Ono–Kondo lattice model for high-pressure adsorption: Pure gases

Fluid Phase Equilibria 299 (2010) 238–251

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Ono–Kondo lattice model for high-pressure adsorption: Pure gases Mahmud Sudibandriyo, Sayeed A. Mohammad, Robert L. Robinson, Jr., Khaled A.M. Gasem ∗ School of Chemical Engineering, Oklahoma State University, Stillwater, OK 74078, United States

a r t i c l e

i n f o

Article history: Received 6 May 2010 Received in revised form 31 July 2010 Accepted 28 September 2010 Available online 3 November 2010 Keywords: Adsorption Coalbed methane CO2 sequestration Ono–Kondo model Lattice theory Model generalization Temperature dependence Adsorption modeling

a b s t r a c t Theoretical models for adsorption behavior are needed to develop optimal strategies for enhanced coalbed methane (CBM) recovery operations. Although several frameworks are available for describing this adsorption phenomenon, the Ono–Kondo (OK) lattice model offers several practical advantages in modeling supercritical, high-pressure adsorption systems. In this study, we evaluated the Ono–Kondo (OK) lattice model for correlating high-pressure, supercritical adsorption encountered in CBM recovery and CO2 sequestration. Specifically, the parameters of the OK model were optimized to obtain reliable representation of pure-gas adsorption on carbon adsorbents. The results were used to develop generalized model parameters, expressed in terms of gas properties and adsorbent characterization which include the temperature dependence of the OK model parameters. The results indicate that the OK monolayer model appears effective in modeling pure-gas adsorption on carbon matrices. The model can represent the adsorption isotherms on activated carbons and coals with about 3.6% average absolute deviation (AAD), which is within the expected experimental uncertainties of the data. The generalized model can predict the adsorption isotherms on activated carbon with about 7%AAD. Moreover, generalized model parameters determined from isotherms of a single gas can be used to predict the adsorption isotherms of other gases. The generalized model also appears effective for pure-gas adsorption on wet coals when the moisture content in the coal is above the equilibrium value. However, at water contents below the saturation value, the model parameters are dependent on the water saturation. © 2010 Elsevier B.V. All rights reserved.

1. Introduction To be useful for CBM applications, an adsorption theory should be capable of handling different adsorbate/adsorbent systems over the full range of operating conditions. Among the theories widely used for adsorption modeling are the Langmuir model [1], ideal adsorbed solution (IAS) theory [2], heterogeneous ideal adsorbed solution [3] (HIAS), vacancy solution model [4–6] (VSM), theory of volume filling micropores (TVFM)[7], two-dimensional equations of state models [8–11] (2D-EOS), and the simplified local density (SLD) model [12–15]. In general, these models work well for lowpressure adsorption; however, few are capable of modeling the high-pressure, supercritical adsorption that is of interest in CBMrelated work. In this study, we evaluate the Ono–Kondo (OK) lattice model for correlating high-pressure, supercritical adsorption isotherms encountered in CBM recovery and CO2 sequestration. In addition to its theoretical origins in lattice theory, the OK model offers several distinct practical advantages. Specifically the model:

∗ Corresponding author. Tel.: +1 405 744 5280; fax: +1 405 744 6338. E-mail address: [email protected] (K.A.M. Gasem). 0378-3812/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2010.09.032

1. Describes monolayer and multilayer adsorption. 2. Has the potential to describe adsorption behavior based on the physical properties of the adsorbates and accessible characterization of the adsorbent. 3. Generates direct estimates for the adsorbed-phase densities, which facilitate reliable predictions of absolute gas adsorption. 4. Is structured to incorporate accurate density calculations from equation-of-state models, which reduces the correlative burden on the adsorption model.

The application of the OK model to pure-gas adsorption on activated carbon has been demonstrated by Aranovich and Donohue [16] and Benard and Chahine [17,18]. Recent work on gas adsorption modeling with variants of the Ono–Kondo lattice theory has also been reported in the literature [19,20]. Critical point corrections to the multilayer Ono–Kondo lattice theory have also been formulated [21,22]. The OK modeling results on activated carbons in the literature [16–18] show that the model is in good agreement with experimental data. However, the regressed model parameters from those studies are not entirely suitable for application to other similar systems in the high-pressure, supercritical region. The results also reveal that, although the OK model has the potential to repre-

M. Sudibandriyo et al. / Fluid Phase Equilibria 299 (2010) 238–251

sent gas adsorption, the model parameters must be determined with care. This includes (a) determination of the optimal number of layers used to describe the adsorbed phase for a specific system, (b) a general formulation that will produce reasonable estimates for each parameter, and (c) describing the parameters in terms of accessible adsorbate and adsorbent characteristics. These capabilities are important for predicting adsorption of systems beyond those where experimental data are available. Moreover, the temperature dependence of the model parameters must also be evaluated. In this study, the parameters of the OK model were evaluated to obtain reliable representation (within the expected experimental uncertainties) of pure-gas, high-pressure adsorption on selected carbon adsorbents for adsorbates in the near-critical and supercritical regions. Generalized model parameters were then developed which can predict the adsorption equilibrium to within twice the expected experimental uncertainties. Extending mathematical models to describe adsorption behavior on coals is complicated by (a) the difficulty in characterizing the coal matrix adequately and (b) assessing the effect of water (found in essentially all coalbeds) on the adsorption behavior. As a result, we decided to perform initial studies (a) on a more readily characterized carbon matrix and (b) in the absence of water (i.e., on dry activated carbons). Our rationale is that useful models should prove capable of fitting data on activated carbons prior to extending them to include the effects of the complex adsorbent structure of coals and/or the presence of water. To apply the OK model, the number of layers must be specified. Therefore, before performing model parameter evaluations, we determined the number of layers required to best describe the systems considered. The multilayer and monolayer adsorption models were compared, and the more appropriate model was used for the rest of this study. The main sections of the manuscript are organized in the following manner: Section 1 provides a discussion on the main aspects of OK model for pure gas adsorption and the determination of number of layers for the OK model used in this study. Section 2 includes the OK modeling results for adsorption on activated carbon and the basis for model generalization. Finally, Section 3 presents the OK modeling results for pure gas adsorption on dry and wet coals. 1.1. Ono–Kondo lattice model for adsorption The OK adsorption model is based on lattice theory and was proposed originally by Ono and Kondo [23] and further developed by Donohue and coworkers [16,21,22]. In the lattice model, the fluid system is assumed to be composed of layers of lattice cells that contain fluid molecules and vacancies. For the case of adsorption, more fluid molecules reside in the cells of the adsorbed-phase layers than in the cells of the bulk-phase layers. Molecular interactions are hypothesized to exist only between nearest neighboring molecules. When equilibrium exists between the gas-phase and the multilayer adsorbed phase, the expression for the thermodynamic equilibrium for pure-component adsorption under the mean-field approximation can be written as [16]: ln

 x (1 − x )  t b xb (1 − xt )

+

z2 (xt+1 − 2xt + xt−1 )εii z0 (xt − xb )εii + = 0 (1) kT kT

for t = 2, 3, . . ., m, number of the layer, and ln

 x (1 − x )  1 b xb (1 − x1 )

+

ε (z1 x1 + z2 x2 − z0 xb )εii + is = 0 kT kT

(2)

for the 1st adsorbed layer. In these equations, xt is the reduced density or fraction of sites occupied by adsorbed molecules in layer t, and xb is the fraction of sites occupied by fluid molecules in the bulk. For a hexag-

