Limitations in Correlating Strongly Nonideal Binary Systems with the NRTL and LEMF Equations Dimiirios Tassios New Jersey lnstifute of Technology, Newark, New Jersey 07 102
An analytical method and one using graphs included herein are presented for establishing the limitations of the NRTL and LEMF equations, whereby parameters obtained by regressing binary experimental data predict partial miscibility for some highly nonideal systems. It is shown that prediction of partial miscibility can be eliminated by changing the value of a but this can lead to poor correlation of the experimental data. On the other hand, the Wilson equation appears to successfully correlate the data for such systems.
Introduction Correlation and prediction of vapor-liquid equilibrium (VLE) data is essential in the design of certain separation processes such as distillation. The correct determination of the ideal number of stages for a binary separation through computer calculations depends, to a great extent, on the accurate correlation of the available experimental VLE data. Typically, the first step in such an effort is the correlation of the binary activity coefficients, which at low pressures can be simply defined by: Y i =-
Yi P
xipis
by such equations as the Wilson, NRTL, or LEMF. In 1964, Wilson, using the concept of local mole fractions, introduced the following expression for y1 and 7 2 . In y1 = -In (Xl
+ 1112x2)
where
This equation successfully correlates binary data but it can not predict partial miscibility. Renon and Prausnitz (19681, combining Wilson's concept of local mole fractions with Scott's (1956) two liquid theory, developed an expression for the excess Gibbs free energy that can correlate both miscible and partially miscible systems. For a binary system:
where 712
= (g12 - gm)/RT
721
=
(g21 - gll)/RT
with g12 = gzl and G12
= exp(-a12712)
G21
= exp(-a12721)
From eq 3, the following expressions are obtained for y1 and Y 2 after the appropriate differentiation: 574
Ind. Eng. Chem., Process Des. Dev., Vol. 15. No. 4, 1976
Renon and Prausnitz recommended use of one of the following values for a: 0.20,0.30, or 0.47. They grouped liquid mixtures into seven categories according to polarity, association, etc., and suggested specific cy values for each. However, a t times, the rules for the selection of the proper value of a are ambiguous and difficult to apply. Marina and Tassios (1971) showed that a single value of a = -1 yielded comparable accuracy for miscible and improved performance for partially miscible systems. Since a negative value of a is inconsistent with the development of the NRTL equation, they introduced the concept of local effective mole fractions (LEMF). Determination of the parameters in eq 2 or 4 is accomplished by regressing the activity coefficients calculated through eq 1. These parameter values are next used with an appropriate bubble point calculation subroutine to determine the calculated X-Y curve. Novak and his coworkers (1972; 1974) have shown, however, that for certain very nonideal systems, the predicted curve can, for certain positive and negative values of a , correspond to a partially miscible system. Such an example is presented in Figure 1 for the system 3-methylpyridine-water ( a = 0.47). Notice that, even though the quality of the fit is fairly good, partial miscibility is predicted as demonstrated by the existence of a maximum in the predicted curve. From thermodynamics, the criterion for complete miscibility is given by: (5)
This criterion is always met by the Wilson equation but not by the NRTL and LEMF equations. Let (Gll)xobe the minimum value of Gll in the interval 0 < X I < 1and Xo the value of X1 where this occurs. Novak and his coworkers (1972) used these values of XOand (Gll)xo,along with the values of the infinite dilution activity coefficients (71"and 72") to establish plots describing the limitations of the NRTL equation. Finally, in a brief note to the editor (1974), they have indicated that the LEMF equation is not applicable to systems where 0.1 < Xo < 0.15. In evaluating G11, and hence Xo, they made direct use of the experimental data. According to Suska et al. (1972),however, the value of Xo cannot be uniquely established by this approach because of uncertainties involved in the determination of GI1 by differentiation of the experimental data. For example, in the system ethanol (1)-n-hep-
1
2
I
c(: 0.47 yI
0.6
-
0.4 -
0.2
0.4
0.8
0.6
1.0
X
Figure 1. System: 3-methylpyridine-water at 89.83 'C: +, calcd a = 0.47.
