ARTICLE pubs.acs.org/EF
OnoKondo Model for High-Pressure Mixed-Gas Adsorption on Activated Carbons and Coals Mahmud Sudibandriyo,† Sayeed A. Mohammad,‡ Robert L. Robinson, Jr.,‡ and Khaled A. M. Gasem*,‡ † ‡
Department of Chemical Engineering, Faculty of Engineering, Universitas Indonesia, Depok, Indonesia 16424 School of Chemical Engineering, Oklahoma State University, Stillwater, Oklahoma 74078, United States ABSTRACT: Theory-based models for adsorption behavior are needed to develop optimal strategies for enhanced coalbed methane (CBM) recovery operations and CO2 sequestration. Although a number of frameworks are available for describing the adsorption phenomenon, the OnoKondo (OK) lattice model offers several practical advantages in modeling supercritical, high-pressure adsorption systems. In a recent work, a generalized OnoKondo model was developed for predicting pure-gas adsorption on activated carbons and coals. The goal of the present work is to utilize the pure-component, generalized OK model to predict, a priori, the mixture adsorption of coalbed gases. Specifically, the OK model parameters obtained from pure-gas adsorption were used to predict mixed-gas adsorption for selected multicomponent adsorption systems. In addition, the ultimate correlative capabilities of the OK model for mixed-gas adsorption were also investigated by using binary interaction parameters.Traditional modeling of mixed-gas adsorption typically involves the equilibrium gas-phase mole fractions as required model input. However, the experimental gas-phase molar fractions are generally not available for coalbed reservoir simulation studies. Therefore, in this work, an iteration function method is developed for mixed-gas adsorption that does not rely on measurements of gas-phase molar fractions and, therefore, is ideally suited for use in coalbed reservoir simulators. The results indicate that the OK model can be used to (a) predict binary gas adsorption within 2 times the experimental uncertainties, on average, based on pure-component model parameters alone and (b) represent total and individual adsorptions to within their expected experimental uncertainties with the use of one binary interaction parameter.
1. INTRODUCTION Our recent studies have indicated that the OnoKondo (OK) lattice model has the capability to represent high-pressure adsorption data for pure-gas adsorbates on activated carbon and coal adsorbents.1 A newly developed generalized model was also presented in an earlier work for predicting high-pressure, pure-gas adsorption on activated carbons and coals. In continuation of our previous work on pure-gas adsorption,1 this paper presents the OK model for highpressure, mixture adsorption of systems encountered in enhanced coalbed methane and CO2 sequestration applications. The goal of this study is to extend the OK modeling approach to mixed-gas adsorption and utilize a pure-component generalized model to predict, a priori, the mixed-gas adsorption on activated carbons and coals. Specifically, we have 1. derived a general equilibrium equation for monolayer, random mixed-gas adsorption 2. evaluated the predictive capability, where the OK model parameters obtained from pure-gas generalized model are used to predict mixture adsorption for selected multicomponent adsorption systems, and 3. investigated the ultimate correlative capability of the model for mixed-gas adsorption when binary interaction parameters (BIPs) are included in the model. There appear to be very few studies in the literature on the OK model approach applied to high-pressure, mixed-gas adsorption on coals. Recently, Ottiger et al.2 have used a density functional theory-based lattice model to investigate the adsorption of methane, nitrogen, and CO2 on a single dry coal. In this work, the OK model approach has been applied to carbonaceous matrices with varied structural complexity and different levels r 2011 American Chemical Society
of moisture content. Specifically, the adsorbents ranged from wellcharacterized dry activated carbons to wet coal samples obtained from the Tiffany and Illinois basins.3 Further, the mixed-gas adsorption (up to ternary gas mixtures) on wet coals has been investigated in this work. To our knowledge, this wider application of the OK model for predicting high-pressure, mixture adsorption of gases on dry carbons and wet coals has not been presented previously in the literature. Traditional methods for mixed-gas adsorption modeling in the literature typically require the experimental gas-phase equilibrium molar fractions for evaluating the amounts adsorbed of each component. Such experimental molar compositions, however, are rarely available in reservoir simulations of enhanced coalbed methane recovery and CO2 sequestration applications. In contrast, the overall or the feed gas compositions are generally available for reservoir simulations. Therefore, in this work, an iteration function method is presented that does not require the experimental gas-phase molar fractions; rather, the method uses the feed gas compositions, thereby limiting the experimental information needed to conduct the mixture adsorption calculations. Thus, the method appears to be ideally suitable and more useful for conducting coalbed reservoir simulations. In the work presented in the following sections, the model development/evaluation was conducted using our newly acquired adsorption data as well as selected data from the open literature. As in the modeling of pure-gas adsorption,1 we first performed studies on dry activated carbons followed by studies Received: April 14, 2011 Revised: May 27, 2011 Published: June 16, 2011 3355
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on wet coals. Literature data on high-pressure multicomponent gas adsorption, however, are not as plentiful as for pure fluids. The following mixture adsorption data on activated carbon were selected for model evaluation: 1. mixture adsorption of methane, nitrogen, and CO2 on Calgon F-400 activated carbon at 318.2 K4 2. mixture adsorption of methane, nitrogen, and CO2 on Norit R1-Extra activated carbon at 298 K5 3. mixture adsorption of methane, ethane, and ethylene on BPL activated carbon at 301.4 K6 A few experimental studies have examined high-pressure multicomponent gas adsorption on coals.712 However, limited information was given on the experimental data (e.g., the expected experimental uncertainties) in most of those references. Experimental data are also available in the dissertation work of Stevenson13 and Clarkson.14 For the purposes of the current work, we elected to use only Oklahoma State University (OSU) mixture adsorption measurements on various wet coals for model evaluation. A future study will involve an even larger database including some of the literature data on multicomponent adsorption mentioned above.
