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Ind. Eng. Chem. Res. 2001, 40, 2942-2946
GENERAL RESEARCH Permeation Time Lag and the Concentration Dependence of the Diffusion Coefficient of CO2 in a Carbon Molecular Sieve Membrane K. Wang, H. Suda, and K. Haraya* National Institute of Advanced Industrial Science & Technology, Central 5, Higashi 1-1, Tsukuba, Ibaraki 305-8565, Japan
The CO2 permeation experiments were performed on a microporous carbon molecular sieve membrane (CMSM) at 25 °C and over a wide pressure range. The permeation time lag of the penetrant was employed to study the concentration dependence of the diffusion coefficient. The analytical solutions are derived for permeation systems with a nonlinear sorption isotherm (Langmuir) and a concentration-dependent diffusivity [with the functional form of Dµ(c) D0µ ) 1/(1 - θ)n, where θ is the ratio of surface coverage and n g 1]. By comparison of the experimental data with the theoretical models, it is found that, in this CMSM, the apparent diffusivity of CO2 takes a concentration dependence stronger than that described by the traditional HIO model [i.e., Dµ(c)/D0µ ) 1/(1 - θ)]. Introduction The gas permeation experiment (time-lag measurement) is a useful technique for the study of diffusion processes in porous media. The permeation time lag is related to the diffusion coefficient of the transient state and, for some functional forms of the concentration dependence, analytical expressions between these two variables can be derived. The elegant method of Frisch1 has been widely used in the time-lag analysis because of its simplicity and its capability of addressing a concentration-dependent diffusion coefficient. Recently, Rutherford and Do2 studied the permeation time lag in the bidispersed sorbent (in which both bulk-phase diffusion and surface diffusion coexist) with nonlinear sorption isotherms and a concentration-dependent surface diffusivity. By comparison of their models with the experimental data of Ash et al.,3 they concluded that the traditional Darken relation is approximately applicable for the functional concentration dependence of the surface diffusivity. Ash et al.4,5 investigated the concentration dependence of the time lag under various operation modes of a permeation process. Thirteen functional forms of the concentration dependence of the diffusion coefficients were investigated in detail for time lags under the four operation modes. Permeation in a microporous carbon molecular sieve membrane (CMSM) has been known to follow a sorption-diffusion mechanism.6 A CMSM can present high permselectivity for gases with a very small difference in their molecular sizes. For example, under specific pyrolysis (thermal treatment) conditions, Kapton polyimide can produce a dense CMSM with an effective pore size below 4 Å which presents a (O2/N2) selectivity of 36.7 Microporous CMSMs can virtually exclude the
Knudsen and/or molecular diffusion processes, which always present in bidispersed sorbents like activated carbons. This greatly facilitates the study of the diffusion of the adsorbed phase which has been known to take a complicated functional concentration dependence for various systems.8,9 This paper will present the experimental results of CO2 permeation on a KP 950 CMSM under a wide pressure range and investigate the concentration dependence of the diffusion coefficient for the system. Theories The diffusion flux, Jµ, in microporous media can be written as
Jµ ) -Dµ
(1)
where r is the coordinate of the diffusion path and Dµ is the “apparent” diffusivity dependent on the sorbed phase concentration, Cµ. The HIO model10 is a popular correlation for this dependence; that is
D0µ D0µ ) Dµ ) 1 - Cµ/Cµs 1 - θ
(2)
where D0µ is the diffusion coefficient at zero loading, Cµs is the maximum sorption capacity, and θ is the fractional surface loading. Equation 2 is equivalent to the Darken relation, Dµ ) D0µ ∂ ln Cµ/∂ ln C, if the gradient of chemical potential is the driving force and a local Langmuir sorption isotherm applies; that is
Jµ ) -D0µCµ * To whom correspondence should be addressed. E-mail:
[email protected]. Fax: 81-298-61-4487. Tel: 81-298-61-4732.