XI
-, exptl;
tane (2), use of the P-X data yields XO= 0.35 while use of the X-Y data gives a value of XO= 0.55. In a more recent publi1 cation (1974), they also used the parameters 712 and ~ 2 obtained by regression of the activity coefficients for the same purpose but only for positive values of a.I t is the purpose of this paper to: (1)present an analytical method for determining if a set of parameters obtained by regressing the available VLE data will lead to the prediction of partial miscibility; (2) present appropriate graphs to be used for the same objective but for specific values of a: (-1; -2; 0.2; 0.3; and 0.47); (3) investigate methods for eliminating the prediction of partial miscibility and assess the effects on the accuracy of the obtained correlation.
Figure 2. Variation of G l 1 with X1
Curve
LY
1
0.30
2
0.47 0.47
3
' 1 2
12'
2.6 2.6 2.9
0.0 2.0 2.6
The Proposed Method The analytical method consists of the following steps: (1) Regress the available experimental activity coefficients through eq 4 to obtain the values for the parameters (912 gll), (g12 - g22). (2) Introduce these d u e s into the following expression for Gll:
.
(3) Determine the value of X1 = XOfor which G11 assumes its minimum value, by solving the following equation:
Figure 3. Miscible (M) and partially (P) miscible regions for three values of a:0.2; 0.3, and 0.47: M, to the left of curves; P, to the right of curves.
6721G2i2(1- G2i)
(4) Insert the value X I = Xo into eq 6 +d determine the value of G11. If GI1 3 0, no partial miscibility will be predicted for the chosen value of a and the values of (g12 - gll) and (g12 g22) obtained by regression of the experimental data. If G11 < 0, partial miscibility will be predicted. An examination of the possible shapes of the GI1 vs. XI curves indicate three possible types (Figure 2 ) . The first exhibits a single minimum (curve 1);the second exhibits a single minimum plus a zero slope inflection point, and the third exhibits two minima and one maximum. Clearly, more than one solution of eq 7 exists for the two last cases and, hence, the real minimum of GI1 may not be found. A close examination of type 3 curves, in Figure 2 and in Novak et al. (1974), for example, suggest that the two minima occur around X1 = 0.25
and X1 = 0.75 and the maximum around X1 = 0.5. For type 2 curves, the minimum occurs around X1 = 0.25 or 0.75 and the zero slope inflection point around XI = 0.5. Use, therefore, of two starting values for XO(Xo = 0.01 and Xo = 0.99) in the subroutine solving eq 7, along with allowing small changes in XO,leads-after comparison of the resulting solutions-to the true minimum of G11. This is apparent for type 1 and 2 curves while for type 3 curves it should be noticed that the restriction of small changes in Xo will lead to the two minima rather than the maximum. If the value of a is set a priori-according to Renon or Marina-use of this elaborate procedure is not warranted. Instead, one can use the graphs presented in this paper showing miscible and partially miscible regions as functions of the values of the dimensonless parameters: 721
- gll = *g12 RT
7
712
g12 - g22 =RT
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4, 1976
575
P
.. SYSTEM: THF
-7
-
- WATER
AT 5 b C
0.05-
0.04-
0.03 -
I
t
-08
-1 2
-0 4
00
0 4
T>!
\
\
\I
Figure 7. Average absolute deviation in Y1 and regions of partial (PI and complete (MI miscibility for the system tetrahydrofuran-
08
12
water.
~
Figure 4. Miscible (M)and partially (P) miscible regions for two values of a:-1.0 and -2.0: M, to the left of curves; P, to the right of curves.
0 02
0.01
1
I
.. .,... M
,P
0.0 -5.0
-4.0
-3.0
-2.0
-1.0
cc
1.0
O.O
Figure 5. Average absolute deviation in Y1 and regions of partial (P) and complete (M) miscibility for the system pyridine-water a t 89.83 "C.
for the following values of a ; 0.2,0.3,0.47 (Figure 3); -1 and -2 (Figure 4). For isothermal data, the use of the graphs is straightforward. For isobaric data, it is recommended that the two extreme temperatures are used. If in either case the re-
suiting dimensionless falls in the miscibility region, final decision must be made on the basis of the G11 vs. X1 plot. Similar plots, using temperature independent parameters ~~2 and 721 and for positive only values of a , are presented by Novak et al. (1974).
