3726
Ind. Eng. Chem. Res. 2005, 44, 3726-3732
Virial Excess Mixing Coefficient Corrections for the Adsorbed Solution Theory Nan Qi and M. Douglas LeVan* Department of Chemical Engineering, Vanderbilt University, Nashville, Tennessee 37235
The virial excess mixing coefficient (VEMC) model is proposed to describe multicomponent adsorption equilibria for nonideal systems. It expresses a nonideal adsorbed mixture by an ideal adsorbed solution with VEMC corrections. It reduces to the ideal adsorbed solution theory (IAST) when depicting an ideal adsorbed mixture. The model is thermodynamically sound and accurate in describing systems. Adsorbed-phase activity coefficients can be evaluated with the model. The VEMC model is used to investigate binary and ternary adsorption of hydrogen sulfide, carbon dioxide, and propane on H-mordenite and binary adsorption of n-hexane and water on BPL activated carbon. Introduction An accurate description of multicomponent adsorption equilibria is crucial for the design and modeling of adsorption separation processes. A common approach is to treat the adsorbed phase as a solution with adsorption equilibrium described in a way similar to that of traditional vapor-liquid equilibrium, with the standard state defined as a pure component at the temperature and spreading pressure of the mixture. This adsorbed solution theory (AST)1 has its thermodynamic basis in the Gibbs adsorption isotherm. Further approximations have been made to develop the ideal AST (IAST) by assuming that the adsorbed phase forms an ideal solution. In the IAST model, purecomponent isotherms can be used in any form, with the pure-component Henry’s law constants of thermodynamic significance. IAST is thermodynamically consistent and describes ideal systems or nearly ideal systems well. Efforts to correct the deviation of a real adsorbed solution from an ideal adsorbed solution have been made by introducing spreading-pressure-dependent activity coefficients2 and by constructing an empirical equation for the excess Gibbs free energy from which all equilibrium properties can be evaluated.3 Though originally developed for adsorption on a homogeneous surface, IAST has been extended to heterogeneous surfaces in two ways. One is by applying heterogeneous pure-component adsorption isotherms within the IAST framework. The other is the heterogeneous IAST (HIAST),4 in which the local adsorption is treated with homogeneous IAST and the overall adsorption is determined by integrating over all sites according to an adsorption energy distribution5 or a pore size distribution.6 These distributions can have large effects on the overall adsorption equilibrium. Also, it is known that HIAST is better than global IAST at describing multicomponent adsorption equilibria.7 Two-dimensional equations of state (EOSs; e.g., see Van Ness8) have been used to correlate9 or predict,10,11 with some empirical mixing rules, both pure-component * To whom correspondence should be addressed. Tel.: (615) 322-2441. Fax: (615) 343-7951. E-mail: m.douglas.levan@ vanderbilt.edu.
and multicomponent adsorption equilibria. EOSs are comparable with IAST for ideal or nearly ideal system descriptions and better than IAST for nonideal systems. The virial mixture coefficient (VMC) model reorganizes a two-dimensional virial EOS and attributes the overall adsorption amounts to contributions from pure components and their mixtures.12 Pure-component contributions can be described totally by pure-component isotherms. Mixture contributions can be calculated with only a few VMCs, which need to be determined from mixture equilibrium data to account for adsorbateadsorbate interactions. With