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Gas permeation in hollow fiber membranes with nonlinear sorption isotherm and concentration dependent diffusion coefficient
Journal of Membrane Science 267 (2005) 99–103
Gas permeation in hollow fiber membranes with nonlinear sorption isotherm and concentration dependent diffusion coefficient Kean Wang School of Chemical and Biomedical Engineering, Nanyang Technological University, Singapore 637722, Singapore Received 25 April 2005; received in revised form 26 July 2005; accepted 28 July 2005 Available online 8 September 2005
Abstract Gas permeation in membranes of cylindrical geometry (hollow fiber) is studied. Mathematical formulation is derived for the time-lag method, assuming Langmuir sorption equilibrium and Darken-type diffusion. Analytical expressions for the steady state concentration profile and the rate of permeation are also provided. It is demonstrated that the use of the traditional time-lag expression for a slab (Lt = ls2 /6D) in the place of a hollow fiber may incur large errors for both adsorbing and weakly adsorbing gases. © 2005 Elsevier B.V. All rights reserved. Keywords: Time-lag; Permeation; Hollow fiber membrane; Diffusion; Adsorption
1. Introduction Hollow fiber membranes (HFM) offer a number of advantages in applications and have attracted much research attention [1–5]. An effective experimental method for membrane characterization is permeation time-lag, which is capable of deriving such important properties as steady state permeation flux, concentration profile, and diffusion coefficient. While the fundamental researches for permeation in membranes of slab geometry are relatively abundant, those for HFMs are rare. Some researchers conveniently applied the mathematical formulations for membranes with slab geometry in the place of HFMs [1,2,6]. Even such an approximation might be applicable (if the thickness of the membrane is very small compared to its radius), the estimation of the errors incurred is necessary. Frisch derived the general equation of permeation timelag in membranes of cylindrical and spherical geometries [7]. The pioneer mathematical work is elegant but the effect of the sorption isotherm was not studied. The concentration dependence of diffusion coefficient was discussed but
E-mail address: [email protected]. 0376-7388/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2005.07.037
no specific function was evaluated. Rutherford and Do gave a comprehensive review on time-lag technology [8] and studied the effect on permeation time-lag of surface heterogeneity, isotherm-nonlinearity, and structural characterization [9]. But their work (some are listed in (8) as references) are mainly on carbon pellets with slab geometry and bi-dispersed porous structure. Ash [10] recently investigated the permeation time-lag and transport properties in a ν-dimensional membrane (ν = 1–3 for slab, cylinder and sphere, respectively). By proposing a specific function of concentration profile in the membrane, he successfully derived the expressions of time-lag and sorption kinetics in membranes of various geometries and at various boundary conditions. The importance of nonlinearity in the sorption isotherm and concentration dependence in diffusion coefficients were discussed but not substantiated with any mathematical formulations. This study will target gas permeation processes in HFMs of which the sorption isotherm is nonlinear (Langmuir isotherm) and the diffusion coefficient is concentration dependent (Darken relation). The membrane is initially free of permeation species. Such a system is of practical importance for the research of gas permeation in inorganic membrane [1,11].
K. Wang / Journal of Membrane Science 267 (2005) 99–103
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2. Theory
the method of Frisch [7] as: (see the Appendix A for more details):
The overall mass balance for gas permeation in a microporous HFM can be written as: ∂C 1 ∂ = (rJ ) (1) ∂t r ∂r where J = −D(C )∂C /∂r is the diffusion flux, r the radial co-ordinate, C the adsorbed phase concentration, and D the diffusion coefficient following the Darken relation: [12] D =
0 ∂ln C D
∂ln C
=
0 Γ D
(2)
0 the diffusion where C is the bulk phase concentration, D ∂ln C coefficient at zero loading and Γ = ∂ln C the thermodynamic correction factor. The Langmuir isotherm is used to relate the bulk phase concentration to the adsorbed phase concentration: bC C = Cs (3) 1 + bC
0 Q = 2πlD Cs k(t − L)
where L is the permeation time-lag and: kr12 4 + k + 2(k + 2) ln rr01 k
2 −4r02 1 + k + k4 − rr01 L= 0 k(k + 2)2 ln r0 4D r1
(6)
(7)
If k → 0, i.e. b is very small for a weakly adsorbing species, Eq. (7) will reduce to the time-lag expression on a cylindrical membrane derived by Frisch [7] and Ash [10], which assumes non-adsorbing species and a constant diffusion coefficient: Lk→0 =
r12 − r02 + (r12 + r02 ) ln(r0 /r1 ) 0 ln(r /r ) 4D 0 1
(8)
Eq. (7) can also be expressed in Taylor series as: 1 k (r0 − r1 )2 + (r − r1 )3 + O[(r0 − r1 )4 ] 0 0 0 6D 24r1 D (9)
where b is the adsorption affinity. The system is subject to the boundary conditions (BCs) as shown in Fig. 1:
L=
• r = r0 , C = C0 (adsorbed phase at bore side, C = C0 ) • r = r1 , C = 0 (adsorbed phase at shell side, C = 0) • t = 0, C(r) = 0 (C (r) = 0, membrane is initially clean).
