I n d . Eng. C h e m . Res. 1988,27, 506-512
506
Sciamanna, S. F.; Lynn, S. "Sulfur Solubility in Mixed Organic Solvents" Znd. Eng. Chem. Res. 1988a, part one of three in this issue. Sciamanna, S. F. Lynn, S. "Solubility of Hydrogen Sulfide, Sulfur Dioxide, Carbon Dioxide, Propane, and n-Butane in Poly(glyco1
ethers)" Ind. Eng. Chem. Res. 198813, preceding paper in this issue.
Received for review May 18, 1987 Revised manuscript received November 10, 1987
GENERAL RESEARCH Extension and Evaluation of the Minka and Myers Theory of Liquid Adsorption Peter E. Price, Jr.,* and Ronald P. Danner Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802
The thermodynamic theory of liquid adsorption developed by Minka and Meyers permits correlation and prediction of liquid-solid adsorption equilibria in microporous systems using only binary parameters. The theory uses activity coefficients to characterize the equilibria between distinct bulk and adsorbed phases. I n this paper, we examine the performance of the theory using the fourparameter Redlich-Kister, NRTL, and UNIQUAC models for the surface-phase activity coefficients. Correlations and predictions for seven ternary systems show that, with several additional assumptions, the theory can be applied t o macroporous systems and t h a t representation of the surface-phase activity coefficients using the four-parameter Redlich-Kister model yields the most accurate results. Minka and Myers (1973) presented a thermodynamic method for predicting multicomponent liquid adsorption equilibria on homogeneous microporous adsorbents. They used the four-parameter Redlich-Kister model to represent the adsorbed-phase activity coefficients. For the systems considered in their paper, the model is quite accurate. In this paper, we examine the method for a number of additional systems using various surface-phase activity coefficient models. Although the Minka and Myers method was developed for microporous systems, a few additional assumptions allow the method to be applied to macroporous systems.
The Method of Minka and Myers The experimental variable of interest in the study of liquid adsorption is the surface excess, which is defined as
n; = no(x? - xi')
(1)
Combining a total mass balance and a mass balance on any component, i, with eq 1 allows the excess adsorption to be expressed as
n: = ns(x;B- xi')
(2)
Furthermore, it can readily be shown that
En: = 0 i
(3)
* Current address: Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, M N 55405. 0888-5885/88/2627-0506$01.50/0
For microporous adsorbents, the number of moles in the surface phase, ns, can be related to the pure-component adsorbent capacities. One assumes that the micropores are always filled and that no volume change occurs when a liquid solution is adsorbed. Then
For macroporous or nonporous adsorbents, the number of moles in the adsorbed phase is not as readily determined. For monolayer coverage in a binary system, Schay (1969)stated that, if all adsorbed molecules adopt the same orientation and the adsorbent surface remains completely covered, then the following relationship is valid: n18a1+ n2sa2= a,
(5)
Everett (1973) extended this treatment to adsorbed layers of molecular thickness t: a1
nlst
+ n2,-a2t
=
as
The pure-component adsorbent capacities may be determined from the estimated molecular areas of the adsorbates and the estimated adsorbent area: mi= t a s / a i
(7)
Everett (1982) described two tests to determine if the layer thickness estimate is thermodynamically acceptable. The first of these requires that the surface phase mole fraction of a given component must always lie in the range of zero to one. Second, the adsorbed phase mole fraction 0 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 3, 1988 507 Table I. Integral Thermodynamic Consistency Test of Binary Adsorption Isotherms adsorbate components +lo - 4Z0, 1 2 3 adsorbate temp, O C system ref cavg -3.49 I 1 benzene ethyl acetate cyclohexane activated C 30 silica gel 0.96 I1 2, 3 chlorobenzene benzene n-heptane 20 -1.51 I11 2, 3 nitrobenzene benzene n-heptane silica gel 20 benzene n-heptane cyclohexane silica gel -4.07 IV 3 20 toluene n-heptane cyclohexane silica gel 20 v 3 -3.58 -3.19 VI 3 p-xylene n-heptane cyclohexane silica gel 20 VI1 4 dioxane n-hexane benzene -6.27 20 silica gel
4Z0 - 4 3 O , cal/g -2.22 -4.47 -4.47 0.00 0.00 0.00 2.86
- $lo, cal/g 5.58 4.31 7.15 3.89 2.98 3.05 3.82
43O
sum, cal/g -0.13 0.80 1.17 -0.18 -0.60 -0.14 0.41
percentb 2.3 17.9 16.4 4.4 16.8 4.4 6.5
aReferences: (1) Minka et al., 1973; (2) Goworek et al., 1985: (3) Jaroniec et al., 1981; (4) Vasili’eva et al., 1970. bPercent = (Isuml/lthe largest binary difference in free energies of immersionl) 100.
