Modeling Gas Adsorption and Transport in Small-Pore Titanium

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Langmuir 2005, 21, 4532-4546

Modeling Gas Adsorption and Transport in Small-Pore Titanium Silicates R. P. Marathe, S. Farooq,* and M. P. Srinivasan Department of Chemical and Biomolecular Engineering, National University of Singapore, 4 Engineering Drive 4, Singapore 117576 Received December 14, 2004. In Final Form: March 4, 2005 Engelhard titanium silicate, ETS-4, is a promising new adsorbent for size-selective separation of mixtures of small gases, a leading industrially important example of which is methane-nitrogen separation. Single component equilibrium and kinetics of oxygen, nitrogen, and methane adsorption in Na-ETS-4 and cationexchanged Sr-ETS-4, measured in an earlier study over a wide range of temperatures and pressures, are analyzed in this study. The adsorbent crystals were synthesized and pelletized under pressure (without any binder), thus giving rise to a bidispersed pore structure with controlling resistance in the micropores. Change in equilibrium and kinetics of adsorption of the aforementioned gases in Sr-ETS-4 due to pore shrinkage with progressively increasing dehydration temperature has also been investigated. Differential uptakes have been measured at various levels of adsorbate loading, which has allowed the elucidation of the nature of concentration dependence of micropore diffusivity. Both homogeneous and heterogeneous models are examined on the equilibrium data, while a bidispersed pore diffusion model is able to capture the differential uptakes very well. On the basis of chemical potential gradient as the driving force for diffusion, the impact of isotherm models on the concentration dependence of micropore diffusivity is also analyzed. It is shown that pore tailoring at the molecular scale by dehydration can improve the kinetic selectivity of nitrogen over methane in Sr-ETS-4 to a promising level. The models investigated are evaluated to identify essential details necessary to reliably simulate a methane-nitrogen separation process using the promising new Sr-ETS-4 adsorbent.

Introduction Adsorption separation forms an important class among separation processes. The more commonly used microporous adsorbents are activated carbon, silica gel, activated alumina, carbon molecular sieves, and zeolites. Microporous solids in general and zeolites in particular possess a very large surface area and find numerous applications. Upgrading natural gas by removing diluent nitrogen from methane is a challenging gas separation problem of considerable industrial importance for which a suitable adsorbent has been sought for a long time. Success in this search would allow the utilization of those natural gas sources that remain unutilized due to high nitrogen content and lack of an economically viable upgrading method.1 Cryogenic distillation technology that is currently available is not always economical. A desirable attribute in a promising adsorbent is high selectivity of nitrogen over methane at well-head pressure. Equilibrium selectivity dictates which component in the feed mixture is more strongly adsorbed in the adsorbent. Kinetic selectivity stems from difference in uptake rates of the components in the feed. In the case of kinetically controlled separation, the adsorbate molecules are nearly the size of the adsorbent pores. While the kinetic selectivity of most known adsorbents such as activated carbon cannot be tailored, it can be modified to a small degree in the case of zeolites and some molecular sieves. Zeolites, with their reasonably uniform pore sizes, are more amenable to this tuning. Some techniques to tune selectivity include pore size control by ion exchange2 and chemical vapor deposition in zeolites,3 and modifying the pore size during the activation step in the case of carbon molecular sieves.4 * Corresponding author. Fax: 65-67791936. Tel: 65-68746545. E-mail: [email protected] (S. Farooq). (1) Yang, R. T. Adsorbents: Fundamentals and Applications; John Wiley & Sons: New York, 2003.

Ackley and Yang5 studied the potential of Clinoptilolite, a naturally occurring zeolite, for separating a methanenitrogen mixture. They had investigated the adsorption and diffusion characteristics of modified and ion-exchanged clinoptilolites and reported equilibrium selectivities over 1 order of magnitude and kinetic selectivities spanning 2 orders of magnitude, thereby highlighting the importance of cation manipulation in achieving specific gas separation applications. More recently, Ambalavanan et al.6 have studied the adsorption characteristics of purified clinoptilolite, its single ion-exchanged forms, and mixed ion-exchanged variants. They have reported that purified and ion-exchanged clinoptilolite, Mg-exchanged in particular, show promise of high selectivity for separating the methane-nitrogen mixture. However, it should be noted that batch to batch reproducibility in terms of composition and adsorption characteristics in naturally occurring minerals and zeolites is always an uncertain factor. Synthetic adsorbents, are therefore, preferred over their natural counterparts due to greater control over the synthesis conditions and hence on the adsorption characteristics. Engelhard titanium silicate (ETS-4), a microporous titanium silicate developed by the Engelhard Corporation is a promising new candidate for methane/nitrogen separation.7,8 Ion-exchanging Na+ with bivalent ions such (2) Ruthven, D. M. Principles of Adsorption and Adsorption processes; John Wiley & Sons: New York, 1984. (3) Niwa, M.; Kato, M.; Hattori, T.; Murakami, Y. Fine control of the Pore-opening Size of Zeolite ZSM-5 by Chemical Vapor Deposition of Silicon Methoxide. J Phys. Chem. 1986, 90, 6233-6237. (4) Ju¨ntgen, H.; Knoblauch K.; Harder K. Carbon molecular sieves: Production from Coal and application in Gas Separation. Fuel 1981, 60, 817-822. (5) Ackley, M. W.; Yang, R. T. Diffusion in ion-exchanged Clinoptilolites. AIChE J. 1991, 37 (11), 1645. (6) Ambalavanan, J.; Maldonado, A. J. H.; Yang, R. T.; Chinn, D.; Munson, C. L.; Mohr, D. H. Clinoptilolites for nitrogen/methane separation. Chem. Eng. Sci. 2004, 59, 2407.

