Revisiting Transport of Gases in the Micropores of Carbon Molecular

Langmuir 2003, 19, 393-405

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Revisiting Transport of Gases in the Micropores of Carbon Molecular Sieves Huang Qinglin, S. M. Sundaram, and S. Farooq* Department of Chemical & Environmental Engineering, National University of Singapore, 4 Engineering Drive 4, Singapore 117576 Received August 22, 2002 Single component differential uptake of oxygen and nitrogen in three samples of carbon molecular sieve adsorbent and carbon dioxide and methane in one of the samples has been measured volumetrically at different temperatures over a wide range of adsorbent loading. The results seem to suggest that the early part of the uptake is controlled by a barrier resistance confined at the micropore mouth while the later part is controlled by a pore diffusional resistance distributed in the micropore interior. A dual resistance model is able to capture the observed single component uptake behavior for all the four gases, which is assuredly less tentative than what the current literature may suggest. Moreover, the extracted micropore transport parameters have much stronger dependence on adsorbent loading than that expected from chemical potential gradient as the driving force for diffusion and the assumption of a constant limiting transport coefficient based on prevailing evidence. An approximate way of accounting for this stronger concentration dependence has been proposed and validated by application to the prediction of experimental integral uptake results.

Introduction Diffusion of gases in carbon molecular sieve (CMS) adsorbents is important since it has direct application in the design of kinetically controlled pressure swing adsorption (PSA) processes, such as production of nitrogen from air. It is also important in view of growing interest in CMS membranes. Bergbau-Forschung GmbH (BF) and Takeda Chemical Industries Ltd. are the two leading manufacturers of CMS used in nitrogen PSA plants all over the world. Shirasagi MSC is the trade name of Takeda CMS products. BF CMS is made from hard coal.1 We have one sample of BF CMS and two samples of Takeda CMS in our laboratory. According to the accompanying material data sheets, both Takeda samples are called Shirasagi MSC 3A. One of the Takeda samples was made from coconut shell, but the starting material for the other sample was not specified. In the following text, we will identify the Takeda sample made from coconut shell as Takeda I and the other Takeda sample as Takeda II. However, products from both manufacturers will be called CMS. CMS is known to have bidisperse pore structure with clearly distinguishable macropore and micropore resistances to the transport of sorbates. Kawazoe et al.2 and Chihara et al.3 measured the diffusion of nitrogen and propylene, respectively, in Takeda MSC 5A using the pulse chromatographic method. They concluded that the micropore resistance was distributed in the micropore interior. Ruthven4,5 and Chen et al.6 gravimetrically * Corresponding author. Fax: 65-6779-1936. E-mail: chesf@ nus.edu.sg. (1) Pilarczyk, E.; Knoblauch, K.; Ju¨ntgen, H. Coal-char gasification with steam and steam-oxygen mixtures. Thermochim. Acta 1985, 85, 315. (2) Kawazoe, K.; Suzuki, M.; Chihara, K. Chromatographic study of diffusion in molecular-sieving carbon. J. Chem. Eng. Jpn. 1974, 7, 151. (3) Chihara, K.; Suzuki, M.; Kawazoe, K. Adsorption rate on molecular sieving carbon by chromatography. AIChE J. 1978, 24, 237. (4) Ruthven, D. M.; Raghavan, N. S.; Hassan, M. M. Adsorption and diffusion of nitrogen and oxygen in a carbon molecular sieve. Chem. Eng. Sci. 1986, 41, 1325. (5) Ruthven, D. M. Diffusion of oxygen and nitrogen in carbon molecular sieve. Chem. Eng. Sci. 1992, 47, 4305.

measured the diffusion of oxygen and nitrogen in BF CMS and were able to fit their data with a pore diffusion model. LaCava et al.7 used gravimetric and batch column adsorption methods to measure diffusion of oxygen and nitrogen, while Srinivasan et al.8 measured the diffusion of the same sorbates using the volumetric method. The manufacturers of the samples used in these studies were, however, not disclosed. In these studies, fit of the barrier resistance model was very clear. Diffusion of oxygen and nitrogen was also measured by Loughlin et al.9 in BF CMS using volumetric, gravimetric, and pulse chromatographic methods and by Rynders et al.10 using the isotope exchange technique in Takeda CMS. While the former study advocated a dual resistance model, the barrier model fitted the data presented in the latter study. Liu and Ruthven11 gravimetrically measured the diffusion of oxygen, nitrogen, argon, methane, and carbon dioxide in BF CMS. They found that for oxygen, nitrogen, and argon, the data were consistent with distributed pore diffusional resistance. The data for methane were consistent with barrier resistance, and carbon dioxide showed a transition from barrier resistance control at lower temperatures to diffusion control at higher temperatures. They concluded that the results perhaps suggested a dual resistance model with varying importance of the two components depending on pressure and temperature. Reid et al.12 have reported a linear driving force (equivalent to barrier) model for the (6) Chen, Y. D.; Yang, R. T.; Uawithya, P. Diffusion of oxygen, nitrogen and their mixtures in carbon molecular sieve. AIChE J. 1994, 40, 577. (7) LaCava, A. I.; Koss, V. A.; Wickens, D. Nonfickian adsorption rate behaviour of some carbon molecular sieves: I. Slit-potential rate model. Gas Sep. Purif. 1989, 3, 180. (8) Srinivasan, R.; Auvil, S. R.; Schork, J. M. Mass transfer in carbon molecular sieves - an interpretation of Langmuir kinetics. Chem. Eng. J. 1995, 57, 137. (9) Loughlin, K. F.; Hassan, M. M.; Fatehi, A. I.; Zahur, M. Rate and equilibrium sorption parameters for nitrogen and methane on carbon molecular sieve. Gas Sep. Purif. 1993, 7, 264. (10) Rynders, R. M.; Rao, M. B.; Sircar, S. Isotope exchange technique for measurement of gas adsorption equilibria and kinetics. AIChE J. 1997, 43, 2456. (11) Liu, H.; Ruthven, D. M. Diffusion in Carbon Molecular Sieves. In Proceedings of the 5th International Conference on the Fundamentals of Adsorption; LaVan, M. D., Ed.; Kluwer Press: Boston, 1996.

