Improved Multisite Langmuir Model for Mixture Adsorption Using

Prediction of mixed gas adsorption by the multisite Langmuir (MSL) theory is significantly improved by incorporating the multiregion adsorption (MRA) ...
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Langmuir 2003, 19, 2776-2781

Improved Multisite Langmuir Model for Mixture Adsorption Using Multiregion Adsorption Theory Runsheng Bai I NET, Tsinghua University, Beijing 102201, People’s Republic of China

Jingguang Deng Department of Biological Science and Biotechnology, Tsinghua University, Beijing 100084, People’s Republic of China

Ralph T. Yang* Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2136 Received October 10, 2002. In Final Form: January 13, 2003 Prediction of mixed gas adsorption by the multisite Langmuir (MSL) theory is significantly improved by incorporating the multiregion adsorption (MRA) theory. The new model, called the multiregion multisite Langmuir (MR-MSL) model, is thermodynamically consistent. MR-MSL uses the same pure-component isotherm parameters as MSL does, and no new parameters are introduced. Eight binary systems and one ternary system are used to test the MR-MSL model. For systems containing like components (such as mixtures of hydrocarbons and that of O2 + N2), MR-MSL yielded only minor improvements over MSL. This is true regardless of the magnitude of the differences in the pure-component MSL parameters among different components. For highly nonideal mixtures, MR-MSL results in very substantial improvements. It is also capable of predicting the azeotropic behavior of the adsorption of CO2 + C3H8 on H-mordenite and the correct azeotropic compositions. For such systems, it is also shown that surface heterogeneity needs to be considered.

Introduction Prediction of the equilibrium adsorption of mixtures from pure-component isotherms is important both in theory of adsorption and for practical applications. Development and improvement of the mixture adsorption theories and models have attracted much attention in the past. The extended Langmuir (EL) model1 and the ideal adsorbed solution (IAS) theory of Myers and Prausnitz2 are two typical such theories/models. They can be used to predict mixture adsorption equilibrium from corresponding pure-component isotherms. However, they are not suitable for mixture systems that show nonideal behaviors. When the energetic heterogeneity of the adsorbent surface and/or interactions between adsorbed molecules are taken into account, improvements can be achieved to some extent. These theories include the real adsorbed solution (RAS) theory (e.g., that of Costa et al.3), the vacancy solution (VS) theory,4,5 the heterogeneous extended Langmuir (HEL) model,6 and the heterogeneous ideal adsorbed solution (HIAS) theory.7 For its simplicity, the extended Langmuir model remains the most widely used isotherm for mixed gas * To whom correspondence should be addressed. E-mail address: [email protected]. (1) Markham, E. C.; Benton, A. F. J. Am. Chem. Soc. 1931, 53, 497. (2) Myers, A. L.; Prausnitz, J. M. AIChE J. 1965, 11, 121. (3) Costa, E.; Sotelo, J. L.; Calleja, G.; Marron, C. AIChE J. 1981, 27, 5. (4) Suwanayuen, S.; Danner, R. P. AIChE J. 1980, 26, 76. (5) Cochran, T. W.; Kabel, R. L.; Danner, R. P. AIChE J. 1985, 31, 268. (6) Kapoor, A.; Ritter, J. A.; Yang, R. T. Langmuir 1990, 6, 660. (7) Valenzuela, D. P.; Myers, A. L.; Talu, O.; Zwiebel, I. AIChE J. 1988, 34, 397.