239

onal configuration of lattice cells, the coordination numbers z0 and z1 are 8 and 6, respectively; and by definition, z2 = (z0 − z1 )/2. The fluid–fluid interaction energy is expressed by εii /kT, and the fluid–solid surface interaction energy is expressed by εis /kT, where k is Boltzman’s constant, and T is the absolute temperature. The analytical expression for the Gibbs excess adsorption from this model is:

 =C

m 

(xt − xb )

(3)

t

where C is a prefactor related to the capacity of the adsorbent for a specific gas, and m is the maximum number of adsorbed layers in an adsorption isotherm. The reduced densities xt and xb are expressed as xt = t /mc and xb = b /mc , where t and b are the adsorbed and the bulk density of the adsorbate at layer t, respectively, and mc is the adsorbed phase density at maximum capacity. For simplicity, we have modeled the adsorption as occurring within a slit. For monolayer adsorption inside a slit, the equilibrium expression can be written as [17]: ln

x

ads (1 − xb )

xb (1 − xads )



+

((z1 + 1)xads − z0 xb )εii εis + =0 kT kT

(4)

The Gibbs excess adsorption then simplifies to:  = 2C(xads − xb ) = 2C



ads

mc

−

b mc

 (5)

1.2. Determination of the appropriate number of layers As described in the previous section, the OK lattice model can be applied to monolayer or multilayer adsorption. We used selected experimental data to evaluate the number of layers required to represent the systems considered adequately. Specifically, our measurements at 318 K for adsorption of pure nitrogen, methane and CO2 on activated carbon [24] were used to represent adsorption in the supercritical region; the CO2 adsorption data on activated carbon measured by Humayun and Tomasko [25] at 304 K were used to represent the adsorption in the near-critical region; and our CO2 adsorption data on a dry coal (Illinois #6) were used to represent an adsorbent with a wide pore-size distribution [26]. Table 1 presents a summary of our model evaluations for monolayer and three-layer models used to correlate the selected data. The model parameters, given in Table 1, were determined by minimizing the sum of squares of weighted absolute errors in the calculated adsorption, ω, for the pure gas of interest. The weights used in the regressions were the expected experimental uncertainties, determined through multivariate error propagation analysis. In addition, Table 1 presents other measures of the quality of the fits, expressed in terms of absolute average percentage deviation (%AAD) and weighted average absolute errors (WAAE). Figs. 1 and 2 illustrate the quality of representation produced by the models. Both models show excellent representation, within the expected experimental uncertainties; as such, comparable representation is provided by the monolayer and multilayer models. This suggests that the simpler monolayer model is appropriate for the systems considered. The results are not surprising, considering that the adsorption of small molecules occurs mostly in the micropore structure [17,27,28]. Perhaps, monolayer adsorption is more likely because the size of the micropores is only several times the diameter of the molecules. Since the monolayer model proved effective and adequate in modeling pure-gas adsorption on carbon matrices, it was used in the subsequent calculations reported here.

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Table 1 Comparison of monolayer and multilayer modeling results for selected systems. ACa

Model parameters

Nitrogen Monolayer model εis /k (K) εii /k (K) mc (g/cc) C (mmol/g) NPTS %AAD WAAE Three-layer model εis /k (K) εii /k (K) mc (g/cc) C (mmol/g) NPTS %AAD WAAE a b c

Methane

CO2

ACb

Dry Illinois #6 Coalc

CO2

CO2

−1032 41 0.67 2.72 22 0.3 0.3

−1385 64 0.34 3.26 18 0.6 0.6

−1690 82 0.98 4.53 52 2.8 0.9

−1610 100 0.98 5.27 28 4.3 0.8

−1170 60 0.95 1.19 11 2.9 0.7

−1020 50 0.67 2.81 22 0.3 0.3

−1380 64 0.40 3.20 18 0.6 0.7

−1650 82 1.02 4.58 52 3.4 1.0

−1570 110 0.99 5.45 28 4.3 0.8

−1160 70 0.98 1.23 11 3.0 0.6

Ref. [24]. Ref. [25]. Ref. [26].

9 .0

CO2 at 304 K 7.0

CO2 at 318 K

6.0

CH4 at 318 K N2 at 318 K

5.0

OK Monolayer OK Multilayer

4.0 3.0 2.0 1.0

8 .0

Linear Region

7 .0 6 .0 5 .0 4 .0 3 .0 2 .0 1 .0

0.0 0.0

Gibbs Adsorption, mmol/g

Gibbs Adsorption, mmol/g AC

8.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

0 .0 0

Pressure, MPa

5

10

15

20

25

G a s D e n s ity , m o l/L Fig. 1. Comparison of monolayer and multilayer (3-layer) OK model representations of pure-gas adsorption on activated carbon (data for CO2 adsorption at 304 K from Humayun and Tomasko [25]).

2. Modeling of pure-gas adsorption on activated carbon (AC) 2.1. Maximum adsorbed-phase density estimate

Gibbs Adsorption, mmol/g Coal

The Ono–Kondo model has four parameters: mc , εii /k, εis /k and C. To reduce the number of parameters in the model, the maximum 2.0 Illinois #6 Coal 1.6

OK monolayer OK multilayer

1.2

0.8

0.4

0.0 0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

Pressure, MPa Fig. 2. Comparison of monolayer and multilayer (3 layer) OK model representation of pure CO2 adsorption on Illinois #6 Coal at 328.2 K.

Fig. 3. Graphical method for estimating adsorbed-phase density: CO2 adsorption on activated carbon at 318 K.

adsorbed-phase density, mc , is typically estimated independently. One commonly used approximation is the liquid density at the normal boiling point, as was done by Arri and Yee [29]. However, examination of the results from the OK model reveals that the regressed adsorbed-phase densities generated by the Ono–Kondo model, as presented in Table 2, are lower than the boiling point estimates and are closer to the reciprocal van der Waals covolume estimates. Moreover, the adsorbed-phase densities generated by the Ono–Kondo model and the reciprocal van der Waals covolumes were also close to the “graphical estimate,” which is based on the Gibbs adsorption definition:  = Vads (ads − gas ), as illustrated by Fig. 3. Here, Vads is the adsorbed-phase volume and ads and gas are the density of the adsorbed phase and the gas phase, respectively, where the gas phase densities were calculated from highly accurate reference equations of state [30–32]. As shown in Fig. 3, if the absolute adsorption Vads ads (which equals the total amount adsorbed) becomes constant at high pressures (i.e., the matrix becomes saturated with adsorbent), then the Gibbs adsorption should show a linear decrease with increasing gas . Extrapolation of this linear relation yields an x-axis intercept where ads = gas = mc . Fig. 3 indicates that ads = 22.5 mol/L or 1.02 g/cm3 . Use of this technique requires sufficient data in the linear (high pressure) region, beyond the maximum in the Gibbs adsorption; thus, an estimate is shown in Table 2 only for carbon dioxide. For the other adsorbates, the available data do not extend into the linear region.