10.1021/la046938d CCC: $30.25 © 2005 American Chemical Society Published on Web 04/13/2005

Gas Transport in Titanium Silicates

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Figure 1. Reproducibility of (a) XRD, (b) TGA, (c) nitrogen isotherm at 283.15 K for Sr190 sample, and (d) nitrogen uptake at 283.15 K in Sr270 sample. Original (0), new (0): immediately after synthesis; original (10): 10 months after synthesis.

as Sr2+, Ba2+, Ca2+, Mg2+, etc. yields improved thermal stability, and these ion-exchanged forms of ETS-4 exhibit the possibility of finer pore size tuning than that possible in zeolites. The pore size reduction in ion-exchanged ETS-4 results from increasing dehydration temperature and can be adjusted within 0.1 Å heralding opportunities for new and exciting gas separation applications, especially of similar sized molecules. Kuznicki7 first reported the synthesis of ETS-4 with TiCl3 as the titanium source and an alkali metal hydroxide such as sodium hydroxide or sodium silicate solution as the source of sodium oxide, Na2O. A recent article highlights the potential of ETS-4 and its ion-exchanged variants as adsorbents for many separation applications.9 Mitariten9 has also reported the successful operation of a PSA process for natural gas upgrading by separating nitrogen from methane using ion-exchanged ETS-4. However, available information on equilibrium and kinetics of gas adsorption in this new material is scarce. In a recent communication,10 we reported systematic equilibrium isotherm and linear range uptake data on Na-ETS-4 and its Sr-exchanged variant (Sr-ETS-4). The equilibrium and kinetic results in Sr-ETS-4 dehydrated at progressively higher temperatures (190, 230, 270, 310, and 330 °C) were consistent with the behavior expected (7) Kuznicki, S. M. Preparation of small-pored crystalline titanium molecular sieve zeolites. U.S. Patent 4,938,939, 1990. (8) Kuznicki, S. M.; Bell, V. A.; Nair, S.; Hillhouse, H. W.; Jacubinas, R. M.; Braunbarth, C. M.; Toby, B. H.; Tsapatsis, M. A titanosilicate molecular sieve with adjustable pores for size-selective adsorption of molecules. Nature 2001, 412 (6848), 720. (9) Mitariten, M.; System removes N2 at low flow rates. Am. Oil Gas Reporter 2001, 103. (10) Marathe, R. P.; Mantri, K.; Srinivasan, M. P.; Farooq, S. Effect of ion-exchange and dehydration temperature on the equilibrium and kinetics of gas adsorption in ETS-4. Ind. Eng. Chem. Res. 2004, 43 (17), 5281.

when the pore size contracts at the fractional angstrom level. In this communication, we report the findings of an extensive modeling study on the pure component equilibrium and uptake of oxygen, nitrogen, and methane adsorption in Na-ETS-4 and Sr-ETS-4 reported earlier. In addition, we also report differential uptakes measured at various levels of adsorbate loading, which are analyzed to establish the nature of the concentration dependence of the controlling micropore diffusivity. Both homogeneous and heterogeneous models have been applied and compared to identify the required features of a process model that will be able to reliably simulate methane-nitrogen separation in the pore-shrunk Sr-ETS-4. Experimental Section Synthesis, Physical Characterization, and Pelletization. ETS-4 samples were synthesized using TiCl3 as the titanium source and sodium silicate solution as the silicon source. The molar composition of the gel was 4.42Na2O/0.95K2O/1TiO2/ 5.71SiO2/81.88H2O. The Na-ETS-4 was exchanged with Sr by boiling the Na-ETS-4 sample in an excess of SrCl2‚6H2O solution. The ion-exchanged product was dried overnight at 100 °C and then calcined at five different temperatures, 190, 230, 270, 310, and 330 °C. These Sr-ETS-4 samples will hereafter be named Sr190, Sr230, Sr270, Sr310, and Sr330, respectively. The scanning electron microscopy (SEM) images of Na-ETS-4 show crystals as clusters of plates/needles. The images are in good agreement with those reported in the literature.11,12 Elemental analysis of Sr-ETS-4 gave the following composition: 64.66% O, 21.35% Si, 7.98% Ti, 6.01% Sr, and 0.001% Na. Strontium (11) Chapman, D. M.; Roe, A. L. Synthesis, characterization and crystal chemistry of microporous titanium silicate materials. Zeolites 1990, 10, 730. (12) Miraglia, P. Q.; Yilmaz, B.; Warzywoda, J.; Bazzana, S.; Sacco, A., Jr. Morphological and surface analysis of titanosilicate ETS-4 synthesized hydrothermally with organic precursors. Microporous Mesoporous Mater. 2004, 69, 71.

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Marathe et al. Table 1. Isotherm Parametersa

Langmuir

sorbent

sorbate

qs (mmol/ cm3)

Na-ETS-4

O2 N2 CH4 O2 N2 CH4 O2 N2 CH4 O2 N2 CH4

5.53 4.53 2.90 6.81 5.54 4.40 5.57 4.52 4.64 5.68 5.23 4.46

Sr190 Sr270 Sr310

a

bo (cm3/ mmol)

-∆U (kcal/ mol)

0.047 0.098 0.244 0.051 0.072 0.049 0.069 0.032 0.356 0.106 0.281 0.238

2.07 2.00 1.76 2.00 2.07 2.89 1.92 2.42 0.89 1.37 1.29 0.88

multisite Langmuir

R2

qs (mmol/ cm3)

bo (cm3/ mmol)

-∆U (kcal/ mol)

a

0.053 0.183 0.147 0.028 0.205 0.633 0.183 0.466 0.045 0.013 0.047 0.001

8.17 6.66 5.27 11.61 9.88 7.94 10.26 9.06 8.56 9.89 9.05 8.12

0.026 0.042 0.291 0.032 0.025 0.012 0.051 0.025 0.215 0.080 0.071 0.112

2.20 2.31 1.40 1.99 2.40 3.50 1.82 2.33 0.91 1.27 1.37 0.93

1.73 2.12 2.68 2.18 2.56 3.18 2.43 2.76 2.90 2.43 2.65 2.96

heterogeneous Langmuir

R2

qs (mmol/ cm3)

bo (cm3/ mmol)

A

p

R2

0.051 0.156 0.098 0.025 0.191 0.362 0.152 0.478 0.050 0.013 0.042 0.003

5.98 4.70 2.87 7.18 6.36 4.58 5.88 4.80 4.80 5.89 5.48 4.50

0.046 0.088 0.238 0.046 0.040 0.027 0.057 0.029 0.346 0.107 0.114 0.237

74.31 40.67 72.11 60.73 28.39 38.31 42.30 39.45 77.30 61.25 52.43 78.05

149.00 80.00 128.00 122.00 64.00 112.96 83.00 98.00 77.00 80.00 68.00 67.00

0.010 0.184 0.135 0.028 0.259 0.539 0.169 0.434 0.042 0.013 0.044 0.001

Note: A and p are the gamma distribution parameters. Mean of gamma distribution ) (p + 1)/A, variance ) (p + 1)1/2/A.