10.1021/la026451+ CCC: $25.00 © 2003 American Chemical Society Published on Web 12/21/2002

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uptake of oxygen, nitrogen, argon, and krypton on a CMS sample received from Air Products and Chemicals Inc. Their uptake data for neon, however, followed a dual resistance model. Rutherford and Do13 found the uptake of carbon dioxide in a sample of Takeda 5A to be controlled by pore diffusional resistance distributed in the micropore interior. In another study, Nguyen and Do14 fitted the uptake of oxygen, argon, and nitrogen in a Takeda 3A sample with a dual Langmuir kinetic model. Kawazoe et al.2 and Chihara et al.3 found the concentration dependence of micropore diffusivity to follow the following model:

d ln p 1 ) Dc0 (for Langmuir isotherm) Dc ) Dc0 d ln q 1-θ (1) which is obtained from chemical potential gradient as the driving force for diffusion, together with the assumption of a constant intrinsic diffusivity (Dc0). Equation 1 is commonly known as Darken’s equation. Ka¨rger and Ruthven15 have discussed the origin of the equation and to whom the original credit should go. Ruthven5 observed similar behavior for the diffusion of oxygen in BF CMS at -80 °C. Rutherford and Do13 also found the uptake of carbon dioxide in Takeda 5A to follow Darken’s relation. Chen and Yang,16 on the other hand, took a kinetic approach based on transition state theory to explain the concentration dependence of micropore diffusivity. According to this theory,

1 Dc ) Dc0 1 - (1 - λ)θ

(2)

where λ is a fitting parameter and is defined as the ratio of the sticking probability of the diffusing molecule on an adsorbed site to the sticking probability on a vacant site. To be physically meaningful, λ should be positive and less than 1. Chen et al.6 obtained λ values of 0.052 82 for oxygen and 0.012 87 for nitrogen from their diffusion data in BF CMS. It is clear from the above discussion that the mechanism of transport of gases in CMS micropores is an area that is not yet well understood. We have undertaken a study in our laboratory in which both single component and multicomponent diffusion of several gases in CMS micropores are systematically investigated over a wide range of adsorbent loading. The pure component results obtained for oxygen, nitrogen, carbon dioxide, and methane in one or more of the three CMS samples mentioned earlier are presented in this communication. Three kinetic models, namely, the pore model, the barrier model, and a dual resistance model, are thoroughly evaluated with the experimental uptakes of all the gases over a wide range of temperature and adsorbent loading. Unlike the published results, the results obtained in this study clearly support dual resistance and the concentration dependence of the transport parameters is consistently much stronger (12) Reid, C. R.; O’Koye, I. P.; Thomas, K. M. Adsorption of gases on carbon molecular sieves used for air separation. Spherical adsorptives as probes for kinetic selectivity. Langmuir 1998, 14, 2415. (13) Rutherford, S. W.; Do, D. D. Adsorption dynamics of carbon dioxide on a carbon molecular sieve 5A. Carbon 2000, 38, 1339. (14) Nguyen, C.; Do, D. D. Dual Langmuir kinetic model for adsorption in carbon molecular sieve materials. Langmuir 2000, 16, 1868. (15) Ka¨rger, J.; Ruthven, D. M. Diffusion In Zeolites and Other Microporous Solids; Wiley: New York, 1992. (16) Chen, Y. D.; Yang, R. T. Predicting binary Fickian diffusivity from pure-component Fickian diffusivity for surface diffusion. Chem. Eng. Sci. 1992, 47, 3895.

Figure 1. Schematic diagram of the constant volume apparatus.

than that expected from eq 1 and its equivalent for the barrier coefficient. Measurement of Adsorption Equilibrium and Kinetics Constant Volume Apparatus. Pure component equilibrium isotherms for oxygen, nitrogen, carbon dioxide, and methane were measured on different CMS samples up to about 8 bar pressure at several temperatures ranging from -20 to 90 °C. Differential uptakes of these gases in the CMS samples were also measured at several levels of adsorbent loading within the pressure and temperature range covered for equilibrium isotherms. Both equilibrium and kinetic measurements were carried out in a constant volume apparatus, which was designed and fabricated in the laboratory. A schematic diagram of the apparatus is shown in Figure 1. The apparatus basically consisted of two cylindrical chambers, called the test chamber (TC) and the dose chamber (DC), connected by an on/off solenoid valve (Asco/Joucomatic, model 71235S) controlled with a dc power supply (Topward Electric Instruments, model TPS-4000). The use of a solenoid valve made the opening very sharp. It helped minimize the dynamics associated with manual opening, which should not have any impact on equilibrium measurement but could affect the early part of the uptake curve. The early part of the uptake data is very crucial for making a distinction between possible transport mechanisms in the micropores. The adsorbent was placed in the test chamber. To facilitate more effective regeneration and to perform equilibrium measurements in the subatmospheric pressure range, a vacuum pump (Edwards, two-stage, model M3), VP, was also connected to the system. Prior to beginning the experiment and between runs conducted at different temperatures, the CMS sample placed in the test chamber was regenerated at 200 °C under high vacuum for 8-10 h. Particular care was taken to purge the system with helium before heating the adsorbent to 200 °C. Our limited experience seems to suggest that adsorption kinetics may be affected by heating at a high temperature in the presence of oxygen. The system was also