2. ONOKONDO LATTICE MODEL FOR MIXTURE ADSORPTION An adsorption model based on the lattice theory was proposed first by Ono and Kondo in 1960.15 The more general formalism was developed further by Donohue and co-workers for the adsorption of solutes in liquid solutions.1619 In an earlier work,1 we adopted the AranovichDonohue (AD) formalism for lattice systems to investigate the pure-gas adsorption of supercritical gases. In this work, following a similar approach, the extension of OnoKondo model to mixtures is presented and applied to high-pressure, mixed-gas adsorption on activated carbons and coals. Further, as was the case with the work on pure-gas adsorption,1 the adsorbent is represented as a rectangular slit pore. In the following, the OK equations for monolayer adsorption of a binary, random mixture are derived and then generalized to a multicomponent system. The configurational Helmholtz free energy of a nonrandom mixture in the lattice system can be written as20 A¼
z0 2 þ
n
n
∑i N i εii þ kT ∑i N i ln xi z0 MT 8
n
n
∑i ∑j Δijxi xj
Z
1=T
this number represents the deviations of a nonrandom mixture from its random limit for which Ψij is unity. For a random, binary gas mixture containing components A and B, the free energy of the system can be written using eq 1 with ψij = ψji = 1 as follows z0 A ¼ ðN A εAA þ N B εBB þ N n εnn Þ þ kTðN A ln xA 2 z0 M ðΔAA xA xA þ ΔAB xA xB þ N B ln xB þ N n ln xn Þ þ 4 þ ΔAn xA xn þ ΔBA xB xA þ ΔBB xB xB þ ΔBn xB xn þ ΔnA xn xA þ ΔnB xn xB þ Δnn xn xn Þ ð2Þ Here, subscript n represents the empty cells. Further, because Δij = 2εij (εii þ εjj), so Δaa = Δbb = Δnn = 0, and because there is no interaction energy between a molecule and an empty cell and between the empty cells, then Δan = Δna = εaa and Δbn = Δnb = εbb. Also εab = εba implies that Δab = Δba. Thus, eq 2 can be simplified as A¼
z0 ðN A εAA þ N B εBB Þ þ kTðN A ln xA þ N B ln xB þ N n ln xn Þ 2 z0 M ½f2εAB ðεAA þ εBB ÞgxA xB εAA xA xn εBB xB xn ð3Þ þ 2
Noting that xi = ni/m and xn = 1 xa xb, this equation can be written as A ¼ kTðN A ln xA þ N B ln xB þ N n ln xn Þ z0 M ðεAA xA 2 þ 2εAB xA xB þ εBB xB 2 Þ þ 2
ð4Þ
or A ¼ xA ln xA þ xB ln xB þ ð1 xA xB Þ lnð1 xA xB Þ MkT z0 εAA 2 2εAB εBB xA þ xA xB þ xB 2 þ ð5Þ 2 kT kT kT The chemical potential for each component in the bulk can be determined using DA DA ¼ ð6Þ μi ¼ DN i T, M, N j6¼i MDxi T, M, N j6¼i which leads to the following equations for the bulk phase for each component (A and B)
ðΨij þ Ψji Þ dð1=TÞ
ð1Þ
μA, b
0
where Ni is the number of molecules of component i, M is the total number of sites including vacancies, xi is the mole fraction of component i in the adsorbed phase, and Ψij is a correlation coefficient that represents the deviations of a nonrandom mixture from its random limit. Other notational details are available in the Nomenclature section. The lattice coordination number, z0, represents the number of primary nearest-neighbor cells in the lattice system. The interaction energy between molecule i and j is expressed by εij. Note that z0Δij/8 is the interchange energy, i.e., the amount of energy that accompanies the exchange of molecule i (from a lattice completely filled with i’s) with a molecule j (from a lattice completely filled with j), where Δij 2εij (εii þ εjj). The correlation coefficient, Ψij, is the ratio of the probability for having a molecule i around an arbitrary molecule j to the probability of molecule i occupies the lattice cell, xi = ni/m. Thus,
and μB, b
" !# εAA εAB xA, b xA, b þ xB, b þ ln ¼ kT z0 ð7Þ kT kT 1 xA, b xB, b " !# εBB εAB xB, b xB, b þ xA, b þ ln ¼ kT z0 ð8Þ kT kT 1 xA, b xB, b
where subscript b represents the bulk-phase properties. The configurational free energy of the first adsorbed layer can be written as20 n n n n z1 εii þ εis þ A1st ¼ N i N i xj, 2nd εij þ kT N i ln xi 2 i i j i Z 1=T z1 MT n n þ Δij xi xj ðΨji þ Ψji Þdð1=TÞ ð9Þ 8 i j 0
∑
∑∑
∑
∑∑
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where z1 is the parallel coordination number representing the number of primary nearest-neighbor cells in the parallel direction (or in one layer). For a binary mixture comprising components A and B as adsorbates, eq 9 can be written as follows after algebraic simplification A1st ¼ ðxA, 1st εAs þ xB, 1st εBs Þ þ xA, 1st xA, 2nd εAA M 1st kT þ xA, 1st xB, 2nd εAB þ xB, 1st xA, 2nd εBA þ xB, 1st xB, 2nd εBB þ kT½xA, 1st ln xA, 1st þ xB, 1st ln xB, 1st þ ð1 xA, 1st xB, 1st Þ lnð1 xA, 1st xB, 1st Þ z1 M 1st ðεAA xA, 1st 2 þ 2εAB xA, 1st x B, 1st þ εBB xB, 1st 2 Þ þ 2 ð10Þ The chemical potential of each adsorbed component in a slit of adsorbent can be derived using eq 10, with the same assumption as in the bulk phase, as follows DA1st DN i, 1st
μi, ads ¼
!
¼ T, M1st , N j6¼i, 1st , N i, 2nd
DA1st M1st Dxi, 1st
!