Figure 2. Model fits to experimental isotherms of oxygen, nitrogen, and methane at 283.15 K in (a) Na-ETS-4 dehydrated at 175 °C, (b) Sr190, (c) Sr270, and (d) Sr310. L, ML, HL denote Langmuir, Multisite Langmuir, and Heterogeneous Langmuir isotherm models, respectively. exchange was, therefore, practically complete. Strontium exchange had no discernible change in the external morphology of the crystal agglomerates. The X-ray diffraction (XRD) pattern of the product ETS-4 matched well with published report13 and showed characteristic signatures of ETS-4 framework. The dried samples were pelletized in a pellet die set using a hydraulic press under a load of 6 metric tons for 10 min before conducting the equilibrium and kinetic measurements. Detailed accounts on synthesis, physical characterization, sample preparation, and experimental procedures for equilibrium and kinetic measurements have been published in our recent communication.10 Adsorption equilibrium and kinetic measurements were carried out by the constant volume method. The measurements were done up to about 10 bar pressure at 263.15, 273.15, 283.15, and 303.15 K. Prior to each experimental run, the samples were

thoroughly regenerated by a systematic sequence of controlled heating under vacuum. The details of the constant volume apparatus, the experimental procedure, and data processing were also elaborated in our previous communication10 and are not repeated here. It is worth mentioning that timely reproducibility checks were done on the results at different stages of the study. The XRD and thermal gravimetric analysis (TGA) results were reproducible as shown in parts a and b of Figure 1, as were the isotherms (Figure 1c) and the uptakes (Figure 1d), including the ones that were measured 10 months after synthesis and after several regeneration cycles. Such checks on the repeatability of results and stability of the material are essential, particularly when developing a new adsorbent.

(13) Philippou, A.; Anderson, M. W. Structural investigation of ETS4. Zeolites 1996, 16 (2/3), 98.

Adsorption Equilibrium. The adsorption equilibrium data were analyzed using three different isotherm models,

Theoretical Section


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Figure 3. Temperature dependence of experimentally measured Henry’s constant for nitrogen and methane in (a) Na-ETS-4, (b) Sr190, (c) Sr270, and (d) Sr310.

namely, Langmuir, multisite Langmuir, and heterogeneous Langmuir. Langmuir Model. Among the models to describe single component adsorption equilibrium, the Langmuir model is more frequently used, which is described by the following equation

q)

qsbc Kc ) 1 + bc 1 + bc

(1)

and

b ) bo exp(-∆U/RgT)

(2)

where q is the adsorbate concentration in solid phase, qs is the temperature-independent saturation capacity of the adsorbate in solid phase, c is adsorbate concentration in gas phase, b is the Langmuir constant, K is the dimensionless Henry’s constant, ∆U is the change in internal energy due to adsorption, and bo is pre-exponential factor in the Arrhenius type temperature dependence of b. The Langmuir isotherm parameters, bo, ∆U, and qs, for each adsorbate gas were extracted by simultaneous nonlinear regression of the experimental isotherm data measured at three temperatures. The isotherm parameters for oxygen, nitrogen, and methane are presented in Table 1, and representative fits of the Langmuir model to the experimental isotherm data are shown in Figure 2. Because the methane capacity was sufficiently different from those of oxygen and nitrogen, qs was allowed to vary independently for each adsorbate. The implication is, for multicomponent equilibrium calculation, the extended Langmuir model will be thermodynamically inconsistent and ideal adsorbed solution (IAS) theory will have to be employed.

Multisite Langmuir Model. The multisite Langmuir model is an extension of the Langmuir model for single component and multicomponent equilibrium on microporous, homogeneous adsorbents that has created the provision for taking the variation of adsorbate size into account. It was derived by Nitta et al.14 using the statistical thermodynamic approach. The model assumes that an adsorbent has a fixed number of adsorption sites (qs) and that an adsorbate molecule, depending on its size and orientation in the adsorbed phase, occupies a certain number of these adsorption sites (ai). Therefore, for thermodynamic consistency

qs ) qsi ai ) constant

(3)

where qsi is the saturation capacity of each adsorbate. For single component equilibrium, it has the following form

bici )

qi/qsi (1 - qi/qsi)ai

(4)

where ai and qsi are independent of temperature while bi has the same Arrhenius form of temperature dependence as in eq 2. It should be noted that the thermodynamic consistency requirement15 given by eq 3 is important for multicomponent equilibrium calculation; if equilibrium data of each gas is individually fitted without the constraint, ai will lose the physical significance as a relative (14) Nitta, T.; Shegetomi, T.; Kuro-oka, M.; Katayama, T. An adsorption isotherm of multisite occupancy model for homogeneous surface. J. Chem. Eng. Jpn. 1984, 17, 39. (15) Rao, M. B.; Sircar, S. Thermodynamic Consistency for Binary Gas Adsorption Equilibria. Langmuir 1999, 15, 7258.

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Figure 4. Dependence of isosteric heat of adsorption (∆Us) on surface coverage for adsorption of oxygen, nitrogen, and methane in (a) Na-ETS-4, (b) Sr190, (c) Sr270, and (d) Sr310.

measure of molecular size in the adsorbed state. It will then become an additional fitting parameter over the Langmuir model. Therefore, single component equilibrium parameters for the multisite Langmuir model were extracted by simultaneous nonlinear regression of oxygen, nitrogen, and methane data at all experimental temperatures, subjected to constraint that aiqsi was equal for all three adsorbates in a given adsorbent. The optimized fits of the multisite Langmuir model are included in Figure 2, and the corresponding model parameters are given in Table 1. Upon comparison of the residual values given in Table 1, the multisite Langmuir model appears to give a somewhat better fit than does the Langmuir model for nitrogen and methane data in Na-ETS-4 and Sr190, and for oxygen in Sr270. The difference is practically insignificant in other cases. The number of adsorption sites (a values in Table 1) occupied by each adsorbate molecule in the ETS-4 samples is in the order oxygen < nitrogen < methane. The relative order is consistent with the order of the size of these molecules based on the Lennard-Jones values: oxygen (3.43 Å) < nitrogen (3.68 Å) < methane (3.82 Å). It may be noted from Table 1 that saturation capacities, qsi, of the three adsorbates are considerably higher than the Langmuir model predictions. Similarly, Huang et al.16 measured equilibrium data at high pressures and found that the loading exceeded predictions by the Langmuir model but were well captured by the multisite Langmuir model. They therefore concluded that the qsi values obtained from the fit of the multisite Langmuir model predictions were not unreasonable. (16) Huang, Q.; Farooq, S.; Karimi, I. A. Binary and Ternary adsorption kinetics in carbon molecular sieves. Langmuir 2003, 19, 5722.