∂Cµ ∂r
∂µ ∂ ln C ∂Cµ ) -D0µ and ∂r ∂ ln Cµ ∂r Cµ ) CµsbC/(1 + bC) (3)
10.1021/ie000946h CCC: $20.00 © 2001 American Chemical Society Published on Web 05/30/2001
Ind. Eng. Chem. Res., Vol. 40, No. 13, 2001 2943
where µ ) µ0 + RT ln C is the chemical potential, b is the sorption affinity, and C is the bulk-phase concentration (or fugacity for a nonideal gas). For physical sorption processes in an ultramicroporous CMSM, the gas molecules are always within the overlapped potential field of the pore walls; thus, C represents the hypothetical bulk-phase concentration (or fugacity), which is in local equilibrium with the sorbed phase. This paper investigates a new type of functional concentration dependence for the diffusivity:
Dµ )
D0µ (1 - Cµ/Cµs)
) n
D0µ (1 - θ)
) D0µF(n) n
(4)
Equation 4 reduces to the traditional HIO model when n ) 1 and presents a stronger functional concentration dependence when n > 1. The overall mass balance equation for the permeation process in a membrane of slab geometry is
∂Jµ ∂Cµ )∂t ∂t
or
f ′(C)
∂Jµ ∂C )∂t ∂r
L)
]
L)
(
)
2 2 b3C03 b4C04 l2 1 bC0 b C0 + + + ... 12 20 30 42 D0µ 6
1 C∞ ) {([1 + bC0)n-1(1 - x) + x]1/(n-1) - 1} (8a) b and the analytical solution for the time-lag expression is
[
l2(n - 1) 1 + D0(w - 1) 2 µ
(n - 1) w2+1/(1-n) - (n - 1)(n - 2 + 3w - 2nw)
where t is the time, l is the thickness of the membrane, and C0 is the upstream gas-phase concentration (fugacity). For a sorption process in which the Langmuir isotherm is applicable and the diffusivity takes the functional dependence of the HIO model (or n ) 1 in eq 4), the steady-state concentration profile, C∞, is found to be (see Appendix b)
(6a)
where x ) r/l is the reduced coordinate of the diffusion path. Using the method of Frisch,1 the time lag, L, for the HIO model is derived as (see Appendix a)
L)
{
[
l2 1 1 1 D0µ ln(1 + bC0) 2 (1 + bC0) ln2(1 + bC0) ln(1 + bC0) - 1 ln2(1 + bC0)
]}
(6b)
Using the Taylor series, the approximate solution for eq 6b is found to be
L)
(
)
2 2 17b3C03 41b4C04 l2 1 bC0 7b C0 + + + ... 24 240 720 2016 D0µ 6 (6c)
It is seen that eq 6c reduces to the traditional time-lag expression L ) l2/6D0µ when the bulk-phase concentration is very low. With the Langmuir isotherm and n ) 2 in eq 4, the steady-state concentration profile in the membrane is
C∞ ) (1 - x)C0 and the corresponding time-lag expression is
]
2
(5b)
1 C∞ ) [(1 + bC0)1-x - 1] b
(7c)
For n > 2, the steady-state concentration profile takes the form of
where Cµ ) f(C) is the isotherm equation. The boundary conditions for eq 5a are
C(0,t) ) C0, C(l,t) ) 0, C(r,0) ) 0
(7b)
The approximate solution for eq 7b is (when bC0 is small)
L) (5a)
[
bC0 - (1 + bC0) ln(1 + bC0) l2 1 + D0µ 2bC0 b3C03
(7a)
(2 - n)(2n - 3)(w - 1)2
(8b) where w ) (1 + bC0)n-1. Equations 6-8 will be used in this paper for the analysis of the permeation time lag and the concentration dependence of the diffusivity. Experiment A CMSM (referred to as KP 950) was prepared from the pyrolysis of the Kapton film at 950 °C for 1 h under high vacuum. The thickness of the CMSM is 23 µm. The elemental analysis reveals that the CMSMs obtained under such conditions contain mainly carbon (Table 1). Molecular probe experiments show that ethane hardly adsorbs on such CMSMs, suggesting that its effective pore size range is below 4 Å. Permeation experiments of gases with different molecular sizes were carried out at 25 oC (Table 2). The permselectivities of each gas to H2 verify that the CMSM is free of pinholes. The detailed procedure for making the CMSM and its structural analysis were described in ref 7. The sorption equilibrium of CO2 was measured on the KP 950 with a standard high-resolution volumetric rig, which is capable of measuring the single-component sorption equilibrium up to 40 atm. The gas permeation experiments were performed on a conventional timelag rig. The CMSM is cleaned under vacuum and high temperature before each permeation experiment was performed. The CMSM used in permeation experiments is circular in shape, with a diameter of approximately 2 cm. The CMSM is fixed on a porous metal base with Table 1. Structural Properties of the KP 950 CMSM density (g/cm3)
C (%)
H (%)
O (%)
2.14
95.2
0.56
2.16
Table 2. Permeation Experiments at 25 °C for Various Gas Molecules H2
He
CO2
O2
N2
Ar
LJ diameter (Å) 2.89 2.6 3.3 3.46 3.64 3.54 Pe (barrer) 67.1 29.9 7.79 1.57 0.084 0.043 permselectivity 1 2.24 8.61 42.7 799 1550 Pe(H2)/Pe
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Figure 1. Sorption isotherm of CO2 on KP 950 CMSM at 25 °C. The line is the optimal fitting from the Langmuir equation with Cµs ) 3.67 mmol/g and b ) 1.223 × 10-2 kPa-1.