T, M1st , N j6¼i, 1st , N i, 2nd
ð11Þ
where a binary interaction parameter Cij was introduced to facilitate calculation of the unlike-molecule interaction energy in cases where it may deviate from the geometric mean relation. In such cases, the value of Cij is determined by regression of the available adsorption data. The Gibbs excess adsorption for each component was calculated using the following expression pure
Γi ¼ 2Ci
μA, ads
and
εBs εBB εAB εBB εAB þ xB þ x A þ z1 xB þ xA kT kT kT kT kT xB ð13Þ þ ln 1 xA xB
μB, ads ¼ kT
The equality of the chemical potential in the adsorbed and the bulk phases for each component leads to the following equilibrium equations for the binary mixed-gas adsorption ln
xA ð1 xA, b xB, b Þ εAA þ ððz1 þ 1ÞxA z0 xA, b Þ xA, b ð1 xA xB Þ kT εAB εAs ððz1 þ 1ÞxB z0 xB, b Þ þ ¼0 ð14Þ þ kT kT
ð18Þ
is the maximum adsorption capacity of the pure where component. The fractional coverage in the bulk phase, xi,b was obtained from the following equation yF xi, b ¼ i b ð19Þ Fmc where the bulk density, Fb, was calculated using the Benedict WebbRubin (BWR) equation of state.21 Because the mixture adsorbed-phase density is generally not available experimentally, the maximum density, Fmc, was estimated using the following ideal mixing rules 1 xAbs xAbs ¼ A þ B Fmc Fmc, A Fmc, B
And noting that xi,1st = xi,2nd (we will use symbol xi only) leads to
εAs εAA εAB εAA εAB þ xA þ x B þ z1 xA þ xB ¼ kT kT kT kT kT kT xA þ ln ð12Þ 1 xA xB
ðxi xi, b Þ
Cpure i
ð20Þ
Abs The absolute adsorbed-phase mole fractions, xAbs A and xB , are used in this equation. These mole fractions are calculated on the basis of absolute adsorbed amounts of each adsorbate rather than the Gibbsian or excess amounts adsorbed. Because the maximum adsorption capacity of a component may well be different in pure and mixture adsorption, a modification can also be introduced to calculate the Gibbs adsorption for each component. In this case, eq 18 becomes
pure
Γi ¼ 2βCi
ðxi xi, b Þ
ð21Þ
where β was evaluated as follows β¼
n
n
Abs ∑i ∑j xAbs i xj Eij
ð22Þ
where an additional binary interaction parameter, Eij, is introduced in this expression in which Eii = Ejj = 1. Note that the Eij is only used to test correlative capabilities of the model and is not needed when the OK model is used in an entirely predictive mode.
and xB ð1 xA, b xB, b Þ εBB þ ððz1 þ 1ÞxB z0 xB, b Þ xB, b ð1 xA xB Þ kT εAB εBs þ ððz1 þ 1ÞxA z0 xA, b Þ þ ¼0 ð15Þ kT kT Thus, a general equilibrium equation for monolayer, random mixed-gas adsorption for each component can be written as ln
xi ð1
ln
n
∑ xj, b Þ j¼1
xi, b ð1
n
∑ xj Þ j¼1
þ
n
ε
ε
∑ ij ½ðz1 þ 1Þxj z0 xj, b þ kTis ¼ 0 j ¼ 1 kT
ð16Þ
where the summation n is over all the components. Further, a geometric combination rule was used to evaluate the interaction energy between molecules i and j; i.e., pffiffiffiffiffiffiffiffi εij ¼ ð1 þ Cij Þ εii εjj ð17Þ
3. ITERATION FUNCTION METHOD (IFM) The Gibbs adsorption can be calculated using a model if the pressure, temperature, and equilibrium composition in the bulk gas phase are known. In a previous study,4 the equilibrium mole fraction in the gas phase was obtained directly from the experimental adsorption data. Although the gas composition obtained from the experiment is adequate to calculate the individual Gibbs adsorption, errors in the gas composition measurement may affect the model representation in some cases. Moreover, (a) using the experimental gas composition facilitates model adsorption calculations only at the conditions where the equilibrium gas composition has been measured and (b) in enhanced coalbed methane production simulations, more information is available on the overall gas composition than the equilibrium gas composition. To overcome these problems, an iteration function method, which is somewhat similar to a flash calculation in vaporliquid equilibrium was used to determine the gas mole 3357
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Table 1. OK Model Predictions of Binary Mixture Adsorption on Dry Activated Carbon at 318.2 K system
NPTS
% AAD
RMSE (mmol/g)
Table 2. OK Model Representations of Binary Mixture Adsorption on Dry Activated Carbon at 318.2 K
WAAE
system
NPTS
% AAD
RMSE (mmol/g)
WAAE
Based on Parameters from the Pure-Adsorption Model
Based on One Regressed Parameter (Cij)
CH4N2
CH4N2
methane
40
2.5
0.080
1.0
methane
40
2.8
0.073
1.1
nitrogen total
40 40
11.7 0.7
0.064 0.029
1.7 0.3
nitrogen total
40 40
5.0 1.8
0.032 0.065
0.7 0.8
methane
40
17.8
0.132
2.1
methane
40
4.8
0.047
0.4
CO2
40
4.1
0.113
1.0
CO2
40
2.7
0.147
0.7
total
40
4.0
0.206
1.1
total
40
2.2
0.152
0.6
nitrogen
40
81.1
0.199
3.5
nitrogen
40
8.2
0.045
0.6
CO2
40
4.6
0.152
1.2
CO2
40
4.4
0.255
1.4
total
40
7.5
0.335
2.4
total
40
4.1
0.262
1.4
CH4CO2
Cij
Eij
0.198
1.0
0.335
1.0
0.658
1.0
0.351
1.078
0.280
0.956
0.446
0.871
CH4CO2
N2CO2
N2CO2
Based on Generalized Pure-Adsorption Model
Based on Two Regressed Parameters
CH4N2
CH4N2
methane
40
3.3
0.096
1.2
methane
40
1.5
0.036
0.6
nitrogen
40
6.9
0.070
1.5
nitrogen
40
2.0
0.020
0.5
total
40
2.9
0.098
1.2
total
40
0.7
0.033
0.3
methane
40
18.7
0.126
1.9
methane
40
4.2
0.033
0.4
CO2
40
3.1
0.130
0.8
CO2
40
2.1
0.101
0.5
total
40
1.5
0.087
0.4
total
40
1.7
0.107
0.5
nitrogen
40
71.0
0.164
2.6
nitrogen
40
17.6
0.044
0.7
CO2
40
3.8
0.127
1.0
CO2
40
2.3
0.130
0.7
total
40
5.6
0.277
1.8
total
40
2.6
0.162
0.8
CH4CO2
CH4CO2
N2CO2
N2CO2
fraction for a given pressure, temperature, feed composition, and specific void volume (void volume per unit amount of adsorbent) of the system. represents the mole fraction of each component i in the If zfeed i in terms of the feed, then by a molar balance, we can express zfeed i other experimentally accessible variables as zfeed ¼ i
ðnGibbs Þi þ V void Fb yi ntotal Gibbs þ V void Fb
ð23Þ
where (nGibbs)i is the Gibbs adsorption of component i, Vvoid is the void volume, Fb is the bulk density, and yi is the gas-phase composition of component i. The component Gibbs adsorption is first calculated using the OK model of eqs 16 and 21. The solution, however, is contingent on equilibrium mole fractions, yi, as they are needed to calculate the gas density and to calculate the individual fractional coverage in the bulk phase, xi,b, as defined in eq 19. The gas mole fractions were initialized with the available experimental values to speed up the calculation (although any reasonable initial values can be used). The next step is to evaluate eq 23 for each component. If eq 23 is not satisfied for each component, then a new set of equilibrium mole fractions is used to calculate the next trial adsorbed amount.
