Ind. Eng. Chem. Res. 1988,27, 630-635
630
1 = level, lb reboiler level T = temperature t = time u = control variable V = vapor flow, V’denotes steam flow to reboiler w = disturbance I = composition, X D distillate, X B bottom product z = feed composition A = denotes deviation from steady state = denotes feedback, e.g., (D T4)denotes that D is used for feedback control of T4
-
-
Literature Cited Bristol, E. H. IEEE Trans. Autom. Control 1966, AC-11, 133. Finnerman, D. H.; Sandelin, P. M. MSc. Theses, Ab0 Akademi, Abo, Finland, 1986. Gustafsson, S. E., Abo Akademi, Abo, Finland, unpublished work, 1984. Hiiggblom, K. E. Tech.Lic. Thesis, Abo Akademi, Abo, Finland, 1986. Haggblom, K. E. “Estimation of Consistent Process Gains for Distillation Control Structures”. Report 87-8,1987a; Process Control Laboratory, Abo Akademi, Abo, Finland. Haggblom, K. E., unpublished results, 1987b. Hiiggblom, K. E.; Waller, K. V. Preprints DYCORD 86: IFAC Symposium on Dynamics and Control of Chemical Reactors and Distillation Columns, Bournemouth, England, 1986, p 243. Haggblom, K. E.; Waller, K. V. “Transformations and Consistency Relations of Distillation Control Structures. I. Theory”. Report 87-6, 1987a; Process Control Laboratory, Abo Akademi, Abo, Finland. Hiiggblom, K. E.; Waller, K. V. “Transformations and Consistency Relations of Distillation Control Structures. 11. Applications”. Report 87-7, 1987b; Process Control Laboratory, Abo Akademi, Abo, Finland. Haggblom, K. E.; Wikman, K. E.; Gustafsson, S. E.; Waller, K. V. Proceedings Automaatiopaivat, Helsinki, Finland, 1984, Vol. 11, p 427.
Hammarstrom, L. G.; Waller, K. V.; Fagervik, K. C. Chem. Eng. Commun. 1982, 19, 77. McAvoy, T. J. Interaction Analysis; Instrument Society of America: Research Triangle Park, 1983. McAvoy, T. J.; Weischedel, K. Proc. 8th IFAC World Congress, Kyoto, Japan, 1981. Rademaker, 0.; Rijnsdorp, J. E.; Maarlevald, A. Dynamics and Control of Continuous Distillation Units;Elsevier: AmsterdamOxford-New York, 1975. Roat, S.; Downs, J.; Vogel, E.; Doss, J. In Chemical Process Control-CPC III; Morari, M., McAvoy, T. J., Eds.; CACHE/ Elsevier: New York, 1986. Ryskamp, C. J. Hydrocarbon Process 1980, 59(6), 51. Shinskey, F. G. Distillation Control, 2nd ed.; McGraw-Hill: New York, 1984. Takamatsu, T.; Hashimoto, I.; Hashimoto, Y. Proc. PSE 82, Kyoto, Japan, Tech. Session 243, 1982. Takamatsu, T.; Hashimoto, I.; Hashimoto, Y. Preprints 9th IFAC World Congress, Budapest, Hungary, 1984, Vol. 111, p 98. Tsogas, A.; McAvoy, T. J. Proc. 2nd World Congress of Chemical Engineering, Montreal, Canada, 1981. Waller, K. V. In Chemical Process Control 2; Seborg, D. E., Edgar, T. F., Eds.; Engineering Foundation/AIChE: New York, 1982; p 395. Waller, K. V. Preprints DYCORD 86: IFAC Symposium on Dynamics and Control of Chemical Reactors and Distillation Columns, Bournemouth, England, 1986, p 1. Waller, K. V.; Finnerman, D. H. Chem. Eng. Commun. 1987,56,253. Waller, K. V.; Finnerman, D. H.; Sandelin, P. M.; Haggblom, K. E. “On the Difference between Distillation Column Control Structures”. Report 86-2, 1986; Process Control Laboratory, Abo Akademi, Abo, Finland. Waller, K. V.; Haggblom, K. E.; Sandelin, P. M.; Finnerman, D. H. “On Disturbance Sensitivity of Distillation Control Structures”. AIChE J . 1988, in press. Received for review April 29, 1987 Revised manuscript received November 16, 1987 Accepted December 7, 1987
A Simple Potential-Theory Model for Predicting Mixed-Gas Adsorption S.J. Doong and R. T.Yang* Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260
The Dubinin-Astakhov equation, which is based on the micropore volume filling theory, is extended to mixed-gas adsorption by using the concept of maximum available pore volume. The proposed model requires only single-gas data for predicting mixed-gas adsorption; no additional equation, such as the Lewis relationship, is needed. The proposed model is not iterative, and the calculation involved is extremely simple. The model has been tested against experimental data for 16 binary mixture systems involving a wide variety of adsorbents and a wide range of pressures, with satisfactory results. It also compares favorably or equally satisfactorily with the ideal adsorbed solution theory, the Grant-Manes model, and an extension of the DA equation proposed by Bering et al. However, all three of these methods require iterative (numerical) computation. The experimental data of many gas-solid systems involving microporous sorbenta such as activated carbon can be fitted well by the Dubinin-Radushkevich (DR) equation ( D u b i n i n , 1966, 1975):
