Ind. Eng. Chem. Res. 1987,26, 1679-1686
1679
Equilibrium Adsorption of Multicomponent Gas Mixtures at Elevated Pressures J. A. Ritter and R. T. Yang* Department of Chemical Engineering, State University of New York ut Buffalo, Buffalo, New York 14260
Equilibrium adsorption of H2,CO, CH4,C 0 2 ,and H2S(single and mixed gases-two to five species) was measured on activated carbon at pressures up t o 400 psia. Temperature-dependent parameters, regressed from single-gas data, were applied t o four theoretical models which predict adsorption from gas mixtures. Deviations between theory (IAS) and experiment increased with pressure and with the number of components in the mixture. Furthermore, none of the theories employing only single-gas data could consistently predict multicomponent adsorption. It was found, however, that multicomponent adsorption could be predicted from the extended Langmuir equation by including a n interaction parameter calculated from only single and binary data. An unusual result of this work was t h a t in the application of mixture adsorption theories, i t was shown t h a t from the same model significantly different results can be obtained depending on the selection of the independent variable set, (T, P, X i )or ( T ,P, Yi}. Information on equilibrium adsorption from mixtures is crucial to the design of commercial adsorbers. The adsorption of single gases has been published extensively, whereas data on the adsorption of mixed gases are lacking. Moreover, high-pressure, multicomponent data are not readily available, except for a few studies (Reich et al., 1980; Lewis et al., 1950; Hyun and Danner, 1982; Miller e t al., 1987), yet these are the conditions encountered in many commercial pressure swing adsorption processes. The primary concern of this research was to add to the paucity of high-pressure, multicomponent data in the literature. A secondary concern was the modeling of mixed gas adsorption based on limited experimental data. Five commercially important gases were studied: methane, carbon monoxide, carbon dioxide, hydrogen, and hydrogen sulfide. The adsorption of single gases and various combinations of these gases was measured on activated carbon. The experimental mixture data were correlated with theoretical models, which predict adsorption from mixtures based on single-gas isotherms. Some models also require binary gas as well as single-gas data for predicting adsorption from multicomponent mixtures (Costa et al., 1981; Cochran et al., 1985; Talu and Zwiebel, 1986). More than a dozen theories on mixed-gas adsorption have been reviewed by Yang (1987);the most prevalent ones are applied and compared here. A new correlation is also presented which significantly improves the correlation of multicomponent data via information from binary experimental data.
Theoretical Considerations Four models were considered in this work: the extended Langmuir (EL) relationship first employed by Markham and Benton (1931), the ideal adsorbed solution (IAS) theory of Myers and Prausnitz (1965), the vacancy solution-Flory-Huggins (VS-FH) model developed by Danner and co-workers (Cochran et al., 1985), and the Lewis correlation (LC) (Lewis et al., 1950). IAS and VS-FH models are based on analogies drawn from solution thermodynamics, whereas EL and LC are not thermodynamically rigorous. However, EL is explicit with respect to the amount adsorbed and therefore is far superior to the thermodynamic models for the modeling of adsorber dynamics. The LC, alone, cannot predict mixed gas adsorption, but it serves as a basic assumption in several models for predicting mixed gas adsorption (Yang, 1987).
The Langmuir isotherm contains two constants. The extended Langmuir equation is Vi bi Yip
e.=-= '
1+
N
C biYiP i=l
The single-gas energy parameter, bi,cannot account for inter-species interactions unless an interaction parameter, vi, is introduced, as first suggested by Schay (1956):
The values of vi depends on all the species present, and it may also depend on temperature and pressure. For application in the design and modeling of adsorbers, it is useful to employ constant vi and different Vmiand bi values. The value of vi for each component can be calculated from mixture adsorption data by (Schay, 1956)
Hence, vi is a fitting parameter for a giving mixture and varies for different mixtures. It is interesting to note that for a reversible, isothermal process, eq 1 as well as eq 2 is thermodynamically consistent if the monolayer values, V,, are equal for all species. Solution Thermodynamic Models. The IAS and VS-FH models are each defined by a set of coupled equations. They have in common