M. Sudibandriyo et al. / Fluid Phase Equilibria 299 (2010) 238–251

241

Table 2 Adsorbed-phase densities estimated by different methods for adsorption of gases on dry activated carbon at 318 K. Adsorbed-phase density (g/cm3 )

Method

Ono–Kondo model Zhou–Gasem–Robinson (ZGR) EOS Liquid density estimate Solid density estimate Reciprocal van der Waals covolume Graphical estimate from the Gibbs adsorption

Methane

Nitrogen

CO2

0.345 0.345 0.421

0.673 0.839 0.808 – 0.725 –

0.977 0.982 – 1.18 1.03 1.02

0.374 –

In our previous publication on the simplified local density (SLD) model [13], a theoretical explanation was offered for the proximity of the adsorbed-phase densities to the equation of state (EOS) reciprocal co-volumes. For the SLD model, the adsorbed-phase density can be shown to approach the reciprocal co-volume at high pressures. 2.2. Fluid-fluid energy parameter estimate In this study, the fluid–fluid energy parameter, εii /k, was estimated as being proportional to the Lennard–Jones well depth energy parameter. For the Lennard–Jones 12-6 potential, the pairwise interaction between two molecules separated by a distance r is given by ∗

˚(r) = 4ε

 12  r

−

  6  r

(6)

where ˚(r) is the potential energy, ε* is the well depth of the potential, and  is the collision diameter, which is defined as the distance at which the potential energy is zero. If the adsorbed molecules are randomly distributed, the total energy of interaction between a molecule and all the surrounding molecules is:

∞ U1 =

˚(r)

N 6 ∗ 2 N 2rdr = − ε  A 5 A

The fluid–fluid energy parameters in Table 1, obtained for gas adsorption on activated carbon, are reasonably close to the ones obtained from the above expression; for example, the regressed value of εii /k for CO2 is 82 K compared to the 84.3 K value estimated using Eq. (12). Further, the value of the calculated Gibbs adsorption is not highly sensitive to small deviations in εii /k obtained from Eq. (12); for example, a 10% deviation in εii /k can still produce reasonable values of the Gibbs adsorption. We also observed that the fluid–fluid energy parameters are positive values, which represents a repulsive energy potential. These results confirm the observations of Benard and Chahine [17] and also agree with molecular simulation results obtained recently by Aranovich and Donohue [34]. This phenomenon is similar to the negative values obtained for parameter ˛ in 2-D equation of state [10] and is not totally unexpected in light of the fact that surface forces are significant in the adsorbed phases. The effect of solid surface potential on adsorbed molecules may affect the molecular energy at conditions such that the interaction between them becomes repulsive. Because the lattice model was also applied to the gas phase, the positive fluid–fluid energy parameter is also repulsive in the gas phase, which might be unreasonable. The use of an accurate equation of state to calculate the chemical potential in the gas phase might be more appropriate; this approach is scheduled for a future study.

(7) 2.3. Two-parameter OK model



where N is the number of molecules on the surface and A is the surface area. The total energy due to the molecular interaction is then simply [33] UT =

N 3 ∗ 2 N 2 U1 = − ε  2 5 A

(8)

In the OK model, the total energy due to molecular interactions is expressed in terms of the coordination number and the fractional coverage. Equating the two total energy expressions, for z0 = 8 (hexagonal configuration), results in UT = −

z0 3 ∗ 2 N 2 Nεii = −4Nεii = − ε  2 5 A

(9)

or canceling the negative sign on both sides gives εii =

3 ∗ 2 N ε  20 A

(10)

When the spherical particles are closely packed, the fraction occupied by those particles is 0.907, and we write [33]:  a2 N = 0.907 4 A

(11)

where “a” is the distance between the two particles at the minimum energy potential, a = (2)1/6 . Combining Eqs. (10) and (11) yields the following estimate for the fluid–fluid energy parameter in the OK model: εii = 0.432ε∗

(12)

As described above, the adsorbed-phase density and the fluid–fluid energy parameter can be estimated from the reciprocal van der Waals co-volume and from a proportional relation to the well depth of the Lennard–Jones 12-6 potential, respectively. For further generalization, the above estimates were applied for modeling of gas adsorption on activated carbon and zeolites reported in the literature and on our gas adsorption measurements on activated carbon. In this case, the fluid–solid energy parameter, εis /k, was regressed for each specific adsorption system and the parameter C was regressed for each adsorption isotherm. Because the available literature data do not usually contain detailed information on experimental uncertainties, the percentage average absolute error of the Gibbs adsorption (%AAD) was used as the objective function to determine the two model parameters. Table 3 presents the results of our model representations of the above selected data. Overall, for 2243 data points, the OK model with two regressed parameters (one common εis /k for each system, and individual C for each isotherm of a system) represents the data with about 3.6%AAD. However, some significantly larger errors were observed, especially for the CO2 and propane isotherms. Also, the percentage deviation is exaggerated when the Gibbs adsorption becomes exceedingly small; i.e., at lower pressure, at high temperature, or at higher pressures when the excess adsorption becomes near zero. The near-zero excess adsorption amounts leads to larger percentage errors, although the uncertainty in terms of amounts adsorbed are small. For the OSU adsorption data set (Systems 4750), the OK model is capable, on average, of representing the data within their expected experimental uncertainties.

242

Table 3 Results of two-parameter OK model for pure-gas adsorption. System

Adsorbent

no.

Adsorbent surface area (m2 /g)

Parameters Gas

NPTs

AC, Columbia Grade L

1152

N2

36

2

AC, Columbia Grade L

1152

CH4

45

3

AC, Columbia Grade L

1152

C2 H6

58

4

Charcoal

1157

CH4

55

5

Charcoal

1157

C3 H8

52

6

AC, BPL

988

CH4

72

7

AC, BPL

988

CO2

60

8

AC, BPL

988

C2 H6

49

9

AC, BPL

988

C2 H4

52

10

AC, PCB-Calgon

1150–1250

CH4

22

11

AC, PCB-Calgon

1150–1250

CO2

12

12

AC, F30/470 Chemviron Carbon

CO2

164

993.5

−εis /k (K)

C (mmol/g)

310.9 338.7 366.5 394.3 422.0 310.9 338.7 366.5 394.3 422.0 310.9 338.7 366.5 394.3 422.0 449.8 477.6 283.2 293.2 303.2 313.2 323.2 293.2 303.2 313.2 323.2 333.2 212.7 260.2 301.4 212.7 260.2 301.4 212.7 260.2 301.4 212.7 260.2 301.4 296 373 480 296 373 480 278 288 298 303 308 318 328

1090

3.071 2.947 2.724 2.583 2.756 3.692 3.546 3.217 2.847 2.737 3.574 3.147 2.865 2.706 2.434 2.381 2.159 5.037 4.834 4.710 4.521 4.332 3.774 3.662 3.592 3.486 3.375 4.515 3.880 3.279 9.753 6.344 5.110 4.090 3.588 3.157 4.387 3.758 3.281 3.994 3.191 2.547 6.123 4.170 2.910 5.741 5.620 5.341 5.326 5.143 4.808 4.598

1410

2190

1330

2490

1320

1520

2000

1935

1380

1600

1625

%AAD

RMSE (mmol/g)

Reference

3.1

0.015

Ray and Box [42]

2.6

0.049

Ray and Box [42]

4.4

0.145

Ray and Box [42]

1.2

0.051

Payne et al. [43]

6.7

0.200

Payne et al. [43]

3.1

0.112

Reich et al. [35]

4.9

0.325

Reich et al. [35]

6.3

0.318

Reich et al. [35]

4.5

0.234

Reich et al. [35]

3.2

0.066

Ritter and Yang [44]

4.6

0.230

Ritter and Yang [44]

1.2

0.138

Berlier and Frére [45]

M. Sudibandriyo et al. / Fluid Phase Equilibria 299 (2010) 238–251

1

T (K)

Table 3 (Continued) System

Adsorbent

no.