Figure 1. CH4 Gibbs adsorption of CH4/CO2 on dry activated carbon at 318.2 K and at different feed compositions.
4. CASE STUDIES Four different modeling scenarios (case studies) were investigated in this work. The scenarios were designed such that both predictive and correlative capabilities of the OK model for mixture adsorption could be tested rigorously. Specifically, the four scenarios were as follows: 3358
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Table 3. OK Model Predictions of Ternary Mixture Adsorption on Dry Activated Carbon at 318.2 K systems
NPTS
% AAD
RMSE (mmol/g)
WAAE
Based on Parameters from the Pure-Adsorption Model methane
11
8.9
0.040
0.7
nitrogen CO2
11 11
540 3.1
0.201 0.107
3.3 0.9
total
11
6.8
0.335
2.2
Based on the Generalized Pure-Adsorption Model
Figure 2. CO2 Gibbs adsorption of CH4/CO2 on dry activated carbon at 318.2 K and at different feed compositions.
methane
11
nitrogen
11
CO2
11
total
11
11.4
0.054
0.9
0.158
2.3
1.7
0.077
0.5
4.9
0.275
1.6
474
Based on Pure and One Binary Interaction Parameter (Cij) methane nitrogen
11 11
2.6 37.6
0.012 0.022
0.2 0.4
CO2
11
4.3
0.290
1.2
total
11
4.2
0.299
1.3
Based on Pure and Two Binary Interaction Parameters methane
11
nitrogen
11
CO2
11
total
11
4.2
0.017
0.5
0.052
0.9
2.5
0.139
0.7
3.2
0.177
0.9
121
Figure 3. Comparison between the gas-phase compositions obtained from experimental measurements and from the iteration function method (IFM) calculations for the adsorption of CH4/CO2 mixture on dry activated carbon at 318.2 K.
1. The regressed pure component model parameters obtained in the work on pure-gas adsorption1 were used to predict the mixture adsorption. 2. The generalized pure component model parameters obtained in the work on pure-gas adsorption1 were used to predict the mixture adsorption. 3. A single binary interaction parameter in the geometric mean combining rule for fluidfluid energy was used to represent or correlate the mixture adsorption data (eq 17). 4 Two binary interaction parameters were used to correlate the mixture adsorption data. One BIP was from scenario 3 above and the other BIP was in the modified equation for excess adsorption (eq 22). The OK model was used in an entirely predictive mode in scenarios 1 and 2 above. Further, the ultimate correlative capabilities of the model were tested in scenarios 3 and 4. The motivation for investigating the representation or correlative capabilities of the model was to facilitate a direct comparison between the correlative and predictive usage of the model. More importantly, this exercise yields an estimate of the loss in accuracy that occurs when the model is used in an entirely predictive mode. For the cases where binary interaction parameters were regressed (scenarios 3 and 4), the weighted sum of squared errors in the calculated adsorption amounts was used as the
Figure 4. OK model predictions of a 10/40/50 mol % CH4/N2/CO2 feed mixture adsorption on dry activated carbon at 318.2 K.
objective function. The weights used were the expected experimental uncertainties of the amounts adsorbed. For the literature data, the experimental uncertainties were not available. Therefore, the average absolute deviation was selected as the objective function. The difference between the regressed and generalized model parameters relates to the manner in which model parameters were estimated. Specifically, for scenario 1 (regressed parameters), the OK model parameters were obtained by direct regressions of the pure gas experimental data. In contrast, for scenario 2 (generalized parameters), the OK model parameters were obtained from generalized expressions that are derived in terms of accessible properties of the adsorbates and the adsorbent structure. Thus, the generalized expressions provide a method to conduct a priori predictions, and they also account for the temperature dependence of the OK model 3359
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Table 4. OK Model Predictions of Binary Mixture Adsorption on Dry Activated Carbon at 298 K (Data from Dreisbach et al.5) system
NPTS
% AAD
Table 5. OK Model Representations of Binary Mixture Adsorption on Dry Activated Carbon at 298 K (Data from Dreisbach et al.5)
RMSE (mmol/g)
system
Based on Two-Parameter Pure-Adsorption Model
NPTS
% AAD
RMSE (mmol/g)
Eij
Based on One Regressed Parameter (Cij)
CH4N2
CH4N2
methane
24
12.2
0.352
methane
24
14.8
0.345
nitrogen
24
22.3
0.181
nitrogen
24
5.8
0.119
total
24
4.3
0.197
total
24
6.4
0.291
methane CO2
24 24
13.4 18.8
0.207 0.569
methane CO2
24 24
6.8 10.1
0.182 0.451
total
24
5.8
0.487
total
24
6.2
0.540
nitrogen
24
0.317
nitrogen
24
17.8
0.087
CO2
24
7.6
0.483
CO2
24
5.7
0.344
total
24
6.0
0.364
total
24
4.8
0.327
CH4CO2
0.610
1.0
0.294
1.0
0.832
1.0
0.665
1.197
0.319
1.206
0.832
0.971
CH4CO2
N2CO2 139
Cij
N2CO2
Based on Generalized Pure-Adsorption Model
Based on Two Regressed Parameters
CH4N2
CH4N2
methane nitrogen
24 24
7.4 12.8
0.240 0.113
methane nitrogen
24 24
11.0 8.2
0.214 0.204
total
24
4.6
0.192
total
24
2.9
0.100
methane
24
10.2
0.319
methane
24
3.7
0.097
CO2
24
31.3
0.571
CO2
24
9.4
0.257
total
24
8.3
0.584
total
24
4.8
0.276
nitrogen
24
38.2
0.100
nitrogen
24
17.2
0.079
CO2
24
13.2
0.597
CO2
24
5.9
0.363
total
24
11.9
0.609
total
24
4.9
0.342
CH4CO2
CH4CO2
N2CO2
N2CO2