adsorption in the design and modeling of adsorber and adsorption processes.8 The drawback of the extended Langmuir model is, as with the other models above, that it does not generally describe the experimental data very well.8 Thus, efforts have been made to modify and improve the model.9-11 The multisite Langmuir (MSL) model is one of them. Instead of assuming that one molecule occupies only one adsorption site, MSL assumes that a molecule, when adsorbed, can occupy more than one site on a homogeneous surface. Nitta et al.12 developed the MSL model using statistical thermodynamics and discussed its applications for both pure-component and mixture adsorption equilibria. A similar expression was derived based on the vacancy solution theory, by Honig and Mueller.13 A multisite model, in the form where interactions between adsorbed molecules are neglected, was suggested earlier by Henry14 based on kinetic theory. MSL uses one more coefficient than the Langmuir equation and is a three-parameter model. It fits experimental data better for pure-component adsorption and consequently also predicts results more accurately for mixture adsorption from the corresponding pure-gas (8) Yang, R. T. Gas Separation by Adsorption Processes; Butterworth: Boston, 1987. (9) Do, D. D. Adsorption Analysis: Equilibrium and Kinetics; Imperial College Press: London, 1998. (10) Myers, A. L. AIChE J. 1983, 29, 691. (11) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: San Diego, 1992. (12) Nitta, T.; Shigetomi, T.; Kuro-Oka, M.; Katayama, T. J. Chem. Eng. Jpn. 1984, 17, 39. (13) Honig, J. M.; Mueller, C. R. J. Phys. Chem. 1962, 66, 1305. (14) Henry, D. C. Philos. Mag. 1922, S6 44, 689.

10.1021/la020838v CCC: $25.00 © 2003 American Chemical Society Published on Web 02/20/2003

Improved Multisite Langmuir Model

adsorption isotherms. This model, however, was not widely adopted, perhaps because it is not explicit and hence it needed iteration and numerical calculation to get the solution for fractional coverage from a known pressure. This is no longer a problem today because of the availability of computing capabilities. More recently, the MSL model has gained attention and application. It has been shown by Sircar15 that thermodynamic consistency can be satisfied for the MSL model by following a site balance constraint. The model has been applied successfully to describe a number of adsorption systems.16-18 The remaining problem in using the MSL model is in selecting proper isotherm parameters when they are forced to obey the thermodynamic consistency constraint. In some cases, for example, for nonideal systems, where isotherm parameters for different components are widely different, large fitting errors for pure-component isotherms would occur. This may seriously influence the prediction accuracy for mixture adsorption. In the last two years, Bai and Yang19,20 have proposed a multiregion adsorption (MRA) theory which was used to improve a number of mixture adsorption models while maintaining thermodynamic consistence for each model. The basis of MRA is the heterogeneity of adsorption surfaces for all real adsorbents. Activated carbons have different surface oxygen and acid functional groups.21 Zeolite surfaces comprise negatively charged oxide anions and dispersed cations that are located on certain isolated sites.22,23 For silica, adsorption on the surfaces is dominated by bonding with hydroxyl groups, and there are different types of such groups.24,25 Different molecules may adsorb on different types of sites. For mixture adsorption, therefore, surfaces can be divided into different regions/ sites. The regions may be considered patches where each patch contains a number of similar adsorption sites. Then the mixture adsorption equilibrium model is used in each region. The MRA theory has been successfully applied to the EL model19 and the HEL model.20 Compared to the respective original models, the modified models showed similar prediction results for ideal mixtures or mixtures containing like components. For nonideal mixtures, however, the improvements made by the MRA theory were proven to be very significant. In this work, the MRA theory is applied to the MSL model. The modification of the MSL model does not introduce any new parameters, nor does it significantly increase the amount of computation. The modified MSL model is thermodynamically consistent. It is expected that this modification will help the MSL model to be more widely and more effectively used for predicting mixture adsorption equilibria. (15) Sircar, S. AIChE J. 1995, 41, 1135. (16) Buss, E.; Heuchel, M. J. Chem. Soc., Faraday Trans. 1997, 93, 1621. (17) Silva, J. A.; Rodrigues, A. E. Ind. Eng. Chem. Res. 1999, 38, 2434. (18) Sundaram, N.; Yang, R. T. Chem. Eng. Sci. 2000, 55, 1747. (19) Bai, R.; Yang, R. T. J. Colloid Interface Sci. 2001, 239, 296. (20) Bai, R.; Yang, R. T. J. Colloid Interface Sci. 2002, 253, 16. (21) Puri, B. R. Surface Complexes on Carbon. In Chemistry and Physics of Carbon; Walker, P. L., Ed.; Marcel Dekker: New York, 1970; Vol. 6. (22) Barrer, R. M. Zeolites and Clay Minerals; Academic Press: New York, 1978. (23) Yang, R. T. Nanostructured Adsorbents. In Nanostructured Materials; Advances in Chemical Engineering, Vol. 27; Academic Press: San Diego, 2001. (24) Iler, R. K. The Chemistry of Silica; Wiley: New York, 1979. (25) Vansant, E. F.; Van Der Voort, P.; Vrancken, K. C. Characterization and Chemical Modification of the Silica Surface; Elsevier: Amsterdam, 1995.

