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SEPARATIONS Frame of Reference Effects on the Performance of Hollow Fiber Membranes Mohammed Al-Juaied† and W. J. Koros* School of Chemical and Biomolecular Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332
Fick’s first law is generally accepted for describing the transfer of gas mixtures through porefree dense polymeric membranes. Typically, the permeation flux is made up of two parts, namely, that resulting from the bulk motion and that resulting from diffusion. For simplicity most hollow fiber permeator models consider only the diffusion part; however, neglecting the frame of reference contribution can be erroneous in the case of mutlicomponent mixtures when the permeation flux of one of the permeants is much higher than the others. A comparative theoretical study is performed on the influence of two transport models on gas separation hollow-fiber modules: the diffusion and the frame of reference models. Simulations are performed for CO2/CH4 separation in 6FDA-TADPO polypyrrolone membranes. The comparison is made in terms of CH4 recovery and membrane selectivity under common design and operating conditions. It is shown that the differences in performance can be as high as 25% for membrane selectivity and 10% for CH4 recovery, especially for high CO2 concentrations and high pressures. 1. Introduction Transport equations describing permeation fluxes across the membrane usually are known to include both molecular diffusion and bulk motion given by eq 1 in the case of binary mixtures of A and B. The diffusion transport through a pore-free polymeric medium can be well-described by the so-called Fick’s first law of diffusion. Equations 1a and 1b shown below are the diffusion and the bulk transport equations for component A, respectively. The effective diffusivity of A in the membrane medium is DAm (cm2/s). The mass flux of permeant i with respect to a fixed frame of reference is ni (g/cm2‚s), and ωi is the mass fraction composition of permeant i in the membrane (g/g). The density of the system comprised of both polymer and sorbed penetrants is F. The mass flux of the polymer, np, is zero at steady state since the membrane is stationary.1 bulk nA ) ndiff A + nA
ndiff A ) -FDAm
dωA dx
) (nA + nB + nP)ωA nbulk A
(1) (1a) (1b)
In simulations of hollow fiber separators the contribution of bulk flow conditions is generally neglected.2-4 This assumption in some cases is completely reasonable * To whom correspondence should be addressed. Tel: 404-385-1838. Fax: 404-385-2683. E-mail: wjk@ chbe.gatech.edu. † Current address: Saudi Aramco, Dharan, Saudi Arabia.
when the sorption amount of penetrants, ωA and ωB, are very low such as the sorption of simple gases H2, He, O2, and N2. In a recent study, Kamaruddin and Koros1 showed that, for CO2/CH4 separation using 6FDATADPO polypyrrolones and phenol/water separation using a polyether-block-polyamide membranes, bulk contributions are significant. Paul and Ebra-Lima5-7 have also shown the importance of the bulk flux term in single-component permeation in a highly swollen membrane. The purpose of this paper is to extend the analysis by Kamaruddin and Koros1 of the single-stage membrane to hollow fiber membranes. The paper discusses the CO2/CH4 separation in 6FDATADPO polypyrrolone membranes where the frame of reference model (FM), which includes both the diffusion and the bulk motion, is compared with the diffusion model (DM). The effects of CO2 mole fraction, feed pressure, membrane area, and thickness on the bulk flux contribution are discussed. Methane recovery and membrane selectivity to CO2 are compared with those calculated by the diffusion model (DM). 2. Model Development The proposed model considers a hollow fiber module as shown in Figure 1 for the countercurrent configuration. For CO2/CH4 separation, the CO2/CH4 feed gas enters the module where it is separated into a permeate stream and a retentate stream. The membrane acts as a CO2 permselective barrier, so CO2 is concentrated in the permeate and CH4 is concentrated in the retentate stream. A one-dimensional mathematical model is solved for the case of the so-called “dual-mode” sorption and transport description of permeation in glassy polymers,
10.1021/ie049059v CCC: $30.25 © 2005 American Chemical Society Published on Web 03/26/2005
Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005 3649
n1 ) -FD1
dω1
nc
+ ω1(
dx
nj + nP) ∑ j)1
(3a)
... ... Figure 1. Simplified block diagram of carbon dioxide/methane separation in a countercurrent hollow-fiber module.