Adsorbent surface area (m2 /g)

Parameters NPTs

1450 1450 1450 3106

N2 CH4 CO2 CH4

10 12 12 122

850

CO2

116

13 14 15 16

AC, Norit R1 Extra AC, Norit R1 Extra AC, Norit R1 Extra AC, Coconut Shell

17

AC, Calgon F-400

18

AC, Norit RB1

1100

CH4

64

19

AC, Norit RB1

1100

CO2

64

20

AC, Coconut Shell

3106

N2

71

21

AC F30/470

993.5

N2

116

22

AC F30/470

993.5

CH4

122

23

AC F30/470

993.5

C3 H8

102

T (K)

−εis /k (K)

C (mmol/g)

298 298 298 233 253 273 293 313 333 303.6 305.2 309.2 313.2 318.2 304.9 311.4 331.3 350.5 305.2 11.2 329.5 348.3 178 198 218 233 258 278 298 303 323 343 363 383 303 323 343 362 383 303 323 343 363 383

1050 1390 1600 1140

3.409 4.335 6.773 11.650 11.221 10.578 9.926 9.301 8.647 4.847 4.843 4.667 4.530 4.443 3.731 3.171 3.024 2.875 5.839 4.788 4.301 3.884 11.026 10.664 9.823 9.000 8.537 7.703 7.424 2.617 2.457 2.284 2.134 2.008 3.625 3.420 3.207 2.987 2.809 3.045 2.877 2.802 2.654 2.487

1700

1480

1655

880

1135

1395

2550

%AAD

RMSE (mmol/g)

Reference

3.3 3.3 9.5 3.0

0.021 0.124 0.267 0.237

Dreisbach et al. [46] Dreisbach et al. [46] Dreisbach et al. [46] Zhou et al. [47]

8.8

0.306

Humayun and Tamasko [25]

1.5

0.030

van Der Vaart et al. [48]

1.7

0.060

van Der Vaart et al. [48]

1.5

0.162

Zhou et al. [49]

2.0

0.044

Frére and De Weireld [50]

2.2

0.094

Frére and De Weireld [50]

2.3

0.173

Frére and De Weireld [50]

M. Sudibandriyo et al. / Fluid Phase Equilibria 299 (2010) 238–251

Gas

243

244

AC Norit R1 AC Norit R1 Zeolite, Linde 13X

1262 1262 525

N2 CO2 N2

31 29 24

27

Zeolite, Linde 5A

∼400

N2

27

28

Zeolite, Linde 5A

∼400

CH4

28

29

Zeolite, Linde 5A

∼400

CO2

41

30

Zeolite, Linde 5A

∼400

C2 H6

27

31

H-Modernite, Z-900H

∼300

CO2

93

32

H-Modernite, Z-900H

∼300

H2 S

69

33

H-Modernite, Z-900H

∼300

C3 H8

92

34

Zeolite G5

430

CH4

51

35

Zeolite G5

430

C2 H6

40

36

Zeolite G5

430

C2 H4

34

37 38 47 48 49

Zeolite 13X Zeolite 13X AC, Calgon F-400 AC, Calgon F-400 AC, Calgon F-400

383 383 850–998 850–998 850–998

CH4 C2 H6 N2 CH4 CO2

12 11 22 40 52 10

50

AC, Calgon F-400 Overall

850–998

C2 H6

21 2243

298 298 298 323 348 298 323 348 298 323 348 298 323 348 298 323 348 283 303 323 283 303 338 368 283 303 324 283 303 283 303 283 303 298 298 318.2 318.2 318. 328.2 318.2

1070 1500 1185

1310

1600

2500

2600

2600

3550

3150

1200 2525 2630 1500 2380 1060 1385 1710 2135

3.655 7.513 1.627 1.480 1.329 1.525 1.403 1.284 1.559 1.449 1.336 2.136 2.060 1.990 1.158 1.098 1.023 1.634 1.548 1.434 2.108 1.998 1.847 1.734 0.618 0.603 0.562 3.352 3.100 1.699 1.620 1.923 1.798 1.699 1.575 2.577 3.164 4.500 4.259 2.835

1.5 3.6 2.2

0.035 0.269 0.042

Beutekamp and Harting [51] Beutekamp and Harting [51] Wakasugi et al. [36]

1.3

0.032

Wakasugi et al. [36]

2.7

0.048

Wakasugi et al. [36]

5.6

0.317

Wakasugi et al. [36]

2.1

0.051

Wakasugi et al. [36]

11.2

0.090

Talu and Zwiebel [52]

2.2

0.032

Talu and Zwiebel [52]

7.2

0.045

Talu and Zwiebel [52]

3.9

0.055

Berlier et al. [53]

1.3

0.044

Berlier et al.[53]

1.1

0.048

Berlier et al. [53]

9.8 2.1 0.4 1.4 4.4 3.8

0.131 0.062 0.012 0.064 0.222 0.206

Beutekamp and Harting [51] Beutekamp and Harting [51] Sudibandriyo et al. [24] Sudibandriyo et al. [24] Sudibandriyo et al. [24]

6.4 3.6

0.153

Sudibandriyo et al. [24]

M. Sudibandriyo et al. / Fluid Phase Equilibria 299 (2010) 238–251

24 25 26

M. Sudibandriyo et al. / Fluid Phase Equilibria 299 (2010) 238–251

245

-0.8

40 30

-1.2

ln (1/C, mmol/g)

% Deviation

20 10 0 -10 -20

-1.6 Ethane Ethylene Methane CO2

-2.0

-30 -40 0.0

2.0

4.0

6.0

8.0

-2.4 200

10.0 12.0 14.0 16.0 18.0 20.0 22.0

Pressure, MPa

220

240

260

280

300

320

T, K Fig. 4. Deviation plot for two-parameter OK model representation of gas adsorption on activated carbons.

Fig. 7. Temperature dependence of the OK model parameter C for on BPL activated carbon [data from Reich et al. [35]].

Gibbs Adsorption, mmol/g AC

14.0 12.0 10.0

CO2 CH4 N2 OK Model

2.4. Basis for generalization

8.0 6.0 4.0 2.0 0.0 0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

Pressure, MPa

C(T ) =

Fig. 5. Ono Kondo model representation of gas adsorption on Norit R1 extra activated carbon at 298 K [data from Dreisbach et al. [46]].

Fig. 4 presents the percentage deviations for the OK model representation of the adsorption data on activated carbon. About 90% of the data can be represented by the model within 8.4%AAD. As mentioned above, the large percentage deviations occur mainly when the Gibbs adsorption values are small at relatively low pressures. Fig. 5 illustrates the OK model representation of System 15 (CO2 on Norit R1 Extra activated carbon), where the percentage deviations are large (9.5%AAD). However, as shown in the figure, the model can actually represent the experimental data reasonably well at pressures above 0.5 MPa. The nitrogen and methane adsorption on the same activated carbon are also shown in the figure for comparison. Fig. 6 illustrates the OK model representation of a system over a wide range of temperatures. As shown in this figure, the model is capable of describing the temperature variation for this system. 18.0

Gibbs Adsorption, mmol/g AC

Based on our preliminary evaluation of the regressed parameter C for a given adsorbate at fixed temperature presented in Table 3, the value of C increases as the surface area of the adsorbent increases. This suggests that the maximum adsorption capacity, C, can be divided into two contribution parts; i.e., the contribution from the adsorbent characteristic, represented by surface area (A, m2 /g), and the contribution from the adsorbate characteristics. In this study, we proposed the following simple relation for the maximum adsorption capacity, C: ACa (T ) 2

(13)

where Ca is the surface adsorbed-phase density (mmol/m2 ), with its value depending only on the adsorbate. C is also temperature dependent. As reported in Table 3, the value of C increases as temperature decreases. This temperature dependence of the maximum adsorption capacity is not uncommon, based on previous studies. Benard and Chahine [17,18] reported the temperature dependence of C, and they proposed an empirical relation between C and temperature T. Similarly, Do [33] noted that the maximum adsorption capacity in the Langmuir model was a function of temperature. He described this temperature dependence as due to the thermal expansion of the adsorbed phase. Because of its theoretical basis, we adopted Do’s approach and used the following thermal expansion expression for evaluating the surface adsorbed-phase density: dCa /dT = −ı Ca

(14)

Integrating Eq. (14) and combining with Eq. (13) results in:

16.0 14.0

ln

12.0 10.0 8.0 233 K 273 K 313 K OK Model

6.0 4.0 2.0

253 K 293 K 333 K

0.0 0.0

2.0

4.0

6.0

8.0

10.0

Pressure, MPa Fig. 6. Ono Kondo model representation of methane adsorption on activated carbon [data from Zhou et al. [47]].