a few parameters, the VMC model can give a good description of multicomponent adsorption for both ideal and nonideal systems. Several other approaches have been developed to treat nonideal adsorption equilibrium. In particular, the Langmuir isotherm has been applied in various ways by allowing for different monolayer capacities for adsorbing species,13 using dual sites,14,15 integrating over full energy distributions,16 or summing over all pores.17 The potential theory has also been applied to multicomponent systems using various assumptions.18-20 The vacancy solution theory21,22 has been extended to heterogeneous surfaces in a modified version that is thermodynamically consistent.23 Besides the classical methods of describing multicomponent adsorption equilibria, other approaches based on molecular theory, such as density functional theory and grand canonical Monte Carlo simulation, have also been developed. In this paper, we propose a new, powerful, and general model to describe multicomponent adsorption equilibria for both nonideal and ideal systems. The model attributes the mixture adsorption equilibrium to contributions from an ideal adsorbed solution and excess surface mixing through the two-dimensional virial EOS approach. The ideal adsorbed solution contribution is from all pure components with ideal surface mixing. The excess surface mixing describes the deviation from an ideal adsorbed solution and is expressed using virial excess mixing coefficients (VEMCs). The model is thermodynamically sound. It is applied to several sets of adsorption equilibrium data for binary and ternary mixtures. The model will be applied to additional sets of mixture adsorption equilibrium data that we measured and compared in performance to the VMC model in a following paper.24
10.1021/ie049110v CCC: $30.25 © 2005 American Chemical Society Published on Web 04/12/2005
Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005 3727
Theory Two-Dimensional Virial EOS. Multicomponent adsorption equilibria on homogeneous surfaces can be expressed by the two-dimensional virial EOS:10
πA RT
)n+
1 A
∑i ∑j
ninjBij +
1 A2
∑i ∑j ∑k
(1)
1 1 πA ) n1 + B11n12 + 2C111n13 + ... + RT A A 1 1 n2 + B22n22 + 2C222n23 + ... + A A 3 3 2 B n n + C n 2n + C n n 2 + ... (2) A 12 1 2 A2 112 1 2 A2 122 1 2 where the summations on the first, second, and third lines denote the contributions to πA/RT of an adsorbed solution from pure component 1, pure component 2, and the mixture, respectively. The surface mixing effect of an adsorbed solution can be considered as arising from an ideal surface mixing contribution and an excess surface mixing contribution. Adding the ideal surface mixing contribution and all pure-component contributions together gives an ideal adsorbed solution contribution to πA/RT of the adsorbed mixture. In other words, ideal surface mixing makes all pure components adsorbed on the surface an ideal adsorbed solution. Thus, the excess surface mixing contribution remains to describe the deviation of a real adsorbed solution from the ideal adsorbed solution. We split each VMC (e.g., Bij) following
(3)
E where Bid ij is an ideal mixing part and Bij is an excess mixing part. We will refer to them as the “virial ideal mixing coefficient” and the “virial excess mixing coefficient”, respectively. Rewriting eq 2, we have
πA 1 1 ) n1 + B11n12 + 2C111n13 + ... + RT A A 1 1 n2 + B22n22 + 2C222n23 + ... + A A 3 3 2 id B n n + Cid n 2n + Cid n n 2 + ... + A 12 1 2 A2 112 1 2 A2 122 1 2 3 3 2 E B n n + CE n 2n + CE n n 2 + ... (4) A 12 1 2 A2 112 1 2 A2 122 1 2 which can be represented by
|
|
|
or
|
|
πA πA πA E ) + RT RT IAS RT mixing
ninjnkCijk + ...