thereby allowing concentration dependence of the effective diffusion and sorption coefficients. The dualmode model has been used to simulate the permeation of multicomponent components in hollow fiber membranes by Thundyil et al.,8,9 Chern et al.,10 and Taveria et al.11 The following assumptions are made: (i) Steady-state operation. (ii) Isothermal operation (T ) 308 K). (iii) Pressure change in the shell side is negligible. (iv) Pressure change in the tube side is given by HagenPoiseulle equation. (v) Resistance of the porous support is negligible. (vi) Resistance of the shell and the tube side boundary layers is negligible. (vii) Uniform flow distribution within the module. (viii) No defects in the separating layer. (ix) Uniform dimensions of all the fibers. (x) Plug flow conditions for the permeate and the retentate sides. (xi) The permeability changes along the fibers are described by the dual-mode transport model.8-11 (xii) Hollow fiber deformation is negligible. (xiii) Negligible plasticization. (xiv) Constant membrane density. The last assumption, constant bulk density, is not seriously in error based on related sorption and dilation data from Fleming and Koros.12 While the other above conditions are important and can cause performance to depart dramatically from expectations, they are not addressed here. According to these assumptions, the steady-state mass-balance equations for species i, on both the retentate and the permeate sides, are given by
nnc ) -FDnc
nP ) -FDP
dωnc
nc
+ ωnc(
dx dωP
nj + nP) ∑ j)1
(3b)
nc
+ ωP(
dx
nj + nP) ∑ j)1
(4)
Clearly, the mass flux of the polymer nP is zero at steady state since the membrane is stationary. 3. Numerical Solution Assuming constant density within the membrane and average effective diffusion coefficients evaluated between the upstream and downstream conditions, the mass flux of component j can be obtained by integrating eq 4 with the following boundary conditions,
x ) 0; ω1 ) ω10 ... ... ωnc ) ωnc0 x ) l; ω1 ) ω1l ... ... ωnc ) ωncl which gives
[ ]
(5)
nc
rj - ωj2
FDDj ln
nc
rj - ωj1
njl )
ri ∑ i)1
nc
ri ∑ i)1
(6)
ri
∑ i)1r
j
where rj and ri are given by
dRRi ) -ni dz
(2a)
dRPi ) -ni dz
(2b)
where RRi and RPi are the axial molar flow rates of species i on the retentate and the permeate sides, respectively, and z is the axial direction. The pressure on the shell side is assumed constant and equal to the feed pressure. The pressure on the permeate side is calculated for each individual stage using the HagenPoiseulle equation.13 The membrane is divided into a predetermined number of stages small enough that the flow properties and, hence, the pressure and composition gradients, are almost constant in each element. This stepwise procedure is generally referred to as succession of states method with fixed length interval.9 At each stage, the radial permeation fluxes of species i, ni, must be determined by solving the multicomponent mixture permeation relations. The multicomponent mixture permeation system comprises “nc” components and the polymer as shown below:
rj )
nj nref
ri )
ni nref
(7)
where nref is the reference component and can be taken to be equal to the mass flux of the slowest mobile component in the mixture. The fraction of the bulk flux contribution of component j, Πbulk , is the ratio of the j mass flux of component j due to bulk flow relative to the total mass flux as shown in eq 8. Since the mass fraction of component j, ωj, is decreasing in the direction of the mass flux, an average mass composition, ωavg j , should be used in eq 8 when estimating the fraction of the bulk flux contribution. nc
ωavg j
∏
bulk j
)
ni ∑ i)1
nj
nc
)
ωavg j
ri
∑ i)1r
j
(8)