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Theory Assuming that a molecule occupies more than one site on a homogeneous surface and that there are no interactions between the adsorbate molecules, the MSL model takes the following forms:12

For pure component: θi (1 - θi)ni

) KiPi

i ) 1,2,...,N

(1)

For mixture: θi (1 -

∑θi)n

i

) KiPi

i ) 1,2,...,N

(2)

where θi ) qi/qsi is the fractional coverage of the surface for component i at pressure Pi, qsi is the saturated adsorption capacity of the component, ni is the number of sites or equivalent spaces occupied by the component, and K is the adsorption equilibrium constant. The parameters qsi and ni are assumed to be independent of temperature. Equation 2 is thermodynamically consistent only when the following relation is obeyed:

qsi ni ) q/i ) constant

i ) 1,2,...,N

(3)

Here q/i is the maximum number of adsorption sites that component i can adsorb on the surface. The above equation stipulates that all adsorbed components in the mixture have the same q*, not influenced by the nature of the adsorbate. Equation 3 is thought to be easy to satisfy.15 Generally MSL is suitable for ideal mixtures or mixtures containing like components where the components have similar properties. For nonideal or highly nonideal systems, different molecules adsorb very differently; so do their MSL parameters. Thus, fitting errors for pure-component isotherms would increase when different components are forced to obey eq 3. Because the accuracy for mixture prediction drops quickly with the increase of the fitting errors, not all systems are suitable to be forced to meet eq 3. In terms of the MRA theory, for mixture adsorption, the surface can be divided into several types of regions. On each type of region, the number of sites is the same for different components. The number of types of regions is determined by the difference in the maximum number of sites that adsorb a component. For a binary mixture, two types of regions are assumed. One type of region can be occupied by both components, whereas the other type can be occupied only by the component with the larger q*. The other component is excluded from this type of region due to size exclusion or lack of competition. Then MSL is used for each type of region. Therefore, the MSL model modified by MRA theory can obey thermodynamic consistency without forcing its coefficients to satisfy the thermodynamic constraint. This is advantageous in improving the prediction results for mixture adsorption, especially for nonideal and highly nonideal systems, as will be shown shortly. For a binary mixture containing components A and B, component A has the larger q* (total sites upon saturation) than component B. We now apply the MRA theory to the MSL model: On Region 1. Both components A and B adsorb on this region where the maximum number of adsorption sites is

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qsBnB. Thus, the saturated adsorption capacities on this region are

qsA,1 )

qsBnB/nA

qsB,1 )

qsB

(4) (5)

Equation 2 (i.e., the MSL model) is then solved to obtain the amount adsorbed for each component at their respective partial pressures. On Region 2. Only component A adsorbs on this region. The saturated adsorption amounts are

qsA,2 ) qsA - qsA,1 qsB,2 ) 0

PB ) 0

(6) (7)

Equation 2 is then used to get the adsorption capacity of component A on the region. As in the previous work,19,20 we assume that the purecomponent parameters for component A are the same on both regions. For more regions, this assumption remains. For a mixture with N components, we rank their maximum numbers of adsorption sites (which are different for all components) from small to large and obtain N adsorption regions. On region j, the saturated adsorption amount of component i is s s ) (qsj nj - qj-1 nj-1)/ni qi,j

Pi ) 0

i ) 1,2,...,N j ) 1,2,...,N (8)

when qsi ni < qsj nj

(8a)