Internal Energy Change in the Linear Range. The internal energy change (∆U) due to adsorption in the linear range of the isotherm was obtained by plotting the Henry’s law constant (K ) bqs) vs inverse of temperature, as shown in Figure 3. The temperature dependence of Henry’s constant is given by

K ) Ko exp(-∆U/RgT)

(5)

The K values from the linear range of the isotherm yielded higher internal energy change (given in Figure 3) than that obtained by nonlinear regression of the isotherm data over a wider pressure range at different temperatures (given in Table 1). The ∆U from nonlinear regression is essentially an average value over the entire isotherm. The difference is particularly significant for adsorption in Sr190 and Sr270 and is an indication of energetic heterogeneity in the adsorbent. When there is an adsorption energy distribution, the high energy sites are occupied first and the lower energy sites are then progressively occupied as the adsorbate partial pressure is increased. To investigate the energetic heterogeneity, isosteric heat analysis was conducted using the isotherm data for oxygen, nitrogen, and methane in all the adsorbent samples under consideration, which is presented next. Estimation of Isosteric Heat. Isosteric heat (∆Us) obtained by differentiating an adsorption isotherm at a constant adsorbate loading, q, is given by17

ln P (∂ ∂T )

∆Us ) RgT2

q

(6)

Subject to the assumption that the isosteric heat (17) Sircar, S. J. Chem. Soc., Faraday Trans. 1985, 1 (81), 1527.


Langmuir, Vol. 21, No. 10, 2005 4537

Figure 5. Distribution of adsorption energy in (a) Na-ETS-4, (b) Sr190, (c) Sr270, and (d) Sr310.

of adsorption is independent of temperature, integration of eq 6 yields

ln P )

∆Us + constant RgT

(7)

A plot of ln P versus 1/T at a given level of adsorbent loading, q, yields a linear isostere with slope equal to ∆Us/ Rg. Isosteric heat of adsorption thus obtained at various levels of adsorbate loading in Na-ETS-4, Sr190, Sr270, and Sr310 for oxygen, nitrogen, and methane (Figure 4) are either constant or decrease with increasing surface coverage. In Na-ETS-4, methane and nitrogen adsorption are practically homogeneous, while the smallest molecule among the three gases, viz., oxygen, experiences some adsorbent heterogeneity (Figure 4a). This suggests that the high-energy sites are not accessible to methane and nitrogen which are the two larger molecules. The common notion that a smaller molecule experiences greater adsorbent heterogeneity is due to adsorbents such as activated carbon where the heterogeneity is in the texture of the pore walls. However, if the heterogeneity is due to a distribution of charge in the framework, then the impact of that distribution is expected to be in the order of increasing polarity or polarizability so long as the adsorption sites are accessible. All three adsorbates investigated here are polarizable, but only nitrogen is polar. Since all three adsorbates experience some degree of heterogeneity, differences in their polarizability and size seem to cogently explain the observed trends. It, therefore, appears that cation exchange with strontium induces charge heterogeneity in the structure but limits the access of methane to the high energy sites (Sr190 in Figure 4b). As a result, despite being the most polarizable of the three gases, methane does not experience much heterogeneity. Be-

tween oxygen and nitrogen, the latter sees more of the heterogeneity due to higher polarizability. The quadrupole interaction of nitrogen may have also played a part. Cation relocation, and pore contraction with progressive dehydration seem to increase the charge heterogeneity (Sr270 in Figure 4c). The effect of this heterogeneous charge distribution is more pronounced on nitrogen and oxygen, and to a smaller extent on the larger methane molecule that does not have access to the higher energy sites. Continued dehydration, however, eventually makes nearly all the high-energy sites inaccessible to all the three adsorbate species and the adsorbent becomes nearly homogeneous (Sr310 in Figure 4d). Heterogeneous Langmuir Isotherm Model. As discussed in the preceding section, analysis of the isosteric heat of adsorption revealed that there exists some degree of energetic heterogeneity in the adsorbent samples. In other words, there is a distribution of adsorption energy unlike a constant value expected for an energetically homogeneous adsorbent. A normalized gamma distribution was used to represent the adsorbent heterogeneity. The probability density function for the distribution of an entity x is defined as

f(x) )

Ap+1 p x exp(-Ax) Γ(p + 1)

(8a)

∫0∞ f(x) dx ) 1

(8b)

where Γ is the gamma function, A and p are the two adjustable parameters of the gamma distribution. We represent a heterogeneous adsorbent as a collection of many homogeneous patches and assume that the Langmuir isotherm given by eq 1 applies to each of these

4538


Marathe et al.

homogeneous patches. An overall isotherm taking the energy distribution into account can then be written as

q)

∫0∞ qlocalf(∆U) d(∆U) ) q b exp(∆U/RgT)C

∫0∞ 1 +s bo

o

f(∆U) d(∆U) (9)

exp(∆U/RgT)C

where the distribution function, f(∆U), follows gamma distribution according to eq 8a. The heterogeneous equilibrium parameters were extracted by minimizing the following objective function

∑i

[(

qi

qs

-

∫0

∞

bo exp(∆U/RgT)Ci 1 + bo exp(∆U/RgT)Ci

)] 2

f(∆U) d(∆U)

(10)

Here, i denotes the number of data points used in the regression. Starting from a set of guessed values for qs, bo, A, and p, the function was minimized to yield the isotherm parameters. The optimized fits of the heterogeneous Langmuir isotherm to the experimental data are included in Figure 2. The model parameters are given in Table 1. This distribution of adsorption energy was used to characterize the energetic heterogeneity of the NaETS-4 and Sr-ETS-4 adsorbents. The adsorption energy distributions for oxygen, nitrogen, and methane on the four samples are presented in Figure 5. It may be noted from the figure that the energy distributions are consistent with corresponding isosteric heat vs fractional coverage plots in Figure 4. For all three gases in Na-ETS-4 and Sr310, and for methane in Sr270, the energy distributions are sharp, as expected for fairly homogeneous adsorbents indicated by a nearly constant isosteric heat. In the case of adsorption of all three gases in Sr190, and oxygen and nitrogen adsorption in Sr270, the distributions are much broader as expected when isosteric heat shows a decline with increasing coverage. It should also be noted that in all the cases, -∆U from homogeneous analysis is in good agreement with the mean of the distribution from heterogeneous analysis. By the measure of the residuals, the heterogeneous Langmuir model has brought only marginal improvement to the fits of the experimental data in a few isolated cases over the homogeneous Langmuir model fit. Adsorption Kinetics. Differential uptake measurements for oxygen, nitrogen, and methane were also carried out on the Na-ETS-4 and Sr-ETS-4 samples at the same temperatures used in equilibrium measurements. The experiments were particle scale measurements, and the amount of adsorbent sample used was kept small enough to prevent bed diffusion from layering of particles. The pore diffusion model applied to analyze the measured uptake data assumes that the gas diffusion is limited by the resistance distributed within the micropore interior. For completeness, diffusional resistance in the adsorbent macropores is also accounted for in the particle mass balance. External film resistance to mass transfer is not relevant since pure adsorbates were used. The following additional assumptions are also made for analyzing the adsorption kinetics: the ideal gas law applies; the system is considered isothermal; only molecular diffusion is assumed for the transport in the macropores; both macroparticles and microparticles are assumed to be spherical; the segment of the isotherm covered in a differential step change is taken as linear and the kinetic parameters are assumed to remain constant over the small segment; the time to open the solenoid valve is assumed