must always increase with increasing bulk liquid mole fraction. Substituting eq 6 in eq 2 gives, after some algebraic manipulation,
Using eq 2, 4, and 11, Minka and Myers derived the following general equation for the surface excess in a multicomponent system: CX/xjl(l - K i j ) nie =
I
(12)
X)Kij
Ej
where the separation factor, The proper value of t is the minimum value for which eq 8 satisfies Everett’s conditions. As the molecular sizes of the various adsorbates differ by increasing amounts, and as the adsorbed phase extends beyond a monolayer, these assumptions become increasingly unsound. To make use of the Minka and Myers theory for macroporous systems, however, these assumptions must be made. Equations for the activity coefficients in the adsorbed phase, where the adsorbent is assumed to be homogeneous, have been derived by several authors, including Larionov and Myers (1971) and Everett (1973). Minka and Myers (1973) considered the case of microporous adsorbents. They defined the free energy of immersion of the adsorbent in the mixture, 4, to be
4 = a,a
(9)
Using the Gibbs adsorption isotherm, the isothermal Gibbs-Duhem equation, and eq 2, Minka and Myers gave the expression for the difference in free energies of immersion for a binary system
1 . and an equation for the surface-phase activity coefficient:
Minka and Myers used the four-parameter RedlichKister model to correlate the adsorbed-phase activity coefficients. In this work, the NRTL and UNIQUAC models were also used. For monolayer or multilayer adsorption on nonporous or macroporous adsorbents, one can assume that all of the adsorbed molecules have the same orientation and that the volume of the adsorbed phase is constant. Under these assumptions, the equations derived by Minka and Myers may be used to calculate the surface-phase activity coefficients. For these systems, the adsorbent capacities are given by eq 7.
~
m, ~isj given ,
by
and has the properties Kii
=1
Kij
=
1/Kji
Kij
= KipKp,
(14)
The adsorbed-phase composition is related to the bulk composition by
where the summations in eq 12 and 15 are over the total number of components. The differences in free energies of immersion, 4 - 4?, are determined by integrating the Gibbs adsorption isotherm (Larionov and Myers, 1971): -d4 = En; db/ i
(16)
Equation 16 can be integrated by using the Gibbs-Duhem equation for an isothermal, isobaric system:
Equations 12-17 can be used to correlate binary adsorption equilibrium data and to predict the adsorption equilibria in multicomponent systems.
Correlation of Binary Adsorption Isotherms Data. A recent review of the multicomponent adsorption theory by Borowko and Jaroniec (1983) lists 25 ternary systems which have been studied over the entire concentration range. Data for three more ternary systems were reported by Goworek et al. (1985), although two of these appear to correspond to systems listed as unpublished in the previous reference. For seven of these systems, sufficient data were available in the literature to test the Minka and Myers model. These data, however, were gathered from various sources and are of varying quality. Of those references that were readily obtainable, only Minka and Myers’ (1973) paper contained sufficient tabular data to apply their method. For six other systems,
508 Ind. Eng. Chem. Res., Vol. 27, No. 3, 1988 Table 11. Estimated Adsorbent Capacities on Silica Gel at 20 o c BET molec. adsorbent area, area, layer capacity, adsorbate mZ/g m2/mmol thickness mmol/g 2 3.58 497 277.6 chlorobenzene 2 4.02 497 247.5 benzene 2.90 2 342.7 n-heptane 497 nitrobenzene benzene n-heptane
497 497 497
283.0 247.5 343.3
6 6 6
10.54 12.05 8.69
benzene n-heptane cyclohexane toluene n-heptane cyclohexane p-xylene n-heptane cyclohexane dioxane n-hexane benzene
292 292 292 292 292 292 292 292 292 365 365 365
247.5 343.3 241.5 313.1 343.3 241.5 333.0 343.3 241.5 255.3 338.4 262.6
3 3 3 2 2 2 2 2 2 3 3 3
3.54 2.55 3.63 1.87 1.70 2.42 1.75 1.70 2.42 4.29 3.24 4.17
the adsorption equilibrium data were either digitized from figures given in the literature or calculated by using a different model which was shown to provide a good representation of the data. Table I shows the sources for the binary data and the results of the integral thermodynamic consistency test described by Sircar and Myers (1971) for each set of binary adsorption data. A set of three binary isotherms is said to be thermodynamically consistent if the sum of the differences in free energies of immersion for the three binary pairs is zero. The final column in the table gives the percentage that the sum is of the largest difference in the free energies of immersion of the binary mixtures. Minka and Myers suggest that if this percentage was within the range of error of the experimental data, then the