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In the case of binary mixture (CO2 and CH4) permeation, the reference term is a function of both the penetrant sorption level and the permeation flux of CO2 and CH4,
( )
bulk
∏ CO
avg ) ωCO 1+ 2
2
1
(8a)
r
bulk
avg ) ωCH (1 + r) ∏ CH
(8b)
4
4
where r is given by
r)
nCO2
(8c)
nCH4
Average mass composition in the membrane of component j can be calculated as follows,
∫0
λ
) ωavg j
ωj(x) dx
(9)
∫0λ dx
where ωj(x) is the mass fraction profile of component j in the membrane and is a function of position that can be calculated by integrating eq 4 with the following boundary conditions:
x ) 0; ω1 ) ω10 ... ... ωnc ) ωnc0 x ) x; ω1 ) ω1(x) ... ... ωnc ) ωnc(x) Integrating, we have
ωj(x) )
1 nc
[
ri
∑ i)1r
] [ ] nc
[
nc
1 - 1 - ωj1
ri
∑ i)1r
j
(10)
ri j
exp
FDDj
(11)
j
When the local mass composition ωj(x) is averaged over the membrane thickness, we obtain
) ωavg j
1 nc
[
1-
ri
∑ i)1r
j
[
1 - ωj1
[ [ ] ]] nc
nc
]∑
ri FDDj
∑ i)1r
j
nc
njl
ri
i)1rj
ri
∑ i)1r
njl
exp
j
FDDj
-1
∏j )
[
[
nc
1 - 1 - ωj1
]∑
ri FDDj
∑ i)1r
j
nc
njl
(12)
An expression for the bulk flux contribution of component j can be obtained by substituting eq 12 into eq 8:
[ [ ] ]]
ri
i)1rj
ri
∑ i)1r
exp
j
FDDj
-1
(13)
The mass fraction can be calculated using the dual-mode model. The dual-mode model idealizes glassy polymers as having two distinct environments, unrelaxed volume (defects), and dense matrix.14-17 The population of the components sorbed in the free volume is referred to as the Langmuir’s population while those occupying the dense matrix are referred to as the Henry’s population. The concentration of component j is shown in eq 14 using the dual-mode transport model.
ωj )
kDjpjMj 22400F
(
FjKj
1+
nc
1+
bif i ∑ i)1
)
(14)
where kDj is the Henry’s law constant of component j which characterizes the sorption in the dense region of the polymer matrix; bj is a constant that is a measure of the affinity of the penetrant to the Langmuir sites. The constant Fj is equal to the ratio of the diffusion coefficients of Langmuir’s population to Henry’s populations of component j; and fj is the fugacity of component j. The gas-phase fugacity of pure and mixed CO2/CH4 can be calculated using the virial equation of state.18,19 The Kj constant can be calculated by the following equation,
Kj )
x ∑ i)1r
nj
bulk
nc
njl
C′Hjbj kDj
(15)
where C′Hj is the Langmuir capacity constant. Different numeric techniques have been proposed in the literature for solving the differential equations of hollow-fiber models: shooting methods with initial value algorithms,13 weighed residues methods,20 and finite difference methods8,9 are the most commonly used. In the present work, finite difference solution technique was used. 4. Simulations Results and Discussion The study used data for CO2/CH4 separation using 6FDA-TADPO polypyrrolones.1 This polymer is known to its unique properties in resisting CO2 plasticization better than most other polymers, so it was a good ideal example to illustrate the bulk flux effects at high CO2 pressure without such plasticization complication. Table 1 gives the dual-mode parameters of the 6FDA-TADPO polypyrrolone membrane at 35 °C. The default parameters for simulation are given in Table 2. 4.1. Effect of CO2 Mole Fraction. The CO2 mole fraction has a significant impact on the bulk flux contribution of CO2 and CH4. Increasing the faster component (CO2) feed-side mole fraction increases the value of r, the mass flux ratio of CO2 to CH4, as can be seen in Figure 2a. Increasing the CO2 mole fraction increases also the CO2 and decreases the CH4 sorption levels in the membrane based on the dual-mode and
Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005 3651 Table 1. Fugacity-Based Dual-Mode and Partial Immobilization Parameters of CO2 and CH4 at 35 °C in 6FDA-TADPO Polypyrrolone1 F (DH/DD) DD (cm2 s-1) kD (cm3 (STP) cm-3 atm-1) CH′ (cm3 (STP) cm-3) b (atm-1)
CO2
CH4
0.084 1.196 × 10-7 1.526 34.084 1.023
0.026 1.12 × 10-8 0.327 22.838 0.160
Table 2. Default Parameters for Simulations parameter
default value
thickness of membrane feed flow rate feed pressure permeate pressure (at exit) temperature feed mole fraction fiber o.d. fiber i.d. fiber active length number of fibers
0.1 µm 50000 SCFH 1000 psia 20 psia 308 K 50/50 CO2/CH4 250 µm 125 µm 100 cm 300000
competitive sorption standpoint (13). Figure 2b,c shows the CO2 and CH4 concentration profile along the membrane for the 10/90, 50/50, and 90/10 CO2/CH4 mixtures. As can be seen in Figure 2b, the CO2 concentration decreases from the feed/permeate side to the residue side because the CO2 is selectively permeating to the low-pressure side of the membrane. From eq 8 and Figure 2, the bulk flux contribution of CO2 and CH4 can be determined. The results are shown graphically in Figure 3. Therefore, increasing CO2 feed-side mole fraction can increase the bulk flux contribution of CO2 but to a lesser degree than CH4. 4.2. Effect of Feed Pressure. As the feed pressure increases, so does the sorption levels of CO2 and CH4, as can be seen from the dual-mode expression described by eq 14. This effect results in an increase in the bulk bulk bulk flux contribution of both components, ΠCO and ΠCH . 2 4 Figure 4a,b shows the effect of increasing the total feed pressure for the 10/90, 50/50, and 90/10 gas mixture on the CO2 and CH4 bulk flux contribution. 4.3. Effect of Membrane Thickness and Area. The bulk flux contribution of the 50/50 gas mixture was analyzed by plotting the required membrane area against the bulk flux contribution of CO2 and CH4. Membrane area can be increased by increasing either the fiber length or the diameter. Simulations were performed here by increasing the fiber lengths. The active length of the fibers is varied between 10 and 250 cm. Clearly, increasing the membrane area debulk bulk creases ΠCO and ΠCH as can be seen in Figure 5. The 2 4 increase in the membrane area decreases the CO2 mole fraction in the feed side, and therefore leads to lower bulk bulk and ΠCH . ΠCO 2 4 This means that nearly all of the faster gas permeates in the first stages of the membrane and, in these conditions, the faster gas mole fraction is relatively small for the rest of the membrane length. In a countercurrent arrangement, this means that the bulk flux contribution is only higher in a small fraction of its length, located at the very end. As a net result, the bulk flux contribution of CO2 and CH4 decreases with the increase of membrane area. On the other hand, lowmembrane areas lead to high CO2 mole fraction in the bulk bulk and ΠCH . This feed side and, therefore, high ΠCO 2 4 suggests that, in the limit of a single stage, the bulk
Figure 2. Effect of CO2 feed mole fraction on (a) r, (b) CO2 average concentration inside the membrane, and (c) CH4 average concentration inside the membrane.
flux contribution of CO2 and CH4 is expected to reach a maximum. This theoretical limit is consistent with observations from Kamaruddin and Koros1 for lab film experiments. One should also notice that increasing the membrane thickness has the same effect as decreasing the membrane area. The justification is analogous for these two cases. Figure 6 shows the effect of increasing the
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Figure 3. Effect of CO2 feed mole fraction on (a) CO2 bulk flux contribution and (b) CH4 bulk flux contribution.