Figure 2. Steady-state concentration profile of CO2 in the sorbed phase in the KP 950 CMSM with different functional concentration dependences in the diffusivity. The upstream CO2 fugacity is 6 atm.
O-rings. The highest experimental pressure in the upstream vessel is about 50 atm. Results and Discussion The sorption equilibrium of CO2 was measured on the KP 950 CMSM at 25 °C. The experimental data are shown in Figure 1 as symbols. Please note that the experimental pressure readings are converted to fugacity in this and subsequent figure(s). Also shown in Figure 1 as a solid line is the optimal fitting results of the Langmuir isotherm equation. The fitting is seen to be reasonable. The optimal isotherm parameters for the Langmuir equation are listed in the caption of Figure 1. Figure 2 shows the simulation results of the steadystate adsorbed phase concentration profile of CO2 in the CMSM under different functional concentration dependences of the apparent diffusivity. The values of n are chosen as 1-4 in eq 4 for the simulation plots. The related bulk-phase concentration profiles are calculated using eqs 6a, 7a, and 8a, respectively. The upstream CO2 fugacity is chosen as 6 atm. It is seen in Figure 2 that when n ) 1, the CO2 concentration profile in the CMSM is relatively linear and, as the value of n increases, this profile becomes steeper and steeper toward the downstream side of the CMSM. This observation indicates that, for this system, the assumption of a linear sorption isotherm (Henry’s law) and a constant diffusion coefficient may significantly deviate from the real permeation process in which a nonlinear
Figure 3. Permeation time lag of CO2 measured on the KP 950 CMSM at 25 °C and various upstream fugacities and the model simulations with the different functional forms of diffusion coefficient [different values of n in eq 4]. The solid line (optimal fitting) is the simulation with n ) 2.06 and D0µ ) 6.37 × 10-12 cm2/s, the dashed line is that with n ) 1 and D0µ ) 6.95 × 10-12 cm2/s, and the dotted line is that with n ) 2 and D0µ ) 6.20 × 10-12 cm2/s.
sorption isotherm and a concentration-dependent diffusivity prevail. Figure 3 presents the permeation time lags of CO2 measured on KP 950 at 25 °C (symbols). It is seen that the permeation time lag is strongly dependent on the upstream CO2 fugacity. The permeation time-lag data are simulated using the model equations developed in this paper (eqs 6-8). The optimal fitting results of the experimental time lags are shown in Figure 3 as solid lines. The optimal values of the fitting parameters are found to be n ) 2.06 and D0µ ) 6.37 × 10-12 cm2/s, which suggest that, under our experimental conditions, the functional concentration dependence of the CO2 diffusivity is much stronger that that described by the HIO model (eq 4 with n ) 1). To clearly show this point, the simulation results with different values of n are also presented in Figure 3. The dotted line represents the simulation results with the HIO model (n ) 1 and D0µ ) 6.95 × 10-12 cm2/s). We see that, with the concentration dependence of the diffusivity being described by the traditional HIO model, the simulated time lag significantly deviates from the experimental data when pressure is high. The dotted line in Figure 3 gives the simulation results with n ) 2 and D0µ ) 6.20 × 10-12 cm2/s; it is seen that this curve is in reasonably good agreement with the experimental data. Considering its closeness to the optimal fitting and the two-parameter Langmuir isotherm equation, the expression Dµ ) D0µ/(1 - θ)2 is a reasonable correlation for the description of the concentration dependence of the diffusion coefficient of CO2 in the KP 950 CMSM at the experimental conditions. The