data set could be considered thermodynamically consistent. Using this criterion, we find that systems I, IV, VI, and VI1 are thermodynamically consistent, while systems 11,111, and V are not. For systems I1 and 111, this is probably due to the use of adsorption data for the mixture of benzene and n-heptane on a somewhat different silica gel than the other mixtures. Use of these data, published by Jaroniec et al. (1981), was necessary because no binary data for this mixture were published by Goworek et al. (1985). For systems IV, V, and VI, it was assumed that there was no preferential adsorption of either component in the mixture of n-heptane and cyclohexane on silica gel. Jaroniec et al. (1981) reported no data for this system, and Sircar and Myers (1971) found no preferential adsorption for this pair on a different silica gel. The differences in free energies of immersion for the binary mixtures of benzene, ethyl acetate, and cyclohexane on activated carbon a t 30 "C are somewhat different than the values originally reported by Minka and Myers (1973). These differences reflect the uncertainty in the evaluation of eq 10. This uncertainty also resulted in slightly different adsorbed-phase activity coefficient model parameters, as shown below. Minka and Myers reported experimentally determined adsorbent capacities. The values for benzene, ethyl acetate, and cyclohexane on activated carbon were found to be 5.48,4.95, and 4.30 mmol/g, respectively. For the remaining systems examined in this work, the adsorbent capacities were determined from BET adsorbent areas and
Table 111. Constants of the Redlich-Kister Equation for the Bulk Liauid Phase Redlich-Kister binary mixture components constants 1 2 temp, "C A B benzene ethyl acetate 30 0.1551 0.0309 cyclohexane 30 0.5328 0.0519 benzene 1.2206 -0.0226 ethyl acetate cyclohexane 30 20 0.1825 -0.1349 chlorobenzene benzene 0.7346 0.0663 chlorobenzene n-heptane 20 0.6173 0.1378 benzene n-heptane 20 nitrobenzene benzene 20 0.4565 -0.4228 2.0627 -0.0818 nitrobenzene n-heptane 20 0.5867 0.0234 20 benzene cyclohexane 0.1176 -0.0224 20 n-heptane cyclohexane 0.4651 20 0.0760 n-heptane toluene 0.4861 -0.0255 20 toluene cyclohexane 0.3607 20 0.0332 n-heptane p-xylene 0.4864 -0.0652 cyclohexane p-xylene 20 n-hexane 1.4096 0.0513 20 dioxane 0.2062 -0.1216 benzene dioxane 20 benzene 0.6423 -0.1166 n-hexane 20
estimated molecular areas of the adsorbates. These adsorbent capacities are listed in Table 11. BET areas were published with the adsorption data for all six of these systems. Molecular area estimates were obtained from data published in the original references and values given by McClellan and Harnsberger (1967). When the objective is to predict ternary adsorption equilibria, the estimated adsorbent capacities for systems with multilayer adsorption on nonporous or macroporous adsorbents must be based on the minimum layer thickness required for all three constituent binary systems to satisfy Everett's conditions for thermodynamic acceptability. This requirement was used to determine the estimated adsorbent capacities for all of the silica gel systems shown in Table 11. Table I11 contains the bulk-liquid-phase Redlich-Kister parameters for the binary systems. The bulk-phase Redlich-Kister constants for all mixtures, except those studied by Minka and Myers (1973) and the mixture of nitrobenzene and n-heptane, were determined from Margules model constants given in the publication by DECHEMA (extant 1982). The parameters given for those systems for which parameters were originally published at a different temperature were obtained by scaling the original parameters in proportion to the inverse of the absolute temperatures. The bulk-phase Redlich-Kister parameters for the binary mixtures of benzene, ethyl acetate, and cyclohexane were given by Minka and Myers (1973). Activity coefficients for the mixture of nitrobenzene and n-heptane were predicted by using the UNIFAC method (Fredenslund et al., 1977). The Redlich-Kister parameters were then regressed from the predicted activity coefficients using a least-squares technique. Table IV gives the Redlich-Kister, NRTL, and UNIQUAC parameters for the adsorbed-phase activity coefficients of the binary mixtures. Optimal parameters were obtained by using a least-squares technique in conjunction with activity coefficients calculated from eq 10 and 11. The NRTL model was found to be considerably more flexible when the restriction that the alj parameter must be positive was removed. Although the elimination of this restriction limits the theoretical validity of the model, it greatly improved the model's usefulness for correlating and predicting liquid adsorption equilibria. UNIQUAC R and Q parameters were taken from the DECHEMA publication (extant, 1982).