Yi&P = y'XiI0 Czi = 1 zi = X i or Yi 1
(4)
(5)
where Yi, Xi, and P are experimental quantities, I o defines the reference state, and 4iis predicted from an equation of state. In this work, the Peng-Robinson (1976) equation of state was used. IAS requires two more relationships, one to define the reference state and the other one to give the total amount adsorbed. The former is Gibb's adsorption isotherm, and the latter is analogous to assuming the partial molar adsorbed areas are additive (thus yi = 1, also). The reference state, Io = Po(rm),is defined as the vapor pressure of the pure adsorbate taken a t the spreading pressure, rm,and the T of the mixture. The
0888-588518712626-1679$01.50/0 0 1987 American Chemical Society
1680 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987
VS-FH model requires four additional relationships (Cochran et al., 1985),plus the definition of the reference state, which is given by IO
=
Note, the primary difference between the U S and VS-FH models is the way the reference states are defined. The assumptions here are that a hypothetical component exists (vacuum entities) which can take into account deviations in the infinitely dilute adsorbed phase via an activity coefficient model. The single-gas adsorption isotherm for this situation is
% V= E
p = -b1 -l - d% exp(&)
CHI
40 165 350 545 760 910 970
45.5 91.5 113 121 125 126 126
Cob
33.6 117 286 527 790 16.1 74.1 250 526 85 100 300 465 648 838 925 8.1 23.1 47.1 98.5 167
24.8 51.8 78.0 97.6 106 60.2 130 183 200 4.2 8.4 13.8 20.4 26.2 32.8 36.6 59.0 111 159 197 208
COP
(7)
which reduces to Langmuir's isotherm when a vanishes. IAS is not restricted to any particular single-gas isotherm relationship. The Langmuir model was used here for IAS because it leads to a simple algebraic expression for r = f ( P ) .Note, a much simpler set of equations results for the VS-FH model if di and/or yi is set to unity. The Lewis Correlation. The LC was developed empirically for binary hydrocarbon adsorption on activated carbon. Later, it was theoretically derived by Lee (1973) and extended for any number of components by Yang (1987) with the assumption that there is no volume change upon mixing (as in the IAS theory) and gases are adsorbed by micropore filling (as in the potential theory). It is given by
Note eq 7 is different from that used in IAS since here Vio (the reference state-single-gas molar volume) is taken a t the T and P of the mixture. Equation 7 has been the basis for many mixed gas adsorption theories since it relates mixture adsorption to single-gas adsorption in a very simple fashion (Grant and Manes, 1966). Alternate Selection of Independent Variables. In adsorption systems there are N 1 degrees of freedom. For a binary system, the three independent variable sets of choice are (T,P, Yl),(T,P, Xl), and (T,P, rm(VJ). After application of an appropriate theory (for the last set, the theory must necessarily be thermodynamic in nature), the three respective dependent variable seta can be solved for and are (Xl, VJ, { Y l ,V,],and {Xl, Yl). In the literature, one set of independent variable sets is selected, and the corresponding results are presented. The vacancy solution theory was typically applied with the adsorbed phase composition known from experiment (Wilson and Danner, 1983). This was seemingly done to avoid a major complication in the numberical solution. y; is strongly dependent upon Xiand thus a second iteration loop is required. The third independent variable set was applied only by Myers (1968),where X1and Yl were solved for. This is not practical and served only as an example since T , is not a priori known from mixture experimental data. The first set was employed the most in the literature (Friederich and Mullins, 1972; Van Ness, 1969; Ruthven et al., 1973; Lee and O'Connell, 1986). It will be shown in this study that different solutions result from the same model depending on which independent variable set is specified. The independent variable sets of interest were ( T ,P, Yi) and (T,P, Xi]. Large differences in rm and Vi will be shown.
+
Table I. Single-Gas Adsorption on PCB-Activated Carbon at 296 K at 373 K at 480 K P V" P V P V
Hz
HzS
70 163 235 290 500 715 893 955 55.6 153 282 446 822 29.1 93.6 260 490
80 180 310 480 680 856 945 13.1 34.6 64.1 106 144 177
25 41.5 50 57 71 78 82.5 84.5 11.9 25.5 38.9 50.4 66.4 23.8 54.3 93.2 119.3 1.7 3.7 6.3 9.3 13.0 16.0 17.5 27.2 55.4 82.2 107 123 133
a P in psia, V in cm3 (STP)/g of carbon. sured at 296, 373, and 473 K.