rather than as the forming of successive layers as assumed in the Langmuir model. The DR equation, however, also fails for many gas-solid systems, and the deviations have been interpreted in terms of pore-size distribution, surface energy distribution, and other factors ( N a k a h a r a et al., 1974; Rand, 1976; Marsh, 1987). A more general equation, proposed b y Dubinin and Astakhov (DA), has been found to be useful (Dubinin,
where V is the volume adsorbed at relative pressures PIP8, Vo is the micropore volume, Eois the characteristic energy of the sorbent, and @ is the affinity coefficient of the sorbate. The DR equation is a result of the pore volume filling theory, in which it is postulated that adsorption in micropores occurs as the filling of the micropore spaces
1975):
0888-5885/88/2627-0630$01.50/0
The exponent n i n the DA equation lies in the range 1-4, 0 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 4, 1988 631 depending on the type (distribution of energies) and pore structure of the sorbent. The value of n accounts for the deviation of adsorption energies from a Gaussian distribution that is assumed for the DR equation. Large n values a.e found for sorbents with narrow distributions of micropores, and small n values are found for sorbents with wide ranges of pore sues, usually containing mesopores and macropores. The DR and DA equations have been used extensively in adsorption and sorbent characterization studies, such as estimation of micropore volume (McEnaney, 1987), description of heterogeneous structure (Huber et al., 1978; Dubinin and Stoeckli, 1980), and adsorption of polar molecules (Rozwadowski and Wojsz, 1981). However, with only one exception, no attempts have been published to extend these useful equations for predicting equilibrium adsorption of mixtures. The only exception k the extended DR equation, proposed by Bering et al. (1963), for binary mixtures. One version of the extended DR equation for binary mixture adsorption is
[(
I')"
V1 + Vz = Vo exp - RT In P1zEo
(3)
bateaorbent interactions energy (DE,), as given by the DA equation. Under the assumption of no inter-species interactions, the parameters n and @Eofor any component are not influenced by the other components. However, the maximum available micropore volume is reduced from Vo to Vo- C Vi where the summation is carried over all other components. Thus, for a binary mixture, the volumetric amount of adsorbate for component 1 is
For the same sorbent, both Vo and n should be fixed, independent of the gas species. However, values of Vo and n calculated from experimental data on different gases do show variations. These variations can be attributed to several origins, e.g., the nondispersive forces between sorbate and sorbent, and the uncertainties in the adsorbate molar volumes especially near or above the critical temperatures. It appears reasonable, as a first approximation, to retain the values of Vo and n for each component for mixed-gas adsorption. Consequently, for binary mixtures,
PSlZ
where Vl and V2 are the volumes adsorbed for each component and PSlzis the saturated vapor pressure for the adsorbed mixture, over which the saturated vapor has the same composition as that of the equilibrium vapor over the sorbent. Another version of eq 3 differs from the above equation in that In PSl2/(P1 + Pz)is replaced by E X i In PSi/Pi(Yang, 1987). The value of B12 may be taken as
Since eq 7a and 7b are linear, explicit solutions for Vl and V, can be obtained as
(4) = XlPl + X2P2 Equation 3 alone is not sufficient for calculating Vl and V2. An additional equation, the Lewis relationship (Lewis et al., 1950), is needed:
where
812
Q1/Q1°
+ 42/42O = 1
(5)
The Lewis relationship can be derived by assuming ideal mixing of adsorbates (Yang, 1987). Equations 3, 4, and 5 will be referred to in this paper as the extended DA model where the exponent in eq 3 is changed from 2 to n. This model can be further extended to an N-component mixture in a straightforward fashion (Yang, 1987). It is important to point out, from a practical viewpoint, that this model, as well as two major theories in the literature, ideal adsorbed solution theory (Myers and Prausnitz, 1965) and Grant-Manes model (Grant and Manes, 1966), is iterative and requires considerable numerical computation in application. In this paper, we present a simple model based on the concept of maximum available micropore volume for predicting mixed-gas adsorption. The only needed data are single-gas parameters for the DA equation. The advantage of this proposed model is its computational simplicity-it requires no iteration. Explicitly simple equations are used for the prediction. The proposed method has been extensively tested, with favorable results, against a wide range of literature data on binary mixture adsorption in activated carbon, molecular sieve zeolite, and molecular sieve carbon. The proposed method is modified further to account for lateral interactions.