Ind. Eng. Chem. Res., Vol. 27, No. 3, 1988 509 Table IV. Constants for t h e Adsorbed-PhaseActivity Coefficient Models Redlich-Kister binary mixture components temp, 1
2
adsorbent
O C
benzene benzene ethyl acetate chlorobenzene chlorobenzene benzene nitrobenzene nitrobenzene benzene benzene benzene n-heptane toluene toluene n-heptane p-xylene p-xylene n-heptane dioxane dioxane n-hexane
ethyl acetate cyclohexane cyclohexane benzene n-heptane n-heptane benzene n-heptane n-heptane n-heptane cyclohexane cyclohexane n-heptane cyclohexane cyclohexane n-heptane cyclohexane cyclohexane n-hexane benzene benzene
activated C activated C activated C silica gel silica gel silica gel silica gel silica gel silica gel silica gel silica gel silica gel silica gel silica gel silica gel silica gel silica gel silica gel silica gel silica gel silica gel
30 30 30 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20
A -0.4547 -0.7838 0.1917 0.0585 -0.0454 -0.3991 0.3523 0.7987 0.0735 -0.3583 0.0634 0.1176 -0.1653 -0.0285 0.1176 -0.5976 -0.2251 0.1176 -0.4851 -0.6690 -0.1126
B 0.1717 0.5861 0.5297 -0.1244 0.1429 0.1168 -0.3544 0.7914 0.3976 0.0573 0.1955 -0.0224 -0.6860 0.1709 -0.0224 -0.1811 0.2748 -0.0224 0.3965 -0.0946 -0.1402
Evaluations. The surface excess, separation factor, and surface-phase mole fractions for a binary system are given by eq 12,13, and 15, respectively. As shown by Minka and Myers (1973), the difference in free energies of immersion for a binary solution can be written as
NRTL
C 0.0701 0.1506 -0.1292 0.1021 0.1887 0.2318 -0.0542 0.2827 -0.0124 0.2419 0.0914
D
712
-0.0411 -0.3063 -0.3762 -0.0163 -0.3437 -0.0674 0.0426 -0.4146 -0.0566 0.1135 -0.0170
O.ooO0
0.OooO
-0.0271 0.1185 0.0000 0.2445 0.1605 0.0000 1.1380 0.4739 0.2189
0.5237 -0.0701 O.oo00 0.0974 -0.2373
-168.14 52.509 2749.9 16.308 9.363 768.63 -10186.5 -180.78 40756.3 44890.3 96121.0 -255.09 27.238 1305.48 -255.09 -8801.56 1293.74 -255.09 1084.70 153.83 -12137.3
O.ooO0
-0.3459 -0.2187 0.0255 1 .o
1
Ai2
Ail
-32.43 -141.97 -1713.2 -7.005 -14.705 -424.85 8335.28 453.83 -44986.5 49392.6 -100072 372.60 -26.971 -842.59 372.60 7706.20 -1970.71 372.60 -234.11 -355.76 10723.0
-0.274 3.447 0.1106 -15.412 -17.611 0.891 -0.0139 -1.152 -0.00135 -0.00113 -0.00024 0.2867 -0.4607 0.2342 0.2867 -0.007 -0.124 0.2867 1.2657 0.7656 -0.00612
-573.61 -562.07 754.19 -100.88 -51.17 692.83 -471.47 495.99 649.26 565.97 630.88 -21.27 -573.30 612.98 -21.27 -580.74 495.26 -21.27 1010.65 -679.17 -426.79
920.34 969.22 -448.17 120.23 35.65 -460.14 994.62 -238.24 -398.08 -394.89 -382.47 39.94 1152.44 -398.73 39.94 1035.86 -359.28 39.94 -523.77 1386.26 662.29
1
1
1
1
.
/ ' \ ,
.
o.9
UNIQUAC a12
721
/', *
1
1
1
I
I
I
1
- ,'UNIQUAC
I, NRTL\\.