parameters. Additional details of this approach are available in our paper on OK modeling of pure-gas adsorption.1 The case studies 14, mentioned above, were conducted on three dry activated carbons and three wet coals. The coal samples were from the Illinois and Tiffany basins.3 The details for these case studies are provided below for these systems: 4.1. Modeling Mixed-Gas Adsorption on Dry Activated Carbons. Methane, Nitrogen, and CO2 Mixture Adsorption on Calgon F-400 (OSU Data). Our measurements on pure and mixture adsorption of methane, nitrogen, and CO2 on activated carbon at 318.2 K and pressures to 13.6 MPa4 were used to evaluate the OK modeling capability. The binary mixture adsorption includes methane/CO2, nitrogen/CO2, and methane/nitrogen systems at molar feed gas compositions of 20, 40, 60, and 80%. Adsorption isotherms were also measured for a methane/nitrogen/CO2 ternary mixture at a feed composition of 10/40/50 mol percent, respectively. The IFM method was used for the results reported for this system. Both the regressed and the generalized model parameters from the pure-gas adsorption modeling work were used to predict the mixed-gas adsorption. The reader is referred to the previous work for details of the generalized OnoKondo model for pure-gas adsorption.1 Table 1 presents the results of the OK model predictions for binary mixture adsorption on dry activated carbon. The results
are based solely on the model parameters derived from the puregas adsorption measurements and, therefore, the OK model has been used here in an entirely predictive mode. As shown in Table 1, the OK model based on either the regressed or generalized pure-gas parameters can predict the binary adsorption data within two times the expected experimental uncertainties. The percentage errors for the lower-adsorbed component adsorption appear large due to the low Gibbs adsorption values; however, the errors in terms of the amounts adsorbed are small for this component. In general, the pure-gas adsorption predictions from a generalized model would be less accurate than direct parameter regressions.1 Nevertheless, the capability of the generalized model to predict both the temperature dependence of pure-gas adsorption (as seen in Sudibandriyo et al.1) and the composition dependence in mixed-gas adsorption (through the OK model mixing theory) is a practically useful and highly desirable feature of a multicomponent adsorption model intended for coalbed methane work. The OK model representations of mixture adsorption when binary interaction parameters (BIPs) were included in the model are presented in Table 2. The use of a single BIP (Cij in eq 17) results in precise model representations (e.g., the weighted average absolute error (WAAE) for the nitrogen component adsorption in the 3360
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nitrogen/CO2 system decreased from 3.5 to 0.6 when compared to model predictions that do not involve any BIPs). For illustrative purposes, Figures 1 and 2 present the OK model predictions and representations of the individual component adsorption of methane/CO2 system. The model is capable of predicting a maximum in the Gibbs adsorption as observed experimentally for the lower-adsorbed component. For completeness, the pure-component adsorption is also included in each figure. Other binary mixtures for this system yielded similar results and are not shown here for brevity.
Figure 3 presents a comparison between the gas-phase compositions obtained from experimental measurements and those obtained from the IFM calculations. The figure shows that the gas compositions obtained from IFM calculations are in excellent agreement with the experimental values. The results for the ternary mixture are presented in Table 3. The direct predictions of ternary data were generally within about 2 times the experimental uncertainties, based on the puregas adsorption model parameters alone. The only exception to this was the nitrogen component adsorption in the ternary mixture. This was partly due to the low-adsorbing nature of nitrogen in the ternary mixture (as illustrated in Figure 4). The inclusion of BIPs produced model representations generally within the experimental uncertainties. Methane, Nitrogen, and CO2 Mixture Adsorption on Norit R1-Extra (Data from Dreisbach et al.5). Pure and mixture adsorption of methane, nitrogen and CO2 on Norit R1-Extra activated carbon at 298 K reported by Dreisbach et al.5 were used to evaluate the OK mixture modeling capability. The OK model parameters for pure component adsorption have been reported in an earlier work1 and those parameters are used here to obtain direct predictions of mixture data. Table 4 presents the prediction of binary mixture adsorption on dry Norit R1-Extra activated carbon at 298 K based on pure component parameters alone. The OK model can predict the total adsorption data within 6% AAD. However, the prediction for individual component adsorption was less accurate and the deviations were larger for the lower-adsorbing component in the mixture. The OK model representations of binary mixture adsorption are presented in Table 5. The inclusion of a single BIP can decrease the model deviations to (roughly) half their values for the prediction case, especially for the adsorption of lower-adsorbing component. However, the inclusion of two BIPs provides no significant improvement over the results obtained with only one BIP in the model.
Table 6. OK Model Predictions of Ternary Mixture Adsorption on Dry Activated Carbon at 298 K (Data from Dreisbach et al.5) system
NPTS
% AAD
RMSE (mmol/g)
Based on Parameters from the Pure-Adsorption Model methane
40
13.4
0.514
nitrogen
40
45.0
0.446
CO2
40
12.5
0.261
total
40
8.7
0.794
Based on Pure and One Binary Interaction Parameter (Cij) methane
40
17.2
0.661
nitrogen CO2
40 40
48.5 14.2
0.5 0.34
total
40
11.9
1.09
Based on Pure and Two Binary Interaction Parameters methane
40
17.9
0.638
nitrogen
40
47.2
0.473
CO2
40
14.9
0.393
total
40
9.1
0.731
Table 7. Comparison of Model Predictions and Representations for Mixture Adsorption of CH4, C2H6, and C2H4 on Dry Activated Carbon at 301.4 K (Data from Reich et al.6) % AAD