Figure 6. Representative results showing the fits of the kinetic model to the linear range uptake in Na-ETS-4, Sr190, Sr270, and Sr310 of (a) oxygen, (b) nitrogen, (c) methane at 283.15 K.

to be negligible compared to the time of observation, and an approximate linear form of a quadratic driving force is assumed to represent the dynamics of the solenoid valve separating the dose and test chambers. Unary uptakes in the adsorbent samples were measured volumetrically. The following relation is used to account for any influence of the solenoid valve on the measured adsorption uptake

dnv ) X(Pd - Pu) dt

(11)

where nv is the number of moles of gas flowing through the valve and X is the valve constant, Pd is the pressure on the dose side, and Pu is the pressure on the test side. Blank experiments (i.e., without any adsorbent on the test side) were conducted with different gases to calibrate the valve constant, X. We obtained a constant value (X ) 0.087), confirming that the linear driving force assumption is reasonable.


Langmuir, Vol. 21, No. 10, 2005 4539

Table 2. Adsorption Kinetic Parameters from Pore Model temp (K)

Dco/rc2 × 103 (s-1)

Dco′/rc2 (s-1)

Ed (kcal/mol)

273.15 283.15 303.15 273.15 283.15 303.15 273.15 283.15 303.15 263.15 273.15 283.15 263.15 273.15 283.15 273.15 283.15 303.15 263.15 273.15 283.15 263.15 273.15 283.15 273.15 283.15 303.15 263.15 273.15 283.15 263.15 273.15 283.15 273.15 283.15 303.15 283.15 283.15

0.22 0.41 0.65 0.044 0.049 0.056 0.030 0.039 0.056 4.71 6.55 9.67 3.31 5.45 6.72 0.013 0.018 0.025 2.18 4.36 6.88 0.10 0.14 0.41 0.008 0.013 0.018 1.596 1.760 2.120 0.090 0.140 0.190 0.0088 0.0100 0.0110 0.160 67.50

5.94

5.46

adsorbent adsorbate Na-ETS-4 (175 °C)

O2 N2 CH4

Sr190

O2 N2 CH4

Sr270

O2 N2 CH4

Sr310

O2 N2 CH4

Sr330

N2 CH4

Dose cell mass balance gives

Vd

dPd dnv ) -RgT dt dt Pd|t)0 ) Pd0+

0.00048

1.29

0.012

3.24

119.14

5.31

81.16

5.26

The macropore mass balance equation is given by

( (

Na-ETS-4 (175 °C) Sr190 Sr230 Sr270 Sr310 Sr330

(14a)

and the boundary conditions are

0.0081

∂cp | )0 ∂R R)0

3.47

27778

8.54

41027

10.43

0.0341

4.50

0.0859

2.09

3.83

5.56

0.000078

1.17

-

-

cp|R)Rp ) c )

temp (K)

KN2/KCH4

DN2/DCH4

kinetic selectivity

273.15 283.65 303.15 273.15 283.65 303.15 283.15 273.15 283.65 303.15 273.15 283.65 303.15 283.15

0.71 0.73 0.76 0.38 0.34 0.36 0.44 1.83 2.11 1.51 1.10 1.14 0.83 1.92

1.44 1.27 1.01 406.71 375.81 511.37 213.01 18.23 31.77 66.34 15.91 19.08 33.91 0.00237

0.85 0.66 0.76 7.85 6.66 8.12 6.38 7.83 11.89 12.34 4.39 4.98 4.86 0.093

(

|

Dp dPu dcp ) RgT -3 V dt Rp p a dR

R)Rp

+

)

dnv dt

(14b)

Pu RgT

(14c)

The average adsorbed phase concentration in the micropore, q j , is related to the adsorbate flux at the micropore mouth according to

∂q ∂q j 3 ) D | ∂t rc c ∂r r)rc

(14d)

The mass balance in the micropore is given by

∂q 1 ∂ 2 ∂q ) r Dc ∂t r2 ∂r ∂r

( (

))

(15a)

with the boundary conditions

The uptake cell mass balance is given by

Vu

))

∂cp 1 - p ∂q ∂cp 1 ∂ j + ) R2Dp ∂t p ∂t R2 ∂R ∂R

Table 3. N2/CH4 Selectivity for ETS-4 Samples adsorbent

(13)

(12)

Pu|t)0 ) Pu0 where Vu is the empty test chamber volume, Va is the adsorbate volume, is the fraction of the test chamber not occupied by the adsorbent, cp is the gas concentration in the macropore, Rp is the particle radius, R is the radial distance in the macroparticle, p is the adsorbent macroporosity, Dp is the molecular diffusivity in the macropore, Rg is the gas constant, T is the temperature, and t is the time.

∂q | )0 ∂r r)0

(15b)

q|r)rc ) f(cp)

(15c)

Uptake in the Linear Range. Figure 6 shows representative results of model fits to experimental uptakes of oxygen, nitrogen, and methane on Na-ETS-4, Sr190, Sr270, and Sr310. The uptake results shown were measured at low surface coverage (θ f 0), the surface coverage being defined as θ ) qc/qs, where qc and qs are the adsorbate concentration in solid phase and the saturation capacity for the adsorbate, respectively. The diffusional time constant in the micropores (Dc/rc2) is assumed to remain constant in the concentration range covered in each differential run, and it is the only fitting parameter extracted by minimizing the residual between the experimental uptake and the model solution. As these measurements were done in the linear range of the isotherm (fractional coverage in the range 0.005-0.031), the extracted diffusivity values are the limiting diffusivities, which are given in Table 2 for the three gases in Na-ETS-4 and Sr-ETS-4 samples. The activation energies for adsorption were calculated from the Eyring equation

Dco 2

rc

)

Dco′ rc2

exp(-E/RgT)

(16)

Figure 7 shows the semilogarithmic plots of Dc/rc2 vs 1/T for nitrogen and methane in the four adsorbent samples. The activation energies obtained from the slopes are given in the respective figures as well as in Table 2.