The difference in free energies of immersion of the mixture and pure component 2 is given by
(4 - 42") = (4 - 4 1 O ) + (@I0 - 42O) (19) where - 420is determined by integrating eq 10 over the entire concentration range. The same set of equations can also be applied to macroporous systems using the surface-phase capacities estimated from the BET theory, as described earlier. Since eq 18 is not explicit in x;, it cannot be integrated directly. The procedure for correlating binary surface excess data is as follows: 1. Obtain estimates for mi. 2. From the experimental adsorption data, determine the differences in the free energies of immersion of the pure components by integrating eq 10 over the entire concentration range. 3. Determine the surface-phase activity coefficients as a function of the surface-phase composition by using eq 2,4,10, and 11, and regress the parameters for the desired surface-phase activity coefficient model (i.e., RedlichKister, NRTL, or UNIQUAC). 4. Equation 18 must be written in differential equation form. Furthermore, it cannot be integrated directly since the value of K12 is a function of the surface-phase composition. The surface-phase composition is, in turn, a function of the bulk-liquid-phase composition. Since the relationship between the bulk- and surface-phase compositions is not known, it is necessary to solve several equations simultaneously a t each step of the integration. Equation 15 is rearranged to give K ~ ~ :
0 . O L - I
0.0 0.1
I I I 0.2 0.3 0.4
I
0.5 0.6 0.7 0.8
I
\
0.9 1.0
LIQUID MOLE FRACTION OF BENZENE
Figure 1. Surface excess correlation of benzene in the mixture of benzene and ethyl acetate on activated carbon at 30 " C .
This equation is then substituted in eq 13 to form one of the simultaneous equations. The other equation is
Ex; = i
1
(21)
By use of the current value of the integral, C$ - C$lo, the solution of these equations is possible. The differential form of eq 18 is integrated by using the fourth-order Runge-Kutta technique (Carnahan et al., 1969). 5 . The result of the integration of eq 18 up to the bulk-phase composition of interest can then be used to determine K~~ from eq 13. The surface excess values are calculated by using eq 12. Figures 1-3 show typical results for the correlation of binary adsorption isotherms using the three different surface-phase activity coefficient models. The results for all of the binary isotherm correlations are summarized in Table V. Values for the isotherm of n-heptane and cyclohexane on silica gel a t 20 "C are not given because it
510 Ind. Eng. Chem. Res., Vol. 27, No. 3, 1988 Table V. Average Absolute Differences between Experimental and Correlated Binary Surface Excess Values Using the Minka and Myers Theory with Three Surface-Phase Activity Coefficient Models av absolute error mixture adsorbent T, " C Redlich-Kister NRTL UNIQUAC benzene + ethvl acetate activated C 30 0.022 0.059 0.082 benzene + cyciohexane activated C 30 0.037 0.209 0.218 0.044 0.093 0.116 30 ethyl acetate + cyclohexane activated C chlorobenzene + benzene silica gel 20 0.008 0.035 0.035 silica gel 0.034 0.055 0.054 20 chlorobenzene + n-heptane silica gel benzene + n-heptane 0.026 0.047 0.044 20 nitrobenzene + benzene silica gel 20 0.052 0.354 0.022 silica gel nitrobenzene + n-heptane 0.488 0.502 20 0.264 benzene + n-heptane silica gel 20 0.027 0.032 0.140 benzene + n-heptane silica gel 20 0.006 0.041 0.048 silica gel 0.017 0.010 benzene + cyclohexane 20 0.002 toluene + n-heptane silica gel 0.005 0.053 0.018 20 silica gel 0.004 0.006 toluene + cyclohexane 20 0.003 p-xylene + n-heptane silica gel 0.002 0.019 0.015 20 silica gel 0.004 0.013 0.021 p-xylene + cyclohexane 20 dioxane + n-hexane silica gel 20 0.060 0.117 0.058 silica gel 0.027 0.053 0.046 dioxane + benzene 20 silica gel 0.048 0.151 0.140 benzene + n-hexane 20
1.1
1.0 *
p
e &
0.9
0.8
v)
3
0.7
0 X
w w
0.6
0.5 IL
K
a 0.4
v)
-0.300
wi
'\
-0.275
-
-0.3251
\
NRTL A N D UNIQUAC
1
I
1
-
O
/
\
1
0.3
/
\
I
I
I
I
r-
0.2 0.1
0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
LIQUID MOLE FRACTION OF DIOXANE
Figure 3. Surface excess correlation of dioxane in the mixture of dioxane and benzene on silica gel at 20 "C.
