Journal of Membrane Science 249 (2005) 83–88
Computational fluid dynamics study on concentration polarization in H2/CO separation membranes Hiromitsu Takaba∗ , Shin-ichi Nakao Department of Chemical System Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, 113-8656 Tokyo, Japan Received 30 January 2004; received in revised form 26 August 2004; accepted 1 September 2004
Abstract Computational fluid dynamics (CFD) was applied to modeling the H2 selective extraction from an H2 /CO gas mixture using a ceramic membrane, e.g. porous glass membrane and porous silica membrane, in the steam reforming process. The calculated separation factor and flux as a function of permeability and flow rate were compared with the result from the conventional plug flow (PF) model calculation. The result of CFD modeling agreed well with that of the PF model when lower permeability was assumed (permeability of H2 less than 10−7 mol m−2 s−1 Pa−1 ), where ideal flow dynamics were achieved. However, the CFD simulation gave a lower separation factor and flux for H2 than those by PFM for the condition where high H2 recovery was expected (the permeability of H2 is greater than 10−6 mol m−2 s−1 Pa−1 ). This shows the importance of considering non-ideal fluid dynamics in the design of a membrane module. © 2004 Elsevier B.V. All rights reserved. Keywords: Gas separation; Computational fluid dynamics; Concentration polarization; Membrane module
1. Introduction The design of a membrane module is of fundamental importance in the development of high efficiency membrane processes. A general model of a membrane module, applicable to pressure driven membrane processes, has been reported based on the permeation model [1–3]; however, most assume ideal gas flow, which will sometimes overestimate, or underestimate, performance. This error is caused partially by non-ideal flow that is a function of scale or the geometry of the module. In addition to non-ideal fluid dynamics, the concentration polarization of preferable permeation gas is an important factor that determines the performance of the module [4,5], which is difficult to consider in general models. Haraya et al. [6] reported experiments showing that the polarization effect in pressure-driven gas permeation in the membrane is significant when the permeability of the membrane used is greater than 2.4 × 10−6 mol m−2 s−1 Pa−1 and ∗
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the separation factor is approximately 2. He et al. [7] reported a theoretical study in which they describe how the concentration polarization becomes significant when the flux is greater than 3 × 10−7 mol m−2 s−1 Pa−1 . To investigate these problems, computational fluid dynamics (CFD) is one technique that can model actual fluid dynamics in complex geometries and consider the concentration polarization effect. Recently, Willey and Fletcher [8] reported a CFD study on pressure-driven membrane processes involving selective component rejection with concentration polarization on the membrane surface. Their CFD study modeled an ultrafiltration membrane for selective removal of components in the feed. Koukou et al. [9] performed a CFD simulation for a concentration polarization on a pressure-driven gas separation processes, with a selective membrane. To the best of our knowledge, their work is the first attempt to deal with gas separation processes with selective membranes using CFD. Their CFD calculation using a two-dimensional model described their experimental data well. The concentration polarization is a function of the feed rate, selectivity, operating pressure, mass transfer coefficient and module geometry. However, a
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full computational study of the conditions of the concentration polarization in gas separation membranes has not been performed. This paper presents the result of testing the validity of the CFD simulator for modeling gas permeation through a membrane, and the condition where the consideration of a concentration polarization becomes important for evaluating membrane performance is discussed. We treated H2 /CO selective membrane, especially used for H2 separation from steam reforming gas.
2. CFD simulation detail The CFD calculation is performed based on the differential equations for all variables, which are as follows [8,10]: ∂(ρΦ) + div(ρ uΦ − ΓΦ grad Φ) = SΦ ∂t
(1)
where ρ is the density of the mixture gas, Φ the dependent variable, u is the velocity vector, Γ Φ is the appropriate coefficient for variable Φ, which in the mass fraction equation is calculated as Γ Φ = ρD, D is the mutual diffusion coefficient for gas mixture, and SΦ is the source-sink term per unit volume for variable ρ is calculated using the ideal gas law. In our calculations, Φ represents the weight fraction of each component of gas mixture and SΦ represents the mass transfer
across the membrane. SΦ is described as follows: SΦ = PC p
(2)
where P is the permeability of each component of mixture gas, C is a constant for unit adaptation, and p is the pressure difference across the membrane. The above set of equations is solved numerically based on the finite volume method embedded in the PHOENICS package [10]. The entire region is discretized into a grid of finite control volume cells. Bi-channel and tube geometry were considered for the model of the membrane module in the CFD calculations, as shown in Fig. 1. In the bi-channel model, the feed mixture gas is introduced to the feed channel and selectively extracted at the membrane on the feed side and injected into the permeate channel on the opposite side of the membrane. The feed and permeate streams flow co-currently. The CFD box consists of the discrete sub-cells of 40 × 20 × 1 with the dimension 0.5 m × 0.005 m × 0.1 m. On the other hand, the tube geometry model represents a quarter of the volume of the permeable section of the entire membrane module as reported by Haraya et al. [6]. The feed mixture gas is introduced to the permeation cell, and selectively permeates from the permeation cell to the capillary tube located at the center of the module. The retentate and permeated gases come from the outlet, which is placed on the opposite side to the inlet. Flow type is concurrent plug flow. The membrane thickness is assumed almost zero for simplification. In the experiment [6], the capillary porous glass
Fig. 1. Schematic representation of geometry models: (a) bi-channel and (b) tube geometry models.