Transport of Gases in Micropores flushed with helium a few times during heating to increase the effectiveness of regeneration. The effect of regeneration temperature on the measured isotherm and kinetics is discussed in the Appendix. In a system of known constant volume containing adsorbent, the response to a step perturbation in pressure by a known quantity contains information on both equilibrium and dynamics. A pressure transducer (Endress & Hauser, model PMP131A2201R4S; range, 0-300 psig), PT, was connected to the dose side to measure the initial and final system pressures, which were necessary to carry out the mass balance and calculate the equilibrium adsorbed amount. In addition to this absolute pressure transducer, there were two differential pressure transmitters (Validyne, model P55D-1-N-2-38-S-4-A; range, 0-8 psig), DPT1 and DPT2, on the dose and the test sides and their reference ports were connected to a common cylinder (RC). The differential pressure transmitters were necessary to track the change in pressure on the two sides of the solenoid valve until the system reached new equilibrium following the introduction of a known pressure step. The pressure transmitters were periodically calibrated using a digital pressure calibrator (Fluke, model 700P07) to ensure accuracy of the pressure signals. The transient pressure signals monitored on the dose and test sides were combined to generate fractional uptake data for kinetic study. The voltage signal from the absolute pressure transducer was directly read on a multimeter (Hewlett-Packard, model 34401A) that could read up to 0.1 mV accurately. The signals from the two differential pressure transmitters were continuously recorded on a two-pen chart recorder (Yokogawa, model LR4220). The chart range was appropriately chosen to capture the change with sufficient resolution. The dose, test, and reference chambers were immersed in a constant-temperature bath to maintain steady temperature at a desired level during measurements. A heavy-duty digital temperature controller (Lauda RK8 KS) was used for the purpose. Most of the available commercial brands of constant-temperature baths do well in maintaining temperature in the range of 10-100 °C. The heating capacity is normally adequate, but the cooling capacity drops drastically for temperatures below zero. With the water bath and the tubing carefully insulated and the laboratory air conditioning maintained at 1820 °C, the present controller could cool the experimental system down to -25 °C and maintain the temperature within (0.1 °C. Water was the heating/cooling medium above 0 °C. 2-Propanol was used as the coolant for attaining lower temperatures. Since any change in the operating temperature had a direct impact on the system pressure, tight control of the temperature at the desired level for the entire period of measurement was very important. To ensure that the system was completely leakproof, it was pressurized or evacuated without absorbent in place and all the outlet valves were closed to see if the pressure level was retained. The criterion used for pressure retention was a constant voltage reading from the absolute pressure transducer for at least 2-3 h. Calculation of equilibrium and kinetic data from constant volume experiments also depended on the measured system volume. Hence, volumes of the dose and test sides including associated tubes were carefully measured and good accuracy was ensured by checking reproducibility for a few times. Volumes of the dose and test sides were 384.4 and 365.8 cm3, respectively. The amount of adsorbent used in constant volume measurements was a compromise between the need to ensure that the change in pressure due to adsorption was measurable with sufficient resolution and yet the adsorbents were sufficiently spread out so that all the particles were uniformly exposed to the gas. This was not important for equilibrium measurement but was a crucial factor in the kinetic measurement in order to ensure a uniform surface boundary condition for all the absorbent particles. The amounts of BF, Takeda I, and Takeda II CMS samples used in the present study were 8.1, 7.9, and 7.7 g, respectively. These weights were taken after regeneration. According to the material data sheet, the densities of the three CMS samples based on external volume (i.e., not including the helium pore volume) were 0.988 g/cm3 (BF), 1.02 g/cm3 (Takeda I), and 1.0 g/cm3 (Takeda II). Experimental Procedures. After regenerating the adsorbent at 200 °C under vacuum, the valve connecting to the vacuum

Langmuir, Vol. 19, No. 2, 2003 395 pump was closed and the system was brought to the desired experimental temperature using the constant-temperature bath. A steady voltage output from the absolute pressure transducer was taken as the measure of thermal equilibrium. By then, the temperature reading of the thermocouple, T, placed in the test chamber also attained that of the constant-temperature bath. At this point, the system pressure and temperature were noted and the solenoid valve, SV, was closed to separate the test side from the dose side. Let us denote this system pressure and temperature by P∞(j - 1)[P∞d (j - 1) ) P∞u (j - 1)] and Ts, respectively. The subscripts d and u represent dose side and test side, respectively, and the superscript denotes complete equilibrium. j in the argument is the pressure step indicator and is introduced to develop a general data processing algorithm in the next section. Valve V3 was also closed to lock the pressure in the reference cell. A known amount of adsorbate gas was then added to the dose side through valve V1, and some time was allowed for the gas to attain the system temperature. Let this new pressure of the dose side be denoted by P0+ d (j) when the temperature stabilized to Ts. The solenoid valve was turned on and left in that position to allow the pressures on the dose and test sides to be equalized and the system to reach new equilibrium. Let the new equilibrium system pressure be P∞(j) [P∞d (j) ) P∞u (j)]. The reading of the thermocouple placed inside the test chamber was found to remain practically unchanged at the set system temperature. The solenoid valve was closed after recording the new equilibrium system pressure. Valve V3 was then opened briefly to equalize the pressure of the reference chamber with that of the dose side. A fresh supply of adsorbate gas in a known amount was added to the dose side, and the rest of the steps were repeated in the same sequence as described above. This was continued until equilibrium system pressure reached the target upper range, which was approximately 8 bar in the present study. Changes of pressure on the dose and test sides [Pd(j) and Pu(j), respectively] following the opening of the solenoid valve were also followed until new equilibrium was reached, when desired, by continuously recording voltage output signals from the two differential pressure transmitters, DPT1 and DPT2, on a chart recorder. The approach to new equilibrium following the pressure perturbation could be a very slow process depending on the system temperature and adsorbate gas in use. Hence, a sufficiently long time was allowed in each run to ensure complete equilibrium. The magnitude of perturbation introduced in each step depended on the amount of fresh gas added each time to the dose side. For equilibrium measurement, the step size was limited by the frequency of data points desired in the experimental pressure range of the isotherm. Hence, step size was limited to ∼0.5 bar when the target was only to get an equilibrium data point. The other objective of the present study was to measure differential uptake of the adsorbate gases at various levels of the adsorbent loadings. Therefore, at approximately chosen equilibrium levels, experiments were conducted with very small step size (0.05-0.1 bar) and pressure transients on the two sides were followed on the chart recorder. A small step size ensured linearity of the segment of the isotherm traveled during the uptake so that the transport parameters could be assumed to remain approximately constant. How small a step size was small enough to ensure linearity depended on the curvature of the isotherm, and hence some degree of trial and error was necessary. Data Processing. Calculating the amount adsorbed was a matter of straightforward material balance. Assuming that the ideal gas law was valid in the pressure range for the gases in question, the following mass balance was applicable for the jth equilibrium step:

Vd ∞ {P0+ ) d (j) - P (j - 1)} RgTs Vu - Va + ∆n(j) (3) RgTs

{P∞(j) - P∞(j - 1)}

∆n(j) ) n(j) - n(j - 1)

(4)

j ) 1, 2, 3, ... In eq 3, Vd, Vu, and Va were volumes of the dose side, test side,