92.5 188 345 538 773 906 970
8.0 17.0 27 36 43 46.5 48
69.5 164 287 477 843 50.5 158 329 533 70 190 320 497 717 892 967 25 58.5 103 150 188
6.29 12.5 19.5 27.6 38.8 10.5 25.8 42.2 57.2 0.5 1.3 2.3 4.0 4.5 6.0 6.5 12.7 25.5 38.8 50.5 58.2
* CO isotherms mea-
Experimental Section A static system was designed and refined (Saunders, 1983, Ritter, 1986) for measuring both single-gas and mixed gas adsorption. The system contained sample (110 cm3) and reservoir (150 cm3) bottles. An additional mixing bottle (500 cm3) was used for the preparation of gas mixtures. Each of the bottles was accessible to pressure measurement, gas sampling and analysis, and evacuation. It was found that 24 h was required for thorough mixing in the mixing bottle. Equilibrium was established by the attainment of a constant gas composition as well as constant pressure in the sample bottle. The equilibration times required in the experiments reported in this work ranged from 48 to 96 h, depending on the specific conditions. The sorbent was a commercial activated carbon (PCB from Calgon Corp.). The gases were CP to ultrahigh purity grade, used without further purification (Ritter, 1986). Results a n d Discussion Experimental Single-Gas a n d Mixed Gas Data. Fifteen single-gas isotherms are given in Table I, and five of these (at 296 K)are shown graphically in Figure 1. The isosteric heats of adsorption calculated from these isotherms generally declined slightly at small amounts adsorbed and leveled off to the following values (in kcal/mol): 4.6 (H,S), 4.9 (CO,), 5.0 (CH,), 3.8 (CO), and 2.8 (HJ. Compared with the heats of evaporation, the above values were only slightly higher for H2S and CO, but more than twice higher for CHI, CO, and H2. Moreover, since these isosteric heats were apparently independent of surface coverage, it was not straightforward to assess the degree of surface heterogeneity from these data. The adsorption of 45 mixtures of these gases, ranging from 2-5 components, was determined. The systems investigated were CH4-CO (6), CH4-C02 (6), CO-CO2 (61,
Ind. Eng. Chem. Res., Vol. 26, No. 8,1987 1681 Table 11. Binary Mixture Adsorption on Activated Carbon
co P, psia 116.1 150.0 190.9 259.0 263.0 353.5
V,, cm3 (STP)/g 90.1 103.9 107.8 142.7 147.4 180.3
Y 0.769 0.723 0.802 0.700 0.755 0.463
293 293 295 294 294 295
P, psia 124.9 183.0 243.1 310.0 364.3 391.1
V,, om3 (STP)/g 75.8 90.9 102.4 133.6 114.1 117.4
Y 0.506 0.644 0.758 0.802 0.696 0.880
expt 1 2
T,K
P, psia
V,, cm3 (STP)/g
293 293
100.0 122.4
174.3 195.4
Y 0.953 0.919
expt 1 2 3 4 5 6
T,K
P, psia 117.1 144.2 181.3 231.1 246.0 328.2
V,, cm3 (STP)/g 109.8 117.7 134.1 136.3 158.8 172.5
Y 0.688 0.716 0.663 0.596 0.558 0.476
expt 1 2 3 4
T,K
P, psia 165.2 234.1 239.1 260.1
V,, cm3 (STP)/g 37.2 47.0 50.9 47.6
Y 0.472 0.382 0.530 0.280
expt 1 2 3 4 5 6
T,K
expt 1 2 3 4 5 6
T,K
297 293 295 294 293 294
COP
X
Y
X
0.230 0.174 0.252 0.154 0.198 0.038
0.231 0.277 0.198 0.300 0.245 0.537
0.770 0.826 0.748 0.846 0.802 0.962
X
Y 0.494 0.356 0.242 0.198 0.304 0.120
co
CHI
0.753 0.826 0.896 0.923 0.867 0.952 COP
HPS
X
X
Y 0.047 0.081
0.820 0.658 CHI
296 296 295 295 294 296
X 0.247 0.174 0.104 0.077 0.133 0.048
0.180 0.342 COP
X
X
Y 0.312 0.284 0.337 0.403 0.442 0.524
0.369 0.416 0.324 0.241 0.200 0.138
0.631 0.584 0.676 0.759 0.800 0.862
co 294 295 290 294
Table 111. Ternary Mixture Adsorption on Activated Carbon V,, cma CHI ' COP exut T,K P, usia (STP)/a Y X Y X 64.1 0.121 0.470 0.032 0.421 1 295 215.0 0.036 0.408 290.8 82.5 0.131 0.505 2 295 61.2 0.111 0.312 0.070 0.627 3 295 315.5 0.201 0.365 0.094 0.598 4 297 335.5 104.9 116.8 0.277 0.428 0.094 0.539 5 295 353.1 297.1 0.183 0.577 nl 293 86.5 2 296 319 96.1 0.233 0.559 49.4 0.158 0.429 1 293 153.7 0.443 2 293 174.2 51.8 0.108 3 293 211.2 56.7 0.124 0.541 244.2 0.159 0.577 4 297 68.2 5 291 284.1 76.4 0.203 0.679 72.3 0.199 0.643 6 291 324.5 0.128 0.489 7 297 343.1 73.1 81.9 0.166 0.530 8 298 379.1