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Table 1 The Flow properties and operating variables Geometry model (×10−6
mol m−2 s−1
Pa−1 )
Permeability of H2 Selectivity (H2 /CO) Pressure in the feed side (MPa) Pressure in the permeate side (MPa) Kinematic viscosity of gas (10−6 m2 s−1 ) Molar ratio of the feed mixture gas (H2 :CO) Feed velocity (m/s) Temperature (K)
membrane having the small permeable area was placed at a point apart from the gas inlet and outlet, to eliminate the influence of velocity disturbance. Although the CFD calculation does not consider the entire geometry of the module, the condition of flow would be similar because the gas velocity given at the inlet in the calculations is uniform and perpendicular to the inlet surface. The flow properties and operating variables are summarized in Table 1. The conditions for tube geometry are the same as those used by Haraya et al. In bi-channel geometry, the operating condition models that of separation of H2 from steam reforming gas (H2 /CO mixture gas), which is carried out under high-temperature and elevated-pressure. The large selectivity for bi-channel geometry is determined by the consideration of selectivity of recently developed microporous silica membranes, which showed 200–800 for the H2 /CO selectivity [11,12]. The flow was assumed to be steady state, laminar and incompressible.
3. Results and discussion 3.1. Bi-channel geometry model The concentration polarization is a function of the feed volume rate (F), the permeance of H2 (P), surface area (S), total pressure difference across the membrane ( p), selectivity for H2 (α) and module geometry. To characterize the CFD result we introduce the dimensionless parameter, θ, defined as θ=
PS p F
Bi-channel
Tube
0.001–10 1000 0.8 0.1 3.49 at 0.8 MPa 0.75:0.25 5.624 × 10−3 to 5.624 × 102 773
2.4 3.74 1.1 0.1–0.6 2.54 at 1.1 MPa 1:1 (2.05–5.55) × 10−2 373
Fig. 2. Calculated YH2 for the membranes showing different P as a function of θ. Key: () PH2 = 10−5 mol s−1 Pa−1 m−2 ; () PH2 = 10−6 mol s−1 Pa−1 m−2 ; () PH2 = 10−7 mol s−1 Pa−1 m−2 ; () PH2 = 10−8 mol s−1 Pa−1 m−2 ; () PH2 = 10−9 mol s−1 Pa−1 m−2 ; the solid line = PF model.
by CFD with those from the PF model, enables discussion of the degree of concentration polarization. The calculated YH2 , for membranes showing different P, are summarized as a function of θ in Fig. 2. In the PF model, where ideal flow was achieved, YH2 decreases slightly as θ increases. Since θ increase corresponds to F decrease if P is constant, the total flux of H2 through the membranes increases
(3)
This dimensionless parameter is useful because membrane performance can be described only a function of θ if ideal flow in the module is assumed. In the following calculations, the module geometry, S, α, and p are constants. Therefore, the concentration polarization in the following calculations is a function of both P and F. The concentration polarization is evaluated by two calculated parameters; the weight fraction of H2 in the outlet-gas in the permeate side, YH2 , and the cut of H2 , RH2 . These parameters are constants for the same θ if ideal flow in the module is assumed. In an ideal flow condition, YH2 and RH2 are estimated based on the plug flow model (PF model) [1]. The comparison of YH2 and RH2 , calculated
Fig. 3. Calculated RH2 for the membranes showing different P as a function of θ. Keys are the same as those in Fig. 3. The solid line is the result from the PF model.