1 C



= ıT − ln Ca,o + ıTo + ln

 A  2

(15)

where To (K) is chosen at the normal boiling point of the adsorbate (triple point for CO2 ), T (K) is the absolute temperature, Ca,o is the maximum surface adsorbed-phase density at To , and ı is the thermal expansion coefficient of the adsorbed phase. For a given system, Eq. (15) yields a linear correlation if ln (1/C) is plotted as function of temperature T. Fig. 7 presents the correlation between ln (1/C) obtained from Table 3 for gas adsorption on BPL activated carbon [35] and temperature T. The linear relationship shown in this figure suggests that Eq. (15) provides a useful representation of the temperature dependence of the maximum adsorption capacity, C. Note that this linear dependence is only

246

M. Sudibandriyo et al. / Fluid Phase Equilibria 299 (2010) 238–251 Table 4 Physical properties of the adsorbatesa and adsorbents.

12 CO2

10

C, mmol/g

Methane Nitrogen

8 6 4 2 0 0

1000

2000

3000

4000

2

Surface Area, m /g Fig. 8. Correlation of Ono Kondo model parameter C with surface area at constant temperature.

Adsorbate/ adsorbent

Normal boiling point (K)

Reciprocal van der Waals co-volume (mol/L)

 × 1010 (m)

ε*/k (K)

H2 N2 H2 S CO2 CH4 C2 H4 C2 H6 C3 H8 i-C4 H10 Carbon O (zeolite)

20.4 77.3 212.8 216.6 b 111.7 169.4 184.6 231.1 261.4

38.16 25.89 23.08 23.34 23.37 17.39 15.41 11.07 8.60

2.827 3.798 3.623 3.941 3.758 4.163 4.443 5.118 5.278 3.4c 3.04d

59.7 71.4 301.1 195.2 148.6 224.7 215.7 237.1 330.1 28c 139.96d

a b c d

2 ˚i,SLP = 4c ε∗iC iC

  10 1 iC 5

z

−

 4  1  iC

2

z

(16)

The potential energy has a minimum at a depth given by: εis =

6 2 c ε∗iC iC 5

(17)

where c = 0.382 atom/Å2 is the area density of carbon atoms in a graphite plane. The adsorbate–carbon collision diameter is estimated as iC = 12 (ii + CC ). The adsorbate–carbon well depth



ε∗ii ε∗CC . potential is estimated as ε∗iC = Generalization of the model parameters was performed by evaluating Ca,o , ı and ε∗CC of the systems studied. All other physical properties of the adsorbate and carbon atom used are listed in Table 4. Five case studies described in Table 5 were explored. For each case, 1520 independent adsorption data points were employed. These data include all the activated carbon systems shown in Table 3 which contain multiple isotherms. The singleisotherm systems were used later to validate the generalized model. In Case 1, all the parameters, Ca,o , ı and ε∗CC , were optimized simultaneously, while the surface area was taken from information provided in the literature. Table 6 shows the parameters obtained in Case 1. The regressed solid–solid energy parameter, ε∗CC /k, varied from 38 to 43 K. For all components studied, except CO2 , the mean thermal expansion coefficient of the adsorbed phase, ı, was approximately 0.0024 K−1 . This is close to the value of 0.0025 K−1 estimated by Wakasugi et al. [36] for all components. However, the regressed ı for CO2 is much higher (0.0039 K−1 ).

0.012

2

0.011

Ca,o , mmol/m

approximate for CO2 . However, from the limited data available, the linearity assumption was adopted for CO2 systems. The values of Ca,o and ı depend only on the adsorbate; therefore, a linear correlation should also be obtained if C is plotted against A at constant temperature. Fig. 8 presents the plots of parameter C obtained in Table 3 against the surface area, A. The values of the surface areas, in most cases, were reported by the investigators. In a few cases, however, they were obtained from the information provided by the adsorbent production company or estimated from the literature. Although the accuracies of the reported surface areas are questionable, Fig. 8 still shows a reasonable linear correlation between the parameter C and the surface area, A. This observation further supports the assumption made in Eq. (13). The fluid–solid energy parameter, εis /k, was generalized based on the interaction of a single molecule with a single lattice plane. If z is the distance between the adsorbate molecule i and the lattice carbon plane, the potential energy can be written as [33]:

Ref [54]. Triple point temperature. Ref. [55]. Ref. [56].

0.01

y = 0.1019x + 0.0034 R2 = 0.99

0.009 0.008 0.007 0.006 0.03

0.04

0.05

0.06 2

0.07

0.08

-2

1/σ , Å

Fig. 9. Variation of parameter Ca,o with the diameter of the molecules, .

The maximum surface adsorbed-phase density at To (normal boiling point), Ca,o , obtained in Case 1 was correlated with the diameter of the adsorbed molecules. Fig. 9 shows a linear correlation between Ca,o and the reciprocal square of the molecule diameter. This finding is not unexpected since for a close-packed hard sphere √ molecule, the surface density will be equal to 2/Nav  2 , where Nav is the Avogadro number. However, Ca,o is much higher for CO2 , and it does not follow the observed linear trend for the other adsorbates. In Case 2, the general linear correlation for Ca,o obtained in Case 1 was applied. A constant ı obtained in Case 1 was also used in Case 2. As shown in Table 5, the overall percentage deviations obtained in Case 1 are practically the same as for Case 2. The overall percentage deviations did not change significantly (about 7%AAD) when the solid–solid energy parameter, ε∗CC /k, was set constant at 40 K in Case 3. Slightly better results were obtained when the surface area was regressed, as in Case 4 (6.4%AAD). In this case, the regressed surface area deviated up to 8% from the reported surface area. Moreover, as shown in Case 5, using the regressed surface area and a constant solid–solid energy parameter, ε∗CC /k = 40 K, the model can predict the adsorption isotherms with an AAD of 7%. Fig. 10 shows the percentage deviation plot for the generalized OK model prediction of the adsorption data on activated carbon (Case 5). About 90% of the data can be predicted within 14.5%AAD. Moreover, the model predictions appear to be fairly accurate at higher pressures, as seen from the figure. Fig. 11 illustrates the comparison of generalized (Case 5) and two-parameter OK model

M. Sudibandriyo et al. / Fluid Phase Equilibria 299 (2010) 238–251

247

Table 5 Description of the cases studied in model generalization. Case no.