Proposed Model The micropore volume filling theory is formulated for mixed adsorbates by assuming that no inter-species lateral interactions exist. The amount of single-gas adsorption is determined by three factors: limiting micropore volume ( Vo), pore-size distribution parameter (n),and the sor-
-(
AI = ex.[
PlEO In
$)]
P2Eo Equation 8 is sufficient for predicting binary mixture adsorption. Additional equations, such as the Lewis relationship that is necessary in the extended DA model, are not needed. Most important, no iteration is required for solving eq 8. Equations 7 and 8 can be extended to a multicomponent mixture in a straightforward fashion. To convert volumetric adsorbed amount ( V ) to molar adsorbed amount ( q ) , the following relationship is used: 4 = V/Va
(9)
where V, is the adsorbate molar volume which can be calculated by a number of methods (Chen and Yang, 1986). The following relationships are used in this work: va
=
Vs,nbp
e
Tnbp
(loa)
(lob)
T L T, (104 where Vs,nbpis the molar volume of the saturated liquid at its normal boiling point (nbp), and V, is the molar volume a t its critical temperature (calculated as the van der Waals constant, RTc/(8Pc)). Equations 10a and 10b were suggested by Reich et al. (19801, and eq 1Oc was Va = VcT$6
632 Ind. Eng. Chem. Res., Vol. 27, No. 4, 1988 Table I. Parameters for the DA Equation for Single-Gas Adsorption adsorbate adsorbent; ref temp, K V,, cm3/g CH, a 293 0.275 0.355 C2H4 CZH6 0.361 C3H6 0.397 C3H8 0.404 N2 b 144 0.198 0 2 0.183 N2 C 144 0.264 0 2 0.228 d 301 0.341 CH4 CH, 260 0.368 213 0.381 CH4 301 0.358 CZH* 260 0.380 CZH4 213 0.417 C2H4 301 0.362 C2H6 213 0.411 C2H6 C2H4 e 303 0.167 0.165 C2H6 0.184 C3H6 0.166 C3H8 0.177 CHSOH 0.177 CH3COCH3 0.182 C6H14 0.180 C6H6
@&I (RT) 3.954 4.386 4.456 5.547 5.568 9.600 6.316 6.944 4.662 3.521 3.608 4.119 4.050 4.538 5.346 4.168 5.778 6.740 6.670 8.410 7.810 4.651 9.774 14.968 11.52
n 2.02 1.86 1.89 1.89 1.85 3.69 3.54 2.00 3.05 1.93 1.72 1.55 1.75 1.60 1.48 1.79 1.56 2.68 2.85 2.78 3.02 1.81 2.00 1.62 1.78
av re1 error 1.38 X 6.40 x 10-3 1.39 X 1.44 X 1.88 X 5.42 x 10-3 3.79 x 10-2 1.95 X 1.09 x 10-1 4.88 x 10-3 1.20 x 10-2 2.55 X 1.33 X 1.67 X 3.58 X 1.50 X 4.00 X 1.67 X 6.79 x 10-3 1.09 x 10-2 3.20 X 1.33 X lo-’ 4.26 x 10-3 2.65 x 10-3 5.58 x 10-3
“ “ i t activated carbon; Szepesy and Illes (1963a-c). *5A molecular sieve; Danner and Wenzel (1969). 1OX molecular sieve; Danner and Wenzel (1969). dBPL activated carbon; Reich et al. (1980). eMolecular sieve carbon; Nakahara et al. (1974), Konno et al. (1985a,b). I
I
I
I
I
I
1
: CA I eq.