HZ
X 0.859 0.904 0.902 0.835
HZ
Y 0.847 0.834 0.819 0.706 0.629 0.813 0.761 0.412 0.458 0.510 0.422 0.499 0.492 0.457 0.401
X
Y 0.528 0.618 0.470 0.720
0.141 0.096 0.098 0.165
co
HZS
X 0.109 0.087 0.061 0.037 0.032 0.080 0.058 0.127 0.095 0.131 0.054 0.130 0.153 0.101 0.072
Y
X
0.004 0.007
0.343 0.382
Y
X
0.429 0.434 0.365 0.420 0.297 0.309 0.415 0.433
0.444 0.462 0.328 0.369 0.191 0.204 0.410 0.398
Table IV. Adsorption of Four- and Five-Component Mixtures on Activated Carbon
V,, cma expt 1 2 3 4 1 2
T,K 293 295 294 295 297 296
P, psia 197 283 285 302 336 402
(STP)/g 123.5 172.7 164.3 137.7 131.3 117.5
co
CHI
Y 0.297 0.334 0.296 0.304 0.186 0.180
X 0.204 0.136 0.204 0.228 0.216 0.218
Y 0.551 0,432 0.500 0.556 0.260 0.260
CO-H2 (4),C02-H2S (2), CH4-CO-H2 (8),CH4-C02-H2 (5), CHI-H2-H2S (2), CH4-C0402-Hfi (4),and CH4-CMO2-H2-Hfi (2). The number of mixtures determined for each system is given in parentheses. The experiments
0.119 0.057 0.128 0.126 0.139 0.103
Y 0.130 0.197 0.193 0.120 0.070 0.066
h25
HZ
COP
X
X 0.356 0.395 0.524 0.345 0.315 0.305
Y 0.466 0.468
X 0.022 0.001
Y 0.023 0.037 0.010 0.020 0.017 0.025
X 0.320 0.412 0.144 0.301 0.308 0.373
were performed at room temperature (290-298 K)and at pressures up to 400 psia. Binary, ternary, and quaternary/quinary data are presented, respectively, in Tables II, III, and IV. Many different pressures and compositions
1682 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 Table V. Single-Gas Isotherm Parameters for Adsorption on Activated Carbon" Langmuir or VS-FH CH4 CO COz H, HzS For Langmuir 11.6 3.73 1.47 8.42 lo%,, l/psi 3.81 918.1 1723 1266 1885 eo, K 1730 -0.433 -0.599 A, cm3 (STP)/g/K -0.281 -0.299 -0.557 378 283 420 216 214 B , cm3 (STP)/g A,cm3 (STP)/g B , cm3 (STP)/g/K i05bo l/psi eo, K ff0 1 0 3 ~ , ,K
V, = A
For 209 -0.253 2.63 1882 0.169 0.630
VS-FH 217 -0.293 8.01 1255 0.469 -0.520
365 -0.499 2.17 2051 0.664 -0.700
290 -0.190 0.235 1344 -0.102 2.32
410 -0.553 5.95 1974 1.231 -2.20
+ BT; b = bo exp(e,/T); cy = a. + a,T.
Table VI. Average Values of 7 from Mixture Data system CH4 CO COZ CHA-CO 0.77 0.93 CHi-CO, 1.42 0.85 co-co2 1.76 0.80 CO-Hz 1.18 0.62 COZ-HZS binary av 1.10 1.30 0.76 CH,-CO-HZ 1.02 1.77 0.80 CH,-CO,-H2 1.02 CH4-Hz-HzS 0.97 CHd-CO-CO2-HZS 1.02 1.38 0.65 CH4-CO-COZ-Hz-H,S 1.39 1.66 0.98
Adsorption H,
HZS
0.80 0.80 0.45 0.99 0.87 0.72
0.26 0.26 0.20 0.13 0.58
were employed to test the range of the equilibrium adsorption theories at widely different conditions; thus, the mixture data cannot be tested for thermodynamic consistency. Data Correlation for Prediction of Multicomponent Adsorption. The 15 single-gas isotherms were fitted to two relationships: the two-constant Langmuir isotherm and the three-constant VS model (eq 6). The method of least squares was used to determine the two Langmuir parameters, b and V,, and a nonlinear, trial-and-error search program was used to regress the three VS-FH parameters, b, V,, and a. Since the mixture data were obtained at slightly different temperatures, the five isothermal single-gasparameters were regressed by the method of least squares to linear temperature-dependent equations which are given in Table V along with the new parameters. These temperature-independent parameters were the only single-gas regression parameters applied to each model. It is important to consider how well the model (employing the regression parameters) can reproduce the experimental data. The solid lines drawn in Figure 1 were calculated by using the Langmuir temperature-dependent
240
I
Pressure, p s j a
Figure 1. Adsorption isotherms of H2 ( O ) , CO ( O ) , CHI (A),CO, (v),and H2S (0)at 295 K on PCB-activated carbon. Temperature-dependent fit of the data by the Langmuir model (solid lines) and the VS-FH model (dashed lines).