Description

Overall %AAD

1

Based on reported surface area, A Ca,o and ı are regressed for specific adsorbate εcc /k is regressed for specific activated carbon

7.25

2

Based on reported surface area, A Generalized Ca,o and ı: For CO2 : Ca,o = 0.0142 mmol/m2 ; ı = 0.0039 K−1 For other components: Ca,o = 0.102/ 2 + 0.0034;  in Å ı = 0.0024 K−1 εcc /k (K) is obtained from Case 1 (varied from 38 to 43)

7.33

3

Based on reported surface area, A Generalized Ca,o and ı: For CO2 : Ca,o = 0.0142 mmol/m2 ; ı = 0.0039 K−1 For other components: Ca,o = 0.102/ 2 + 0.0034;  in Å ı = 0.0025 K−1 εcc /k (K) = 40

7.44

4

Surface area, A, is regressed Generalized Ca,o and ı: For CO2 : Ca,o = 0.0142 mmol/m2 ; ı = 0.0039 K−1 For other components: Ca,o = 0.102/ 2 + 0.0034;  in Å ı = 0.0023 K−1 εcc /k (K) is obtained from Case 1 (varied from 38 to 43)

6.34

5

Surface area, A, is regressed Generalized Ca,o and ı: For CO2 : Ca,o = 0.0142 mmol/m2 ; ı = 0.0039 K−1 For other components: Ca,o = 0.102/ 2 + 0.0034;  in Å ı = 0.0023 K−1 εcc /k (K) = 40

7.00

40 30

% Deviation

20 10 0 -10 -20 -30 -40 0.0

2.0

4.0

6.0

8.0

10.0 12.0

14.0 16.0

18.0 20.0

22.0

Pressure, MPa Fig. 10. Deviation plot for the generalized OK model predictions of gas adsorption on activated carbon (case 5 from Table 5).

representation of methane adsorption on activated carbon (System 22). As illustrated in the figure, the generalized model predicts the adsorption isotherms with about twice the error of the twoparameter OK model. The results of the five cases above suggest that the gas adsorption can be predicted using the generalized OK model (Case 5) with the average deviation of about 7%AAD. Several gas adsorption systems were used subsequently to validate this generalized OK model. Fig. 12 shows the generalized OK model prediction of our gas adsorption measurements on activated carbon at 318.2 K. A surface area of 920 m2 /g and the solid–solid energy parameter, ε∗CC /k, of 39 K, were obtained from the best fit of the CO2 adsorption data. Using this information and the generalized Ca,o and ı,the adsorption isotherms for the other three gases were then predicted. A similar prediction procedure was applied for the other systems, confirming that the gas adsorption on activated carbon can be predicted using the generalized OK model with about 7%AAD or about twice the error of the two-parameter OK model.

Gibbs Adsorption, mmol/g AC

6.0

3. Modeling of pure-gas adsorption on coals 3.1. Two-parameter OK model for pure-gas adsorption on coals 4.0

Similar to our treatment of pure-gas adsorption on activated carbon, we modeled pure-gas adsorption on coals using the 303 K 323 K 343 K 362 K 383 K Two-Parameter OK Model Generalized OK Model

2.0

Table 6 Parameters used in case 1 of model generalization.

0.0 0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

Pressure, MPa Fig. 11. Comparison of the generalized predictions and two-parameter OK model representations of methane adsorption on activated carbon [Frere and De Weireld [50]].

Component

Ca,o (mmol/m2 )

ı (K−1 )

N2 CH4 C2 H4 C2 H6 C3 H8 CO2 Activated carbon

0.0105 0.0106 0.00945 0.00831 0.00735 0.0142

0.0024 0.0024 0.0024 0.0024 0.0024 0.0039

εcc /k (K)

38–43

248

M. Sudibandriyo et al. / Fluid Phase Equilibria 299 (2010) 238–251

Table 7 Results of two-parameter OK model for pure-gas adsorption on coals. System no.

Adsorbent

Adsorbate

55a 56a 57a 55b 56b 57b 61 62 63 67 68 69 74 75 76 77 78 79 80 81 82 83 84 85 86

Wet Fruitland Coal #1 Wet Fruitland Coal #1 Wet Fruitland Coal #1 Wet Fruitland Coal #2 Wet Fruitland Coal #2 Wet Fruitland Coal #2 Wet Illinois #6 Coal Wet Illinois #6 Coal Wet Illinois #6 Coal Wet Tiffany Coal Wet Tiffany Coal Wet Tiffany Coal Wet LB Fruitland Coal Wet LB Fruitland Coal Wet LB Fruitland Coal Dry Illinois #6 Coal Dry Beulah Zap Coal Dry Wyodak Coal Dry Upper Freeport Coal Dry Pocahontas Coal Wet Illinois #6 Coal Wet Beulah Zap Coal Wet Wyodak Coal Wet Upper Freeport Coal Wet Pocahontas Coal Overall

N2 CH4 CO2 N2 CH4 CO2 N2 CH4 CO2 N2 CH4 CO2 N2 CH4 CO2 CO2 CO2 CO2 CO2 CO2 CO2 CO2 CO2 CO2 CO2

Parameters NPTS

T (K)

−εis /k (K)

C (mmol/g)

%AAD

RMSE (mmol/g)

WAAE

20 20 14 20 20 20 20 20 20 21 22 16 16 16 29 11 22 11 11 11 13 11 12 12 12 360

319.3 319.3 319.3 319.3 319.3 319.3 319.3 319.3 319.3 327.5 327.5 327.5 319.3 319.3 319.3 328 328 328 328 328 328 328 328 328 328

610 990 1340 575 960 1280 450 780 1100 530 930 1385 530 815 1300 1250 1375 1270 1480 1540 950 1060 670 1310 1370

0.572 0.650 0.853 0.495 0.663 0.887 0.331 0.468 0.735 0.287 0.356 0.433 0.213 0.327 0.354 1.204 1.3540 1.5039 0.6768 0.7979 0.8427 0.3482 0.6332 0.6277 0.7199

1.3 0.8 5.1 1.8 0.8 6.7 2.4 1.7 3.2 3.3 3.4 4.2 4.3 2.6 5.4 5.1 3.3 2.9 2.2 1.8 1.8 7.2 14.7 4.7 3.4 3.8

0.005 0.004 0.091 0.004 0.005 0.087 0.002 0.005 0.033 0.003 0.012 0.022 0.004 0.007 0.023 0.067 0.071 0.064 0.026 0.023 0.010 0.057 0.022 0.034 0.019

0.2 0.2 0.5 0.2 0.4 0.7 0.2 0.3 0.5 0.4 0.6 0.6 0.4 0.5 0.6 0.9 0.5 0.5 0.4 0.3 0.2 0.4 0.3 0.4 0.3 0.4

adsorbed-phase density and the fluid–fluid energy parameters estimated from the reciprocal van der Waals covolume and from the Lennard–Jones 12-6 potential, respectively. Our gas adsorption measurements on several coals were used to evaluate the fluid–solid energy parameter, εis /k, and the parameter C in the OK model. The weighted average absolute error of the Gibbs adsorption (WAAE) was used as the objective function to determine the two model parameters. The weights used were the expected experimental uncertainties determined through multivariate error propagation analysis. Table 7 presents the results of our model representation of the above selected data. Overall, for 308 data points, the OK model with two regressed parameters per isotherm can represent the data within the expected experimental uncertainties, which corresponds to 3.8%AAD. Specifically, the two-parameter OK model representation of gas adsorption on wet Fruitland coal gives comparable results to three-parameter models such as the 2-D EOS used by Zhou et al. [10].