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Figure 7. Temperature dependence of micropore diffusivity for nitrogen and methane in (a) Na-ETS-4, (b) Sr190, (c) Sr270, and (d) Sr310.

In all four adsorbent samples, oxygen diffusion is fastest followed by nitrogen and methane, which is in the expected order according to the Lennard-Jones diameters of the gases. Upon Sr exchange, the uptakes of oxygen and nitrogen become significantly faster, while that of methane remains slow. Changes in ion occupancy and relocation of cations due to Sr exchange are the governing factors for this phenomenon.18 Following ion exchange, there is continuous reduction in the uptake rates for all three gases with dehydration at progressively higher temperatures, as may be seen from Figure 6 and the diffusional time constants compiled in Table 2. This is a clear indication of transport channel contraction in Sr-ETS-4. It is, however, interesting to note that in these contracting pores, the activation energies first increase (see Sr190 and Sr270 data) and them decline with further contraction (see Sr270 and Sr310 data). While the initial increase is expected, the subsequent decrease is somewhat counterintuitive. A complex interplay of the effects of pore contraction and cation relocation may be responsible for this behavior. Selectivity for Methane-Nitrogen Separation. As in any separation, selectivity or separation factor is an important parameter for preliminary comparative assessment of the potential adsorbents for the separation of a mixture by adsorption. Selectivity is generally defined as2

ηAB )

q j A/cA q j B/cB

(17)

where A and B denoted the components to be separated. For a kinetically controlled separation, eq 17 may be

manipulated to the following form

( ) ( )

mt qA* m∞ A cA ηK ) mt qB* m∞ B cB

(18a)

It is clear from the above equation that kinetic selectivity is time dependent and will ultimately approach the equilibrium selectivity. Assuming uncoupled equilibrium and kinetics, Ruthven et al.19 showed that for short contact time and for pore diffusion controlled uptake, the kinetic selectivity is given by

ηKAB )

x

KA KB

(Dc0)A (Dc0)B

(18b)

A summary of the impact of increasing dehydration temperature on the equilibrium and transport of methane and nitrogen in Sr-ETS-4 is presented in Table 3. Significant diffusivity difference between methane and nitrogen upon exchanging the sodium ion in Na-ETS-4 with the bivalent strontium cation is evident. Methane adsorption is stronger than nitrogen in Na-ETS-4 by virtue of its higher polarizability, and the difference increases upon ion exchange with strontium (Sr190) due to an increase in charge heterogeneity (see parts a and b in (18) Nair, S.; Tsapatsis, M.; Toby, B. H.; Kuznicki, S. M. A study of heat-treatment induced framework contraction in strontium-ETS-4 by powder diffraction and vibrational spectroscopy. J. Am. Chem. Soc. 2001, 123, 12781-12790. (19) Ruthven, D. M.; Farooq, S.; Knaebel, K. S. Pressure Swing Adsorption; John Wiley & Sons: New York, 1994.


Langmuir, Vol. 21, No. 10, 2005 4541

In contrast to the above result obtained in this study, Ambalavanan et al.6 have reported an equilibrium selectivity of ∼13 with relatively fast diffusion rates for both methane and nitrogen. A much stronger nitrogen isotherm (than that obtained in the present study) with fast uptake rate and nearly complete exclusion of methane has also been reported.7,8 In the absence of explicit details on the synthesis procedure and dehydration conditions adopted in these studies, it is not possible to shed any light on the difference between our results and those reported by others. Uptake at High Adsorbate Loading. In addition to linear range measurements, differential uptake measurements were also conducted at higher adsorbate loading for the three gases on the ETS-4 samples. Some representative fits of the pore model to the differential uptakes of oxygen, nitrogen, and methane at higher levels of loading are shown in Figure 8. The micropore diffusivities (Dc/rc2) extracted from the fit of the pore model to the experimental uptake data were then analyzed to establish the pattern of concentration dependence of the micropore diffusivity. Examples abound in the literature where it has been shown that concentration dependence of gas diffusivity in zeolite micropores obeys Darken’s equation which follows from the chemical potential gradient as the driving force for diffusion20

Dc d ln c ) Dco d ln qc

(19)

where the limiting diffusivity, Dco, the diffusivity in the linear range of the isotherm (θ f 0), is assumed to be independent of concentration. The Darken’s equation takes the following form for the Langmuir isotherm

Dc 1 ) Dco 1 - θ

(20a)

For the multisite Langmuir model, it takes the form

Dc 1 + (a - 1) ) Dco 1-θ

Figure 8. Representative results showing the fits of the kinetic model to differential uptakes of (a) oxygen (in Na-ETS-4), (b) nitrogen (in Sr190), and (c) methane (in Sr270) measured at different adsorbent loadings.

Figure 4). Pore contraction with increasing dehydration temperature also decreases pore potential for adsorption of gas molecules, and greater drop in pore potential for (marginally bigger) methane molecules eventually leads to a reversal of adsorption affinity. However, the large diffusivity difference reduces progressively with pore contraction. It is clear from Table 3 that the initially stronger methane equilibrium adversely affects its kinetic selectivity over nitrogen in Sr-ETS-4 despite very large difference in their diffusivities. Maximum diffusivity ratio (DN2/DCH4) is obtained in Sr190, while the maximum equilibrium reversal (KN2/KCH4) is seen in Sr270. As such, maximum kinetic selectivity in this study is ∼12 in Sr270. A small increase in kinetic selectivity with temperature is observed in Sr190 and Sr270. The kinetic selectivity of nitrogen over methane would improve significantly if the optimum in equilibrium and diffusivity ratios could be synchronized.