RMSE (mmol/g)
system
NPTS
Langmuira
2D-EOSa
2D-EOS (Cij)
methane
14
36.7
39.8
19.5
ethane
14
4.4
2.4
2.3
total
14
5.8
7.2
3.5
methane
15
28.9
33.9
8.8
ethylene
15
5.4
2.9
3.3
total
15
5.8
6.3
2.7
ethane
12
5.2
4.6
5.0
ethylene
12
8.3
6.8
5.7
total
12
6.4
5.6
5.1
methane ethane
14 14
59.3 3.7
51.2 3.5
33.5 4.8
ethylene
14
4.9
4.4
total
14
9.5
8.4
a
OK (Cij)
OK (Cij & Eij)
35.2
23.8
23.7
0.329
0.169
0.168
3.8
2.5
2.4
0.142
0.086
0.084
4.5
2.3
2.2
0.226
0.125
0.153
29.9
15.2
14.0
0.297
0.110
0.104
3.8
3.2
1.9
0.148
0.113
0.088
3.1
2.8
2.2
0.172
0.126
0.141
4.3
5.1
4.1
0.099
0.110
0.084
5.7
5.0
3.3
0.187
0.185
0.116
3.8
3.3
1.3
0.193
0.176
0.080
52.2 5.6
39.1 4.7
38.1 5.4
0.486 0.122
0.303 0.093
0.285 0.102
5.5
4.9
4.9
5.5
0.133
0.154
0.131
5.5
5.7
3.9
3.9
0.420
0.334
0.287
OK
OK
OK (Cij)
OK (Cij & Eij)
CH4C2H6
CH4C2H4
C2H6C2H4
CH4C2H6C2H4
a
The results presented in the third to fifth columns are adopted from Zhou et al.22 3361
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Table 8. Binary Interaction Parameters Used in OK Model for Mixture Adsorption of CH4, C2H6, and C2H4 on Dry Activated Carbon at 301.4 K (Data from Reich et al.6) system
Cij
Eij
One Binary Interaction Parameter CH4C2H6
0.569
1.0
CH4C2H4
0.550
1.0
C2H6C2H4
0.037
1.0
Two Binary Interaction Parameters CH4C2H6 CH4C2H4
0.577 0.642
0.972 0.890
C2H6C2H4
0.037
1.088
Figure 6. C2H6 adsorption of CH4/C2H6 on dry activated carbon at 301.4 K and at different feed compositions. (Data from Reich et al.6)
Figure 5. CH4 adsorption of CH4/C2H6 on dry activated carbon at 301.4 K and at different feed compositions. (Data from Reich et al.6)
Table 6 presents the OK model predictions for the ternary mixture. The individual component adsorption was predicted with a % AAD ranging from about 13% to 45% on the basis of only the pure-gas parameters. Note that similar deviations were obtained by Dreisbach et al.5 for ternary mixtures using a dualsite Langmuir model. The inaccuracies in the experimental gas-phase mole fractions may have contributed to the larger deviations in the mixture adsorption predictions. In his study, Dreisbach et al.5 used two different methods to obtain the gas-phase compositions. For the binary mixtures, they used an equation of state to infer both the gas-phase composition and density from pressure, temperature measurements, and system volume calibrations. For the ternary mixtures, they used a gas chromatograph to measure the gasphase composition. Methane, Ethane, and Ethylene Mixture Adsorption (Data from Reich et al.6). The mixture adsorption of methane, ethane, and ethylene on BPL activated carbon at 301.4 K has been reported by Reich et al.6 The OK model was used to obtain mixture predictions for these data, and the results were compared with those reported by Zhou et al.,22 who used a two-dimensional equation of state (2-D EOS) to investigate the adsorption behavior of this system. Table 7 presents the comparison of model predictions and representations for mixture adsorption of methane, ethane, and ethylene on dry BPL activated carbon at 301.4 K. For individual component adsorption, the OK model predictions have lower errors than the Langmuir model and they give comparable results
Figure 7. OK model-predicted CH4 adsorbed mole fractions for CH4/ C2H4 mixture adsorption on BPL activated carbon at 301.4 K. (Data from Reich et al.6)
to the 2-D EOS model.22 For the total adsorption, the OK model predictions are more accurate than both the Langmuir and 2-D EOS models (on average, the % AADs are 3.8, 6.0, and 6.4 for the OK, Langmuir, and 2-D EOS models, respectively). These results also suggest that the use of more than one BIP does not necessarily lead to improved representations with the OK model. The BIPs for this system are reported in Table 8. Figures 5 and 6 illustrate the OK model results for methane/ ethane mixture adsorption on BPL activated carbon. The model underpredicts the methane adsorption in the methane/ethane mixture at the higher pressures (Figure 5). The OK modelpredicted adsorbed molar fractions for the binary mixture methane/ethylene are shown in Figure 7. The figure illustrates the methane molar composition in the adsorbed phase as a function of pressure and feed gas composition. The selectivity is higher for ethylene, which is in agreement with experimental data for this system from Reich et al.6 4.2. Modeling of Mixed-Gas Adsorption on Coals. Data Employed in This Work. The pure and mixture adsorption of methane, nitrogen, and CO2 on wet Fruitland coal12 and wet Illinois#6 coal3 have been measured at OSU at 319.3 K and pressures to 12.4 MPa. The mixture data include methane/CO2, nitrogen/ CO2, and methane/nitrogen adsorption isotherms at molar feed gas compositions of 20, 40, 60, and 80%. The coal samples varied in their moisture content from 5% to 23%. Adsorption isotherms for these 3362
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Table 9. OK Model Predictions of Binary Mixture Adsorption on Wet Fruitland Coal at 319.3 K system
NPTS
% AAD
RMSE (mmol/g)
Table 10. OK Model Predictions of Binary Mixture Adsorption on Wet Illinois #6 Coal at 319.3 K
WAAE
system
NPTS
% AAD
RMSE (mmol/g)
Based on Two-Parameter Pure-Adsorption Model
Based on Two-Parameter Pure-Adsorption Model
CH4N2
CH4N2
WAAE
methane
40
3.1
0.014
0.3
methane
40
12.6
0.018
1.0
nitrogen total
40 40
16.4 5.1
0.019 0.023
0.9 0.6
nitrogen total
40 40
85.2 13.5
0.013 0.030
0.8 1.2
methane
40
8.9
0.012
0.5
methane
40
17.6
0.017
1.1
CO2
40
5.7
0.036
0.8
CO2
40
9.8
0.054
1.2
total
40
4.7
0.040
0.7
total
40
8.7
0.058
1.3
nitrogen
40
0.026
1.0
nitrogen
40
44.3
0.004
0.5
CO2
40
7.1
0.059
0.9
CO2
40
9.5
0.045
1.6
total
40
11.3
0.078
1.2
total
40
7.6
0.042
1.3
CH4CO2
CH4CO2
N2CO2 152
N2CO2