where O[(r0 − r1 )4 ] is the order of truncation. It is seen that, 2 L ≈ (r0 −r01 ) as r1 → r2 (i.e. a cylindrical HFM approaches 6D
∂C
At steady state, ∂t = 0, solving Eq. (1) with the BCs generates the steady state concentration profile for the adsorbed phase: C∞ (r) = Cs [1 − (1 + bC0 )−(ln(r/r1 )/ln(r0 /r1 )) ] k r = Cs 1 − r1 0) where k = − ln(1+bC ln(r0 /r1 ) > ation, N,∞ (mol/s), is:
0 N,∞ = −2πlrD Γ
(4)
0. The steady state rate of perme-
0 C ln(1 + bC ) 2πlD ∂C∞ (r) s 0 =− ∂r ln(r0 /r1 )
0 = 2πlD Cs k
(5)
where l is the length of the HFM. The amount of permeation species collected at the downstream is derived following
the geometry of a slab). If the HFM in the above permeation system is replaced by a microporous membrane of slab geometry, the steady state s concentration profile for the adsorbed phase, C∞ (corresponding to Eq. (4)), and the time-lag expression, Ls (corresponding to Eq. (7)), will take the forms of: s C∞ (x) = Cs [1 − (1 + bC0 )(x−1) ]
Ls =
(10)
ls2 0 ln(1 + bC ) D 0
1 1 ln(1 + bC0 )−1 × − − 2 (1 + bC0 ) ln2 (1 + bC0 ) ln2 (1 + bC0 ) (11)
where x is the reduced diffusion path (with respect to ls , the thickness of the membrane). Eqs. (10) and (11) were derived in previous studies by Wang et al. [11], Strano and Foley [13] for carbon molecular sieve membranes (CMSMs).
3. Discussion
Fig. 1. The cross section of a HFM.
Eqs. (4)–(7) are applicable to gas permeation processes in microporous HFMs to study such system characteristics as steady state concentration profile, diffusion coefficient, and rate of permeation, provided the system follows the Langmuir isotherm and Darken-type diffusion coefficient. In the following section, parametric study will be carried out to investigate
K. Wang / Journal of Membrane Science 267 (2005) 99–103
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Table 1 Isotherm parameters for CO2 and N2 Cµs (mmol/g) b (kPa−1 )
N2 35 ◦ C (KP 800)
CO2 25 ◦ C (KP 950)
0.2500 1.255 × 10−3
3.670 1.223 × 10−2
the effect of curvature (represented by the ratio of r1 /r0 ) and isotherm nonlinearity on the steady state concentration profile and time-lag in a HFM of fixed thickness of 30 m. The maximum curvature ratio of the HFM is taken as r1 /r0 = 3.0, which is close to the ratios of the HFMs synthesized/tested by Tanihara et al. [14] (r1 /r0 = 2.33 for polymeric and r1 /r0 = 2.12 for carbon membranes). Comparison will also be made between the characteristics obtained on a HFM to those obtained on a slab membrane of the same thickness and under the same operation/boundary conditions. The isotherm data are taken from the previous publications: (1) CO2 on KP 950 CMSM at 25 ◦ C [11] and (2) N2 on KP 800 CMSM at 35 ◦ C [15]. The isotherm data (upto 40 atm.) were fitted to the Langmuir isotherm equation and the optimal isotherm parameters are listed in Table 1. Fig. 2a shows the steady state CO2 concentration profile (C∞ , lines) in the HFM with the curvature ratios of 1.01, 1.67, 2.34, and 3, respectively. The upstream permeation pressure is C0 = 100 kPa. The dots represent the s ) in a slab corresponding CO2 concentration profile (C∞ membrane (thickness of 30 m). It is seen that, as r1 /r0 increases, the concentration profiles in the HFM deviates more from that in a slab membrane. This deviation between s C∞ and C∞ is evaluated using the relative error, defined as s |C∞ − C∞ |/C∞ . The maximum relative error is ∼17%, which approximately locates in the middle layers of the HFM. The effect of the isotherm linearity is shown in the permeation of N2 at 35 ◦ C and 100 kPa. Fig. 2b reveals that the curvature ratio plays an important role even for the weakly adsorbing N2 gas. The maximum relative error of N2 concentration profile in the HFM (relative to that in a slab) is around 16%. Fig. 3 shows the 3D permeation time-lag of CO2 (at 25 ◦ C) as the function of upstream permeation pressure and the