Gibbs Adsorption, mmol/g AC

8.0 CO2 CH4 N2 C2H6 OK Model

7.0 6.0 5.0

3.2. Model generalization for pure-gas adsorption on coals The parameter C in the gas adsorption model on coals was determined by the same approach used for activated carbon. Specifically, we used a generalized Ca,o and ı obtained from the modeling on activated carbon. The surface areas of the coals, however, were regressed from the experimental adsorption data, since the surface area is very specific to the coal samples and is also dependent on the wetness of the coals studied. The regressed surface areas of wet coals should be viewed as “accessible” surface area for each gas, rather than the actual surface area of the coal surface. The fluid–solid energy parameter, εis /k, for adsorption on coals, as shown in Table 7, is much lower than that on activated carbon. This lower value may be attributed to the mean position of a molecule in the structure of the coals and the presence of water. Fig. 13 illustrates the molecule positions in the slit of the activated carbon and coals. In activated carbon, the distance between a molecule and surface plane, z, is approximately equal to the collision diameter of the two atoms, i.e., z =  ic . Due to wider pores of the coals [27], the distance between a molecule and surface plane of the coal might be slightly higher than  ic . Therefore, the fluid–solid energy for pure adsorption on coals was calculated according to Eq. (16) instead of Eq. (17). Further, the chemical structure of the coal

Coal

AC

4.0 3.0 2.0

0.0 0.0

L

L

1.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

z

Pressure, MPa Fig. 12. OK generalized model predictions of gas adsorption on activated carbon at 318.2 K [data from Sudibandriyo et al. [24]].

Fig. 13. Molecule positions in the slit of the activated carbon and coal.

z

M. Sudibandriyo et al. / Fluid Phase Equilibria 299 (2010) 238–251

is much more complex than that of activated carbon. The wetness of the coal may also affect the regressed overall fluid–solid energy. Therefore, we proposed a modification of Eq. (16):

z

−

iC

2

z

Pure-gas adsorbed Methane

(18)

A correction, f was introduced to account for the effect of adsorbent chemical structure on the adsorbate. This value is close to zero for adsorption on activated carbon, and was generalized based on the chemical composition of the coals. A moisture correction effect, ϕc , was calculated as a function of the ash-free coal not occupied by equilibrium moisture content. A more rigorous way to handle the water effect would be to use a multicomponent model and to treat the water as an additional adsorbed component. In our current stage of study, however, such a model is under development. For simplicity, therefore, the correction above was introduced in this study. In summary, the model generalization for gas adsorption on coals was performed in the following sequence: 1. Parameter C was calculated using Eq. (13), where generalized Ca,o and ı as shown in Table 5, Case 2, were applied. The surface area, A, was optimized for each coal. 2. The fluid–solid energy parameter, εis /k, was calculated using Eq. (18), where the distance between a molecule and surface plane, z, was optimized for each coal, and the correction f was optimized for a specific adsorbate-coal system. The solid–solid energy parameter, ε∗CC /k, was set at 40 K. 3. The correction f was generalized by the following contribution method: f = RVC (axO + bxH + cxN + dxS + e)

Model parameters

(19)

where RVC is the ratio of the volatile to the fixed carbon content (on dry basis) and xO , xH , xN and xS are the oxygen, hydrogen, nitrogen and sulfur contents in the coals (on dry basis), respectively. Several researchers have attempted to correlate the gas adsorption capacity with the carbon content or the ratio of volatile to fixed carbon content and the oxygen content [37–41]. Their hypothesis is that the chemical contents of the coal might produce a specific available surface area of the coal. However, the chemical contents also reflect the types of functional groups attached in carbon matrices, which affect the fluid–solid interaction energy. We do not evaluate the chemical contents effect on the surface area, and the surface area is regressed in this study. However, we used an empirical correlation shown by Eq. (19) to account for the effect of chemical content to the fluid–solid interaction energy. Our pure-gas adsorption data on activated carbon, Fruitland coal, Lower Basin (LB) Fruitland coal and Illinois #6 coal were employed to evaluate the correction function in Eq. (19). Because of the limited available data, only three factors in Eq. (19) were used for evaluation, i.e., the ratio of the volatile to the fixed carbon contents, the oxygen content and the hydrogen content (the coefficients c and d are assumed zero). Based on our evaluation, we found that the coefficients in Eq. (19) are a = 0.141, b = 0, e = 0.075 for CO2 ; a = 0, b = 0.166, e = 0.033 for methane; and all are zero for nitrogen. Tables 8 and 9 show the parameters used in the generalized OK model for pure-gas adsorption on wet and dry coals. The generalized OK model can represent the data with about 5.3%AAD and 3.0%AAD for the gas adsorption on wet and dry coals, respectively. Figs. 14–17 present the comparison of the two-parameter OK model representations and generalized OK model predictions for gas adsorption on wet coals (Figs. 14–16) and CO2 adsorption on dry coals (Fig. 17). As shown in the figures, the generalized model can predict the data almost as well as the two-parameter model does. Larger deviations, however, are obtained for the CO2 adsorption

Wet Fruitland Coal (ϕc = 0.97) A (m2 /g) z NPTS 20 %AAD 3.2 RMSE (mmol/g) 0.019 Wet Illinois #6 Coal (ϕc = 0.97) A (m2 /g) z NPTS 20 %AAD 2.6 RMSE (mmol/g) 0.008 Wet LB Fruitland (ϕc = 0.95) A (m2 /g) z NPTS 16 %AAD 4.9 RMSE (mmol/g) 0.019

Nitrogen

CO2

194 1.296  ic 20 4.8 0.013

20 8.5 0.087

150 1.448  ic 20 4.4 0.004

20 3.5 0.037

77 1.296  ic 16 6.4 0.007

29 8.3 0.036

0.4

Gibbs Adsorption, mmol/g Coal

εis 2 ϕc (1 + f ) = 4c ε∗iC iC 5 k

 4  1 

Table 8 Summary results for the OK model of pure-gas adsorption on wet coals.

Fruitland Illinois #6 LB-Fruitland

0.3

2 Parameter OK Model Generalized OK Model

0.2

0.1

0.0 0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

Pressure, MPa Fig. 14. OK model representation of nitrogen adsorption on wet coals at 319.3 K.

1.0

Gibbs Adsorption, mmol/g Coal

˚i,SLP =

   1 siC 10

249

0.8

Fruitland Illinois #6 LB-Fruitland 2 Parameter OK Model Generalized OK Model

0.6

0.4

0.2

0.0 0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

Pressure, MPa Fig. 15. OK model representation of methane adsorption on wet coals at 319.3 K.

in the region above 8 MPa. The low-adsorbing nature of coal and the high uncertainty of the OK model calculated CO2 bulk density may amplify the relative deviations in this region. The possibility of coal swelling when CO2 is adsorbed at high pressure may also contribute to this large deviation. Pure-gas adsorption on Tiffany coal at 327.6 K was used to validate this generalized OK model. Fig. 18 shows the generalized OK model prediction of our gas adsorption measurements on Tiffany coal. The surface area (=100 m2 /g) and the distance between a

250

M. Sudibandriyo et al. / Fluid Phase Equilibria 299 (2010) 238–251

Table 9 Summary results for the OK model of CO2 adsorption on dry coals. Model parameters 2

Gibbs Adsorption, mmol/g Coal

A (m /g) z NPTS %AAD RMSE (mmol/g)