where π is the spreading pressure of the adsorbed mixture at temperature T, A is the specific surface area of the adsorbent, ni is the adsorbed-phase concentration of component i, and Bij, Cijk, ... are the virial coefficients. For a mobile adsorbed phase, virial coefficients describe the adsorbate-adsorbate molecular interactions, and the Henry’s law constants are determined by the adsorbate-adsorbent interactions.25 For the case of binary mixture adsorption equilibrium, rearranging eq 1 by separating pure-component terms from mixture terms gives12
E Bij ) Bid ij + Bij
|
πA πA πA πA id πA E ) + + + (5) RT RT pure1 RT pure2 RT mixing RT mixing
(6)
where πA/RT|pure1 and πA/RT|pure2 signify the contribuid signifies tions from the pure components, πA/RT|mixing the ideal surface mixing contribution as given by the E signifies the excess third line in eq 4, πA/RT|mixing surface mixing contribution as given by the fourth line in eq 4, and πA/RT|IAS signifies the total contribution from an ideal adsorbed solution, i.e., the summation of the first three lines of eq 4 or the first three terms on the right-hand side of eq 5. Thus, the final twodimensional virial EOS becomes
|
2 3 πA πA ) + BE n n + C E n 2n + RT RT IAS A 12 1 2 A2 112 1 2 3 E C n n 2 + ... (7) 2 122 1 2 A Multicomponent Adsorption Equilibrium. Classical thermodynamics is used to derive adsorption equilibria from the EOSs.8-10,12 An appropriate thermodynamic relation obtained from the Helmholtz free energy is given by10
ln
fi f0i
)
∫A∞
[(
)
∂(πA/RT) ∂ni
-1 T,A,nj
]
dA A
(8)
where fi is the fugacity of component i in a gas mixture and f0i ) ni/KiA is the reference fugacity of i with fi f f0i as A f ∞. Ki is Henry’s law constant. Substitution of eq 6 into eq 8 gives
ln
fi f0i
)
∫A∞
[(
)
∂(πA/RT|IAS) ∂ni
∫A∞
]
-1
T,A,nj
(
dA + A
)
E ∂(πA/RT|mixing ) ∂ni
T,A,nj
dA (9) A
The first integral in eq 9 is the evaluation of ln(fi/f0i ) for component i as in an ideal adsorbed solution. Upon cancelation of the reference state on both sides, eq 9 becomes
ln fi ) ln fIASi +
(
∫A∞
)
E ∂(πA/RT|mixing ) ∂ni
dA T,A,nj A
(10)
where fIASi is the fugacity of component i in a gas mixture that would be in equilibrium with an ideal adsorbed solution. Assuming ideal gas behavior, binary adsorption equilibria can be obtained by substituting the virial EOS E into eq 10 and integrating. expression for πA/RT|mixing This gives
2 3 E n1n2 + ln p1 ) ln pIAS1 + BE12n2 + 2C112 A A 3 E C122n22 + ... (11) 2A2
3728
Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005
2 3 E ln p2 ) ln pIAS2 + BE12n1 + 2C122 n1n2 + A A 3 E C112n12 + ... (12) 2A2 where pIASi, the pressure for component i in equilibrium with an ideal adsorbed solution, can be calculated from the IAST. Ternary adsorption equilibria can be derived similarly. The partial pressures for three components are analogous, with the pressure of component 1 given by
2 2 3 E n1n2 + ln p1 ) ln pIAS1 + BE12n2 + BE13n3 + 2C112 A A A 3 E 3 E 3 E C nn + C n 2+ C n 2+ 2 113 1 3 2 122 2 2 133 3 A 2A 2A 3 E C123n2n3 + ... (13) A2 Thus, as shown by eqs 11-13, multicomponent adsorption equilibria can be expressed by the IAST with VEMC corrections. The VEMC model is thermodynamically consistent. Pure-component isotherms are used to describe the adsorbate-adsorbent interactions. The IAST and VEMC corrections are used to describe the adsorbate-adsorbate interactions. Efficient and accurate pure-component isotherm equations, free to be in various forms, are used in the IAST part of the calculations. Mixture adsorption equilibrium data are used to determine the VEMCs. Compared with the IAST, the VEMC model can be used to describe a real adsorbed solution precisely. The VEMC model automatically reduces to the IAST for an ideal adsorbed mixture and to a pure-component isotherm if the system has only a single adsorbed component. Activity Coefficients. The equilibrium equation for multicomponent adsorption is given by1
pi )
γi xi p0i (π)
(constant T)
(14)