Fig. 3. Permeation time lag of CO2 (25 ◦ C) vs. permeation pressure and l2
s curvature ratio (r1 /r0 ). Top plane is the traditional time lag, Lt = 6D , middle surface is L for HFM (Eq. (7)), and lower surface is Ls (Eq. (11)). D = D0 = 6.20 × 10−12 cm2 /s.
curvature ratio. The three meshed surfaces represent three simulated time-lags: (a) top plane, the traditional time-lag in a slab, Lt = ls2 /6D, which assumes non-adsorbing species and a constant diffusion coefficient, (b) middle surface, the time-lag L of Eq. (7) in a HFM and (c) lower surface, the time lag Ls of Eq. (11) in a slab membrane (with Langmuir isotherm and Darken diffusion). The diffusion coefficients (D 0 ) are taken as 6.20 × 10−12 cm2 /s. The three time lags andD are termed as Lt , L, and Ls , respectively. It is seen that, at the fixed membrane thickness, time lag Lt is independent of the permeation pressure, time lag Ls is a strong function of pressure, and time lag L is a strong function of both pressure and curvature ratio. As curvature ratio increases, the difference between L and Ls becomes more prominent and this difference is further enhanced by the increase in the permeation pressure. The three time lags are also compared quantitatively via the relative errors (RE), defined as: RE1 = |L − Ls |/L, and RE2 = |L − Lt |/L. Computation shows that, for CO2 per-
Fig. 2. (a) Steady state concentration profile of CO2 in HFM, with C0 = 100 kPa and r1 /r0 = 1.01–3.00. (b) Steady state concentration profile of N2 in HFM, with C0 = 100 kPa and r1 /r0 = 1.01–3.00.
K. Wang / Journal of Membrane Science 267 (2005) 99–103
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meation at 25 ◦ C, using the expression of traditional time lag Lt in the place of a HFM will incur a large error of 18%, even at C0 = 100 kPa and r1 /r0 = 1.5. At a higher pressure of C0 = 500 kPa, this error may exceed 47%. Therefore, the traditional time-lag expression for a slab, Lt = ls2 /6D, should not be used in the study of a HFM for adsorbing species. It is interesting to see that L can be either higher than, lower than, or even equal to the traditional time lag Lt , depending on the permeation pressure, isotherm linearity and curvature. A simple calculation for Fig. 3 shows that, at r1 /r0 = 3.0, the two time lags possess the same value at C0 ≈ 82 kPa. However, Ls is in reasonably good agreement with L at lower curvature ratio or lower pressure. For example, the relative error is ∼7.5% at C0 = 1000 kPa and r1 /r0 = 2. The weakly adsorbing N2 got an isotherm of more linearity, with L and Ls being less concentration dependent than those of CO2 . However, the curvature may still play an importance role. The use of Ls (Eq. (11)) in the place of L (Eq. (7)) only cause an relative error of ∼5.8% at C0 = 1000 kPa and r1 /r0 = 3. The use of Lt in the place of L, however, is again not recommended as the resulted error is ∼11% at C0 = 500 kPa and r1 /r0 = 1.5. 