Then the MSL model, eq 2, is used to calculate the adsorbed amount for each component on region j. The total adsorbed amount of each component is the sum of its amounts adsorbed on all regions. Equations 2, 8, and 8a constitute the multiregion multisite Langmuir (MR-MSL) model. MR-MSL is thermodynamically consistent, and it does not require any additional parameters. Like the original MSL model, however, the MR-MSL model cannot be solved directly and iteration is required. Convergence is fast using any standard numerical method such as the Newton method. Results and Discussion Bai and Yang19,20 studied eight binary systems and one ternary system in their previous work. These systems ranged from similar-component systems (CH4 and C2H6) to highly nonideal systems (e.g., CO2 and C3H8). Adsorbents included activated carbon, silica gel, and zeolite molecule sieves. These systems are representative of gassolid systems to some certain extent. So they are also used in this work to test the MR-MSL model. In addition, the O2/N2 binary mixture, which is important for air separation, is also included in this work. The pure-component MSL isotherm parameters are given in Table 1. They are fitted in the full ranges of experimental data. H2S and CO2 on H-mordenite have the larger values of n, but the sizes of these molecules are not larger than that of C3H8. This problem will be addressed in a later part. The maximum numbers of adsorption sites for each pure component are also included in this table. Comparisons of the predictions by the original MSL model and the MR-MSL model are given in Table 2 for similar-component mixtures and in Table 3 for nonideal systems. The comparisons are made based on the average

Table 1. Best Fits for MSL Isotherm Parameters for Pure-Component Adsorption compd C2H6a CH4 C3H8b C2H4 O2c N2 CO O2 N2 CO O2 N2 CO CO2d C2H4 C3H8e H2S CO2

adsorbent

TK

n

act carbon

293.15 3.686 3.277 silica gel 313.15 3.312 2.958 10X zeolite 273.15 1.862 2.925 3.550 227.59 1.862 2.925 3.550 172.04 1.862 2.925 3.550 13X zeolite 298.15 3.584 4.036 H-mordenite 303.15 4.591 6.548 6.782

K 1/kPa 0.034 98 0.001 821 0.005 972 0.003 496 0.000 819 8 0.003 341 0.018 23 0.001 393 0.008 671 0.038 26 0.008 455 0.050 07 0.348 9 0.227 2 0.415 9 0.846 9 1.563 0.100 7

qs mol/kg

D %

7.242 4.8 6.764 2.0 4.175 5.9 3.849 6.4 2.103 7.2 2.571 4.1 2.771 9.8 3.662 5.3 3.827 5.7 4.468 12.2 5.909 7.0 6.071 6.0 6.258 4.0 6.139 7.0 4.332 3.3 1.702 5.1 4.524 1.7 5.261 7.7

q* mol/kg 26.7 22.2 13.8 11.4 3.9 7.5 9.8 6.8 11.2 15.9 11.0 17.8 22.2 22.0 17.5 7.8 29.6 35.7

a Szepesy and Illes (1963) (ref 26). b Lewis et al. (1950) (ref 27). Nolan et al. (1981) (ref 28). d Hyun and Danner (1982) (ref 29). e Talu and Zwiebel (1986) (ref 30). c

Table 2. Average Deviation from Experiment of MSL and MR-MSL Predictions of the Amount Adsorbed of Each Component for Like Component Systems D% system

adsorbent

TK

C2H6 + CH4 C3H8 + C2H4 O2 + N2 O2 + N2 O2 + N2 O2 + CO O2 + CO O2 + CO overall avg

act carbon silica gel 10X zeolite 10X zeolite 10X zeolite 10X zeolite 10X zeolite 10X zeolite

293.15 313.15 273.15 227.59 172.04 273.15 227.59 172.04

MSL MR-MSL no. points 4.8 5.3 10.6 13.6 27.2 17.3 11.5 12.5 13.3

4.8 5.7 9.1 11.3 20.5 15.0 9.6 7.6 10.6

14 12 22 26 18 24 28 32

Table 3. Average Deviation from Experiment of MSL and MR-MSL Predictions of the Amount Adsorbed of Each Component for Nonideal Systems D% system

adsorbent

TK

CO2 + C2H4 13X zeolite 298.15 H2S + C3H8 H-mordenite 303.15 CO2 + C3H8 H-mordenite 303.15 overall av

MSL MR-MSL no. points 23.5 44.7 49.6 41.1

16.0 17.6 15.6 16.4

12 16 18

percent deviation from the experimental data, for the amount adsorbed of each component:

D)

1

qcal,i - qexp,i | × 100 qexp,i

Nt

∑

Nti)1

|

(9)

where Nt is the number of data points and qi is the amount adsorbed of component i in a mixture. Like-Component Mixtures. The mixtures of C2H6 + CH4 and C3H8 + C2H4 are nearly ideal. The data of Szepesy and Illes26 on adsorption of C2H6 + CH4 on activated carbon at 293.15 K and that of Lewis et al.27 on adsorption of C3H8 + C2H4 on silica gel at 313.15 K have been widely used for testing theories. For each mixture, the maximum numbers of adsorption sites (q*) of the components are not substantially different (see Table 1). The predictions from MSL and MR-MSL, as seen in Table 2, are quite (26) Szepesy, L.; Illes, V. Acta Chim. Hung. 1963, 35, 245. (27) Lewis, W. K.; Gilliland, E. R.; Chertow, B.; Bareis, D. J. Am. Chem. Soc. 1950, 72, 1160.