T :293K
I
1 1
I
2
1
3
I
4
I 5
6
P ,a t m
Figure 1. Adsorption isotherms of C2Hs on Nuxit-AL activated carbon. Experimental data are from Szepesy and Illes (1963a-c).
suggested empirically by Chen and Yang (1986). Moreover, the adsorbate molar volume is assumed to be the same in both single- and mixed-gas adsorption. The vapor pressure (P,) are calculated by using the reduced Kirchhoff equation and extrapolated to obtain “saturation pressuren whenever the temperature is above T,.The uncertainty in P,is also absorbed by the DA parameters. The nonideality of the gas phase is accounted for by replacing pressures with fugacities.
Results and Discussion Single-Gas Adsorption. Equilibrium data for 25 gas-solid systems, involving five different sorbents, are fitted by the DA equation, eq 2, using nonlinear regression. The resulting DA parameters, along with average relative errors, are given in Table I. The deviations between the fitted DA equation and the experimental data are within 4%, except for 02-10X zeolite where the deviation is 10%. Figure 1shows a representative DA fitting of these data. For the same sorbent, small deviations in Voand n are exhibited among different gases (Table I). The value for Votends to be higher for stronger adsorptives and for the same adsorptive a t lower temperatures. This behavior can be partially attributed to the presence of mesopores and macropores, which can become more accessible to stronger adsorptives and at lower temperatures due to capillary condensation and “cooperative effects” (Marsh, 1987). The particularly low values of Vo for CH, are probably the
consequence of the high values of the adsorbate molar volume, which is extrapolated (by eq 1Oc) at temperatures well above the critical temperature. The above reasonings for the variation of Voare also supported by the variations of the n values (Table I). For the same carbon sorbent, the value of n decreases and becomes substantially less than 2 a t higher Vo values, indicating more accessible mesopores and macropores. The n values for the molecular sieve zeolites and molecular sieve carbon are greater than 2, indicating the dominance of small pores and a narrow pore size distribution. However, the n value for molecular sieve carbon can be less than 2 (Table I), because of the presence of an appreciable amount of macropores. The presence of macropores in the same carbon was concluded by Sircar (1987) as well. It should be pointed out that an experimental error was involved in the experiments of Szepesy and Illes (1963a-c) for both single and mixed gases (Yang, 1987). The error is, however, relatively small as compared to the deviations between experimental data and model predictions for mixed-gas adsorption as will be shown shortly. Binary Mixture Adsorption: Test of Proposed Model. The proposed model is tested against experimental data as well as the ideal adsorbed solution (IAS) theory and the Grant-Manes (GM) model. In addition, the extended DA model (eq 3-5) where the exponent in eq 3 is changed from 2 to n, proposed by Bering et al. (1963),is also compared with the proposed model. Average values of Voand n between the pair components are used in the extended DA model; this is not necessary for the proposed method (eq 7). The literature data on 16 binary mixture systems containing 158 equilibrium mixture data points are used. In order to compare the four theoretical models on an equal basis, the DA eqbation (eq 2) with the same DA parameters (Table I) is used in all four models. The IAS calculations were performed by using an iterative Newton-Raphson routine. For binary mixtures, this became a one-dimensional search method. The spreading pressures were calculated by integrating from 0 to P,O where Pi0 = Pi/Xi. The upper limit for the DA equation is the micropore volume, a t Pi = Psi. At P = 0, the slope of the DA equation is zero (Toth, 1984) and an inflection