with the pure-component standard state p0i (π) defined as the pure-component pressure evaluated at the temperature and spreading pressure of the real adsorbed mixture (corresponding to πA/RT). xi and γi are respectively the mole fraction and activity coefficient of component i in the adsorbed phase. Negative virial coefficients will reduce an activity coefficient; also, activity coefficients of less than unity can be an indication of adsorbent heterogeneity.26 With accurately measured adsorption equilibrium data for pure components and mixtures, activity coefficients can be precisely evaluated using the VEMC model. The procedure for binary systems is as follows. (1) Describe the adsorption equilibrium for each pure component using an accurate isotherm equation. Isotherm equations for all components need not be the same, e.g., the Langmuir equation for component 1 and the Toth equation for component 2. (2) Fit the experimental multicomponent adsorption equilibrium data to the VEMC model (e.g., eqs 11 and 12) and determine sufficient VEMCs with which mixture adsorption equilibria can be precisely described. In this step, values of pIASi are determined from the experi-
mental loadings using the IAST model. πA/RT|IAS values for adsorbed mixtures are also calculated using the IAST model. (3) Calculate πA/RT for the real adsorbed solution from eq 7. (4) Calculate n0i (πA/RT), the loading of component i at the pure-component standard state with the spreading pressure of the real adsorbed solution (corresponding to πA/RT), by solving
πA ∫0n ni d ln pi ) RT 0 i
(15)
where ni and pi are related through the pure-component adsorption isotherm. (5) Calculate p0i (n0i ) from the pure-component adsorption isotherm. (6) Calculate xi from ni. (7) Calculate the activity coefficient γi from eq 14, with pi either measured experimentally or calculated using the VEMC model (e.g., eqs 11 and 12). Multicomponent Adsorption Equilibrium Data Analysis. The VEMC model is general and can be applied to multicomponent systems with any number of species. Pure-component adsorption isotherms are used to calculate the properties of the ideal adsorbed solution. Mixture equilibrium data are used to determine the VEMCs. The VEMCs of binary mixtures are retained and used in the ternary mixtures. The ternary data are only used to correlate those VEMCs describing the three-component interactions and so forth for additional components. Thus, the number of parameters needed to accurately describe the system is kept to a minimum. Four binary sets and one ternary set of multicomponent adsorption equilibrium data ranging from a nearly ideal system to highly nonideal systems are analyzed here with the VEMC model. The VEMCs were determined by minimizing the objective function e:
e)
exp 2 ∑ ∑(ln pcal m i - ln pm i) m i
(16)
where pexp m i is the experimentally measured partial pressure of component i with data index m. pcal m i is the corresponding model-calculated partial pressure from loadings measured in the experiments. The inner summation is over all species, and the outer summation is over all of the data set. The least-squares subroutine lmdif1 in MinPack was used to minimize the objective function e. The normalized error , used to evaluate model description efficiency, is
)
1 xe × 100 MI
(17)
where M and I are the total data point number and the total component number, respectively. Results and Discussion Coadsorption on H-Mordenite. Talu27 and Talu and Zwiebel2 measured adsorption equilibria for hydrogen sulfide, carbon dioxide, and propane on H-mordenite at 30 °C as pure components, binary mixtures, and a ternary mixture. Their data are often used by research-
Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005 3729
Figure 2. Ternary adsorption equilibria of H2S-CO2-C3H8 on H-mordenite at 30 °C using the VEMC model through C terms. Table 1. Parameters for the VEMC Model through C Termsa mixture
BE12
CE112
CE122
H2S-CO2 H2S-C3H8 CO2-C3H8 H2S-CO2-C3H8 n-hexane-water
-0.9129 -1.496 -1.762
0.4684 0.1484 0.7081
0.6576 -0.2828 0.1268
-0.07926
0.1089
-0.004029
CE123
0.4380
BEij and CEijk have units of m2/mol and m4/mol2, respectively. a
Table 2. Comparison of Model Accuracies IAST
Figure 1. Binary adsorption equilibria of H2S-CO2 on Hmordenite at 30 °C using the VEMC model through (a) the B term and (b) the C terms.