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and adsorbent particles, respectively. ∆n(j) was the number of moles adsorbed by the adsorbent particles as a result of pressure perturbation in step j. n(j) was the total number of moles adsorbed up to the jth step and was in equilibrium with the adsorbate at pressure P∞d (j) [)P∞u (j)]. Since the material data sheet provided adsorbent density based on contour volume, Va calculated from the measured mass and given density, in fact, gave external particle volume (i.e., helium pore volume was not included). Hence, n(j) included the moles of gas that were occupying the macropores of the adsorbent. Although CMS has a bidisperse pore structure with clearly distinguished macropore and micropore resistances to the transport of sorbates, the macropores do not contribute to the adsorption capacity of this adsorbent. At equilibrium, the macropore voids have the same gas as in the bulk phase. Since the focus of the present study was transport of gases in the micropores, it was more convenient for the modeling study to use equilibrium adsorbed concentration expressed on the basis of microparticle volume. Equilibrium adsorbed concentration per unit microparticle volume, qc, was related to adsorbed concentration per unit macroparticle volume, qp, by the following equation:

qp ) (1 - p)qc + pc

(5)

In the above equation, p is the macropore voidage and c is the gas-phase concentration with which the adsorbent had attained equilibrium. In this study, macropore voidage of 0.33 was assumed for all the three CMS samples. In terms of the variables used to describe equilibrium data processing, qp ) n(j)/Va and c ) P∞(j)/RgTs. All the isotherms measured in this study have been presented as qc versus c plots. Brandani17 has presented a detailed analysis of kinetic measurement in a constant volume apparatus for the case of pore diffusion control. It has been shown that the presence of a maximum in the pressure versus time plot for the test side is clear evidence of kinetic control, which has led to the conclusion that the pressure response of the test side should be used for extracting kinetic data. We do not disagree with this conclusion and had used the suggested criterion to confirm that our experiments, particularly for the relatively faster diffusing oxygen, were indeed in the kinetically controlled region. However, we decided to use fractional uptake plots by combining the pressure responses from the two sides according to the following equation: ∞ mt {P0+ d (j) - Pdt(j)} - {Put(j) - P (j - 1)}(Vu - Va)/Vd (j) ) ∞ ∞ ∞ m∞ {P0+ d (j) - P (j)} - {P (j) - P (j - 1)}(Vu - Va)/Vd

(6) Fractional uptake, mt/m∞, plotted against the square root of time had more distinct attributes to distinguish among possible transport mechanisms.

Adsorption Equilibrium Single component isotherms of four gases on CMS samples at several different temperatures and the fit of the data to the Langmuir model are shown in Figure 2. Isotherms of nitrogen on a modified 4A zeolite (RS-10) were also measured and are included in Figure 2. In the pressure range covered in this study, carbon dioxide had the highest adsorption capacity. However, the fit of the Langmuir model to this data set is less satisfactory compared to that of the other gases. The choice of a suitable equilibrium model to accurately represent carbon dioxide isotherm data merits further investigation. Compared to the isotherms of oxygen and nitrogen on the two Takeda CMS samples, the isotherms on BF CMS were comparatively lower. In all three CMS samples, both oxygen and nitrogen appeared to have very close adsorption capacity (17) Brandani, S. Analysis of the piezometric method for the study of diffusion in microporous solids: isothermal case. Adsorption 1998, 4, 17.

Table 1. Langmuir Isotherm Parameters adsorbent BF CMS Takeda I CMS

Takeda II CMS 4A zeolite

adsorbate

b0 × 10-3 (cc/mmol)

(-∆U) (kcal/mol)

qsc (mmol/cc)

O2 N2 O2 N2 CO2 CH4 O2 N2 N2

3.86 2.80 3.48 4.02 0.22 0.73 3.15 2.42 0.85

3.90 4.26 3.98 4.01 6.73 5.97 4.02 4.33 4.74

3.80 2.87 4.34 3.61 6.30 3.74 4.44 3.55 4.29

at low coverage but the isotherms became clearly distinguishable as the surface coverage was increased. The Langmuir isotherm has the following form:

qc bc ) qsc 1 + bc

b ) b0e-∆U/RgT

(7)

The model parameters were obtained separately for each gas by simultaneous nonlinear regression of data at all temperatures, and the results are summarized in Table 1. It is clear from Table 1 that the saturation capacities for different gases extracted from the fit of the Langmuir model to experimental data were different. Even for oxygen and nitrogen, saturation capacity differed by 17-25% (based on oxygen value) in the three CMS samples. Since the concentration dependence of micropore transport parameters depend significantly on isotherm curvature, an accurate fit of the isotherm data to the model is important. Since the extended Langmuir model is thermodynamically consistent only when the constituents of the mixture have the same qsc,18 in the present situation, one choice would be to use ideal adsorbed solution (IAS) theory for multicomponent equilibrium prediction. The alternative is to use other isotherm models that give thermodynamically consistent multicomponent extension even when the saturation capacities of the constituents are different. The multisite Langmuir model is one such isotherm. The issue of binary equilibrium prediction and its impact on the calculation of binary uptake will be one of the topics of our next communication. Adsorption Kinetics Model Assumptions. Although the gas diffusion in CMS is practically controlled by the resistance in the micropores, for the sake of completeness, macropore diffusional resistance was also taken into account in the model used in this study. Since pure gas was used, there was no external film resistance to gas transport. The following assumptions were made in the model development: (1) The ideal gas law applied and the system was considered isothermal. (2) Only molecular diffusion was assumed for the transport in the macropores. (3) Both macroparticles and microparticles were assumed to be spherical. (4) The segment of the isotherm covered in a differential step change was taken as linear and the kinetic parameters were assumed to remain constant over the small segment. (5) The opening time of the solenoid valve was assumed to be negligible compared to the time of observation. (6) An approximate linear form of a quadratic driving force was assumed to represent the dynamics of the solenoid valve separating the dose and test chambers. Model Equations. Single component uptakes in the adsorbent samples were measured by the constant volume (18) Rao, M. B.; Sircar, S. Thermodynamic Consistency For Binary Gas Adsorption Equilibria. Langmuir 1999, 15, 7258.