use of the traditional HIO model or the Darken relation would fail to describe the pressure dependence of the time lag for the system. The diffusion coefficient of CO2 at zero loading (D0µ) is on the order of 10-12 cm2/s, and the apparent diffusion coefficient is on the order of 10-10-10-11 cm2/s. Compared with other studies, these values are significantly smaller. For example, the D0µ derived from the permeation time lag of CO2 on the Carbolac carbon pellet is around 10-5 cm2/s,2 while the apparent diffusivities of N2 and O2 on a PM800 CMSM prepared from the copolyimide were found to be on the order of 10-8 cm2/ s.6 This suggests that the high permselectivity on the KP 950 CMSM (a factor of 19 for O2/N2 at room
Ind. Eng. Chem. Res., Vol. 40, No. 13, 2001 2945
temperature and normal pressure) is achieved at the great tradeoff of the permeation flux. The anomalous concentration dependence of the apparent diffusivity of CO2 deserves some discussion. From the results of the permeation experiment of various gases (Table 2) and the molecular probing experiment, we can safely assume that sorption-diffusion is the predominant mass-transfer mechanism for CO2 permeation in the KP 950. The CMSMs are also well-known for their good mechanical stability.11 Two causes are then considered here for the anomalous concentration dependence: (1) The “random hopping” mechanism of the HIO model (proposed by Higashi et al.10 and further verified by Okazaki et al.12) is insufficient to describe the surface flow in our system. This point is highly speculative at this stage and requires further fundamental study. (2) Because time lag is a measure of the average transport property of the system between the transient and steady states, this anomaly arises from the intrinsic structure of the CMSM, which is known to be turbostratic and possibly contains a portion of dead-end (blind) pores. It has been demonstrated that blind pores can result in anomalous behaviors in the transient diffusion coefficient and the discrepancy between the diffusion coefficients of the transient and the steady states3,6,13 (although this has been questioned by Petropoulos et al.,14 who favor a position-dependent diffusion coefficient). If the time for the blind pores to be filled is a strong function of pressure (fugacity) in our CMSM, the overall time lag (from which the apparent diffusivity is derived) can present a strong pressure dependence as well. In this regard, the anomalous concentration dependence is a reflection of the structural complexity/heterogeneity of the CMSM. Nevertheless, regardless of its true mechanism, such an anomalous behavior in the apparent diffusion coefficient was rarely reported in the past for CMSM and deserves attention in future study.
Following the method of Frisch,1 the general form of the amount of adsorbates permeated to the downstream vessel, QL, is
∫0
L QL ) [ A
C0
[
Dµf ′(C) dC] t -
]
∫01∫x1f(C) dz dx ∫0C Dµf ′(C) dC
l2
0
(a2)
where A is the membrane area available for permeation. Appendix b The steady-state concentration profile of the adsorbate in the CMSM can be obtained from the rearrangement and the integral of the flux expression, eq a1, that is
∫0LJµ dx ) -∫C0 Dµf ′(C) dC ) ∫0C Dµf ′(C) dC
(a3)
∫0C Dµ(C) f ′(C) dC ) -Dµ(C) f ′(C) ∂C ∂r
(a4)
0
0
Jµ )
1 L
0
Integrating with the respect to r from r to L, we have
∫0C Dµ(C) f ′(C) dC ) L L- r∫0C Dµ(C) f ′(C) dC ∞
0
(a5)
With eqs a1, a2 and a5, the steady-state concentration profile and the time-lag expression under different functional concentration dependences in apparent diffusivity can be developed. Notation
Funding of this work and a research grant by the Japanese Society for the Promotion of Science (JSPS) are gratefully acknowledged.