was assumed that there is no preferential adsorption of either component in this mixture. In general, the accuracy of the correlations is in direct proportion to the accuracy of the surface-phase activity coefficient correlations. For all of the mixtures examined, the four-parameter Redlich-Kister model provided the most accurate correlations of the surface-phase activity coefficients. This is to be expected since this model has more adjustable parameters than the other two. Neither the NRTL nor the UNIQUAC model is capable of representing the highly nonideal behavior of the surface phase. Thus, the binary isotherm correlations using these models are significantly less accurate than the correlations made with the Redlich-Kister model. The results also suggest that neither the NRTL model or the UNIQUAC model is distinctly superior to the other. It is clear from the table that the model provided a very poor correlation for the isotherm of nitrobenzene and n-heptane on silica gel a t 20 "C. The two most likely
sources of error in this correlation are the larger layer thickness value and the bulk-phase activity coefficients, which were based on values predicted by using UNIFAC. We suspect that the latter uncertainty, however, is less significant for the following reason. Activity coefficients for the mixture of nitrobenzene and n-hexane were also predicted by using UNIFAC. These were in excellent agreement with activity coefficients calculated from thermodynamically consistent VLE data for this mixture contained in DECHEMA. It would therefore appear that the layer thickness used for this system is too large. Prediction of Ternary Adsorption Equilibria Data. Minka and Myers (1973) presented the most complete, tabular set of liquid adsorption data for a ternary mixture and its constituent binary mixtures that has appeared in the literature to date. The paper contained 11 points of ternary adsorption equilibrium data for the
Ind. Eng. Chem. Res., Vol. 27, No. 3, 1988 511 Table VI. Experimental and Predicted Adsorption Equilibria for Ternary Mixtures of Benzene (l),Ethyl Acetate (2), and Cyclohexane (3) on Activated Carbon at 30 "C. Experimental Data from Minka and Myers (1973) surface excess of ethyl acetate, mmol/g surface excess of benzene, mmol/g liq mol fraction calcd calcd XI1 Xll exptl Redlich-Kister NRTL UNIQUAC exptl Redlich-Kister. NRTL UNIQUAC 0.087 0.130 0.176 0.252 0.291 0.410 0.510 0.615 0.719 0.759 0.857
0.841 0.075 0.675 0.515 0.115 0.199 0.126 0.230 0.117 0.157 0.072
0.576 0.960 0.708 0.705 0.947 0.718 0.720 0.487 0.387 0.313 0.218
0.597 1.004 0.744 0.741 0.947 0.731 0.694 0.483 0.407 0.320 0.211
0.565 1.045 0.755 0.795 1.179 0.918 0.892 0.560 0.454 0.338 0.202
0.546 1.090 0.700 0.730 1.150 0.863 0.844 0.523 0.427 0.325 0.196
-0.582 0.283 -0.656 -0.450 0.154 -0.058 -0.033 -0.250 -0.107 -0.172 -0.094
-0.622 0.480 -0.710 -0.501 0.270 0.016 0.020 -0.262 -0.101 -0.192 -0.075
-0.625 0.303 -0.677 -0.507 0.146 -0.038 -0.012 -0.227 -0.101 -0.176 -0.082
-0.719 0.512 -0.738 -0.495 0.281 0.026 0.021 -0.260 -0.103 -0.194 -0.078
Table VII. Average Absolute Differences between Experimental and Predicted Ternary Surface Excess Values Using the Minka and Myers Theory with Three Surface-Phase Activity Coefficient Models av absolute error mixture adsorbent TOC Redlich-Kister NRTL UN IQUA C 0.075 0.070 activated C 30 0.020 benzene + ethyl acetate + cyclohexane silica gel 20 0.031 0.028 0.039 chlorobenzene-+ benzene + n-heptane 20 0.127 0.314 0.227 silica gel nitrobenzene + benzene + n-heptane 0.057 20 0.042 0.292 silica gel benzene + n-heptane + cyclohexane 0.044 0.106 20 0.035 silica gel toluene + n-heptane + cyclohexane 20 0.024 0.063 0.100 silica gel p-xylene + n-heptane + cyclohexane 20 0.641 0.762 0.680 silica gel dioxane + n-hexane + benzene
system of benzene, ethyl acetate, and cyclohexane on activated carbon a t 30 "C. A recent paper by Goworek et al. (1985) contained tabular ternary adsorption data for the mixtures of chlorobenzene and nitrobenzene with benzene and n-heptane on silica gel at 20 "C. The paper also contained tabular data for the system of analine with benzene and n-heptane on silica gel at 20 OC. However, the binary data necessary to apply the Minka and Myers method were not available for this mixture. Adsorption data for the ternary systems of benzene, toluene, and p-xylene in n-heptane and cyclohexane on silica gel a t 20 "C appeared in papers by Borowko et al. (1979) and Jaroniec et al. (1981). The