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Figure 2. Measured equilibrium isotherms of pure gases in CMS and zeolite adsorbents and fits of the Langmuir model: oxygen (solid symbols) and nitrogen (open symbols) in (a) BF CMS, (b) Takeda I CMS, and (c) Takeda II CMS; (d) carbon dioxide (solid symbols) and methane (open symbols) in Takeda I CMS; (e) nitrogen in a modified 4A zeolite called RS-10. The solid and dotted lines are fits of the Langmuir isotherm model for the solid and open symbols, respectively.

method. To account for the influence of the solenoid valve, if any, on the measured adsorption kinetics, the following equation was used:

dnv ) X(Pd - Pu) dt

(8)

where nv is the number of moles of gas flowing through the valve and X is the valve constant. Blank experiments (i.e., without any adsorbent on the test side) conducted with helium, oxygen, and nitrogen to calibrate the valve constant, X, gave a fairly constant value, which was an indication that the assumption of a linear driving force was reasonable. The pressures on the two sides were functions of time in the volumetric measurement and are given by the following mass balance equations between the two sides of the apparatus. Test or uptake cell mass balance:

(

|

Dp dPu dcp ) RgT -3 pVa Vu dt Rp drp 0 Pu|t)0 ) Pu Dose cell mass balance:

rp)Rp +

)

dnv dt

}

dPd dnv ) -RgT dt dt Pd|t)0 ) P0+ d

Vd

(10)

In the above equations, Pu, Pd, P0u, and P0+ d have the same meaning as Pu(j), Pd(j), P∞u (j - 1), and P0+ d (j), respectively, introduced in the section on data processing. The macropore mass balance equation and the boundary conditions have the following form:

( (

))

∂cp ∂cp 1 - p ∂q 1 ∂ j + ) 2 R2Dp ∂t p ∂t R ∂R ∂R ∂cp ∂R

|

)0

(11)

(12)

R)0

cp|R)Rp ) c ) (9)

}

Pu RgT

(13)

The boundary condition at the center of the macroparticle follows from geometrical symmetry. q j is the average adsorbed phase concentration in the micropore, which is related to the adsorbate flux at the micropore mouth by

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the following equation:

|

∂q j 3 ∂q ) D ∂t rc c ∂r

(14)

r)rc

The mass balance in the micropore presents different expressions depending on different transport mechanisms. Here, the experimental results are thoroughly analyzed using three different models, namely, the barrier model, the pore model, and the dual model. 1. Barrier Model. The barrier model assumes that gas diffusion was controlled by the barrier resistance confined at the micropore mouth. The mass balance across the micropore mouth is

dq j ) kb(q* - q j) dt

(15)

where kb is the coefficient of barrier resistance. Equation 15 replaces eq 14. q* in eq 15 is the equilibrium adsorbed phase concentration based on micropore volume corresponding to the macropore gas concentration, cp:

q* ) f(cp)

(16)

where f(cp) refers to the isotherm model chosen to represent the measured equilibrium data. 2. Pore Model. The pore model is based on the assumption that gas diffusion was controlled by resistance distributed in the micropore interior. The mass balance equation for micropore diffusion is

∂q 1 ∂ 2 ∂q ) r Dc ∂t r2 ∂r ∂r

( (

))

(17)

Boundary conditions:

|

∂q ∂r

r)0

)0

q|r)rc ) f(cp)

(18) (19)

3. Dual Model. In the dual model, it is assumed that gas diffusion was controlled by a combination of barrier resistance confined at the micropore mouth followed by a distributed pore interior resistance acting in series. The micropore diffusion equation (eq 17) is also applicable for the dual model, but the boundary condition at the mouth of the micropore, that is, eq 19, changes to the following equation:

|

3 ∂q D rc c ∂r

r)rc

) kb(q* - q)|r)rc

(20)

The barrier and pore models are actually two extreme cases of the dual model. The dual model solution reduces to that of the barrier model when a large value is assigned to the micropore diffusivity and vice versa. The model equations were numerically solved by the method of orthogonal collocation. Extraction of Transport Parameters and Impact of Valve Resistance. To understand the gas transport mechanism in CMS micropores and extract the transport parameters from the experimental uptake results, it was necessary to quantify valve resistance and investigate its impact on the shape of uptake. The resistance of the solenoid valve between the dose and test chambers was measured by conducting blank runs (i.e., without any adsorbent in the test chamber). The valve constant, X,

Figure 3. Fits of (a) the dual model and (b) the pore model with and without valve resistance to pure gas uptakes in Takeda I CMS and a modified 4A zeolite (RS-10), respectively.

was extracted by fitting the model solution [eqs 8-10 for ) 1 and Dp ) 0] to the blank experimental results. The values obtained from different runs were nearly constant around a mean of 0.04 mol s-1 bar-1. The valve resistance thus calculated was used in modeling the adsorbing experiments. In the differential uptake measurements, the transport parameters were assumed to remain constant over the concentration range covered in each run and were the only fitting variables to match the model solutions to the measured uptake data. The representation of the valve resistance is mathematically similar to the linear driving force representation of the barrier resistance used in this study. The solution of both of these models has the sigmoidal signature in the early part of the uptake. As such, there was a need to evaluate the contribution of the valve resistance relative to the extracted barrier resistance, to establish the reliability of the latter. Representative oxygen and nitrogen uptakes compared with fits of the dual model with and without valve resistance are shown in Figure 3. A large value was assigned to the valve constant (X ) 10 mol s-1 bar-1) to simulate negligible valve resistance. The two theoretical curves are practically indistinguishable, which is a clear indication that the valve resistance calibrated from blank runs was negligible compared to the barrier resistance encountered in this study. To further verify that the initial sigmoidal signature observed for the uptake of gases in CMS samples was unambiguous proof of barrier resistance and not due to the valve resistance in the constant volume apparatus, nitrogen uptake in a modified 4A zeolite (RS-10) sample, which has been well established to show pore diffusional behavior for oxygen and nitrogen,19,20 was measured in the same setup. A representative experimental result for nitrogen in RS-10 is also shown in Figure 3 together with the fits of the pore model with and without valve resistance. The good fit of the pore model particularly in the early (19) Ruthven, D. M. Principles of adsorption and adsorption processes; John Wiley & Sons: New York, 1984. (20) Farooq, S. Sorption and diffusion of oxygen and nitrogen in molecular sieve RS-10. Gas Sep. Purif. 1995, 9, 205.