Effect on Accuracy An inspection of the plots in Figures 3 and 4 suggests that a change in the value of cy may lead to parameters that do not predict partial miscibility. Petrie (1975) has shown that this does occur but at the expense of the quality of the obtained fit of the experimental data. The effect of the value of a on the prediction of complete miscibility and the quality of the corresponding fit were investigated in this study for three systems: pyridine(P)-water, 3-methylpyridine(3MP)-water,and tetrahydrofuran(THF)-water. These systems approach partial miscibility, to various degrees, and therefore its prediction can be expected. Figures 5, 6, and 7 present plots of the average absolute deviation in Y N
C I Y ,(exptl) - Y ,(calcd) 1 JAY(,, = i = l
N
vs. the corresponding value of cy for these three systems. The regions of complete (M) and partial (P)miscibility prediction
r
.' I
P T A R T I A L MISCIBILITY
-1
0.02
M:MISCIBLE W: WILSON EOUATION
...
-3.0
-2.0
-1.0
oc
0.0
Figure 6. Average absolute deviation in Y1 as a function of 01, and with the Wilson equation, along with regions of partial (P) and complete (M) miscibility for the system: 3-methylpyridine-water at 89.83 "C. 576
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4, 1976
Table I: Performance of the NRTL/LEMF and Wilson Equations NRTL/LEMF System P-w 3MP-W
THF-W
Wilson
cy
TypeQ
IAYl.3.
I.?rYIrnax
IAYIm
IAYlrnax
Ref
-2.80 0.65
M M
0.004 0.004
0.010 0.012
-
-
A
0.70 0.52 -2.80
M P P
0.024 0.006 0.007
0.048 0.023 0.022
0.007
0.028
A
0.53 0.44 -1.10
M P P
0.017 0.004 0.004
0.032 0.009 0.013
0.005
0.020
B
Type: type of prediction; P: partial miscibility; M: complete miscibility. A: Andon et al. (1957). B: Matous et al. (1972).
o L 0.5
I.o
XI
Figure 8. Experimental and calculated vapor phase compositions for the system pyridine-water with the optimum value of 01 = 0.65 or -2.80: -, exptl; +, calcd.
are also included. In regressing the available experimental data, zero starting values along with the following minimization function were used:
s=
E 1[
1=1
Ylicalcd)
- '?l(obsd)]:+
Yl(obsd)
[
YY(ca1cd)
- YZiobsd)
YZiobsd)
If1
For the pyridine-water system, very good results are obtained both in the positive and negative regions of a values (Figure 5). This is further demonstrated in Figure 8, where the best results for cy > 0 are presented. For the 3MP-water and THF-water systems on the other hand, good fit leads to prediction of partial miscibility as shown in Figures 6 and 7; conversely, complete miscibility is predicted a t the expense of the quality of the obtained fit as shown in Figure 9 where the best fit leading to complete miscibility for the THF-water system is presented. I t should be mentioned, however, that even though Matous et al. (1972) claim that the system THF-water is completely miscible a t 50 "C, their X-Y dataas shown in Figure 9-suggest the existence of partial miscibility. Following the difficulties encountered in correlating the last two systems with the NRTL/LEMF equations, the Wilson equation was used to correlate the data since it can not predict partial miscibility. The results obtained for these two systems, along with those obtained for all three with the N R T L L E M F equations, are presented in Table I. The improved accuracy obtained by the use of the Wilson equation is apparent. I t should be mentioned, finally, that Matous et al. (1972) obtained results comparable to those of the Wilson equation by using the modified Redlich-Kister equation for the THFwater system.
5
Figure 9. Experimental and calculated vapor phase composition for the system tetrahydrofuran-water with the optimum value of cy = 0.53: -, exptl; +, calcd.