4. Conclusion Permeation time lag in membranes of cylindrical geometry (hollow fibers) has been investigated for a system observing Langmuir isotherm and Darken-type diffusion coefficient. Mathematical formulations are derived for time lag, steady state concentration profile and the rate of permeation. The effects of curvature ratio and isotherm nonlinearity are studied and compared to those on a slab membrane under corresponding system/boundary conditions. The use of traditional time lag expression for a slab in the place of a hollow fiber membrane may incur large errors for adsorbing and weakly adsorbing gases. Appendix A The derivation of Eq. (7) using the method of Frisch [7,10] is briefly given here. First, Eq. (1) is re-arranged and integrated from r to r1 : r1 ∂C r dr = rJ (r)|r=r1 − rJ (r)|r=r ∂t r ∂C 0 = rD Γ + r1 J (r1 ) (A.1) ∂r r
Eq. (A.1) is multiplied by 2πl/r on both sides and then integrated from r0 to r1 . Please note that N (r1 ) = 2πr1 lJ (r1 ) is not a function of r. We have: 0 r1 2πl r1 ∂C 0 dz dr = 2πL z D Γ dC r ∂t C0 r0 r r1 +N (r1 ) ln r0
(A.2)
Nomenclature b C0 C Cs C∞ s C∞ 0 D, D J, J k l ls L Lt Ls N Q r
r0 , r1 t x z
sorption affinity (kPa−1 ) permeation pressure (kPa) adsorbed phase concentration (mmol/g) adsorption capacity (mmol/g) steady state concentration in a hollow fiber (mmol/g) steady state concentration in a slab (mmol/g) diffusion coefficient (cm2 /s) flux (mmol/cm2 /s) coefficient defined as ln(1 + bC0 )/ln(r1 /r0 ) length of a membrane (cm) the thickness of a slab membrane (m, cm) time-lag in a cylindrical membrane (s) traditional time-lag in a slab (s) time-lag in a slab with sorption (s) rate of permeation (mmol/s) amount of permeation species (mmol) radial coordinate of a cylindrical membrane (cm) the inner and outer diameter of a hollow fiber (cm) time (s) reduced coordinate in a slab membrane integration parameter
Greek letters Γ thermodynamics correction factor θ surface coverage Subscript 0, 1 inner and outer side of the cylindrical membrane adsorbed phase where z is the integration parameter. Eq. (A.2) is integrated with t from 0 to t, under steady state:
0 t 0 Γ dC 2πl C0 D Q= N (r1 ) dt = − ln(r /r ) 1 0 0
r1 dr r1 zC dz ,∞ r r r (A.3) × t − 0 0 0 Γ dC D C0 It is seen that Q =
t 0
N (r1 ) dt is the amount
of permeation
species collected at downstream and L =
r1 dr r0 r 0
C0
r1
r
zC,∞ dz
0Γ D
dC
is the permeation time lag. References [1] S. Lagorsse, Magalhaes, A. Mendes, Carbon molecular sieve membranes sorption, kinetic and structural characterization, J. Membr. Sci. 241 (2004) 275.
K. Wang / Journal of Membrane Science 267 (2005) 99–103 [2] T. Nozomu, S. Hiroshi, H. Yusei, N. Shunsuke, Y. Toshimune, K. Yoshihiro, Gas permeation properties of asymmetric carbon hollow fiber membranes prepared from asymmetric polyimide hollow fiber, J. Membr. Sci. 160 (1999) 179. [3] T. Centeno, A. Fuertes, Carbon molecular sieve membranes derived from a phenolic resin supported on porous ceramic tubes, Sep. Purif. Technol. 25 (2001) 379. [4] B. Mark B, H. Foley, Reproducible production of nonporous carbon membrane, Carbon, 39, 1421. [5] T. Dominic, G. William, J. Koros, Formation of defect-free polyimide hollow fiber membranes for gas separations, J. Membr. Sci. 167 (2000) 79. [6] J. Qin, T. Chung, C. Cao, R. Vora, Effect of temperature on intrinsic permeation properties of 6FDA-durene/1,3-phenylenediamine (mPDA) copolyimide and fabrication of its hollow fiber membranes for CO2/CH4 separation, J. Membr. Sci. 250 (2005) 95. [7] H.L. Frisch, The time lag diffusion IV, J. Phys. Chem. 63 (1959) 1249. [8] S.W. Rutherford, D.D. Do, Review of time lag permeation technique as a method for characterization of porous media and membranes, Adsorption 3 (1997) 283.