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Figure 1. Comparison of experiment with predictions by MSL and MR-MSL for CO2 + C2H4 adsorption on molecular sieve 13X at 298.15 K and 137.8 kPa.

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Figure 2. Comparison of experiment with predictions by MSL and MR-MSL for H2S + C3H8 adsorption on H-mordenite at 303.15 K and 8.11 kPa. (See the text for a description of normal and larger n.)

similar. Both agree with experimental data fairly well, and the deviations are relatively small. The numbers of data sets for calculating the percent deviations are also listed in Table 2. O2 + N2 and O2 + CO systems are mixtures of similar components. The data of Nolan et al.28 on adsorption of O2 + CO on molecular sieve 10X at three different temperatures are used. For these systems, the maximum numbers of adsorption sites of different components differ significantly at each temperature. For example, q/CO is more than double that of O2. Despite these large differences, the MR-MSL predictions are not very different from those of MSL for both systems. However, the MR-MSL predictions are slightly better than those of MSL at all three temperatures. The average percent deviation from experimental data, averaged over the many sets of data points, shows that for these eight like-component systems, the predictions by both MSL and MR-MSL are fairly good, while those by MR-MSL are consistently better. Nonideal Binary Systems with Azeotropic Behavior. The adsorption of CO2 + C2H4 on 13X zeolite molecular sieve at 298.15 K is highly nonideal and exhibits azeotropic behavior.29 The difference in their maximum numbers of adsorption sites is obvious but not large (Table 1). As seen in Figure 1, predictions of MR-MSL are better than those of MSL in the x-y diagram, although both fail to predict the azeotropic behavior. For predictions on the total amounts adsorbed, MR-MSL is not satisfactory; MSL is actually better. The total amounts are, however, misleading. The amounts adsorbed of each component are meaningful. The average deviations from experimental data for the amounts adsorbed of each component are shown in Table 3. The MR-MSL results are clearly better than those of MSL, because the deviations are reduced by about 1/3. The amounts adsorbed for each component can be obtained readily from Figures 1-3.

The binary systems of H2S + C3H8 and CO2 + C3H8 on H-mordenite at 303.15 K are also highly nonideal.30 For each system, the maximum numbers of sites are significantly different for the individual components (Table 1). Figures 2 and 3 show the comparisons of predictions between MSL and MR-MSL. As seen, the MR-MSL model shows excellent results. For H2S + C3H8, MR-MSL yields

(28) Nolan, J. T.; McKeehan, T. W.; Danner, R. P. J. Chem. Eng. Data 1981, 26, 112.

(29) Hyun, S. H.; Danner, R. P. J. Chem. Eng. Data 1982, 27, 196. (30) Talu, O.; Zwiebel, I. AIChE J. 1986, 32, 1263.

Figure 3. Comparison of experiment with predictions by MSL and MR-MSL for CO2 + C3H8 adsorption on H-mordenite at 303.15 K and 40.86 kPa. (See the text for a description of normal and larger n.)

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Table 4. Normal MSL Parameters for H2S and CO2a component

n

K 1/kPa

qs mol/kg

D%

H2S CO2

3.622 3.767

1.461 0.08688

3.275 4.036

2.8 9.0

a

Such that their n values are less than that of C3H8.