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Figure 4. Single component uptakes in the three CMS samples at low coverage.

part is the most convincing direct evidence that the barrier resistance seen for gas uptake in CMS samples is indeed genuine and not an experimental artifact due to valve resistance. These results also reinforce the conclusion of negligible contribution from valve resistance. Uptake at Low Surface Coverage. The experimental uptake results of O2 and N2 in the three CMS samples and CO2 and CH4 in Takeda I CMS in the linear range (θ f 0) at least at three temperatures were compared with the three models. Representative results are shown in Figure 4. It is seen that in all the cases, the barrier model can be made to fit the early part well but it fails in the later part of the uptake. On the other hand, the solution of the pore model is distinctly different from the shape of the experimental uptake in the early part, but it fits the later part of the data. The inabilities of the pore and barrier models to fit the experimental data are more severe for methane uptake, while the fit of the dual model is excellent. In the case of faster diffusing components such as oxygen and carbon dioxide, runs were conducted at subzero temperatures to make the superiority of the dual model clearer. Undoubtedly, the dual model gives the best fit of the entire uptake data in all the cases. The dual model parameters, namely, barrier coefficient and pore diffusional time constant, were extracted by minimizing the residual of model fit to the experimental

Figure 5. A representative residual surface resulting from the direct search optimization of the dual model transport parameters.

data. A direct search was conducted by systematically varying both the transport parameters over a wide range of making very small increments. A typical residual surface is shown in Figure 5. The optimum fits gave the limiting transport parameters, since θ values for these runs varied in the range 0.01-0.03. The two transport coefficients for

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oxygen, nitrogen, carbon dioxide, and methane in Takeda I CMS plotted against the inverse of temperature are shown in Figure 6 according to the following equations:

Dc0

D′c0 -Ed/RgT e rc2

(21)

kb0 ) k′b0e-Eb/RgT

(22)

2

rc

)

The plots are linear as expected for activated transport processes. A summary of all the pore and barrier coefficients and transport activation energies measured in this study is given in Table 2. It is apparent from Figure 6 that the rates of adsorption of gases in Takeda I CMS are in the order oxygen > carbon dioxide > nitrogen > methane. The sizes of these molecules based on the Lennard-Jones values are in the order carbon dioxide (4.00 Å) > methane (3.82 Å) > nitrogen (3.68 Å) > oxygen (3.43 Å). However, the activation energy for transport across the pore mouth barrier is in the order methane > carbon dioxide > nitrogen > oxygen, and that for transport in the micropore interior is in the order methane ≈ nitrogen > carbon dioxide > oxygen. This shows that the size of gas atoms and molecules is not the only reason for the difference in adsorption kinetics. Reid et al.12 suggested that the adsorption kinetics was determined by interaction of the adsorbate with the adsorbent surface and the relative size of the adsorbate in relation to the pore size distribution of the adsorbent, particularly in the selective part of the porous structure. Uptake at High Surface Coverage. Differential uptake measurements were also conducted at various levels of adsorbent loading outside of the linear range. Representative results for O2 and N2 in all three CMS samples and for CO2 and CH4 in Takeda I CMS are compared with the three models in Figure 7. The observations made in the previous section with respect to the experimental results at low coverage are also valid for these runs. Of course, the pore and dual models seem to become closer as the adsorbent loading is increased. The two models nearly overlap at high coverage in the case of oxygen. This suggests that the importance of barrier resistance across the micropore mouth diminishes with increasing adsorbent loading. Although to a lesser degree,

Figure 6. Temperature dependence of (a) diffusivity in the micropore interior and (b) the barrier coefficient of the dual model for oxygen, nitrogen, carbon dioxide, and methane in Takeda I CMS.

this trend of diminishing importance of barrier resistance with adsorbate loading is also seen for nitrogen and carbon dioxide. However, the three model curves are still clearly distinguishable in the case of methane uptake even at θ ) 0.71. On the basis of these observations, the obvious conclusion is that the dual model provides a unified approach to capture the uptake behavior of all the gases in the CMS adsorbent samples in the entire range investigated in this study. Concentration Dependence of Transport Parameters. The transport parameters for the dual model at various levels of adsorbent loadings were extracted by fitting the model solutions to the experimental uptake

Table 2. Transport Parameters of Gases in Three CMS Samples adsorbent BF CMS

model

adsorbate

dual

O2 N2

Takeda I CMS

dual

O2 N2 CO2 CH4

Takeda II CMS

dual

O2 N2

temp (K)

Dc0/rc2 × 103 (s-1)

302.15 267.65 253.15 302.15 283.15 275.15 267.65 253.15 302.15 283.15 273.15 303.15 283.15 263.15 333.15 303.15 283.15 302.15 283.15 253.15 302.15 283.15 273.15

6.84 3.27 1.85 0.43 0.18 0.13 4.78 2.80 0.58 0.22 0.13 2.10 1.06 0.42 0.0077 0.0026 0.0008 12.19 5.51 1.60 0.28 0.13 0.067

D′c0/rc2 (s-1)

Ed (kcal/mol)

βp

5.70

4.01

0

83.94

7.32

0.61

55.63

4.98

0

706.31

8.42

0.61

81.43

6.35

1.06

2.77

8.43

2.31

386.94

6.25

0

138.65

7.86

0.61

kb0 × 102 (s-1)

k′b0 (s-1)

Eb (kcal/mol)

βb

24 7.57 4.57 1.06 0.51 0.44 8.1 4.45 0.88 0.39 0.26 5.72 1.23 0.57 0.017 0.0033 0.0012 12.48 7.42 2.77 0.53 0.30 0.14

919.38

5.01

5.56

121.61

5.62

2.07

28.62

5.57

5.56

819.83

6.88

2.07

7.25

6.58

468.74

9.85

2.68

299.92

4.67

5.56

7.33

2.07

5607.6

1143.9


Langmuir, Vol. 19, No. 2, 2003 401

Figure 7. Single component uptakes in the three CMS samples at various levels of adsorbent loading.

curves. The transport parameters are plotted as a function of surface coverage (θ) in Figure 8. Equation 1 relates the concentration dependence of the diffusivity in the micropore interior to the occupancy of the adsorption sites. This equation is derived from the chemical potential gradient theory as the driving force with adsorption equilibrium represented by the Langmuir isotherm. In this derivation, the limiting diffusivity, Dc0, is assumed to be independent of θ. Starting from the same principle, a similar equation, having the following form, may also be derived for the barrier coefficient:8

kb 1 ) kb0 1 - θ

(23)