ers in multicomponent adsorption equilibrium modeling and are studied again in this paper. Pure-component adsorption equilibria were described using the Toth isotherm
p)
[
]
b θ -1 -t
1/t
(18)
where b and t are constants and θ ) n/ns, with ns being the saturation loading. The Toth isotherm parameters from Valenzuela and Myers28 for H2S, CO2, and C3H8 were used in this example. Comparisons of partial pressures measured experimentally and those calculated using the VEMC model are given in Figures 1 and 2. Binary mixture data were E and through analyzed through the B term using B12 E E E the C terms using B12, C112, and C122. The H2S-CO2 system is shown in Figure 1; similar behavior occurs for the H2S-C3H8 and CO2-C3H8 systems. The VEMC model describes these binary adsorption equilibrium data very well, with the virial series through the C terms having significantly greater accuracy than the series through the B term. The ternary H2S-CO2-C3H8 system is shown in Figure 2. This system is well
VEMC
mixture
no terms
B term
C terms
H2S-CO2 H2S-C3H8 CO2-C3H8 H2S-CO2-C3H8 n-hexane-water
9.0 64 39 43 13
5.9 5.6 8.4
2.9 2.2 4.2 2.8 6.6
13
described with three sets of binary virial parameters through the C terms and one ternary virial parameter, E . The VEMCs for these binary and ternary systems C123 are given in Table 1. The normalized errors, , were calculated for the VEMC and IAST models and were used for comparison purposes. The values of are given in Table 2. From the values, it can be seen that the VEMC model gives much more accurate descriptions for the three binary systems and one ternary system than the IAST model. The three binary systems are better described through C terms than through the B term by the VEMC model, as expected. Adsorbed-phase activity coefficients were calculated using the VEMC model for the three binary adsorption systems (H2S-CO2, H2S-C3H8, and CO2-C3H8) on H-mordenite at 30 °C. Comparisons of activity coefficients calculated from the model and from experimental paths27 are shown in Figure 3 in constant-pressure planes. The VEMC model is effective in describing the change in the activity coefficients with the vapor-phase mole fractions. Coadsorption of n-Hexane and Water on Activated Carbon. Adsorption equilibria of n-hexane and water on BPL activated carbon at 25 °C by Rudisill
3730
Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005
Figure 4. Binary adsorption equilibria of n-hexane-water on BPL activated carbon at 25 °C using the VEMC model through (a) the B term and (b) the C terms.
croporous adsorbent was used to describe the pure water adsorption isotherm
p)
()
HΨ Ψ exp 1 + kΨ ns
(19)
where H, k, and ns are constants. Ψ is given by
Ψ)
-1 + x1 + 4kζ 2k
(20)
nsn ns - n
(21)
with
ζ) Figure 3. Adsorbed-phase activity coefficients for (a) H2S-CO2, (b) H2S-C3H8, and (c) CO2-C3H8 on H-mordenite at 30 °C in constant-pressure planes.
etal.29 were also studied. The pure n-hexane adsorption isotherm was described using the Toth isotherm, eq 18, with b ) 0.069 81 (kPa)t, t ) 0.1923, and ns ) 4.408 mol/kg. The adsorption equilibium model by Talu and Meunier30 for self-associating molecules on a mi-
For the pure water adsorption equilibrium data of Rudisill et al.,29 we have H ) 1.431 kPa‚kg/mol, k ) 0.7585 kg/mol, and ns ) 25.13 mol/kg. The binary data were analyzed by the VEMC model through the B term and C terms. The results are shown in Figure 4. The model gives an excellent description for the system, with the series through the C terms being significantly more accurate. We also notice that n-hexane pressures are described very accurately over
Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005 3731
a wide range with the C terms. Comparatively, water pressures are not as well described by the model. VEMCs are given in Table 1. Accuracies of the various models are given in Table 2. The VEMC model involves the IAST calculation, where pure-component isotherms are integrated. IAST calculations are very sensitive to the pure-component adsorption isotherm equations used, and these purecomponent isotherms have a profound effect on the regression of parameters in the VEMC model. Of the same importance, accurately measured pure-component adsorption equilibrium data, especially at the low loadings, are crucial for the implementation of the VEMC model. Conclusions The VEMC model has been developed to describe multicomponent adsorption equilibria for nonideal systems. It expresses the real adsorbed solution as an ideal adsorbed solution with an excess surface mixing correction. The model describes highly nonideal systems with high accuracy using a small number of parameters. It