membrane thickness on the bulk flux contribution of CO2 and CH4. 4.4. Diffusion and Frame of Reference Models Comparisons. Methane recovery is represented in Figure 7 for the two permeation models for three different concentrations. Some conclusions can be taken from Figure 7, apart from whichever permeation model is used. The effect of feed pressure is clearly seen: increasing the feed pressure increases the CO2 and CH4 permeation flux due to higher CO2 and CH4 driving across the membrane. Therefore, increasing feed pressure at fixed permeate pressure decreases methane recovery. The effect of not considering the frame of reference transport model is also illustrated. Recovery is overestimated by the diffusion model by 10%. As already referred, the diffusion permeability model underestimates permeation fluxes. This difference is more significant for high CO2 partial pressures, due to the higher bulk flux contribution of CO2 and CH4. Figure 8 shows the separation factor plotted against the feed pressure for three different permeate pressures. In the case of the 10/90 CO2/CH4 mixtures neglecting the bulk flux contribution is a good approximation because CO2 is present in small concentration. As the CO2 concentration increases, the contribution of bulk flux contribution of both components increases and this increases the difference between the two models prediction as can be seen for the 50/50 CO2/CH4
Figure 4. Bulk flux contribution of (a) CO2 and (b) CH4 in 10/90 CO2/CH4, 50/50 CO2/CH4, and 90/10 CO2/CH4 feed as a function of total feed pressure.
Figure 5. Effect of membrane area on bulk flux contribution of CO2 and CH4 in 50/50 CO2/CH4 feed.
case in Figure 8. Ignoring the bulk flux contribution could lead to incorrect results about the membrane
Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005 3653
Figure 6. Effect of membrane thickness on bulk flux contribution of CO2 and CH4 in 50/50 CO2/CH4 feed.
Figure 7. Simulation of CH4 recovery as a function of feed pressure, in a hollow-fiber module, operating in countercurrent, considering the frame of reference model (FM) and the diffusion model (DM) for 10/90 CO2/CH4 feed, 50/50 CO2/CH4 feed, and 90/10 CO2/CH4 feed.
performance. The difference also increases as the pressure increases because of the increase in the bulk contribution of CO2 and CH4 as noted earlier. Under these conditions, the diffusion permeability model can overestimate selectivity up to 25%sa rather serious error. For a vacuum permeate pressure, the selectivity isotherms, for material exhibiting dual-mode behavior, would show a uniform decreasing trend with total feed pressure as seen in Figure 8. In the case of nonzero permeate pressure, there is predicted to be a trade-off between the effects of competition and the effect of permeate pressure on the Langmuir concentration on the permeate side. At low feed pressures, the depression in separation factor caused by the permeate pressure will dominate. It is clear that permeate pressure is predicted to have the effect of reducing the selectivity of the membrane significantly. This observation has been demonstrated both experimentally and theoretically by Thundyil et al.8
Figure 8. Simulation of CO2/CH4 selectivity as a function of feed pressure for different permeate pressures, in a hollow-fiber module, operating in countercurrent, considering the frame of reference model (FM) and the diffusion model (DM) for (a) 10/90 CO2/CH4 feed and (b) 50/50 CO2/CH4 feed.
5. Conclusions When modeling hollow-fiber membrane modules, it is common to neglect the bulk term in the transport equations. This work shows that this simplification may imply incorrect estimation of the membrane modules performance particularly for systems with high CO2 partial pressure in the feed side. Adding effects of plasticization is expected to be necessary for cellulose acetate and some polyimides at high CO2 feed pressures. In comparison to the frame of reference model formulation, the diffusion model overestimates CH4 recovery and membrane selectivity. The bulk flux contribution was shown to increase as the CO2 mole fraction increases, feed pressure increases, membrane thickness increases, and membrane area decreases. In the case of a single stage, the bulk flux contribution of CO2 and CH4 is expected to reach a maximum. This situation in fact applies in most laboratory characterizations of membrane properties with mixed gases.