data for these ternary systems were given in graphical form as plots of the surface excess versus bulk liquid mole fractions of benzene, toluene, or p-xylene with the ratio of the mole fractions of n-heptane to cyclohexane reportedly held constant. Thus, surface excess data could be digitized for only one of the three components. Furthermore, the paper by Goworek et al. (1985), referenced earlier, reported that the ratio of the bulk mole fractions of two of the three components was held constant. From tabular data published in this paper, however, it is clear that this ratio varied considerably. This suggests that the ratios of the bulk liquid mole fractions may have also varied for the data by Borowko et al. (1979) and Jaroniec et al. (1981). For our calculations, we used bulk compositions which were in the proportions reported in the references. Adsorption data for the system of dioxane, n-hexane, and benzene on silica gel a t 20 "C were presented graphically by Vasil'eva et al. (1970). Data for the ternary mixture were presented in the form of adsorption excess triangles for each of the three components. These data were digitized for the purpose of this analysis. Evaluations. For ternary solutions, the surface excesses for two components are determined by eq 12. The third excess is f i e d by eq 3. The separation factors, ~ ~arej given , by eq 13. The bulk-liquid and surface-phase compositions are then related by eq 15. The differences in the free
energies of immersion are calculated for a constant ratio, xJ/x?, according to
4 - 41° RT dx,'
1 - Xl'
II /
m2
The other differences in free energies of immersion are calculated from
4 - 4 2 O = (4 - @I0) + (@I0
- 42")
(23) (24)
Because eq 22 is not explicit in x t , it cannot be integrated directly. The calculational procedure for predicting the surface excess is as follows: 1-3. Steps 1-3 are the same as for binary mixtures. 4. As before, eq 22 must be treated as a differential equation and cannot be integrated directly because the K~;S are functions of the surface-phase composition which is an unknown function of the bulk-liquid composition. Therefore, it is necessary to solve several equations simultaneously a t each step of the integration. The expressions for the K ~ ~given ' s , by eq 13, are substituted into the equations for the surface-phase mole fractions, from eq 15. The resultant expressions, and the requirement that the s u m of the surface-phase mole fractions must be equal one, give three equations to be solved a t each step of the integral. By use of the current value of the integral, the three surface-phase mole fractions are unknowns, and a solution is possible. 5. Once the surface-phase mole fractions have been determined, the surface excess values can be calculated by using eq 12 and 13.
512 Ind. Eng. Chem. Res., Vol. 27, No. 3, 1988
The method detailed above was used to predict the adsorption equilibria for the seven ternary systems for which sufficient experimental data were available. Table VI contains the experimental and predicted ternary adsorption equilibria for the system of benzene, ethyl acetate, and cyclohexane on activated carbon a t 30 "C. The results for the Redlich-Kister model are in agreement with the values originally given by Minka and Myers (1973). The results for all seven systems are summarized in Table VII, which shows the average absolute difference between the experimental and predicted surface excess values for the seven ternary systems. As for the binary correlations, use of the four-parameter Redlich-Kister model for the surface-phase activity coefficients gave the most accurate predictions for a majority of the systems. The NRTL model performed slightly better than the Redlich-Kister model for the system of chlorobenzene, benzene, and n-heptane on silica gel a t 20 OC. The accuracy of the predictions for the chlorobenzene, benzene, n-heptane system is surprising considering the fact that the binary adsorption isotherms, upon which these predictions were based, were clearly not thermodynamically consistent. Predictions for the mixtures of nitrobenzene, benzene, and n-heptane on silica gel at 20 "C and dioxane, n-hexane, and benzene on silica gel at 20 "C were, in general, in gross disagreement with the experimental values. The average absolute errors for these systems are of the same order of magnitude as the experimental data. For the first system, a significant degree of error was expected since the