parameters-an excellent fit resulted. The VS-FH temperature-dependent curves (dashed lines) nearly coincided with the solid lines. However, the average values of (V, - V,)/V, for the Langmuir model and of (P, - Pp)/Pefor the VS-FH model, for all data at room temperature, were respectively 4% and 11%. Thus, the Langmuir model could reproduce the original data very accurately and much better than the VS-FH relationship. The failure of the VS-FH model to fit the twice regressed data may be due to trying to fit three parameters to as few as four data points. Applying eq 3 to the experimental component amounts adsorbed, VI, and to the single-gas parameters made it possible to determine an interaction parameter, vL,which is specific to each component of each system. This changed the EL model into a data fitting model rather than a predictive one-unless binary data are used to predict multicomponent adsorption. Table VI lists the average values of v i obtained from each experimental system. Average values were reported since the variation of q1 with pressure and composition was not great, and furthermore, constant v i values are desirable for practical design purposes. Prediction of Multicomponent Adsorption. Extensive comparisons were made between the experimental mixed gas adsorption data and the aforementioned models employing only single-gas isotherms. The results are summarized in Table VI1 for VS-Y, IAS-Y, and EL (the Y indicates the independent variable of choice). This table gives the combined average percent error of the predicted component amounts adsorbed and the predicted adsorbed phase mole fractions for each system investigated. Typical plots of the predicted vs. experimental adsorbed phase mole fractions and of the component amounts adsorbed are displayed in Figure 2 and 3, for the EL model. Al-
Table VII. Average Percentage Error" between ExDeriment and Model Predictions* svstem M vs-Y IAS-Y EL EL-B CH4-CO 6 3.1 8.8 9.8 4.6 6 19.8 29.1 11.6 6.8 34.5 51.6 24.0 17.9 16.6 16.8 14.6 17.5 31.8 33.8 11.8 28.7 40.6 29.8 28.8 42.9 7.5 12.1 13.6 11.7 38.1 41.3 38.7 13.9 34.2 51.4 11.3 35.2 27.2 23.1 32.2 28.7
+
I
EL-M 2.8 8.0 9.6 16.2 6.7 15.5 6.1 16.3 7.6 12.6
vs-x 10.2 20.6 26.6 13.8 45.3 23.0 14.8 16.3 40.0 41.4
IAS-X 6.4 4.8 12.7 12.5 36.3 31.1 7.5 16.1 29.2 32.7
'av % error = (lOO/M)~:,M.,[l/m~;E~(l(V,, - VLP)/VLell(ZLe - Z,p)/ZLel)], where N = no. of components; M = no. of data points/system Z = X or Y . bVS-Y = VS-FH using ( T ,P, Y,]. IAS-Y = IAS using ( T ,P, Y,). EL = extended Langmuir model. EL-B = EL using binary 7. P, X ! } . EL-M = EL using multicomponent 7. VS-X = VS-FH using IT, P, X,}. IAS-X = IAS using (T,
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1683
I " " " " " ' I l'O
I1
.a
5 .- 0.5 .a
g
0.5
0
7
x
0
0
0.5
\ Ob.
-
*.
.\o
'9t!
X , , experimental
01
Figure 2. Extended Langmuir prediction of adsorbed phase mole fractions from binary (e),ternary (O), quaternary ( O ) ,and quinary (A)gas mixtures on PCB-activated carbon. Pressure range, 100400 psia; temperature range, 291-297 K. IAS and VS-FH gave similar predictions.
0
'
'
'
'
'
0.5
o o
" \
A
t
1.0
0
0
O .
O
I
'
'
I
'
1.o
I
'
I
N
z wv,"
,=2
Figure 4. Lewis correlation of amount adsorbed from binary (a), ternary (O), quaternary ( O ) ,and quinary (A)gas mixtures on PCBactivated carbon. Pressure range, 100-400 psia; temperature range, 291-298 K.
P
4 .-.a "0
30 60 Experimental Component Amt. Ads., ccSTP/g
0.5
90
Figure 3. Extended Langmuir prediction of amount adsorbed from ternary (O), quaternary ( O ) , and quinary (A)gas mixtures on PCBactivated carbon. Pressure range, 200-400 psia; temperature range, 291-298 K. Binary data (not shown) fall within dashed lines. IAS and FH-VS gave very similar fits.
though they are not displayed, similar trends resulted for IAS-Y and VS-Y; however, the spread about the diagonal was slightly less, as indicated by the average predictions being better for IAS-Y and VS-Y, as shown in Table VII. The same single-gas Langmuir constants were used for both the EL and IAS predictions. Table VI1 shows that IAS improved the predictions considerably. The threeconstant VS model made no improvements over IAS predictions. However, VS predictions may be improved if the regression of the single-gas parameters could be improved to better than 11%fit. Table VI1 also shows that the VS-Y, IAS-Y, and EL models cannot consistently predict multicomponent adsorption from single-gas isotherms. However, the predictions were quite good for a few systems. Comparison of the mixture data with the Lewis correlation is displayed in Figure 4. The data are plotted as the ratio of the volume of gas i adsorbed from the mixture to the volume of the single gas adsorbed at the same T and P of the mixture vs. the same ratio summed over all components except the ith. the Lewis correlation predicts a slope of -1.0 and an intercept of 1.0. Figure 4 shows that the results are similar to those obtained from the three theories discussed above. This is not surprising, especially when compared to IAS. Firstly, they are both based on ideal solution behavior (ri= 1and free energy of mixing = 0), and secondly, the same Langmuir single-gas regression parameters were used. The EL model was investigated further by incorporating an interaction parameter determined from mixture ad-
2.0
-
0
OO
30
60
90
Experimental Component Amt. Ads., ccSTP/g
Figure 6. EL-M prediction of amount adsorbed from ternary (O), quaternary ( O ) , and quinary (A) gas mixtures on PCB-activated carbon. Pressure range, 200-400 psia; temperature range, 291-297 K. Binary data (not shown) fall within dashed lines.