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Acknowledgment The authors gratefully acknowledge the financial support of DOE DE-FG03-95ER14538 and the Separations Research Program (SRP) at The University of Texas at Austin. The first author (M. Al-Juaied) appreciates also the scholarship support from Saudi Aramco. Literature Cited (1) Kamaruddin, H. D.; Koros, W. J. Some observations about the application of Fick’s first law for membrane separation of multicomponent mixtures. J. Membr. Sci. 1997, 135, 147-159. (2) Kowali, A. S.; Vemury, S.; Krowidi, K. R.; Khan, A. A. Models and Analyses of Membrane Gas Permeators. J. Membr. Sci. 1992, 73, 1. (3) Lipscomb, G. G. Design of Hollow Fiber Contactors for Membrane Gas Separations. l7ze 1996 Membr. Technol. Rev.; Mulloy, D., Ed.; Business Communications Co.: Norwalk, CT, 1996; p 23. (4) Shindo, Y.; Hakuta, T.; Yoshitome, H.; Inoue, H. Calculation Methods for Multicomponent Gas Separation by Permeation. Sep. Sci. Technol. 1985, 20, 445. (5) Ebra-Lima, O. M.; Paul, D. R. Hydraulic permeation of liquids through swollen polymers networks. I. Poly(vinylalcohol)water. J. Appl. Polym. Sci. 1975, 19, 19. (6) Paul, D. R.; Paciotti, J. D.; Ebra-Lima, O. M. Hydraulic permeation of liquids through swollen polymers networks. II. Liquid mixtures. J. Appl. Polym. Sci. 1975, 19, 1837. (7) Paul, D. R.; Ebra-Lima, O. M. Hydraulic permeation of liquids through swollen polymers networks. III. A generalized correlation. J. Appl. Polym. Sci. 1975, 19, 2759. (8) Thundyil, M. J.; Jois, Y. H.; Koros, W. J. Effect of permeate pressure on the mixed gas permeation of carbon dioxide and methane in a glassy polyimide. J. Membr. Sci. 1999, 152, 29. (9) Thundyil, M. T.; Koros, W. J. Mathematical modeling of gas separation permeators for radial crossflow, countercurrent, and
cocurrent hollow-fiber membrane modules, J. Membr. Sci. 1997, 125, 275. (10) Chern, R. T.; Koros, W. J.; Fedkiw, P. S. Simulation of a hollow fiber gas separator: effects of process and design variables. Ind. Eng. Chem. Prod. Des. Dev. 1985, 24, 1015. (11) Taveira, P.; Cruz, P.; Mendes, A.; Costa, C.; Magalha˜es, F. Considerations on the performance of hollow-fiber modules with glassy polymeric membranes. J. Membr. Sci. 2001, 188, 263. (12) Fleming, G. K. Dilation of silicone rubber and glassy polycarbonates due to high-pressure gas sorption. Disseration, The University of Texas at Austin, 1988. (13) Pan, C.-Y.; Habgood, H. W. Gas separation by permeation: Part II. Effect of permeate pressure drop and choice of permeate pressure. Can. J. Chem. Eng. 1978, 56, 210. (14) Koros, W. J. Sorption and transport in glassy polymers. Ph.D. Thesis, The University of Texas at Austin, 1977. (15) Vieth, W. R.; Howell, J. M.; Hsieh, J. H. Dual-sorption theory. J. Membr. Sci. 1976, 1, 177. (16) Hopfenberg, H. B.; Stannet, V. In The Physics of Glassy State; Haward, R. N., Ed.; Appl. Sci. Publ. Ltd.: London, 1973; Chapter 9. (17) Koros, W. J. Model for sorption of mixed gasses in glassy polymers. J. Polym. Phys. Ed. 1980, 18, 981. (18) Reid, R. C.; Sherwood, T. K.; Prausnitz, J. M. The Properties of Gases and Liquids; McGraw-Hill: New York, 1977. (19) Prausnitz, J. M.; Lichthenthaler, R. N.; de Azevedo, E. G. Molecular thermodynamics for fluid-phase equilibria, 2nd ed.; Prentice Hall Inc.: Englewood Cliffs, NJ, 1986. (20) Tessendorf, S. Modelling analysis and design of membranebased gas separation system. Ph.D. Thesis, Department of Chemical Engineering, Technical University of Denmark, Denmark, 1998.
Received for review September 27, 2004 Revised manuscript received February 6, 2005 Accepted February 25, 2005 IE049059V