Table 5. Average Deviation of Amount Adsorbed for Each Component by MSL and MR-MSL Based on Normal Parameters for H2S and CO2 D% system

MSL

MR-MSL

no. points

H2S + C3H8 CO2 + C3H8 CO2 + H2S + C3H8

38.5 42.1 37.6

31.2 20.3 33.2

16 18 30

major improvements over MSL in both the x-y diagram and the total adsorbed amounts. It reduces the average deviation from experiment by over 60% (see Table 3). As with other models, however, it fails to predict the azeotrope.7,19,20,31 For CO2 + C3H8, the improvements of MR-MSL over MSL are still greater. The average errors are reduced by more than 2/3. For this system, the MRMSL model predicts not only an azeotrope but also the correct azeotropic composition. The adsorption of binary H2S + C3H8 and CO2 + C3H8 mixtures on H-mordenite has been widely used for testing various models. Here we also use these systems to discuss two problems. One is the choice of the parameter n (in MSL), and the other is the effect of the constraint (by forcing the MSL parameters to be thermodynamically consistent) on the prediction results. The parameter n is the number of adsorption sites occupied by one molecule, which is equivalent to a space size on a homogeneous surface. So a larger molecule should have a larger n on the same surface. Thus, the values of n for H2S and CO2 should be less than that for C3H8. However, those used in this work are not so. In Table 1, the values of n for H2S and CO2 are larger than that of C3H8. Larger values of parameter n for H2S and CO2 (than that for C3H8) are obtained by minimizing the fitting errors for the pure-component isotherm data. When the fitting errors are relaxed, however, the values can be made “normal”, that is, smaller than that of C3H8. The values of normal n and the prediction results based on them are listed in Table 4 and Table 5. Compared to “larger n” in Table 1, normal n leads to larger fitting errors for the two pure components. As a result, the MR-MSL predictions for mixture adsorption based on normal n are significantly less satisfactory than those based on larger n, although the improvements of MR-MSL over MSL are still obvious in this case. Furthermore, heterogeneity of the adsorbent surface sites may also affect the value for n. Nitta et al.12 found that for carbon dioxide adsorption on molecular sieve 5A, the experimental adsorption capacities in the low-pressure range were higher than the calculated ones. So they thought that there were an appreciable fraction of strong adsorption sites on the surface. In our previous work, Bai and Yang19,20 discussed surface heterogeneity in detail and showed that it was heterogeneity that resulted in the thermodynamic inconsistency that existed for many models. H2S has a permanent dipole (dipole moment ) 0.97 D, compared to 1.85 for H2O). CO2 has a strong quadrupole (quadrupole moment ) -4.3 esu). C3H8 is nonpolar. (31) Hu, X.; Do, D. D. AIChE J. 1995, 41, 1585.

Table 6. Average Deviation for Amounts Adsorbed of Each Component and the Total Amount for the Ternary Mixture CO2 + H2S + C3H8 on H-Mordenite at 303.15 K model

D %, each component

DT %, total amount

MSL MR-MSL

41.4 19.1

32.4 6.4

Consequently, H2S and CO2 adsorb more strongly than C3H8 on zeolites due to strong electric field-dipole and field gradient-quadrupole interactions. Also due to these interactions, H2S and CO2 find more adsorption sites on zeolites than does C3H8. These additional interactions and additional adsorption sites result in larger values of both q* (saturated amount adsorbed) and n (sites), although more so for q* (Table 1). Force-fitting pure-component isotherms to lower the n values for H2S and CO2 leads to decreased mixture predictions, as shown above. Compared to models using normal parameters, the original MSL model based on larger n generates greater deviations from experimental data, while MR-MSL using the same larger coefficients yields better results (see Figures 2 and 3). This demonstrates that accurate predictions for mixture adsorption require a better physical understanding of adsorption behavior, for example, the adsorbent structure and adsorbate properties. Nonetheless, MR-MSL yields substantially improved mixture results than the original MSL for all values of parameters, both normal and larger. The MSL model can be forced to be thermodynamically consistent. We have set the maximum numbers of sites to be 10, 20, and 30 (in eq 3) and then optimize other coefficients for each case. The results are poor for all three cases. The improvements on average deviations are very small. Ternary System. The ternary system of H2S + CO2 + C3H8 on H-mordenite is a highly nonideal system (as discussed by Talu and Zwiebel and others).30 For the ternary mixture, the adsorption surface is divided into three regions for the MR-MSL model. All three components are calculated for the first region. Components H2S and CO2 are included for the second region. Only component CO2, which has the maximum number of adsorption sites, adsorbs on the third region. The calculated results of MSL and MR-MSL are listed in Table 6. As seen, the improvement of MR-MSL over MSL is significant, especially on the total adsorbed amount, with an average deviation of only 6.4%. The average deviation on the total adsorbed amount, DT, is defined as