also proposes a way to evaluate adsorbed-phase activity coefficients from multicomponent adsorption equilibrium measurements. The VEMC model is thermodynamically consistent and can be used in systems consisting of any number of components. In the VEMC approach, pure-component adsorption isotherms are utilized in IAST calculations. Accurate pure-component adsorption equilibrium data and isotherm models, especially at low pressures, are necessary for an accurate description of multicomponent systems. The VEMC model was applied to two systems: binary and ternary adsorption equilibria of H2S, CO2, and C3H8 on H-mordenite at 30 °C and binary adsorption of n-hexane and water on BPL activated carbon at 25 °C. The VEMC corrections can greatly improve the accuracy of the system description over the leading IAST model term. The VEMC model shows a significantly better performance when applied using the virial series through C terms rather than just the B term. Acknowledgment We gratefully acknowledge financial support from the National Aeronautics and Space Administration for this research (Grant NCC 2-1127). Notation am ) area change upon mixing, m2/mol A ) specific surface area of the adsorbent, m2/kg b ) constant in the Toth isotherm equation, (kPa)t Bij, Cijk, ... ) virial mixture coefficients E , ... ) virial excess mixing coefficients BEij , Cijk id id , ... ) virial ideal mixing coefficients Bij , Cijk e ) least-squares error f ) fugacity, Pa f 0 ) reference fugacity, Pa H ) Henry’s law constant in the Talu-Meunier association model, kPa‚kg/mol I ) total component number k ) constant in the Talu-Meunier association model, kg/ mol K ) Henry’s law constant in mixture adsorption m ) data index M ) total data point number n ) adsorption loading, mol/kg
ns ) saturated loading in the Talu-Meunier association model, mol/kg p ) partial pressure, kPa P ) total pressure of the gas mixture, kPa R ) gas constant, Pa‚m3/mol‚K t ) constant in the Toth isotherm equation T ) temperature, K x ) adsorbed-phase mole fraction y ) vapor-phase mole fraction Greek Letters γ ) adsorbed-phase activity coefficient ) normalized error ζ ) parameter in the Talu-Meunier association model, mol/kg θ ) fractional loading µ ) chemical potential π ) spreading pressure, Pa m Ψ ) spreading-pressure group for water, mol/kg Subscripts i, j, k ) component indices IAS ) ideal adsorbed solution mixing ) surface mixing pure ) pure component Superscripts cal ) calculated value from the model E ) excess property exp ) experimental value id ) property of the ideal adsorbed solution 0 ) property of standard state
Literature Cited (1) Myers, A. L.; Prausnitz, J. M. Thermodynamics of mixedgas adsorption. AIChE J. 1965, 11 (1), 121-127. (2) Talu, O.; Zwiebel, I. Multicomponent adsorption equilibria of nonideal mixtures. AIChE J. 1986, 32 (8), 1263-1276. (3) Siperstein, F. R.; Myers, A. L. Mixed-gas adsorption. AIChE J. 2001, 47 (5), 1141-1159. (4) Valenzuela, D. P.; Myers, A. L.; Talu, O.; Zwiebel, I. Adsorption of gas mixtures: Effect of energetic heterogeneity. AIChE J. 1988, 34 (3), 397-402. (5) Moon, H.; Tien, C. Adsorption of gas mixtures on adsorbents with heterogeneous surfaces. Chem. Eng. Sci. 1988, 43 (11), 29672980. (6) Qiao, S.; Wang, K.; Hu, X. Using local IAST with micropore size distribution to predict multicomponent adsorption equilibrium of gases in activated carbon. Langmuir 2000, 16 (3), 12921298. (7) Hu, X.; Do, D. D. Comparing various multicomponent adsorption equilibrium models. AIChE J. 1995, 41 (6), 1585-1592. (8) Van Ness, H. C. Adsorption of gases on solids. Ind. Eng. Chem. Fundam. 1969, 8 (3), 464-473. (9) Haydel, J. J.; Kobayashi, R. Adsorption equilibrium in the methane-propane-silica gel system at high pressures. Ind. Eng. Chem. Fundam. 1967, 6 (4), 546-554. (10) Taqvi, S. M.; LeVan, M. D. Virial description of twocomponent adsorption on homogeneous and heterogeneous surfaces. Ind. Eng. Chem. Res. 1997, 36 (6), 2197-2206. (11) Zhou, C.; Hall, F.; Gasem, K. A. M.; Robinson, R. L., Jr. Predicting gas adsorption using two-dimensional equations of state. Ind. Eng. Chem. Res. 1994, 33 (5), 1280-1289. (12) Appel, W. S.; LeVan, M. D.; Finn, J. E. Nonideal adsorption equilibria described by pure component isotherms and virial mixture coefficients. Ind. Eng. Chem. Res. 1998, 37 (12), 47744782. (13) LeVan, M. D.; Vermeulen, T. Binary Langmuir and Freundlich isotherms for ideal adsorbed solutions. J. Phys. Chem. 1981, 85 (22), 3247-3250. (14) Mathias, P. M.; Kumar, R.; Moyer, J. D., Jr.; Schork, J. M.; Srinivasan, S. R.; Auvil, S. R.; Talu, O. Correlation of multicomponent gas adsorption by the dual-site Langmuir model.