correlation of the binary isotherm of nitrobenzene and nheptane did not accurately represent the experimental data. The correlations of the constituent binary isotherms for the second system were, however, reasonably accurate. One possible source of error for this system may have been the bulk-phase activity coefficients. The bulk-phase activity coefficient model parameters for two of the three constituent binary mixtures were based on VLE data that passed only one of DECHEMA's two thermodynamic consistency tests. Summary The above results indicate that the adsorption theory of Minka and Myers (1973) provides a useful and flexible method for correlating binary liquid adsorption data and predicting ternary liquid adsorption equilibria based on parameters obtained from the constituent binary mixtures. It has been shown that the method, with several additional assumptions, can be applied to systems involving macroporous adsorbents. For the systems examined, correlation of the surface-phase activity coefficients using the fourparameter Redlich-Kister model gave the most accurate correlations of the binary adsorption equilibria. The advantage of the four-parameter Redlich-Kister model in correlating the binary isotherms was expected. The Redlich-Kister model may, however, be less physically meaningful than the NRTL and UNIQUAC models. Hence, it is not obvious that the Redlich-Kister model should give the best ternary predictions. For the systems examined herein, the predicted ternary adsorption equilibria were, in general, most accurate when the RedlichKister model was used. Acknowledgment We gratefully acknowledge the financial support pro-
vided by the Frontiers of Separations Program of the Exxon Research and Engineering Company. Nomenclature ai = molar area of adsorbate, m2/mmol a, = specific surface area of the adsorbent, m2/g
A , B , C, D = constants for the Redlich-Kister equation Aij = UNIQUAC parameters mi = adsorbent capacity, mmol/g no = moles of liquid per gram of adsorbent prior to contact of liquid and adsorbent, mmol/g n; = surface excess of component i , mmol/g n5 = total number of moles in the adsorbed phase per gram of adsorbent, mmol/g n; = amount of component i in the adsorbed phase, mmol/g R = gas constant t = layer thickness T = temperature, K xi0 = mole fraction of component i in liquid mixture prior to contact of liquid and adsorbent x> = mole fraction of component i in the bulk liquid phase after contact of liquid and adsorbent xis = mole fraction of component i in the adsorbed phase Greek Symbols ai'
= NRTL nonrandomness parameter
ri( = bulk-phase activity coefficient of component i
yis = adsorbed-phase activity coefficient of component i ~ i= , Mmka and Myers theory separation factor, dimensionless
= surface tension qj = NRTL parameter (r
4 = free energy of immersion in the mixture = free energy of immersion in pure liquid i Subscripts and Superscripts i = ith component
1 = bulk liquid phase o = pure component s = adsorbed phase
Literature Cited Borowko, M.; Jaroniec, M. Adu. Colloid Interface Sci. 1983,19,137. Borowko, M.; Jaroniec, M.; Oscik, J.; Kusak, R. J. Colloid Interface Sci. 1979, 69, 311. Carnahan, B.; Luther, H. A., Wilkes, J. 0. Applied Numerical Methods; Wiley: New York, 1969; p 361. DECHEMA, Deutache Gesellschaft fur Chemisches Apparatewesen Vapor-Liquid Equilibrium Data Collection; Behrens, D., Eckermann, R., Eds.; Chemistry Data Series, Schon & Wetzel Gmbh: Frankfurt/Main, F.R.G., extant 1982; Vol. 1. Everett, D. H. In Colloid Science: Specialist Periodical Reports; The Chemical Society: London, 1973; Vol. 1, Chapter 2. Everett, D. H. In Adsorption at the Gas-Solid and Liquid-Solid Interface; Rouquerol, J., Sing, K. S. W., Eds.; Elsevier Scientific: Amsterdam, 1982. Fredenslund, A.; Gmehling, J.; Rasmussen, P. Vapor-Liquid Equilibria Using UNZFAC; Elsevier Scientific: New York, 1977. Goworek, J.;Oscik, J.; Kusak, R. J . Colloid Interface Sci. 1985, 103, 392. Jaroniec, M.; Oscik, J.; Derylo, A. Acta Chim. 1981, 106, 257. Larionov, 0.G.; Myers, A. L. Chem. Eng. Sci. 1971, 26, 1025. McClellan, A. L.; Harnsberger, H. F. J. Colloid Interface Sci. 1967, 23,577. Minka, C.; Myers, A. L. AZChE J. 1973, 19, 453. Schay, G. Surf. Coll. Sci. 1969, 2, 155. Sircar, S.;Myers, A. L. AIChE J. 1971, 17, 186. Vasil'eva, V. S.;Davydov, V. I.; Kiselev, A. V. Dokl. Akad. Nauk SSSR 1970, 192, 1299.
Received for review June 26; 1986 Revised manuscript received August 27, 1987 Accepted October 15, 1987