A ) membrane area available for permeation (cm2) b ) sorption isotherm parameter (kPa-1) CMSM ) carbon molecular sieve membrane C ) bulk-phase concentration or fugacity (mmol/g, kPa) Cµ ) adsorbed phase concentration (mmol/g) Cµs ) saturation sorption capacity (mmol/g) D0µ ) diffusivity at zero loading (cm2/s) Dµ(Cµ) ) apparent diffusivity at concentration Cµ (cm2/s) F(n) ) functional dependence form on loading f(C) ) isotherm equation Jµ ) diffusion flux (mmol/cm2/s) l ) thickness of the membrane (cm) L ) time lag (s) Pe ) permeability [1 barrer ) 1 × 10-10 cm3(STP)‚cm/cm2/ s/cmHg] QL ) amount of adsorbates in the downstream vessel (mmol/s) r ) diffusion coordinate (cm) t ) time (s) x ) normalized coordinate in the diffusion path θ ) surface coverage µ ) chemical potential (kJ/mol)
Appendix a
Literature Cited
The expression of the surface flux in eq 1 can be written in terms of the bulk-phase concentration as
(1) Frisch, H. L. The Time Lag in Diffusion. J. Chem. Phys. 1957, 61, 93. (2) Rutherford, S. W.; Do, D. D. Permeation Time Lag with Nonlinear Adsorption and Surface Diffusion. Chem. Eng. Sci. 1997, 52 (5), 703. (3) Ash, R.; Barrer, R. M.; Pope, C. G. Flow of Adsorbable Gases & Vapors in a Microporous Media. I. Single Adsorbates. Proc. R. Soc. London 1963, A271, 1. (4) Ash, R.; Espenhan, S. E. Transport through a Slab Membrane Governed by a Concentration-dependent Diffusion Coef-
Conclusions The analytical mathematical expressions of permeation time lag are derived for systems with a Langmuir isotherm and a concentration-dependent diffusion coefficient. By comparison of the theoretical models with the permeation time-lag data of CO2 measured on the KP 950 CMSM, it is found that, under our experimental conditions, the apparent diffusivity of CO2 takes a functional concentration dependence stronger than the traditional HIO model. This new functional form can be approximately denoted as Dµ(C) ) D0µ/(1 - θ)2. Acknowledgment
Jµ ) -Dµf ′(C)
∂C ∂r
(a1)
where Cµ ) f(C) is the sorption isotherm and Dµ ) D0µF(n) is the apparent diffusivity.
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ficient. Part 1. The Four Time-lags: Some General Considerations. J. Membr. Sci. 1999, 154, 105. (5) Ash, R.; Espenhan, S. E.; Whiting, D. E. G. Transport through a slab Membrane Governed by a Concentration-dependent Diffusion Coefficient. Part 2. The Four Time-lags for some Particular D(C). J. Membr. Sci. 2000, 166, 281. (6) Zimmerman, C. M.; Singh, A.; Koros, W. J. Diffusion in Gas Separation Membrane Materials: A Comparison and Analysis of Experimental Characterization Techniques. J. Polym. Sci., Part B 1998, 36, 1747. (7) Suda, H.; Haraya, K. Gas Permeation through Micropores of Carbon Molecular Sieve Membranes Derived from Kapton Polyimide. J. Phys. Chem. B 1997, 101 (20), 3988. (8) Gilliland, E. R.; Baddour, R. F.; Perkinson, G. P.; Sladek, K. J. Diffusion on Surface I. Effect of Concentration on the Diffusivity of Physically Adsorbed Gases. Ind. Eng. Chem. Fundam. 1974, 13, 95. (9) Kapoor, A.; Yang, R. T.; Wong, C. Surface Diffusion. Catal. Rev. Sci. Eng. 1989, 31, 129-214. (10) Higashi, K.; Ito, H.; Oishi, J. Surface Diffusion Phenomena
in Gaseous Diffusion I. Surface Diffusion of Pure Gases. J. At. Energy Soc. Jpn. 1963, 5, 846-853. (11) Koresh, J. B.; Soffer, A. The Carbon Molecular Sieve Membrane. General Properties and the Permeability of CH4/H2 Mixture. Sep. Sci. Technol. 1987, 22, 973. (12) Okazaki, M.; Tamon, H.; Toei, R. Interpretation of Surface Flow Phenomenon of Adsorbed Gases by Hopping Model. AIChE J. 1981, 27, 262. (13) Ash, R.; Baker, R. W.; Barrer, R. B. Sorption and Surface Flow in Graphitized Carbon Membranes. II. Time Lag and Blind Pore Character. Proc. R. Soc. London 1968, A304. 407. (14) Tsimillis, K.; Petropoulos, J. H. Experimental Study of a Simple Anomalous Diffusion System by Time Lag and Transient State Kinetic Analysis. J. Phys. Chem. 1977, 81, 23.
Received for review November 7, 2000 Revised manuscript received April 2, 2001 Accepted April 10, 2001 IE000946H