103
[9] S.W. Rutherford, D.D. Do, Permeation time lag and heterogeneity in adsorbed phase transport, Chem. Eng. Sci. 55 (2000) 3542. [10] R. Ash, Transport with time-dependent boundary conditions in a ν-dimensional membrane, J. Membr. Sci. 232 (2004) 9. [11] K. Wang, H. Suda, K. Haraya, The permeation time lag and the concentration dependence of the diffusion coefficient of CO2 in a CMS membrane, Ind. Eng. Chem. Res. 40 (2001) 2942. [12] J. K¨arger, D. Ruthven, Diffusion in Zeolites and Other Microporous Solids, Wiley, New York, 1992. [13] M.S. Strano, H.C. Foley, Temperature and pressure dependent transit analysis of single component permeation through nanoporous carbon membranes, Carbon 40 (2002) 1029. [14] N. Tanihara, H. Shimazaki, Y. Hirayama, S. Nakanishi, et al., J. Membr. Sci. 160 (1999) 179. [15] C. Nguyen, D.D. Do, K. Haraya, K. Wang, The structural characterization of carbon molecular sieve membrane (CMSM) via gas adsorption, J. Membr. Sci. 220 (2003) 177–182.
Gas permeation in hollow fiber membranes with nonlinear sorption isotherm and concentration dependent diffusion coefficient Kean Wang School of Chemical and Biomedical Engineering, Nanyang Technological University, Singapore 637722, Singapore Received 25 April 2005; received in revised form 26 July 2005; accepted 28 July 2005 Available online 8 September 2005
Abstract Gas permeation in membranes of cylindrical geometry (hollow fiber) is studied. Mathematical formulation is derived for the time-lag method, assuming Langmuir sorption equilibrium and Darken-type diffusion. Analytical expressions for the steady state concentration profile and the rate of permeation are also provided. It is demonstrated that the use of the traditional time-lag expression for a slab (Lt = ls2 /6D) in the place of a hollow fiber may incur large errors for both adsorbing and weakly adsorbing gases. © 2005 Elsevier B.V. All rights reserved. Keywords: Time-lag; Permeation; Hollow fiber membrane; Diffusion; Adsorption
1. Introduction Hollow fiber membranes (HFM) offer a number of advantages in applications and have attracted much research attention [1–5]. An effective experimental method for membrane characterization is permeation time-lag, which is capable of deriving such important properties as steady state permeation flux, concentration profile, and diffusion coefficient. While the fundamental researches for permeation in membranes of slab geometry are relatively abundant, those for HFMs are rare. Some researchers conveniently applied the mathematical formulations for membranes with slab geometry in the place of HFMs [1,2,6]. Even such an approximation might be applicable (if the thickness of the membrane is very small compared to its radius), the estimation of the errors incurred is necessary. Frisch derived the general equation of permeation timelag in membranes of cylindrical and spherical geometries [7]. The pioneer mathematical work is elegant but the effect of the sorption isotherm was not studied. The concentration dependence of diffusion coefficient was discussed but
E-mail address: [email protected]. 0376-7388/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2005.07.037
no specific function was evaluated. Rutherford and Do gave a comprehensive review on time-lag technology [8] and studied the effect on permeation time-lag of surface heterogeneity, isotherm-nonlinearity, and structural characterization [9]. But their work (some are listed in (8) as references) are mainly on carbon pellets with slab geometry and bi-dispersed porous structure. Ash [10] recently investigated the permeation time-lag and transport properties in a ν-dimensional membrane (ν = 1–3 for slab, cylinder and sphere, respectively). By proposing a specific function of concentration profile in the membrane, he successfully derived the expressions of time-lag and sorption kinetics in membranes of various geometries and at various boundary conditions. The importance of nonlinearity in the sorption isotherm and concentration dependence in diffusion coefficients were discussed but not substantiated with any mathematical formulations. This study will target gas permeation processes in HFMs of which the sorption isotherm is nonlinear (Langmuir isotherm) and the diffusion coefficient is concentration dependent (Darken relation). The membrane is initially free of permeation species. Such a system is of practical importance for the research of gas permeation in inorganic membrane [1,11].