(20b)

In eq 20b, a is the adsorbate size factor from the multisite Langmuir isotherm equation. According to the chemical potential gradient theory, the concentration dependence of micropore diffusivity comes from the isotherm curvature and can be predicted from isotherm parameters. Equations 20a and 20b are the predictive equations for Langmuir and multisite Langmuir isotherms, respectively. A similar equation for the heterogeneous Langmuir isotherm can also be derived. For a heterogeneous system, the macroscopically observed flux is related to the local flux by the following integral equation

J)-

D h o′ (∇µ) RgT

f(∆U) d(∆U) ∫0∞ qlocal(∆U) exp(R∆U RgT )

(21)

where for a single component system

∇µ )

RgT dCg′ dq Cg′ dq dr

(22)

with µ being the chemical potential and Cg′ the imaginary (20) Ka¨rger, J.; Ruthven, D. M. Diffusion in zeolites and other microporous solids; John Wiley & Sons: New York, 1992.

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Figure 9. Concentration dependence of micropore diffusivity in Na-ETS-4 for (a) oxygen, (b) nitrogen, and (c) methane. L, HL, and ML denote Langmuir, Heterogeneous Langmuir, and Multisite Langmuir isotherm model, respectively.

gas-phase concentration in equilibrium with the local adsorbed phase concentration. The activation energy distribution for diffusion is related to the distribution in internal energy change by the following linear relationship

E ) -R∆U

(23)

The appropriateness of the above linear relationship will be discussed later. Substituting eq 23 in eq 21 and then comparing the resulting equation to Fick’s law, the following defining equation for effective micropore diffusivity in a heterogeneous system is obtained

dCg′ dq

h o′ D hc ) D

f(∆U) d(∆U) ∫0∞ qlocal(∆U) exp(R∆U RgT )

[

( )

dCg′ dq ) 1/ ) dq dCg′ exp(∆U/RgT) ∞ f(∆U) d(∆U) (25) boqs 0 [1 + boCg′ exp(∆U/RgT)]2

∫

Substituting eq 25 and the appropriate expression for

∫

∞

0

D hc ) D h o′

∫

exp[(1 - R)∆U/(RgT)] exp[∆U/(RgT)]

∞

0

f(∆U) d(∆U)

1 + boCg′ exp[∆U/(RgT)]

{1 + boCg′ exp[∆U/(RgT)]} 2

f(∆U) d(∆U)

(26)

It is further assumed that

lim D h c ) Dco

(27)

Cg′ f 0

[

]/

i.e., the limiting diffusivities from the two models converge.

(24)

For a single component system

]

qlocal in eq 24, we get

D hc ) Dco

exp[(1 - R)∆U/(R T)]

∫0∞ 1 + b C ′ exp[∆U/(Rg T)]f(∆U) d(∆U) o

∫0

∞

[

g

g

exp[∆U/(RgT)]

f(∆U) d(∆U) {1 + boCg′ exp[∆U/(RgT)]}2

]

∫0∞ exp[(1 - R)∆U/(RgT)]f(∆U) d(∆U) ∫0∞ exp[∆U/(RgT)]f(∆U) d(∆U)

(28)

The simple relationship between E and ∆U suggested by


Langmuir, Vol. 21, No. 10, 2005 4543

Figure 10. Concentration dependence of micropore diffusivity in Sr190 for (a) nitrogen and (b) methane. L, HL, and ML denote Langmuir, Heterogeneous Langmuir, and Multisite Langmuir isotherm model, respectively.


eq 23 has been widely used in the literature for diffusion of gases in activated carbon21-23 with R values in the range of 0.3-0.8 to explain the effect of adsorbent heterogeneity on adsorption kinetics. Theoretical studies by Bhatia et (21) Kapoor, A.; Yang, R. T. Surface diffusion on energetically heterogeneous surfaces. AIChE J. 1989, 35 (10), 1735.

al.,24 supported by simulation, have shown that the assumed simple relationship is theoretically not valid in molecularly sized pores where the effective diffusional (22) Do, D. D.; Hu, X. J. An energy-distributed model for adsorption kinetics in large, heterogeneous microporous particles. Chem. Eng. Sci. 1993, 48, 2119.

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Marathe et al.


Figure 13. (a) Isotherms and (b) linear range uptakes of nitrogen and methane in Sr330 at 283.15 K. Lines represent the least square fits.

activation energy is dominated by high kinetic energy trajectories. The strong role of kinetic energy is perhaps the reason E is greater than (-∆U) in zeolite and carbon molecular sieve micropores. Hence, in the case of diffusion in ETS-4, the assumed linear relationship between E and ∆U given by eq 23 is an empirical approximation. As a first guess, we have further assumed that R is equal to the ratio of the activation energy to the change in internal energy measured in the linear range of the isotherm. Using the gamma distribution parameters that were derived from the regression of the experimental isotherm data, the expected concentration dependence of micropore diffusivity including the effect of adsorbent heterogeneity was calculated as a function of θ from eq 28. The prediction of concentration dependence of micropore diffusivity for (23) Hu, X. J.; Rao, G. N.; Do, D. D. Effect of energy distribution on sorption kinetics in bidispersed particles. AIChE J. 1993, 39, 249. (24) Bhatia, S. K.; Jepps, O.; Nicholson, D. Tractable molecular theory of transport of Lennard-Jones fluids in nanopores. J. Chem. Phys. 2004, 120 (9), 4472.

oxygen, nitrogen, and methane in Na-ETS-4 and for nitrogen and methane in three Sr-ETS-4 samples corresponding to the three equilibrium isotherms are compared with the measured experimental data in Figures 9-12. Except for nitrogen transport in Sr190 and Sr270, the predictions for the three isotherms are equally good. The latter is consistent with the fact that all the adsorbents are nearly homogeneous and the residuals of the isotherms, fitted to the experimental equilibrium data, are only marginally different from one another. In the two exceptional cases, the distributions of adsorption energy are indeed very broad (see parts b and c in Figure 5), which explain the improvement due to heterogeneous analysis. It is, however, somewhat unexpected that despite methane adsorption in Sr190 having a broad energy distribution like that of nitrogen there is no noticeable difference in the three predictions in Figure 10b. From an overall perspective, the modest heterogeneity seen in the isosteric heat plots of adsorption in Sr190 and