Beulah Zap

Wyodak

Illinois #6

Upper Freeport

Pocahontas

298 1.657  ic 22 3.3 0.068

332 1.653  ic 11 2.9 0.058

251 1.530  ic 11 4.4 0.062

149 1.277  ic 11 2.5 0.025

177 1.163  ic 11 1.8 0.024

1.6 Fruitland Illinois #6 LB-Fruitland 2 Parameter OK Model Generalized OK Model

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

Pressure, MPa Fig. 16. OK model representation of CO2 adsorption on wet coals at 319.3 K.

adsorption on coal can be predicted using the generalized OK model within the expected experimental uncertainties. Application of the generalized model to our recent CO2 adsorption on wet Illinois #6 Argonne Coal, however, gave unsatisfactory results. About 11%AAD was obtained using this generalized model. Specifically, the generalized model over-predicted the fluid–solid energy parameter, which resulted in over-predicted Gibbs adsorption at lower pressures. These results suggest that the water effect on the fluid–solid energy parameter is somewhat more complicated than shown in Eq. (19). As stated earlier, the adsorption on the wet substrate should be treated as a multicomponent adsorption, where water is an adsorbed component. Further study on the modeling of effects of water on gas adsorption behavior is needed. In order to support such a study, additional data for other adsorbates (methane and nitrogen) on wet coals at water contents less than equilibrium moisture content are needed. In addition, the measurements of pure water adsorption on coals would be needed.

Gibbs Adsorption, mmol/g Coal

2.0

4. Summary

1.8 1.6 1.4 1.2 1.0 0.8

W yodak Beulah Zap Illinois #6 Pocahontas Upper Freeport 2 Parameter OK Model Generalized OK Model

0.6 0.4 0.2 0.0 0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

Pressure, MPa

Gibbs Adsorption, mmol/g Coal

Fig. 17. OK model representation of CO2 adsorption on dry coals at 328.2 K.

1.0 CO2 CH4 N2 Generalized OK Model OK Model Prediction

0.8

0.6

0.4

0.2

0.0 0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

Pressure, MPa Fig. 18. OK model prediction of pure-gas adsorption on wet tiffany coal at 327.6 K.

molecule and surface plane, z = 1.29 ic , are obtained from the best fit for the CO2 adsorption. Then, using this CO2 -based information and the generalized Ca,o , ı and f ,the nitrogen and methane gas adsorption isotherms are predicted. As shown in this figure, the gas

We have shown in this study that the OK monolayer model appears effective in modeling pure-gas adsorption on carbon matrices. On average, the OK model with two regressed parameters (one common εis /k for each system, and individual C for each isotherm) can represent the isotherm adsorption on activated carbon with about 3.6%AAD. Specifically, for the OSU adsorption data set, the two-parameter OK model is capable, on average, of representing the data within their experimental uncertainties. The adsorbedphase density and the fluid–fluid energy parameters in the OK model can be estimated from the reciprocal van der Waals covolume and from a proportional relation to the well depth of the Lennard–Jones 12-6 potential, respectively. Generalization of the model parameters was performed using adsorbent surface area and physical properties of adsorbates. The results show that the generalized OK model can predict the adsorption isotherms on activated carbon with about 7%AAD or twice the error of the two-parameter OK model. Moreover, the model can also predict the adsorption isotherms of other gases based on parameters (surface area and the solid–solid energy, ε∗CC /k) obtained from one gas adsorption isotherm. The generalized OK model also appears effective for the pure-gas adsorption on wet coals when the moisture content of the coal is above its equilibrium moisture content (EMC). However, the model parameters in this condition (relative to the dry coal) are affected by the presence of water. Moreover, the generalized model was unable to predict the adsorption on a wet coal, which has moisture content less than its EMC, as was observed for wet Illinois #6 coal. These results suggest that the water effect on the fluid–solid energy parameter is more complicated than that shown in the generalized model. Therefore, the adsorption on the wet substrate should be treated properly as multicomponent adsorption, where water is recognized as one of the adsorbed components. List of symbols A surface area per unit mass of adsorbent C maximum adsorption capacity in OK model

M. Sudibandriyo et al. / Fluid Phase Equilibria 299 (2010) 238–251

Ca Ca,o m T To xads xb xt z z0 z1

surface adsorbed-phase density maximum surface adsorbed-phase density at To number of layers in the lattice model temperature normal boiling point of the adsorbate (or triple point for CO2 ) fractional coverage of adsorbate in the monolayer lattice model fraction of sites occupied by the molecules in the bulk layer of the lattice model fraction of sites occupied by the molecules in the tth adsorbed layer of the lattice model distance from a molecule to the surface lattice coordination number parallel coordination number representing the number of primary nearest-neighbor cells in parallel direction

Greek symbols ı thermal expansion coefficient of the adsorbed phase ε* well-depth of the Lennard–Jones 12-6 potential εii fluid–fluid interaction energy parameter in the OK model fluid–solid interaction energy parameter in the OK model εis  Gibbs excess adsorption per unit mass of adsorbent mc adsorbed-phase density corresponding to the maximum adsorption capacity c density of atoms or molecules on the surface  diameter of a molecule References [1] I. Langmuir, Journal of American Chemical Society 40 (1918) 1361–1403. [2] A.L. Myers, J.M. Prausnitz, AIChE Journal 11 (1965) 121–127. [3] A.L. Myers, Molecular thermodynamics of adsorption of gas and liquid mixtures, in: A.I. Liapis (Ed.), Fundamentals of Adsorption, Engineering Foundation, New York, 1987. [4] S. Suwanayuen, R.P. Danner, AIChE Journal 26 (1980) 68–76. [5] S. Suwanayuen, R.P. Danner, AIChE Journal 26 (1980) 76–83. [6] T.W. Cochran, R.L. Kabel, R.P. Danner, AIChE Journal 31 (1985) 268–277. [7] M.M. Dubinin, Chemistry, in: P.L. Walker (Ed.), Physics of Carbon, Marcel Dekker, New York, 1966, pp. 51–120. [8] S.E. Hoory, J.M. Prausnitz, Chemical Engineering Science 22 (1967) 1025–1033. [9] F. Hall, C. Zhou, K.A.M. Gasem, R.L. Robinson Jr., Adsorption of pure methane, nitrogen, and carbon dioxide and their binary mixtures on wet fruitland coal, in: SPE Eastern Regional Conference and Exhibition, SPE Paper 29194, November 8–10, Charleston, SC, 1994. [10] C. Zhou, F. Hall, K.A.M. Gasem, R.L. Robinson Jr., Industrial and Engineering Chemistry Research 33 (1994) 1280–1289. [11] K.A.M. Gasem, Z. Pan, S.A. Mohammad, R.L. Robinson Jr., Two-dimensional equation-of-state modeling of adsorption of coalbed methane gases, in: M. Grobe, J.C. Pashin, R.L. Dodge (Eds.), Carbon Dioxide Sequestration in Geological Media – State of the Science, AAPG., Studies 59, American Association of Petroleum Geologists, 2009, pp. 1–23. [12] J.H. Chen, D.S.H. Wong, C.S. Tan, R. Subramanian, C.T. Lira, M. Orth, Industrial and Engineering Chemistry Research 36 (1997) 2808–2815. [13] J.E. Fitzgerald, M. Sudibandriyo, Z. Pan, R.L. Robinson, K.A.M. Gasem, Carbon 41 (2003) 2203–2216. [14] J.E. Fitzgerald, R.L. Robinson, K.A.M. Gasem, Langmuir 22 (2006) 9610–9618. [15] S.A. Mohammad, J.S. Chen, R.L. Robinson, K.A.M. Gasem, Energy and Fuels 23 (2009) 6259–6271.

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