Discussion and Conclusions Novak and his coworkers (1972) report that for the system tetrahydrofuran-water, the NRTL equation is unable to predict both miscibility and the experimental infinite dilution activity coefficient with a single value of cy. Table I and Figure 9 lead to the same conclusion: complete miscibility can be predicted only a t the expense of the accuracy of the predicted results. The same conclusion is reached when the two methods are compared for the 3-methylpyridine-water system. The assumption of ideal vapor behavior for the systems presented in this paper has been substantiated by Tsonopoulos (1975).Using his correlation (1974),he found that for the systems pyridine-water and THF-water the maximum deviation from ideality was 2.4%. Figure 3 indicates that for symmetric systems with cy > 0.426 partial miscibility can be predicted which is contrary to the statement of Renon and Prausnitz. Figure 10, presenting a plot of G" vs. X1 for a = 0.45 and 7 1 2 = 7 2 1 = 4.0, further substantiates this observation. This prediction of partial miscibility a t a > 0.426 has been discussed extensively by Heidemann and Mandhane (1973). The results of this paper also indicate that the NRTL/ LEMF equations must be treated as empirical ones and that best results are obtained when the value of a is obtained by regression of the available experimental data. For example, use cy = -1 and 0.47 for the pyridine-water system results in values of / A Y l a v = 0.018 and 0.015, respectively, as against 0.004 for a = -2.8 or 0.65. Finally, as shown in Figures 5 and 6, care should be exercised for a < 0.0 in choosing the starting value of cy, for the wrong minimum could be obtained. Ind. Eng. Chem., Process Des. Dev., Vol. 1 5 , No. 4, 1976 577
0.0
Nomenclature
- g 2 2 ) = constants in eq 4,cal/g-mol ( g 1 2 - gll), GM = Gibbs free energy of mixing G E = excess free energy of mixing N = number of experimental data in a binary system P = total pressure, mmHg Ps = saturation vapor pressure, mmHg R = gas constant, cal/g-mol K T = temperature, absolute, K V, = liquid volume of pure i, cm3/g-mol X , = liquid phase mole fraction, component i X O = value of XI where GI1 assumes its minimum value
AG' R T
-0.0
(G1l)Xo
Y , = vapor phase mole fraction, component i Greek Letters a 1 2 , a = constant in eq 3
-0.10
0.5
0.U
F i g u r e 10. V a r i a t i o n
of AGM with
X I
X If o r a = 0.45 a n d ~~2 = T~~
=
y L = liquid phase activity coefficient, component i y l m = infinite dilution activity coefficient, component All = constants in eq 2 (Al, - A l l ) = constants in eq 2 , cal/g-mol 712, 7 2 1 = constants in eq 3
i
4.0.
In conclusion, the graphs presented here provide a simple and safe method for establishing if the NRTL or LEMF parameters obtained by regressing a set of binary VLE data will in turn predict partial miscibility. The use of an analytical approach is possible if the subroutine searching for X o is carefully chosen and two starting values are used. If partial miscibility is indeed predicted, it can be eliminated by changing the value of a. This, however, can lead to a very poor correlation of the experimental data. In such a case, use of the Wilson equation can provide accurate representation of the data.
Acknowledgment The author wants to acknowledge the assistance of Mr. Norman Silverman of Exxon Chemical Company, U.S.A., Baton Rouge, La., for correlating the data with the Wilson equation and of Dr. Costas Tsonopoulos of Exxon Research and Engineering for some interesting comments on the subject matter.
578
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4. 1976
Literature Cited Andon, R. J. L., Cox, J. D., Herington, E. F. G., Trans. Faraday SOC., 53, 410
(1957). Heidemann, R. A., Mandhane, M. M., Chem. Eng. Sci., 28, 1213 (1973). Marina, J.. Tassios. D. P., lnd. Eng. Chem., Process Des. Dev., 12, 67
(1973). Matous, J., Novak, J. P., Sobr, J., Pick, J.. Collect. Czech. Chem. Comm., 37,
2664 (1972). Novak, J. P., Suska, J., Matous, J.. 4th international Congress of Chemical Engineering, CHlSA 72,Praha, Czechoslovakia, 1972. Novak, J. P., Suska, J., Matous, J., Collect. Czech. Chem. Comm., 39, 1943
(1974). Novak, J. P., Suska, J., Matous, J., lnd. Eng. Chem., ProcessDes. Dev., 13, 198
(1974). Petrie. R., M.S. Thesis, N.J. Institute of Technology, 1975. Renon, H., Prausnitz, J. M., A.I.Ch.E. J., 14, 135 (1968). Scott, R. L., J. Chem. Phys., 25, 193 (1956). Suska, J.. Novak, J. P., Matous, J., Pick, J., Collect. Czech. Chem. Cornrn., 37,
2664 (1972). Tsonopoulos, C.. Exxon Research and Engineering, private communication, July
11, 1975. Tsonopoulos, C., A./.Ch.E.J., 20,263 (1974): Wilson, G . M., J. Am. Chem. SOC., 86, 127 (1964).
Received for review December 22, 1975 A c c e p t e d June 7, 1976
