wilson constants evaluated by the two processes described above and of those given by Hirata et al. (1976) was calculated by the following error function:
E, = I5rd - r’ifPl/rifP+ Ird - rPl/rPl/n i=l
From the analysis of Table I, it can be inferred that the accuracy of the proposed method is at least as good as the other methods. On the other hand, the obtained constants A- are consistent with the complete experimental data set. %inally, it should be mentioned that the method is in principle applicable to any equation for the excess Gibbs energy correlation, though conditioned to the intrinsic fitting capability of such an equation.
Literature Cited Apelblat, A.; Wisniak, J. A Simple Method for Evaluating the Wilson Constants. Ind. Eng. Chem. Res. 1989,28, 324-328. Davison, R. R. Vapor-Liquid Equilibria of Water-Diisopropylamine and Water-Di-n-Propylamine. J. Chem. Eng. Data 1968, 13, 348-351. Hanson, D. 0.;Van Winkle, M. Alteration of the Relative Volatility of n-Hexane-l-Hexene by Oxygenated and Chlorinated Solvents. J. Chem. Eng. Data 1967,12,319-325. Hirata, M.; Ohe, S.; Nagahama, K. Computer Aided Data Book of Vapor-Liquid Equilibrium; Kodansha-Elsevier: Tokyo, 1976. Smith, V. C.; Robinson, R. L. Vapor-Liquid Equilibria at 25 “C in the Binary Mixtures Formed by Hexane, Benzene, and Ethanol. J. Chem. Eng. Data 1970,15,391-395. Taesios, D.; Van Winkle, M. Prediction of Binary Vapor-Liquid Equilibria. J. Chem. Eng. Data 1967, 12, 555-561. Van Ness, H. C.; Soczek, C. A.; Peloquin, G. L.; Machado, R. L. ThermodynamicExcess Properties of Three Almhol-Hydmxrbon Systems. J. Chem. Eng. Data 1967,12,217-224.
Maria R. Gennero de Chialvo, A b 1 C. Chialvo* Programa de Electroquimica Apljcada e Ingenierja
Electroquimica-(PRELINE) Facultad de Ingenieria Quimica (UNL) Santiago del Estero 2829, 3000 Santa Fe, Argentina Received for reuiew December 10, 1990 Accepted May 2, 1991
(11)
CORRESPONDENCE Comments on “Tuning Controllers on Distillation Columns with the Distillate-Bottoms Structure” Sir: A recent paper by Papastathopoulou and Luyben (1990) discusses modeling and controller tuning of distillation columns using the DB scheme. We would like to point out that their main result, which is to show how models for the DB scheme may be derived from more conventional control structures provided the liquid flow dynamics are included, is taken from the work of Skogestad and co-workers (1989a,b, 1990a,b). There are also two misconceptions in their paper that deserve comments. 1. In the Introduction they claim that for the case of perfect level control the DB scheme is equivalent to the RR-BR scheme (also denoted the LID-V/B confiiation; i.e., using reflux ratio to control top composition and boilup ratio to control bottom composition). However, as shown by Skogestad et al. (1990a),these configurations behave entirely differently even when level control is perfect. For example, while the DB scheme works only when both loops are closed, the (LID)(V / B )configuration performs reasonably well even when both loops are in manual. 2. The transfer function 1/(1-gL)in eq 24 contains a pure integrator. This was shown previously by Skogestad 0888-5885/91/2630-2019$02.50/0
and co-workers (1989b, 199Ob). In eq 29 Papastathopoulou and Luyben (1990) derive an expression for 1/(1- gL), which contains the integrator, but which is otherwise incorrect (except for the case with only one tray, i.e., NT = 1).
Nomenclature B = bottoms flow rate D = distillate flow rate gL(s) = transfer function for liquid lag through column L = reflux flow rate L / D = RR = reflux ratio in top NT = number of trays in column V = boilup flow rate V / B = RB = “reflux” ratio in bottom
Literature Cited Papaetathopolous, H. S.; Luyben, W. L. Tuning Controllers on Distillation Columna with the DiatillateBottoms Structure. Znd. Eng. Chem. Res. 1990,29, 1859-1868.
0 1991 American Chemical Society