3732
Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005
Application to nitrogen/oxygen adsorption on 5A-zeolite. Ind. Eng. Chem. Res. 1996, 35 (7), 2477-2483. (15) Bai, R.; Yang, R. T. A thermodynamically consistent Langmuir model for mixed gas adsorption. J. Colloid Interface Sci. 2001, 239 (2), 296-302. (16) Kapoor, A.; Ritter, J. A.; Yang, R. T. An extended Langmuir model for adsorption of gas mixtures on heterogeneous surfaces. Langmuir 1990, 6 (3), 660-664. (17) Nguyen, C.; Do, D. D. Multicomponent supercritical adsorption in microporous activated carbon materials. Langmuir 2001, 17 (5), 1552-1557. (18) Doong, S. J.; Yang, R. T. Adsorption of mixtures of water vapor and hydrocarbons by activated carbon beds: Thermodynamic model for adsorption equilibrium and adsorber dynamics. AIChE Symp. Ser. 1988, 83 (259), 87-97. (19) Dobruskin, V. K. Physical adsorption of a multicomponent mixture in micropores. Langmuir 1996, 12 (4), 987-993. (20) Sundaram, N. Equation for adsorption from gas mixtures. Langmuir 1995, 11 (8), 3223-3234. (21) Suwanayuen, S.; Danner, R. P. Vacancy solution theory from gas mixtures. AIChE J. 1980, 26 (1), 76-83. (22) Talu, O.; Myers, A. L. Letter to the Editor. AIChE J. 1988, 34 (11), 1931-1932. (23) Ding, L. P.; Bhatia, S. K. Vacancy solution theory for binary adsorption equilibria in heterogeneous carbon. AIChE J. 2002, 48 (9), 1938-1956.
(24) Qi, N.; LeVan, M. D. Coadsorption of organic compounds and water vapor on BPL activated carbon. 5. Methyl ethyl ketone, methyl isobutyl ketone, toluene, and modeling. Ind. Eng. Chem. Res. 2005, 3733-3741. (25) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: Oxford, U.K., 1974. (26) Myers, A. L. Activity Coefficients of Mixtures Adsorbed on Heterogeneous Surfaces. AIChE J. 1983, 29 (4), 691-693. (27) Talu, O. Thermodynamics of multicomponent gas adsorption equilibria of nonideal mixtures. Ph.D. Dissertation, Arizona State University, Tempe, AZ, 1984. (28) Valenzuela, D. P.; Myers, A. L. Adsorption Equilibrium Data Handbook; Prentice Hall: Englewood Cliffs, NJ, 1989. (29) Rudisill, E. N.; Hacskaylo, J. J.; LeVan, M. D. Coadsorption of hydrocarbons and water on BPL activated carbon. Ind. Eng. Chem. Res. 1992, 31 (4), 1122-1130. (30) Talu, O.; Meunier, F. Adsorption of associating molecules in micropores and application to water on carbon. AIChE J. 1996, 42 (3), 809-819.
Received for review September 13, 2004 Revised manuscript received February 25, 2005 Accepted March 8, 2005 IE049110V