Like Dc0, in the above equation the limiting barrier coefficient, kb0, is also assumed to be independent of θ. The same form of concentration dependence may also be obtained from the Langmuir kinetics model,11 which is a good approximation of the form obtained from the slit potential model.7 It is obvious that both of the micropore transport parameters are increasing functions of adsorbed phase concentration. For oxygen, the concentration dependence of micropore diffusivity follows Darken’s equation. In all other cases, the dependence is much stronger than that expected from Darken’s equation (or its equiva-

lent for the barrier coefficient). For oxygen, nitrogen, and carbon dioxide, the concentration dependence of the barrier coefficient is stronger than that for micropore diffusivity. The two trends are, however, comparable for methane. Despite any seemingly evident correlation between molecular size and observed trends, it is perhaps somewhat comforting to note that the concentration dependences of the transport parameters for oxygen in the three CMS samples are practically inseparable, even though the resistances are higher in Takeda II. The same observation is also true for nitrogen. Nitrogen uptake in the modified 4A zeolite sample was also measured at various levels of adsorbent loading. The extracted pore diffusivity values are plotted as a function of surface coverage in Figure 9. It is clear that the concentration dependence of nitrogen diffusivity obeys Darken’s relation reasonably well. This serves to unambiguously establish that the stronger concentration dependence of the transport parameters in the CMS samples is indeed genuine. An Empirical Approach To Account for the Stronger Concentration Dependence. In eqs 1 and 23, the limiting transport parameters are assumed to be independent of fractional coverage of the adsorption sites, θ. In the empirical approach proposed here, the stronger concentration dependence is viewed as coming from

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Figure 8. Concentration dependence of the two transport parameters of the dual model in the three CMS samples. The dotted line is the plot of Darken’s equation or its equivalent for the barrier coefficient. The continuous lines are the optimized fits of eqs 26 and 27 with βp and βb as the fitting parameters.

Figure 9. Concentration dependence of nitrogen micropore diffusivity in a modified 4A zeolite (RS-10). At 274.15 K, Dc0/rc2 ) 5.5 × 10-5 s-1.

additional dependence of the limiting transport parameters on θ. In the studies on zeolite diffusion, Dc0 has commonly been reported to remain constant and in some exceptional cases was found to decrease with coverage.15 However, in adsorbents with a micropore size distribution like CMS, where the pore connectivities are not fully understood, the possibility of the limiting transport parameters being different in different pores is not unrealistic. Hence, it is proposed that

θ Dc0 ) D/c0 1 + βp 1-θ

(24a)

kb0 ) k/b0

(24b)

( ) (1 + β 1 -θ θ) b

where βp and βb are the fitting parameters which account for the pore size distribution effect experienced by different adsorbates. The form of the concentration dependence chosen above ensures that Dc0 (or kb0) ) D/c0 (or k/b0) as θ f 0. But there are many other forms that could satisfy this limiting constraint. In that sense, the chosen form is arbitrary. The proposed empiricism may be viewed as a homogeneous approximation of a heterogeneous behavior at the expense of introducing a fitting parameter. It is also implicitly assumed that the pores are filled in the order of increasing Dc0 and kb0. βp was extracted separately for each component by optimizing the fit of eqs 1 and 24a to the experimental Dc/D/c0 versus θ data shown in Figure 8. βb was similarly obtained by optimizing the fit of eqs 23 and 24b to the experimental kb/k/b0 versus θ data. The fits are also shown in Figure 8, and the extracted values are given in Table 2. βp ) 0 for oxygen, since it follows Darken’s equation. On the basis of the optimized βp and βb values, the concentration dependence of the barrier coefficient is in the order carbon dioxide > oxygen > methane > nitrogen and that for micropore diffusivity is in the order methane > carbon dioxide > nitrogen > oxygen. The result for carbon dioxide might have been somewhat affected by the less satisfactory fit of its equilibrium data. Prediction of Integral Uptake. In this section, we examine the impact of the very strong concentration dependence of the micropore transport parameters on integral uptake and the ability of the proposed empirical model to predict the observed impact. To resolve these


Langmuir, Vol. 19, No. 2, 2003 403

crucial issues, integral uptakes of oxygen and nitrogen in Takeda I CMS particles for a step change of 0-10 bar at the surface were measured using the differential adsorption bed (DAB) technique. The DAB method of uptake measurement has been discussed in several publications.6,21,22 The setup used and the procedures adopted here are similar to those in the publications. Equations 11, 12, 14, 17, 18, and 20 and the following boundary condition at the macropore surface constitute the model equations:

cp|R)Rp ) c0 )

P0 RgT

(25)

where P0 is the constant pressure of the feed after introducing the step change. In eqs 17 and 20, Dc, kb, and q* are given by the following equations for a large step change in boundary concentration:

Dc )

D/c0 θ 1 + βp 1-θ 1-θ

)

(26)

kb )

k/b0 θ 1 + βb 1-θ 1-θ

)

(27)

(

(

bcp q* ) qsc 1 + bcp

(28)

The following equation was used to calculate the fractional uptake from the model solution:

mt ) m∞ p

∫0 3cp(R/Rp)2 d(R/Rp) + (1 - p)∫0 3qj (R/Rp)2 d(R/Rp) 1

1

qscbc0 pc0 + (1 - p) 1 + bc0 (29) where

q j)

∫013q(r/rc)2 d(r/rc)

(30)

The intrinsic transport parameters of the dual model, D/c0 and k/b0, and the fitted parameters βp and βb of the empirical models (proposed to account for the very strong concentration dependence of the micropore transport parameters) used in the simulations were established from differential uptake experiments in the previous sections. These values are given in Table 2. The equilibrium isotherm parameters, qsc and b, obtained from the measured isotherms are given in Table 1. The experimental results and model predictions (referred to as dual model 2) are shown in Figure 10. The dual model was also solved for βp ) βb ) 0. βp ) 0 reduces eq 26 to Darken’s equation. Similarly, βb ) 0 reduces eq 27 to the analogue of Darken’s equation for the barrier coefficient. These results are also included in the figure as dual model 1. The quantitative difference between experimental data and prediction by dual model 1 is indeed very large. It is, (21) Carlson, N. W.; Dranoff, J. S. In Fundamentals of Adsorption; Liapis, A. I., Ed.; Engineering Foundation: New York, 1987. (22) Do, D. D.; Hu, X.; Mayfield, P. L. J. Multicomponent adsorption of ethane, n-butane and n-pentane in activated carbon. Gas Sep. Purif. 1991, 5, 35.