K. Wang / Journal of Membrane Science 267 (2005) 99–103
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2. Theory
the method of Frisch [7] as: (see the Appendix A for more details):
The overall mass balance for gas permeation in a microporous HFM can be written as: ∂C 1 ∂ = (rJ ) (1) ∂t r ∂r where J = −D(C )∂C /∂r is the diffusion flux, r the radial co-ordinate, C the adsorbed phase concentration, and D the diffusion coefficient following the Darken relation: [12] D =
0 ∂ln C D
∂ln C
=
0 Γ D
(2)
0 the diffusion where C is the bulk phase concentration, D ∂ln C coefficient at zero loading and Γ = ∂ln C the thermodynamic correction factor. The Langmuir isotherm is used to relate the bulk phase concentration to the adsorbed phase concentration: bC C = Cs (3) 1 + bC
0 Q = 2πlD Cs k(t − L)
where L is the permeation time-lag and: kr12 4 + k + 2(k + 2) ln rr01 k
2 −4r02 1 + k + k4 − rr01 L= 0 k(k + 2)2 ln r0 4D r1
(6)
(7)
If k → 0, i.e. b is very small for a weakly adsorbing species, Eq. (7) will reduce to the time-lag expression on a cylindrical membrane derived by Frisch [7] and Ash [10], which assumes non-adsorbing species and a constant diffusion coefficient: Lk→0 =
r12 − r02 + (r12 + r02 ) ln(r0 /r1 ) 0 ln(r /r ) 4D 0 1
(8)
Eq. (7) can also be expressed in Taylor series as: 1 k (r0 − r1 )2 + (r − r1 )3 + O[(r0 − r1 )4 ] 0 0 0 6D 24r1 D (9)
where b is the adsorption affinity. The system is subject to the boundary conditions (BCs) as shown in Fig. 1:
L=
• r = r0 , C = C0 (adsorbed phase at bore side, C = C0 ) • r = r1 , C = 0 (adsorbed phase at shell side, C = 0) • t = 0, C(r) = 0 (C (r) = 0, membrane is initially clean).
where O[(r0 − r1 )4 ] is the order of truncation. It is seen that, 2 L ≈ (r0 −r01 ) as r1 → r2 (i.e. a cylindrical HFM approaches 6D
∂C
At steady state, ∂t = 0, solving Eq. (1) with the BCs generates the steady state concentration profile for the adsorbed phase: C∞ (r) = Cs [1 − (1 + bC0 )−(ln(r/r1 )/ln(r0 /r1 )) ] k r = Cs 1 − r1 0) where k = − ln(1+bC ln(r0 /r1 ) > ation, N,∞ (mol/s), is:
0 N,∞ = −2πlrD Γ
(4)
0. The steady state rate of perme-
0 C ln(1 + bC ) 2πlD ∂C∞ (r) s 0 =− ∂r ln(r0 /r1 )
0 = 2πlD Cs k
(5)
where l is the length of the HFM. The amount of permeation species collected at the downstream is derived following
the geometry of a slab). If the HFM in the above permeation system is replaced by a microporous membrane of slab geometry, the steady state s concentration profile for the adsorbed phase, C∞ (corresponding to Eq. (4)), and the time-lag expression, Ls (corresponding to Eq. (7)), will take the forms of: s C∞ (x) = Cs [1 − (1 + bC0 )(x−1) ]
Ls =
(10)
ls2 0 ln(1 + bC ) D 0
1 1 ln(1 + bC0 )−1 × − − 2 (1 + bC0 ) ln2 (1 + bC0 ) ln2 (1 + bC0 ) (11)
where x is the reduced diffusion path (with respect to ls , the thickness of the membrane). Eqs. (10) and (11) were derived in previous studies by Wang et al. [11], Strano and Foley [13] for carbon molecular sieve membranes (CMSMs).
3. Discussion
Fig. 1. The cross section of a HFM.