Based on the Generalized Pure-Adsorption Model
Based on the Generalized Pure-Adsorption Model
CH4N2
CH4N2
methane
40
3.1
0.012
0.3
methane
40
26.5
0.039
2.0
nitrogen
40
29.4
0.023
1.5
nitrogen
40
31.9
0.005
0.6
total
40
4.0
0.016
0.4
total
40
16.8
0.038
1.5
methane
40
29.5
0.036
1.3
methane
40
21.2
0.021
1.5
CO2
40
9.5
0.049
1.2
CO2
40
8.3
0.045
1.0
total
40
4.2
0.034
0.6
total
40
8.0
0.053
1.2
nitrogen
40
0.016
0.6
nitrogen
40
0.018
1.6
CO2
40
6.2
0.045
0.6
CO2
40
11.9
0.054
1.9
total
40
5.9
0.053
0.6
total
40
6.8
0.036
1.1
CH4CO2
CH4CO2
N2CO2 130
N2CO2
binary mixtures on wet Tiffany coal were measured at 327.6 K and pressures to 13.8 MPa in a previous study.3 The measurements on Tiffany coal were conducted for a single molar feed composition for each mixture at a moisture content of about 11%. A methane/ nitrogen/CO2 ternary mixture was also measured on wet Tiffany coal at 327.6 K and pressures to 13.8 MPa. The molar feed composition was 10/40/50 and the sample contained about 10% moisture by weight. The detailed adsorption data for these measurements and the structural characterization information of these coals are reported elsewhere.3 The IFM method was used in all the results reported below on wet coals. Water in Coals and Coal Swelling. In this work, the water present in coals has been treated in simplified form as a “pacifier” of the coal matrix. In other words, water in coals is not considered as an active adsorptive component. Rather, the effect of water on gas adsorption is implicit in the OK model parameters derived from adsorption data on wet coals. This has been the traditional modeling approach for adsorption on wet coals.7,10,23 Another aspect of adsorption modeling of coals is the potential swelling of coals when exposed to adsorbates such as CO2. Some investigators suggest that adsorption of gases such as CO2 (and to a lesser extent methane) can alter the pore structure of the coal significantly, and they have attempted to account for swelling of the coal matrix.2426 In our experimental work at OSU, we have not observed any irreversible effects of coal swelling. This finding
207
Table 11. OK Model Predictions of Binary Mixture Adsorption on Wet Tiffany Coal at 327.6 K system
NPTS
methane
11
nitrogen total
% AAD
RMSE (mmol/g)
WAAE
6.9
0.018
1.0
11 11
5.3 6.5
0.003 0.020
0.3 1.0
methane
11
45.7
0.055
4.5
CO2
11
16.9
0.072
2.5
total
11
3.5
0.020
0.6
nitrogen
11
0.015
1.5
CO2
11
7.8
0.049
1.0
total
11
5.9
0.036
0.9
CH4N2
CH4CO2
N2CO2 156
is in agreement with Day et al.,24 who found the coal swelling to be entirely reversible on release of gas pressure. Therefore, the experimental data used in this work and the OK model discussed here do not account for possible coal swelling and volumetric changes of the coal matrix. 3363
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Table 12. OK Model Representations of Binary Mixture Adsorption on Wet Fruitland Coal at 319.3 K system
NPTS % AAD RMSE (mmol/g) WAAE
Cij
Table 13. OK Model Representations of Binary Mixture Adsorption on Wet Illinois #6 Coal at 319.3 K
Eij
system
NPTS % AAD RMSE (mmol/g) WAAE
Based on One Regressed Parameter (Cij)
Based on One Regressed Parameter (Cij)
CH4N2
CH4N2
methane
40
3.1
0.015
0.3
nitrogen total
40 40
17.7 4.6
0.018 0.020
1.0 0.5
methane
40
8.8
0.013
0.4
CO2
40
5.6
0.040
total
40
3.8
0.035
nitrogen
40
67.1
0.014
0.6
CO2
40
8.1
0.073
total
40
9.9
0.073
0.099 1.0
methane
40
11.1
0.021
0.9
nitrogen total
40 40
27.3 9.4
0.005 0.023
0.5 0.8
methane
40
18.7
0.015
0.9
0.7
CO2
40
10.4
0.060
1.2
0.6
total
40
8.0
0.055
1.1
nitrogen
40
34.1
0.003
0.4
1.0
CO2
40
9.2
0.040
1.5
1.1
total
40
8.3
0.042
1.4
CH4CO2
Cij
Eij
1.094 1.0
CH4CO2 0.126 1.0
N2CO2
0.180 1.0
N2CO2 0.438 1.0
Based on Two Regressed Parameters
0.306 1.0
Based on Two Regressed Parameters
CH4N2
CH4N2
methane
40
3.4
0.011
0.3
nitrogen
40
10.5
0.012
total
40
1.7
0.011
methane
40
7.8
0.017
0.5
CO2
40
2.7
0.017
total
40
2.1
0.018
nitrogen
40
82.3
0.010
0.4
CO2
40
4.3
0.034
total
40
3.5
0.035
0.241 0.825
methane
40
3.2
0.006
0.2
0.6
nitrogen
40
54.5
0.007
0.6
0.2
total
40
4.2
0.010
0.4
methane
40
12.0
0.010
0.6
0.3
CO2
40
7.9
0.034
0.9
0.3
total
40
5.7
0.032
0.8
nitrogen
40
86.0
0.006
0.5
0.5
CO2
40
1.3
0.009
0.2
0.4
total
40
1.4
0.010
0.2
CH4CO2
0.403 0.740
CH4CO2 0.012 0.891
N2CO2
0.104 0.799
N2CO2 0.152 0.681
OK Modeling Results. The pure-gas adsorption model parameters for these coals are reported in an earlier work.1 The parameters from that work are used here to predict mixture adsorption on the three wet coals mentioned above. Tables 911 present the results of the OK model predictions for the binary mixtures on the selected wet coals. The OK model can predict the binary adsorption within 2 times the expected experimental uncertainties, on the basis of only the parameters derived from generalized pure-gas adsorption model. Thus, the generalized model appears to be quite capable of accurate a priori predictions of mixture adsorption on coals, based on only the pure-gas adsorption parameters at a single temperature. The only exception was the methane/nitrogen mixture adsorption on Tiffany coal. The OK model yielded predictions with weighted errors of up to 4.5 for this binary mixture. Similar deviations were obtained for this system in an earlier study that used the simplified local-density adsorption model.23 Tables 1214 summarize the OK model representations of binary mixture adsorption using binary interaction parameters. Significant improvement was obtained over the predictions case, especially for the adsorption of lower-adsorbed component. For example, a reduction in WAAE from 4.5 to 1.0 is observed with the use of one binary interaction parameter, Cij, for methane component adsorption in the methane/CO2 mixture on wet Tiffany coal.