Sr270 do not seem to impact the equilibrium and kinetics in a major way and homogeneous analysis appears quite adequate in capturing the equilibrium isotherms and concentration dependence of micropore diffusivity. Exclusion of Methane from the Micropore. Equilibria and uptakes of nitrogen and methane were also measured in Sr-ETS-4 dehydrated at 330 °C (Sr330) to examine the possibility of completely excluding methane from the adsorbent micropores. The results are shown in Figure 13. It is observed that the uptake of nitrogen was slower than that in Sr310 indicating a further shrinkage in pore size. However, methane uptake became very fast. The equilibrium capacities for both nitrogen and methane also decreased further. The isotherms were practically linear and the K values corresponding to the least-squares fits are given in Figure 13a. Extremely fast methane uptake suggests exclusion from the micropores and therefore only macropore filling where the diffusional resistance is minimal. However, the equilibrium capacity of methane plotted on the basis of micropore volume, qc, still showed some degree of adsorbate concentration beyond macropore filling. If the micropores were really inaccessible, the obvious question is where was this capacity coming from? It is not clear if the crystal surfaces were contributing to this capacity, which merits further investigation. Conclusions In this communication, the results of a detailed modeling study of equilibrium and kinetics have been presented for oxygen, nitrogen, and methane adsorption in Na-ETS-4 and samples of Sr-ETS-4 dehydrated at different temperatures. Equilibrium isotherms measured in an earlier study over a wide pressure range at several temperatures were analyzed in order to establish a suitable isotherm model. Differential uptakes measured at various levels of adsorbate loading provided the database for investigating the concentration dependence of micropore diffusivity. A bidispersed pore diffusion model was able to fit all the uptake results. Assuming molecular diffusion in the macropores, the micropore diffusivity was the only fitting parameter that was unambiguously extracted from the experimental uptake data. The linear range equilibrium and kinetic data were analyzed to quantify the impact of cation exchange and dehydration on the selectivity of nitrogen over methane in ETS-4. It was found that the exchange of sodium cation with bivalent strontium cation resulted in a rapid uptake of nitrogen, thus creating a large difference in the diffusional rates of the two molecules. However, due to strong methane adsorption, the large diffusivity ratio in favor of nitrogen did not translate into a high kinetic selectivity. While progressive pore contraction with increasing dehydration temperature resulted in equilibrium reversal in Sr-ETS-4, it also reduced the diffusivity ratio since pore contraction (and associated ion relocation due to dehydration) appeared to have a larger effect on nitrogen kinetics. The maximum kinetic selectivity obtained in this study was therefore limited to ∼12 in Sr270. This value compares well with the values reported for oxygennitrogen separation in various carbon molecular sieve samples19 used in kinetically controlled industrial pressure swing adsorption processes for nitrogen production from air. However, it is quite clear that much higher kinetic selectivity for methane-nitrogen separation in Sr-ETS-4 would be achieved if the optimum in the change in equilibrium and diffusivity ratios with dehydration temperature could be synchronized. In this regard, the role of partial cation exchange is currently being explored.

Langmuir, Vol. 21, No. 10, 2005 4545

Two homogeneous equilibrium isotherm models, namely, Langmuir isotherm and multisite Langmuir isotherm, and a heterogeneous model based on a local Langmuir isotherm were used to analyze the measured equilibrium data. According to the chemical potential gradient theory as the driving force for diffusion, the concentration dependence of micropore diffusivity is related to the isotherm curvature. Therefore, experimentally measured concentration dependence of micropore diffusivity provided an additional reference for a comparative evaluation of the three isotherm models. Although heterogeneous analysis brought small improvements in a few isolated cases, from an overall perspective, the homogeneous analysis appears quite adequate in capturing the experimental equilibrium data and concentration dependence of the micropore diffusivity. Hence, a Langmuir isotherm with independent saturation capacity or the multisite Langmuir isotherm and a bidispersed pore diffusion model including concentration dependence of micropore diffusivity according to chemical potential as the driving force for diffusion should be able to provide reliable simulation of a methanenitrogen separation process using Sr-ETS-4. It should be noted that in view of independent saturation capacity, the use of Langmuir isotherm will necessitate the use of ideal adsorbed solution (IAS) theory for binary equilibrium calculations. The multisite Langmuir model, on the other hand, has an implicit form of multicomponent extension. Glossary a A b bo c cp Cg′ Dc Dco Dco′ D h o′ D hc E j J K Ko n nv n p P Pd Pu Pd0+ Pd∝ Pu0 Pu∝

sites occupied by each adsorbate molecule in the adsorbed phase parameter of gamma distribution Langmuir constant pre-exponential constant for temperature dependence of b gas-phase concentration, mol/cm3 gas concentration in the macropores, mol/cm3 imaginary gas-phase concentration in the micropore, mol/cm3 micropore diffusivity, cm2/s limiting micropore diffusivity, cm2/s pre-exponential factor for temperature dependence of diffusivity in eq 16, cm2/s pre-exponential factor in eq 26, cm2/s micropore diffusivity from heterogeneous model in eq 26, cm2/s activation energy of adsorption, kcal/mol jth step of equilibrium measurement diffusion flux, gmol/cm2, s dimensionless Henry’s constant pre-exponential constant for temperature dependence of K total number of moles of adsorbate adsorbed by adsorbent number of gas moles flowing through the valve number of moles adsorbed by adsorbent at step j parameter of gamma distribution pressure, atm pressure in the dose chamber, atm pressure in the test (uptake) chamber, atm initial pressure in the dose chamber, atm final equilibrium pressure in the dose chamber, atm initial pressure in the test chamber, atm inal equilibrium pressure in the test chamber, atm

4546 P∝ q, qc qlocal qp qs q j q* r rc Rg Rp t T ∆U ∆Us Va Vd Vu X

Langmuir, Vol. 21, No. 10, 2005 final pressure in the constant volumetric system ()Pd∝ ) Pu∝), atm adsorbed phase concentration based on microparticle volume, mol/cm3 local adsorbed phase concentration, mol/cm3 adsorbed gas-phase concentration based on particle, mol/cm3 adsorbate saturation capacity in the solid phase, mol/cm3 average adsorbate concentration in the micropore, mol/cm3 equilibrium value of q radial distance coordinate of the microparticle, cm microparticle radius, cm universal gas constant adsorbent pellet radius, cm time, s system temperature, K internal energy change due to adsorption, kcal/mol isosteric heat of adsorption, kcal/mol volume of adsorbent, cm3 volume of dose cell, cm3 volume of uptake cell, cm3 valve constant, mol/s, bar

Marathe et al. Greek Letters R constant in eq 23 bed voidage particle void fraction p η selectivity µ chemical potential θ fractional coverage of the adsorption sites Subscript A B i E K

component A component B ith component equilibrium kinetic

Acknowledgment. R.P.M. acknowledges the Research Scholarship awarded by NUS. Note Added after ASAP Publication. This article was published ASAP on April 13, 2005. Equations 2 and 5 have been modified. The correct version was posted on April 27, 2005. LA046938D

Modeling Gas Adsorption and Transport in Small-Pore Titanium

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