Eqs. (4)–(7) are applicable to gas permeation processes in microporous HFMs to study such system characteristics as steady state concentration profile, diffusion coefficient, and rate of permeation, provided the system follows the Langmuir isotherm and Darken-type diffusion coefficient. In the following section, parametric study will be carried out to investigate
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Table 1 Isotherm parameters for CO2 and N2 Cµs (mmol/g) b (kPa−1 )
N2 35 ◦ C (KP 800)
CO2 25 ◦ C (KP 950)
0.2500 1.255 × 10−3
3.670 1.223 × 10−2
the effect of curvature (represented by the ratio of r1 /r0 ) and isotherm nonlinearity on the steady state concentration profile and time-lag in a HFM of fixed thickness of 30 m. The maximum curvature ratio of the HFM is taken as r1 /r0 = 3.0, which is close to the ratios of the HFMs synthesized/tested by Tanihara et al. [14] (r1 /r0 = 2.33 for polymeric and r1 /r0 = 2.12 for carbon membranes). Comparison will also be made between the characteristics obtained on a HFM to those obtained on a slab membrane of the same thickness and under the same operation/boundary conditions. The isotherm data are taken from the previous publications: (1) CO2 on KP 950 CMSM at 25 ◦ C [11] and (2) N2 on KP 800 CMSM at 35 ◦ C [15]. The isotherm data (upto 40 atm.) were fitted to the Langmuir isotherm equation and the optimal isotherm parameters are listed in Table 1. Fig. 2a shows the steady state CO2 concentration profile (C∞ , lines) in the HFM with the curvature ratios of 1.01, 1.67, 2.34, and 3, respectively. The upstream permeation pressure is C0 = 100 kPa. The dots represent the s ) in a slab corresponding CO2 concentration profile (C∞ membrane (thickness of 30 m). It is seen that, as r1 /r0 increases, the concentration profiles in the HFM deviates more from that in a slab membrane. This deviation between s C∞ and C∞ is evaluated using the relative error, defined as s |C∞ − C∞ |/C∞ . The maximum relative error is ∼17%, which approximately locates in the middle layers of the HFM. The effect of the isotherm linearity is shown in the permeation of N2 at 35 ◦ C and 100 kPa. Fig. 2b reveals that the curvature ratio plays an important role even for the weakly adsorbing N2 gas. The maximum relative error of N2 concentration profile in the HFM (relative to that in a slab) is around 16%. Fig. 3 shows the 3D permeation time-lag of CO2 (at 25 ◦ C) as the function of upstream permeation pressure and the
Fig. 3. Permeation time lag of CO2 (25 ◦ C) vs. permeation pressure and l2
s curvature ratio (r1 /r0 ). Top plane is the traditional time lag, Lt = 6D , middle surface is L for HFM (Eq. (7)), and lower surface is Ls (Eq. (11)). D = D0 = 6.20 × 10−12 cm2 /s.
curvature ratio. The three meshed surfaces represent three simulated time-lags: (a) top plane, the traditional time-lag in a slab, Lt = ls2 /6D, which assumes non-adsorbing species and a constant diffusion coefficient, (b) middle surface, the time-lag L of Eq. (7) in a HFM and (c) lower surface, the time lag Ls of Eq. (11) in a slab membrane (with Langmuir isotherm and Darken diffusion). The diffusion coefficients (D 0 ) are taken as 6.20 × 10−12 cm2 /s. The three time lags andD are termed as Lt , L, and Ls , respectively. It is seen that, at the fixed membrane thickness, time lag Lt is independent of the permeation pressure, time lag Ls is a strong function of pressure, and time lag L is a strong function of both pressure and curvature ratio. As curvature ratio increases, the difference between L and Ls becomes more prominent and this difference is further enhanced by the increase in the permeation pressure. The three time lags are also compared quantitatively via the relative errors (RE), defined as: RE1 = |L − Ls |/L, and RE2 = |L − Lt |/L. Computation shows that, for CO2 per-
Fig. 2. (a) Steady state concentration profile of CO2 in HFM, with C0 = 100 kPa and r1 /r0 = 1.01–3.00. (b) Steady state concentration profile of N2 in HFM, with C0 = 100 kPa and r1 /r0 = 1.01–3.00.