sorption data. The results are presented in Figures 5-8 and in Table VII. For the first two figures, each system used the average qi given for each component of each system (Table VI). This was referred to as EL-M, where the M indicates .that multicomponent interactions specific to each system were accounted for. The results improved significantly compared to Figures 2 and 3. Table VI1 also illustrates remarkable improvement-average errors dropped to within 10% of the experimental data. Yon and
1684 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987
1 t
J
I
i
0 Y,, experimental
Figure 9. IAS prediction of gas phase mole fractions, using the alternate independent variable set IT,P, Xi), from binary (e),ternary (O), quaternary (a),and quinary (A) gas mixtures on PCB-activated carbon. Pressure range, 100-400 psia; temperature range, 291-297 K. VS-FH gave similar predictions. J
I
l
I
I
I
,
1
1
,
,
,
1
,
20i
0 0
-0
30 60 Experimental Component Amt. Ads., ccSTP/g
90
Figure 8. EL-B prediction of amount adsorbed from ternary (O), quaternary (O),and quinary (A)gas mixtures on PCB-activated carbon. Pressure range, 200-400 psia; temperature range, 291-297 K. Binary data (not shown) fall within dashed lines.
Turnock (1971) employed a similar method where they determined specific 7s; only for those components which deviated substantially from the EL model. Their results also showed significant improvements for the C02-H2Snatural gas-zeolite system. However, they fitted the single-gas data to the three-parameter EL model ( b , Vmi,and ni,where n, is the exponent of Pi) and then determined vi. Thus, four parameters were used, whereas only three were used here. EL-M requires multicomponent data in order to determine the 7, specific to each component of each system. A more general approach was to use binary 7,’s to predict multicomponent adsorption. This was investigated by calculating “binary averaged” vi’s which are given in the middle of Table VI. These were determined for C02,for example, by taking the average of the vi values listed for the binary systems containing COB. The result gives 0.76. This was referred to as EL-B, where the B indicates that only binary interactions were accounted for. Table VI shows that the average vi values do not change very much when comparing binary to multicomponent systems. Thus, the averaging process can be somewhat justified. The results of EL-B are shown in Figures 7 and 8 and are also included in Table VII. Average deviations were cut in half (from 30% to 15%)compared to the predictions obtained from the single-gas isotherms (EL). This result is indeed encouraging since only 5 of the 10 possible binary systems were investigated experimentally. Effects of Pressure and Number of Components on the Model Predictions. The model predictions obtained from IAS-Y are compared to the experimental results here since the IAS theory represents a thermodynamically
sound, ideal system, and the pure component data were well represented by the Langmuir equation. Among the 10 binary and multicomponent systems investigated, only CH,-CO, CH4-C02, and CO-C02 demonstrated that deviations increased with pressure, over the range 100-400 psia. Average deviations increased from 5% to lo%, from 5% to 12%,and from 23% to 40%, approximately linearly with pressure, for the three mixtures, respectively. The effects of pressure for the multicomponent mixtures were not seen since only narrow and intermediate pressure ranges were investigated. In the same range of total pressure, however, generally the model predictions for multicomponent mixtures were worse than those for binary mixtures. The effects of pressure are expected since increasing pressure necessarily increases the surface coverage and hence increases the nonideality of the adsorbed phase. The reason for the effects of the number of components was, however, not clear. The latter effects have also been observed previously by Wilson and Danner (1983). It is interesting to compare their results to ours since they investigated binary systems identical with ours. They compared the adsorption of CH,-CO, CH4-C02,and COC02on activated carbon to the VS and IAS models. Their results were, for the most part, better than those displayed in Table VII. For example, our CH4-CO and CH4-C02 systems compared well with both theories, whereas they reported excellent agreement. For the CO-C02 system, they reported excellent agreement with IAS and fair agreement with VS. Table VI1 shows only a fair agreement with both models. These comparisons suggest that the differences between their results and ours were possibly due to our higher total pressures. Alternate Variable Selection: Effects on Spreading Pressure and Amounts Adsorbed. The results presented thus far for the VS-FH and IAS theories have exclusively employed (T,P, Yi)as the independent variable set. As noted eariler in the paper, both IT, P, Yi)and ( T , P, Xi{have been used in the literature as the independent variable set for the predicting mixture adsorption. Results are presented in Figure 9-11 and in Table VI1 for using { T , P, Xi). These predictions are compared with those from { T ,P, Yi) for both IAS and VS-FH models. Figures 9 compares Yipredicted from IAS-X to the experimental values. The spread in this figure is very similar to that in Figure 2. Thus, the predictions of the mole fractions are similar for both IAS-X and IAX-Y.