0.618 0.715
For illustrative purposes, Figure 8 presents the OK model results for the methane/nitrogen mixture adsorption on wet Tiffany coal. As shown in the figure, the OK model can predict the mixture adsorption within 2 times the experimental uncertainties on the basis of the pure component parameters alone. Table 15 presents the model predictions for the ternary mixture adsorption on wet Tiffany coal. The predictions based on pure-gas parameters alone produce deviations less than about 3 times the experimental uncertainties. The inclusion of BIPs in the model can predict the ternary mixture adsorption within the experimental uncertainties. Note that the BIPs were determined on the basis of only the binary mixture data and the ternary mixture was then predicted. Figure 9 presents the OK model predictions for the ternary mixture adsorption on wet Tiffany coal. As shown in the figure, the OK model can predict the total and individual component adsorption within 3 times the experimental uncertainties, on the basis of the knowledge of only the pure-gas adsorption isotherms for this system. Binary Interaction Parameters. Large binary interaction parameters (BIPs) values (in magnitude) were generally obtained for Cij in the mixtures that contained CO2 as one of the components. This could be related to the significant quadrupole moment of CO2 or the large differences that exist in the fluidsolid interaction energies of CO2 and methane/nitrogen. Further, it appears that 3364
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Table 14. OK Model Representations of Binary Mixture Adsorption on Wet Tiffany Coal at 327.6 K system
NPTS % AAD RMSE (mmol/g) WAAE
Cij
Table 15. OK Model Predictions of Ternary Mixture Adsorption on Wet Tiffany Coal at 327.6 K
Eij
system
CH4N2 11
7.0
0.019
1.0
nitrogen total
11 11
8.1 5.0
0.004 0.016
0.4 0.8
% AAD
RMSE (mmol/g)
WAAE
Based on Parameters from Pure-Adsorption Model
Based on One Regressed Parameter (Cij)
methane
NPTS
0.364 1.0
methane
11
34.0
0.010
0.7
nitrogen CO2
11 11
85.8 19.3
0.037 0.073
2.6 2.1
total
11
5.1
0.028
0.6
Based on Pure and One Binary Interaction Parameters (Cij)
CH4CO2 methane
11
12.5
0.021
1.0
CO2
11
12.9
0.049
2.0
total
11
7.0
0.030
1.2
0.692 1.0
N2CO2 nitrogen
11
52.5
0.005
0.5
CO2
11
6.0
0.036
0.8
total
11
5.1
0.031
0.7
methane
11
7.0
0.002
nitrogen
11
40.7
0.017
0.1 1.2
CO2
11
15.3
0.051
1.6
total
11
8.5
0.034
1.0
Based on Pure and Two Binary Interaction Parameters
1.239 1.0
Based on Two Regressed Parameters
methane nitrogen
11 11
29.2 48.4
0.010 0.021
0.6 1.4
CO2
11
13.1
0.042
1.4
total
11
7.5
0.030
0.9
CH4N2 methane
11
2.7
0.005
0.2
nitrogen
11
6.6
0.003
0.3
total
11
2.6
0.006
0.4
methane
11
8.9
0.015
0.7
CO2
11
4.4
0.017
0.7
total
11
1.8
0.007
0.4
nitrogen
11
53.1
0.005
0.5
CO2
11
6.1
0.037
0.8
total
11
5.2
0.032
0.7
0.108 1.179
CH4CO2 1.074 0.764
N2CO2 1.230 1.036
Figure 9. OK model predictions of a 10/40/50 mol % CH4/N2/CO2 feed mixture adsorption on wet Tiffany coal at 327.6 K.
Figure 8. Gibbs adsorption of a 50/50 mol % CH4/N2 feed mixture on wet Tiffany coal at 327.6 K.
Potential Applications to Coalbed Reservoir Simulations. The OnoKondo modeling approach discussed in this work appears to be an attractive option for coalbed reservoir simulator work. The structure-based-generalization capability offered by theory-based adsorption models such as the OnoKondo model offer a distinct advantage over rudimentary adsorption models that have been used typically in coalbed reservoir simulators. In fact, the iteration function approach, the temperature-dependence of model parameters, and accurate a priori predictions of mixture adsorption from generalized pure-gas model parameters appear to be promising developments that can prove beneficial to reservoir simulations of enhanced coalbed methane recovery and CO2 sequestration in coalbeds.
nitrogen/CO2 mixtures exhibit the largest BIPs and, consequently, these systems contain the largest deviations from the geometric mean combining rule for the fluidfluid interaction energy. The inclusion of surface heterogeneity effects and application of an accurate equation of state to the gas phase may provide improvements and would be considered in a future study.
5. SUMMARY The OK model for pure-gas adsorption presented earlier has been extended to mixture adsorption of gases on activated carbons and coals. The OK mixture model has been shown capable of predicting binary gas adsorption within 2 times the experimental uncertainties, on average, based solely on the information available 3365
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Energy & Fuels from the generalized pure-component adsorption model. Further, the OK model is capable of predicting ternary mixture adsorption within 3 times the experimental uncertainties, when only the purecomponent model parameters are available. The iteration function approach developed in this work provides a robust method to perform mixed-gas adsorption calculations without the need for experimental gas-phase mole fraction measurements. The OK model predictions for mixed-gas adsorption on structurally varied adsorbents (dry activated carbons and wet coals) have demonstrated the viability of the approach presented in this work.
’ AUTHOR INFORMATION Corresponding Author
*Phone (405) 744-5280. Fax: (405) 744-6338. E-mail: gasem@ okstate.edu.
’ NOMENCLATURE A = Helmholtz free energy of the lattice system = maximum adsorption capacity of a component i in its Cpure i pure state Cij = binary interaction parameter for fluidfluid interaction energy between unlike molecules Eij = binary interaction parameter for the modified Gibbs adsorption equation M = total number of lattice sites including vacancies m = number of layers in the lattice model Ni = number of molecules of component i n = number of components (nGibbs)i = Gibbs adsorption of component i = absolute adsorbed mole fraction of component i xAbs i xads = fractional coverage of adsorbate in the monolayer lattice model xi = mole fraction of component i in the adsorbed phase xi,b = fraction of sites occupied by the molecule i in the bulk layer of the lattice model xi,t = fraction of sites occupied by the molecule i in the tth adsorbed layer of the lattice model yi = mole fraction of component i in the gas phase = feed mole fraction Zfeed i z0 = lattice coordination number z1 = parallel coordination number representing the number of primary nearest-neighbor cells in parallel direction AAD = average absolute deviation WAAE = weighted average absolute deviation RMSE = root mean squared error Greek Symbols
εij = fluidfluid interaction energy parameter in the OK model between molecule i and j εis = fluidsolid interaction energy parameter in the OK model Γ = Gibbs excess adsorption per unit mass of adsorbent Fb = gas-phase density Fmc,i = maximum adsorbed-phase density for component i Ψij = correlation coefficient in the lattice model representing the deviations of a nonrandom mixture from its random limit μi = chemical potential for component i
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