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meation at 25 ◦ C, using the expression of traditional time lag Lt in the place of a HFM will incur a large error of 18%, even at C0 = 100 kPa and r1 /r0 = 1.5. At a higher pressure of C0 = 500 kPa, this error may exceed 47%. Therefore, the traditional time-lag expression for a slab, Lt = ls2 /6D, should not be used in the study of a HFM for adsorbing species. It is interesting to see that L can be either higher than, lower than, or even equal to the traditional time lag Lt , depending on the permeation pressure, isotherm linearity and curvature. A simple calculation for Fig. 3 shows that, at r1 /r0 = 3.0, the two time lags possess the same value at C0 ≈ 82 kPa. However, Ls is in reasonably good agreement with L at lower curvature ratio or lower pressure. For example, the relative error is ∼7.5% at C0 = 1000 kPa and r1 /r0 = 2. The weakly adsorbing N2 got an isotherm of more linearity, with L and Ls being less concentration dependent than those of CO2 . However, the curvature may still play an importance role. The use of Ls (Eq. (11)) in the place of L (Eq. (7)) only cause an relative error of ∼5.8% at C0 = 1000 kPa and r1 /r0 = 3. The use of Lt in the place of L, however, is again not recommended as the resulted error is ∼11% at C0 = 500 kPa and r1 /r0 = 1.5. 4. Conclusion Permeation time lag in membranes of cylindrical geometry (hollow fibers) has been investigated for a system observing Langmuir isotherm and Darken-type diffusion coefficient. Mathematical formulations are derived for time lag, steady state concentration profile and the rate of permeation. The effects of curvature ratio and isotherm nonlinearity are studied and compared to those on a slab membrane under corresponding system/boundary conditions. The use of traditional time lag expression for a slab in the place of a hollow fiber membrane may incur large errors for adsorbing and weakly adsorbing gases. Appendix A The derivation of Eq. (7) using the method of Frisch [7,10] is briefly given here. First, Eq. (1) is re-arranged and integrated from r to r1 : r1 ∂C r dr = rJ (r)|r=r1 − rJ (r)|r=r ∂t r ∂C 0 = rD Γ + r1 J (r1 ) (A.1) ∂r r
Eq. (A.1) is multiplied by 2πl/r on both sides and then integrated from r0 to r1 . Please note that N (r1 ) = 2πr1 lJ (r1 ) is not a function of r. We have: 0 r1 2πl r1 ∂C 0 dz dr = 2πL z D Γ dC r ∂t C0 r0 r r1 +N (r1 ) ln r0
(A.2)
Nomenclature b C0 C Cs C∞ s C∞ 0 D, D J, J k l ls L Lt Ls N Q r
r0 , r1 t x z
sorption affinity (kPa−1 ) permeation pressure (kPa) adsorbed phase concentration (mmol/g) adsorption capacity (mmol/g) steady state concentration in a hollow fiber (mmol/g) steady state concentration in a slab (mmol/g) diffusion coefficient (cm2 /s) flux (mmol/cm2 /s) coefficient defined as ln(1 + bC0 )/ln(r1 /r0 ) length of a membrane (cm) the thickness of a slab membrane (m, cm) time-lag in a cylindrical membrane (s) traditional time-lag in a slab (s) time-lag in a slab with sorption (s) rate of permeation (mmol/s) amount of permeation species (mmol) radial coordinate of a cylindrical membrane (cm) the inner and outer diameter of a hollow fiber (cm) time (s) reduced coordinate in a slab membrane integration parameter
Greek letters Γ thermodynamics correction factor θ surface coverage Subscript 0, 1 inner and outer side of the cylindrical membrane adsorbed phase where z is the integration parameter. Eq. (A.2) is integrated with t from 0 to t, under steady state:
0 t 0 Γ dC 2πl C0 D Q= N (r1 ) dt = − ln(r /r ) 1 0 0
r1 dr r1 zC dz ,∞ r r r (A.3) × t − 0 0 0 Γ dC D C0 It is seen that Q =
t 0
N (r1 ) dt is the amount
of permeation
species collected at downstream and L =
r1 dr r0 r 0
C0
r1
r
zC,∞ dz
0Γ D
dC
is the permeation time lag. References [1] S. Lagorsse, Magalhaes, A. Mendes, Carbon molecular sieve membranes sorption, kinetic and structural characterization, J. Membr. Sci. 241 (2004) 275.
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