Figure 10. Single component integral uptakes of (a) oxygen and (b) nitrogen in Takeda I CMS.

therefore, clear that the very strong concentration dependence of the transport parameters has a large impact on the integral uptake and the proposed empirical models predict the observed behavior very well for both oxygen and nitrogen. The applicability of the proposed empirical models in predicting integral uptake will be further verified by comparing with more experimental data and extended to column dynamics calculations in our next communication. Conclusions The equilibrium and kinetics of adsorption for oxygen and nitrogen in the three CMS samples (BF and Takeda I and II CMS) and carbon dioxide and methane in a Takeda I CMS sample have been investigated at several temperatures over a wide range of adsorbent loading using a constant volume apparatus. Similar measurements were also carried out for nitrogen in a modified 4A zeolite (RS10) sample to replicate known behavior and thus authenticate the measurement and analytical technique used. The Langmuir model fitted all the isotherm data well. Transport of gases in the micropores of the three CMS samples used in this study seems to be controlled by a combination of barrier resistance at the micropore mouth followed by a distributed pore interior resistance acting in series. The proposed dual model provides a unified approach that is able to capture uptake results in the entire range covered in this study. The extracted micropore transport parameters have much stronger dependence on adsorbent loading than that expected from Darken’s equation (or its equivalent for the barrier coefficient) for the Langmuir isotherm. A simple empirical approach has been proposed to account for this stronger concentration dependence. The proposed approach has been verified by applying it to the prediction of experimental integral uptakes over a step change of 0-10 bar. Extension of this approach to the prediction of binary and ternary uptakes is currently being investigated. Appendix: Effect of Regeneration Temperature on Measured Isotherms To decide on a suitable regeneration temperature for CMS, oxygen and nitrogen isotherms were measured on

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of oxygen might have enlarged the pores somewhat to affect the uptake. In this study, regeneration II was strictly followed in the regeneration of all CMS samples. b b0 c c0 cp D Dc Dc0 D/c0 D′c0 Figure 11. Equilibrium isotherms of (a) oxygen and (b) nitrogen on Takeda II CMS showing the effect of regeneration temperature: solid symbols, regenerated at 473.15 K; open symbols, regenerated at 373.15 K.

Dp Eb Ed j K kb kb0 k/b0 k′b0 mt

Figure 12. Carbon dioxide uptakes at low surface coverage and -10 °C in BF CMS showing the effect of regeneration conditions.

Takeda II CMS after regenerating the adsorption sample at 100 and 200 °C. The results are shown in Figure 11. The isotherm measured at 29 °C did not seem to have any significant impact of the regeneration temperature. However, a clearly distinguishable capacity increase was observed for the higher regeneration temperature when the isotherms were measured at 10 °C. These observations were true for both oxygen and nitrogen. Consequently, a regeneration temperature of 200 °C was chosen for this study. It was also observed that heating in the presence of oxygen at 200 °C affected the uptake results. Uptakes of carbon dioxide at low pressure and -10 °C in two differently regenerated BF CMS samples are shown in Figure 12. The adsorbent came in contact with room air while loading into the test chamber. For the convenience of easy reference, the two regeneration conditions are called regeneration I and regeneration II. In regeneration I, the adsorbent was heated at 200 °C for 8-10 h first and then subjected to alternative evacuation and helium purge for a few times for cleaning the desorbed gases. In regeneration II, the test chamber containing the adsorbent was alternatively evacuated and flushed with helium a few times to ensure that there was no residual oxygen in the adsorbent. The sample was then regenerated at 200 °C for 8-10 h under high vacuum. It is clear from Figure 12 that the uptake was faster in the case of regeneration I. It appears that high temperature heating in the presence

m∞ n nv ∆n P Pd P0 Pu P0+ d P∞d P0u P∞u P∞ q, qc qp qsc q j q* r rc R Rg

Notation Langmuir constant in the equilibrium isotherm equation pre-exponential constant for temperature dependence of b gas-phase concentration constant feed concentration in the DAB uptake experiments gas concentration in the macropores diffusivity micropore diffusivity limiting diffusivity limiting diffusivity in the smallest accessible pore defined in eq 24a pre-exponential constant for temperature dependence of diffusivity macropore diffusivity activation energy for diffusion across the barrier resistance at the pore mouth activation energy for diffusion in the micropore interior jth step of equilibrium measurement Henry’s constant barrier coefficient limiting barrier coefficient limiting barrier coefficient at the pore mouth defined in eq 24b pre-exponential constant for temperature dependence of the barrier coefficient mass of adsorbate adsorbed by adsorbent up to time t mass of adsorbate adsorbed by adsorbent at equilibrium 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 pressure pressure in the dose chamber constant feed pressure in the DAB uptake experiments pressure in the test chamber initial pressure in the dose chamber final equilibrium pressure in the dose chamber initial pressure in the test chamber final equilibrium pressure in the test chamber final pressure in the constant volumetric system ()P∞d ) P∞u ) adsorbed phase concentration based on microparticle volume adsorbed gas-phase concentration based on particle volume saturation capacity from the Langmuir model average adsorbate concentration in the micropore adsorbed amount in equilibrium with the macropore gas concentration radial distance coordinate of the microparticle microparticle radius radial distance coordinate of adsorbent pellet universal gas constant

Transport of Gases in Micropores Rp t T Ts ∆U Va Vd Vu X

adsorbent pellet radius time temperature system temperature energy of adsorption or internal energy volume of adsorbent volume of dose cell volume of uptake cell constant of the solenoid valve

Langmuir, Vol. 19, No. 2, 2003 405 Greek Letters β p , βb λ θ p

fitting parameters in the proposed empirical models defined by eqs 24a and 24b fitting parameter in eq 2 fractional coverage of the adsorption sites bed voidage particle void fraction LA026451+

Revisiting Transport of Gases in the Micropores of Carbon Molecular

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