DT )

1

Qcal,i - Qexp,i | × 100 Qexp,i

NT

∑

NTi)1

|

(10)

where NT is the number of data points and Q is the total amount adsorbed. Compared to MSL, MR-MSL reduces the average deviation in the amount adsorbed for each component by more than 1/2 and that on the total amount adsorbed by as high as 80%. The average deviations in the adsorbed amounts for each component based on the normal values of n are compared in Table 5. The improvements using the MRA theory are very limited, much less than those based on larger n. Comparison of Multiregion Theory Applied to Various Models. As mentioned, the MRA theory has been applied previously to the EL and HEL models.19,20 Table 7 compares the overall average deviations on all systems studied. The results from EL and HEL as well as their modified models using MRA theory are included. The N2

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Table 7. Overall Average Deviations from Experiment Predicted by Different Models and Their Multiregion (MR) Counterparts for Eight Binary Systems and One Ternary System Listed in Table 1 D% model ELa HELb MSLc

original

MR theory

31.1 20.9 23.6

19.0 13.3 12.6

a Extended Langmuir model and MR-EL by Bai and Yang (ref 19). b Heterogeneous extended Langmuir model and MR-HEL by Bai and Yang (ref 20). c O2 + N2 systems used in this work are excluded in order to compare the models using the same data sources.

+ O2 system used in this work is excluded in order to compare only predictions using the same sources of data. The comparison is clear. All models improved by the MRA theory are substantially better than the corresponding original models. The two three-parameter models (i.e., HEL and MSL) both have less average errors than the two-parameter EL model, and this is true for both the original and the MR-modified models. As three-parameter models, MSL and HEL are similar to each other, and so are MR-MSL and MR-HEL. Conclusion Applying the MRA theory to the MSL model yields the MR-MSL model. The MR-MSL is thermodynamically consistent. It uses the same three parameters as MSL for each component of the mixture. No additional parameters are introduced. With the best fitted pure-component parameters, the new model shows excellent improvements for predicting mixture adsorption. For mixtures of hydrocarbons, O2 + CO2, and O2 + N2 mixtures, the components are similar. Whether the differences in their maximum numbers of adsorption sites (eq 3) are large or small, the predicted results by MSL and MR-MSL are close, and the prediction errors are relatively small in most cases while MS-MSL makes only small improvements. For highly nonideal systems such as C3H8 + H2S, C3H8 + CO2, and C3H8 + H2S + CO2 on H-mordenite, the differences in their maximum numbers of adsorption sites are large. For these systems, the surface heterogeneity needs to be considered. Although the actual sizes of the

H2S and CO2 molecules are smaller than that of C3H8, their larger values of n (obtained by best fit of the purecomponent data) are more suitable for predicting mixture adsorption. With normal parameters (i.e., lower values of n for H2S and CO2), MR-MSL also improves the predictions, although to a lesser degree. With the larger values of n, however, the improvements are very substantial. The model not only reduces prediction deviations from experimental data substantially but also is capable of predicting the azeotropic behavior exhibited by C3H8 + CO2 and its correct azeotropic compositions. For the ternary system CO2 + H2S + C3H8, the results are similar to those for the two binary systems. The significant improvement of MR-MSL over MSL is reached based on the larger values of n for H2S and CO2. The average error is reduced by more than half. With the normal values of n, the improvements are much less. Notation D DT K N NT Nt n P Q q qs q* T x y q

average percent deviation from experiment for the amount adsorbed of each component average percent deviation from experiment for the total amount adsorbed adsorption equilibrium constant, 1/kPa number of components number of data points for the total amount adsorbed number of data points for the adsorbed amount of each component number of sites that a molecule occupies pressure, kPa total adsorption amount, mol/kg amount adsorbed for each component, mol/kg saturated adsorption capacity of a pure component, mol/kg maximum number of adsorption sites where component i can adsorb per kilogram of sorbent, mol/kg temperature, K mole fraction (adsorbed phase) mole fraction (gas phase) fractional coverage LA020838V