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1685
z---------
o A " " ' ~ 30 ' " " ' l ' ' 6 l0 "
90
Experimental Component Amt. Ads., ccSTP/g
Figure 10. IAS predictions of amount adsorbed, using the alternate from ternary (o),quaternary (01, independent variable set IT,P, Xi}, and quinary (A) gas mixtures on PCB-activated carbon. Pressure range, 100-400 psia; temperature range, 291-297 K. VS-FH gave similar predictions. 400
r
I
I
I
I
I
IAS 3 0 0
VS-FH
5
A
A
I
0
I
100
-
-
-
+
0
-+
0'
0
-
-
I
A
N 2 3 4
to different dependent variables, especially if the differences in the predictions were large. There is nothing inherently wrong with the models; the problem is definitely computational. For example, when Yiis selected as one of the independent variables, a certain Xiresults, and when Xi is used, a certain Yiresults. Since the Xi's are not necessarily equal (the former is predicted; the latter is experimental), neither are the Y;s. This also explains why the errors involved in the predictions of the mole fractions were similar for both IAS-X (VS-X) and IAS-Y (VS-Y). On the other hand, the major improvements in the Vipredictions seen for the X Y approach (compared to the Y X approach) resulted from using the experimental adsorbed phase mole fractions directly to calculate V, (and thus Vi).For the Y X method, the predicted Xi's are used to calculate V,, and thus larger errors for the V ipredictions result. For the X Y approach, the predicted Yi'sare not used to calculate any other dependent variables. As a result, the X Y approach can be misleading. It is not uncommon to see graphs of Vi vs. Xiin the literature. If the X Y approach is used, then the Xi's are experimental and only Viis predicted. However, if the Y X method is used, then both the Xi's and the Vi's are predicted and compared in the plot. The problem with this is that the former example should and will give better results. In conclusion, the directional differences associated with the selection of the independent variable set give rise to significantly different standard states and adsorbate loadings even though the same model was used. This unusual result was totally unexpected. The origin of this phenomenon is seemingly computational, and thus the models as well as the experimental data are not in any way at fault. I t is suggested that since X iis never a priori known, except from mixed gas adsorption studies, Yi should be used exclusively to predict mixed gas adsorption. This would eliminate some of the ambiguity involved in comparing and contrasting mixed gas adsorption predictions reported from different laboratories.
1
I
#
200
1
J
300
Y-x IAS Spreadlog Pressure or VS-FH Total Amt. Ads., ccSTP/g
Acknowledgment
Figure 11. Solutions to IAS (rm)and VS-FH (V,) for the independent variable sets IT, P, Xi}and IT, P, Yil.
This work was supported by the Deparment of Energy under Grant DE-AC21-83MC20183.
Figure 10 compares the amounts adsorbed by IAS-X. It shows significant improvements over that by IAS-Y, which is shown in Figure 3. (The VS-FH results were nearly identical with those in Figure 9 and 10.) Table VI1 (last two columns) compares the average predictions by IAS-X and VS-X with those from the Y X methods. The differences do not appear great since Yiwas included in the averages, but significant differences existed for certain mixtures. For systems where experimental data agreed well with predictions (within 5%), both Y X and X Y predictions were in close agreement with the data. The predictions by the two approaches differed widely, however, when both deviated from the experimental data. Figure 11 compares the dependent variables (spreading pressure, a,, for IAS and total amount adsorbed, V,, for VS-FH) predicted by using the Y X and X Y approaches. Disagreements between the two approaches are clearly shown by the scatter about the diagonal. These directional differences occurred because for each experimental mixture a different set of algebraic equations (having the same parameters but a different independent variable set) must be solved. Since a different independent variable set was used, the equations naturally converged
Nomenclature A = specific surface area of sorbent b = Langmuir constant EL = extended Langmuir model EL-B = EL using binary 4 EL-M = EL using multicomponent 4 I" = standard reference state IAS = ideal adsorbed solution theory IAS-X = IAS using (T,P, X,) IAS-Y = IAS using (T,P, Yi] M = number of experiments or mixtures N = number of components in mixture P = total pressure P" = vapor pressure of pure adsorbate at the same spreading pressure and temperature of the mixture S = selectivity T = temperature V = amount adsorbed V , = "monolayer" or saturated amount adsorbed V o = single-gas adsorption at same T and P of mixture VS-FH = vacancy solution theory using the Flory-Huggins activity model VS-X = VS-FH using ( T ,P, X i ) VS-Y = VS-FH using (T, P, Yi) X = mole fraction in the adsorbed phase
+
+
-
+
-
Ind. Eng. Chem. Res. 1987, 26, 1686-1691
1686
Y = mole fraction in the gas phase mole fraction
z =
Greek Symbols = parameter describing nonideality in the adsorbed phase induced by interaction with surface (vacancy) y = activity coefficient 7 = lateral interaction parameter 0 = fractional amount adsorbed based on a monolayer coverage T = spreading pressure 4 = fugacity coefficient a!