Ind. Eng. Chem. Res., Vol. 27, No. 4, 1988 633 Table 11. Comparisons of Predictions of Binary Mixture Adsorption on BPL-Activated Carbon (Reich et al., 1980) by Models av re1 errors of moles adsorbed for each component, % system temp, K no. of data pta proposed method Grant-Manes IAS extended DA 301 16 22.87 35.89 36.58 10.85 CHI C2H4 2.26 1.24 3.63 4.81 av 12.57 18.57 20.10 7.88 301 12 30.00 32.24 37.51 18.18 CHI 5.29 C2H6 3.38 3.48 3.56 av 16.69 17.86 20.53 11.73 CZH6 301 12 19.67 18.73 11.83 18.43 C2H4 14.39 17.5 7.72 8.36 av 17.03 18.12 9.77 13.40 CH4 260 4 21.45 6.27 28.2 17.71 C2H4 2.35 1.55 6.73 7.93 av 11.90 3.91 17.47 12.83 C2H6 213 6 9.15 5.57 1.65 8.03 C2H4 13.73 3.54 2.46 14.26 av 11.44 4.55 2.05 11.15 Table 111. Comparison of Predictions of Binary Mixture on Molecular Sieve Carbon 5A (Nakahara et al., 1981, 1982; Konno et al.. 1986a) av re1 errors of moles adsorbed for each component, % system temp, K no. of data pts proposed method Grant-Manes IAS extended DA C2H4 303 10 70.91 26.11 41.15 73.16 13.79 12.37 5.82 20.63 C3HB av 41.64 27.47 46.90 15.97 CZH6 303 22 47.08 36.58 53.87 26.82 12.61 10.04 14.35 18.51 C3HB av 18.44 29.85 25.47 36.16 CZH4 303 8 39.16 163.49 30.89 61.97 2.09 3.65 2.73 16.61 C3H6 av 21.40 16.49 90.05 32.35
0.2
0. 0
02
06
04
OB
10
YI
Figure 2. Binary mixture adsorption equilibria on Nuxit-AL activated carbon a t 293 K and 1 atm (Szepesy and Illes, 1963a-c). (a, left) (A)C2H6-CH4, (0) C2H6-C3H8, (0) CzH6-CzH,. (b, right) (A) CzH4-CHI, (0) C2H4-C3H6,(0) C3H8-C3H,. (-) Proposed method, (- - -) Grant-Manes, IAS, (- -) extended DA. Mole fraction is for the first component of the pair. (e-)
-
point exists a t PIPs C Thus, using the DA equation in the IAS calculations will cause an error a t the lower integration limit. However, this error is negligibly small. The experimental data and theoretical predictions of the four models are compared in Figures 2-4 and Tables 11and 111. Figure 2 gives the comparisons for binary mixtures of CHI, C2H4, C2H6 C3H6, and C3Hs on Nuxit activated carbon (Szepesy and Illes, 1963a-c). The results for an 02-N2 mixture on zeolite types 5A and 1OX (Danner and Wenzel, 1969) are shown in Figure 3. The equilibrium mixture data by Reich et al. (1980) on BPL-activated carbon were not obtained at constant total pressures; thus, it is not possible to compare them in X-Y diagrams in a figure form. Consequently, their data are compared with model predictions in Table 11. For C2H4,C2H6, C3H6,and
0.4
0.6
0.8
1.0
Yl Figure 3. Binary mixture adsorption equilibria on molecular sieve zeolites 5A and 1OX at 144 K and 1atm (Danner and Wenzel, 1969). (A) N2-02-10X, (0)O2-N2-5A. (-1 Proposed method, (- - -) Grant-Manes, (-.) IAS, extended DA. Mole fraction is for the first component of the pair. (-e-)
C3H6 on molecular sieve carbon, Table 111summarizes the average relative errors of the predictions, and the results for the two pairs C2H4-C3H6and C3H8-C2H6are shown in Figure 4. The comparisons of the literature data given in Figures 2-4 and Tables I1 and I11 are not exhaustive but have covered a broad range of important sorbents and a wide pressure range. These comparisons show that the IAS theory, the G M model, and the proposed model give fair predictions, whereas the predictions given by the extended DA model (Bering et al., 1963) are inferior. The extended DA model fails completely for zeolites (5A and l o x , Figure 3), while the proposed method works surprisingly well. Both the extended DA model and the proposed model are based on the potential theory, where the dispersion forces are assumed to be dominating in the sorbate-sorbent in-
634 Ind. Eng. Chem. Res., Vol. 27, No. 4,1988 10
08
06
xi 04
02
(3'
1 02
0
06
04
08
10
0
Yq Figure 4. Binary mixture adsorption equilibria on molecular sieve carbon 5A at 303 K and 100 Torr. (A)C,H8-C2H6 (Nakahara et al., C2H4-C3H6(Nakahara et al., 1982). 1981), (0)
x1
10
I l l l l r i l l l l 0
02
06
04
08
10
Yl
Figure 5. Adsorption equilibria of acetone-hexane mixture on molecular sieve carbons 5A a t 303 K and 4 kPa (Konno et al., IAS, -) 1985a,b). (-) Proposed method, (- - -) Grant-Manes, extended DA. Mole fraction refers to acetone. (e..)