Subscripts
i = species or component m = mixture t = total v = vacancy e = experimental quantity p = predicted quantity Superscripts s = surface phase m = limiting or maximum value Registry No. H2, 1333-74-0; CO, 630-08-0; CH,, 74-82-8; COP, 124-38-9; HzS, 7783-06-4; C, 7440-44-0.
Literature Cited Cochran, T. W.; Kabel, R. L.; Danner, R. P. AZCHE J . 1985,31,268. Costa, E.; Sotela, J. L.; Calleja, G.; Marron, C. AZCHE J . 1981,27, 5.
Friederich, R. 0.;Mullins, J. C. Znd. Eng. Chem. Fundam. 1972, 11, 439. Grant, R. J.; Manes, M. Ind. Eng. Chem. Fundam. 1966, 5, 490. Hyun, S. H.; Danner, R. P. J . Chem. Eng. Data. 1982,27, 196. Lee, A. K. K. Can. J . Chem. Eng. 1973, 51, 688. Lee, C. S.; O’Connell, J. P. AIChE J . 1986, 32, 96. Lewis, W. K.; Gilliland, E. R.; Chertow, B.; Cadogan, W. P. Znd. Eng. Chem. 1950, 42, 1319. Markham, E. D.; Benton, A. F. J. Am. Chem. Soc. 1931, 53, 497. Miller, G. W.; Knaebel, K. S.; Ikels, K. G. AIChE J . 1987, 33, 194. Myers, A. L. Ind. Eng. Chem. 1968, 60, 45. Myers, A. L.; Prausnitz, J. M. AIChE J . 1965, 11, 121. Peng, D.-Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976,15,59. Reich, R.; Zeigler, W. T.; Rogers, K. A. Ind. Eng. Chem. Process Des. Deu. 1980, 19, 336. Ritter, J. A. M. S. Thesis, State University of New York, Buffalo, 1986. Ruthven, D. M.; Loughlin, K. F.; Holborow, K. A. Chem. Eng. Sci. 1973, 28, 701. Saunders, J. T. M. S. Thesis, State University of New York, Buffalo, 1983. Schay, G. J. Chem. Phys. Hungary, 1956,53, 691. Talu, 0.; Zwiebel, I. AZChE J. 1986, 32, 1263. Van Ness, H. C. Ind. Eng. Chem. Fundam. 1969,8, 464. Wilson, R. J.; Danner, R. P. J . Chem. Eng. Data 1983, 28, 14. Yang, R. T. Gas Separation by Adsorption Processes; Butterworth Boston, 1987; Chapter 3. Yon, C. M.; Turnock, P. H. AZChE Symp Ser. 1971, 67(117), 3.
Received for review October 28, 1986 Revised manuscript received April 24, 1987 Accepted May 1, 1987
Evaluation of an Equation of State Method for Calculating the Critical Properties of Mixtures J. Richard Elliott, Jr. Department of Chemical Engineering, The University of Akron, Akron, Ohio 44325
Thomas E. D a u b e r t * Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802
The accuracy of an equation of state method for predicting critical properties of mixtures is evaluated and compared t o several empirical methods. The Soave equation of state is used with binary interaction coefficients predicted from vapor-liquid equilibrium data. An extensive data base of about 1500 points each for critical temperature and critical pressure is used in the evaluation. These data include hydrocarbon mixtures and hydrocarbon-non-hydrocarbon mixtures with H2, N2,CO, C 0 2 ,and H2S. The equation of state method is determined t o be more accurate than the empirical methods for critical temperature and critical pressure and slightly less accurate for critical volume. Furthermore, the equation of state method predicts anomalous trends in critical loci which were previously noncalculable by empirical methods. When phase equilibria a t high temperatures and pressures are considered, knowledge of the critical properties of mixtures is essential. Many processes involving high pressure are designed specifically to take advantage of the unique phase behavior in the critical region. Enhanced oil recovery with carbon dioxide and supercritical extraction provide two examples of such processes. Accurate knowledge of the critical properties of the mixtures is especially important for these types of processes. Many correlations have been proposed for predicting the critical properties of mixtures. Most of these correlations have been empirical, and they have been limited in the types of systems which they could represent. Empirical correlations were evaluated by Spencer et al. (1973), and the most accurate methods were recommended. One 0888-5885/87/2626-1686$01.50 f 0
modification has been made to the Chueh and Prausnitz (1967) method in the API book (1986) as described later. Since that time, the calculation of the critical properties via an equation of state by applying the rigorous thermodynamic criteria a t the critical point has become more practical for common application. Peng and Robinson (1977) evaluated their equation of state for 30 mixture critical points by this method. Heidemann and Khalil (1980) published an improved algorithm which was considerably more rapid and more robust than its predecessors. Michelsen and Heidemann (1981) have improved the computation speed of this latter algorithm. Considering these developments, a thorough evaluation of the merits of a rigorous method relative to the empirical methods was undertaken. 0 1987 American Chemical Society