(-e
teractions. The reason that the former fails for zeolites while the latter does not appears to be the use of the simple mixing rule for calculating the mixed affinity coefficient (eq 4)in the former model. In the proposed method, the affinity coefficient for each component is retained. As pointed out by Mehta and Danner (1985), the affinity coefficient contains factors that account for polar sorbent-sorbate interactions, in addition to the dispersive forces. The use of average Vovalues in the extended DA model also causes errors, especially for mixtures involving CHI, which has exceptionally low V , values. Figures 5 and 6 show the experimental data and model predictions for the adsorption of acetone-hexane and methanol-hexane on molecular sieve carbon (Konno et al., 1985a,b). Both mixtures show an azeotropic behavior which cannot be predicted by any of the four models. Moreover, the predictions by all four models are poor, with the model proposed here being the closest to the experimental data. The azeotropic behavior was attributed by Konno et al. (1985a,b) to the interactions between the adsorbate and pore walls (rather than the lateral interactions). The culprit for the unusual behavior of these two binary mixtures seems to be hexane, which has a size
02
04
06
08
10
Yl
Figure 6. Adsorption equilibria of methanol-hexane mixture on molecular sieve carbon 5A at 303 K and 4 kPa (Konno et al., IAS, (-.-) 1985a,b). (-) Proposed method, (- - -) Grant-Manes, extended DA. Mole fraction is for methanol. (.e.)
nearly as large as that of the average size of the pores (5-A mean size). In this case, the pore volume filling concept should be more appropriate than the concept of two-dimensional spreading pressure on which the IAS theory is based. Consequently, both the proposed model and the G-M model are better than IAS for these two mixtures. This is not so for the methanol-acetone mixture on the same molecular sieve carbon, where IAS gives as good predictions as the other two models. The relatively good predictive power of the proposed model for hexane mixtures could be attributed to the fact that the parameter P, which accounts for the interactions between the hexane molecule and the pore walls, is retained in the mixture calculations (eq 8). The reason for the large discrepancy between the IAS theory and the G-M model is not clear. Both methods are based on the assumption of an ideal adsorbed phase, i.e., Raoult's law, with the only differences being in the standard state (Sircar and Myers, 1973). Under certain special conditions, all four models encounter a common problem. When the partial pressure of the strongly adsorbed component approaches its saturation value, the mixture spreading pressure can exceed the saturation spreading pressure of the weak component, thus resulting in the failure of IAS. A similar problem arises in the G-M model when the mixture adsorbed volume is greater than the limiting adsorption volume for any component in the mixture. Sircar and Myers (1973) suggested the use of reduced spreading pressure to alleviate this problem. This modified IAS method has been used in this paper whenever the aforementioned problem arises. The Lewis relationship (used in the extended DA model) also encounters a difficulty when the total pressure exceeds the saturation pressure of one of the components. In the proposed model, if the partial pressure of component 2 approaches its saturation pressure and Vol < Voz,the calculated volume adsorbed for component 1, Vl in eq 8, can be negative. In this case, the problem is circumvented by using the average limiting volume (Vo)and recalculating the values of RT/@Eand n by fitting the single-gas data. This procedure is used for the 0,-N2-5A zeolite system, and the results are shown in Figure 3. The basic concept of available pore volume proposed in this work could also be applied to all other isotherms, e.g., the Toth equation. (This was suggested by an anonymous reviewer.) Thus, for a binary mixture,
Ind. Eng. Chem. Res., Vol. 27, No. 4, 1988 635
Greek Symbol
0 = affinity coefficient and explicit simple equations for V, and Vz can be derived. These equations reduce to the extended Langmuir model when Vol = Voz and tl = t2 = 1. These equations could be used for simplicity in computation. The physical meaning of available pore volume is, however, lost in this application, and it is only meaningful when applied to the potential pore filling theory. Inter-Species Lateral Interactions. For simplicity in computation (and application), the effects of the lateral interactions between different components may be accounted for (or empirically fitted) by the change of the adsorbate volume (AVJ upon mixing. Hence, eq 7 becomes Vi = (Vo1- V, - AVJA,
(124
V2 =
(12b)
(Vo2 -
Vi - AVz)Az
An example for the application of eq 1 2 is given for the C2H6(1)-CH4(2)-Nuxit carbon system. Here, by empirically using AV, = 0 and AV2 = 0.04 cm3/g, the average relative errors for prediction by the proposed method are reduced from 17.77% to 6.51% for the number of moles/gram adsorbed. It remains to be seen if AV, can be calculated from the thermodynamic properties of the mixtures. However, the simplicity and usefulness of eq 12 is suggested here. Acknowledgment This work was supported by the U.S. Army Chemical Research, Development and Engineering Center under Contract DAAA- 15-85C-0046. Nomenclature Eo = characteristic energy of the sorbent, cal/mol n = constant in the DA equation P = pressure, atm P, = saturation pressure, atm q = amount adsorbed, mol/g R = gas constant T = temperature, K T,= reduced temperature V = volume adsorbed, cm3/g Vo = limiting adsorption volume, cm3/g V , = adsorbate molar volume, cm3/mol X = mole fraction in the adsorbate phase Y = mole fraction in the gas phase
Subscripts
c = critical value i = component in the mixture nbp = normal boiling point s = saturation state Literature Cited Bering, B. P.; Seruinsks, V. V.; Surinova, S. I. Dokl. Akad. Nauk. SSSR 1963, 153, 1 2 9 Chen. W. N.: Yam. R. T. In Recent DeueloDments in SeDaration Science; Li, N. i., Calo, J. M., Eds.; CRC:. Cleveland, OH, 1986. Danner, R. P.; Wenzel, L. A. AIChE J. 1969, 15, 515. Dubinin, M. M. In Chemistry and Physics of Carbon;Walker, P. L., Ed.; Marcel Dekker: New York, 1966; Vol. 2. Dubinin, M. M. In Progress in Surface Science and Membrane Science; Cadenhead, D. A,, Danielli, J. F., Rosenberg, M. D., Eds.; Academic: New York, 1975; Vol. 9. Dubinin, M. M.; Stoeckli, H. F. J. Colloid Interface Sci. 1980, 75,34. Grant, R. J.; Manes, M. Ind. Eng. Chem. Fundam. 1966, 5, 490. Huber, U.; Stoeckli, F.; Houriet, J.-P. J. Colloid Interface Sci. 1978, 67, 195. Konno, M.; Shibata, K.; Saito,S. J. Chem. Eng. Jpn. 1985a, 18,394. Konno, M.; Shibata, K.; Saito, S. J. Chem. Eng. Jpn. 1985b, 18,398. Lewis, W. K.; Gilliland, E. R.; Chertow, B.; Cadogan, W. P. Ind. Eng. Chem. 1950,42, 1319. Marsh, H. Carbon 1987, 25, 49. McEnaney, B. Carbon 1987,25, 69. Mehta, S. D.; Danner, R.P. Ind. Eng. Chem. Fundam. 1985,24,325. Myers, A. L.; Prausnitz, J. M. AIChE J. 1965, 11, 121. Nakahara, T.; Hirata, M.; Ohmori, T. J. Chem. Eng. Data 1974,19, 310. Nakahara, T.; Hirata, M.; Ohmori, T. J. Chem. Eng. Data 1981,26, 161. Nakahara, T.; Hirata, M.; Ohmori, T. J. Chem. Eng. Data 1982,27, 317. Rand, B. J. Colloid Interface Sci. 1976, 56, 337. Reich, R.; Ziegler, W. T.; Rogers, K. A. Ind. Eng. Chem. Process Des. Deu. 1980, 19, 336. Rozwadowski, M.; Wojsz, R. Carbon 1981,19, 383. Sircar, S. Carbon 1987, 25, 39. Sircar, S.; Myers, A. L. Chem. Eng. Sci. 1973, 28, 489. Szepesy, L.; Illes, V. Acta Chim. Hung. 1963a, 35, 37. Szepesy, L.; Illes, V. Acta Chim. Hung. 1963b, 35, 53. Szepesy, L.; Illes, V. Acta Chim. Hung. 1963c, 35, 245. Toth, J. In Fundamentals of Adsorption; Myers, A. L., Belfort, G., Eds.; Engineering Foundation: New York, 1984; p 657. Yang, R. T. Gas Separation by Adsorption Processes;Butterworths: Boston, 1987. Received for review June 17, 1987 Revised manuscript received December 2, 1987 Accepted December 16, 1987