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as θ increase. In that condition, the partial pressure difference of H2 across the membrane will decrease and, conversely, the partial pressure of CO at the membrane in the feed side will become greater, which decreases YH2 . By comparison of the results for different P, the changes of YH2 in the range of P from 10−9 to 10−7 mol Pa−1 m−2 s−1 are almost same as those of PF model, although significant differences of YH2 are for larger P, which is notable when θ is larger than 0.1. A similar trend is observed in the results for RH2 as shown in Fig. 3. The difference between RH2 calculated by CFD and that for the PF model gradually increases as θ and P increase. Consequently, YH2 and RH2 become smaller than expected from the PF model when P and F are large, when the permeance rate of H2 through the membranes is relatively large. A possible reason for this discrepancy is a concentration polarization at the membrane. Fig. 4 presents the distribution of the partial pressures of H2 (pH2 ) and CO (pCO ) in the geometry model. pH2 at the membrane shows no distribution when P is 10−8 mol Pa−1 m−2 s−1 . Conversely, when P is 10−6 mol Pa−1 m−2 s−1 , it gradually decreases from the inlet to the outlet and shows a definite distribution in the direction perpendicular to the surface, whereas pCO gradually increases from the inlet to the outlet. This means that the concentration polarization occurs at the membrane surface in some CFD calculations. This effect results in the drastic decreases of YH2 and RH2 as shown in Figs. 2 and 3.
Fig. 4. Pictures of the calculated partial pressure distribution: (a) and (b) for H2 and CO, respectively, when PH2 is 10−8 mol m−2 s−1 Pa−1 (θ = 0.01); (c) and (d) for H2 and CO, respectively, when PH2 is 10−6 mol m−2 s−1 Pa−1 (θ = 1.0). Feed velocity is 5.62 × 10−1 m/s. Color spectrum: unit is Pa, and red color means the higher-pressure region, blue region means the lowerpressure region, and yellow region means the medium between them.
Fig. 5. Pictures of the calculated partial pressure distribution: (a) H2 and (b) CO. Bottom figure enlarges the section near the membrane of (a). Unit of color spectrum is Pa.
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Fig. 6. The profiles of pH2 at different slice positions in the module. The zero slice position is at the inlet. The zero position of the x-axis is at the center of the capillary tube.
3.2. Tube geometry model To test the validity of the CFD for quantitative estimation of concentration polarization, the CFD calculation was performed using the same conditions as the experiment described by Haraya et al. [6]. In the experiment, the flux and selectivity were measured with the change of the feed pressure, the size of the permeation cell, and the membrane length. Calculation was performed in one of those conditions; the radius of the permeation cell was 0.97 × 10−2 m, the flux 0.41 × 10−2 m3 m−2 s−1 , and the membrane length 0.45 × 10−2 m. Fig. 5 shows the partial pressure distribution when the feed velocity was 4.10 × 10−2 m/s. pH2 gradually decreases as the membrane surface is approached, whereas that for the CO increases. This means that concentration polarization occurs in this condition. Fig. 6 shows the profile of the pH2 in a slice through the module when the feed velocity was 4.10 × 10−2 m/s; concentration polarization is observed in these profiles. The pH2 in the feed side gradually decreases close to the membrane, although they are constant away from the membrane. In comparison to the slice position, the decline of pH2 becomes large close to the outlet, because the total flux of H2 becomes large close to the outlet and the pH2 decrease is not compensated with the mass transport by H2 diffusion perpendicular to the membrane. In the permeate side, pH2 decreases as the slice position nears the outlet, and at slice positions greater than 0.1 × 10−2 m a dramatic decline of pH2 at the membrane is observed, this results from the high flux of CO at those positions. The estimated thickness of the concentration boundary layer from these profiles is approximately 1 × 10−3 m, which is consistent with the experimental value of (0.3–1) × 10−3 m [6], although its estimation in the experiment included some error because of the uncertain assumption of the estimation of mean radius of the concentration boundary layer. The concentration polarization is a function of F. The CFD result for changing the feed velocity is shown in Fig. 7. The
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Fig. 7. Changes of the profiles of pH2 at the slice positions of 0.12 × 10−2 m for different of the feed velocity. The zero position of the x-axis is at the center of the capillary tube.
slice positions of the indicated profiles are 0.12 × 10−2 m. The decline of pH2 at the membrane is observed in all profiles, which means that a concentration polarization occurs, although the profiles of pH2 for the feed velocity of 4.10 × 10−2 and 3.07 × 10−2 m/s are similar. For the profile of 2.05 × 10−2 m/s, pH2 the decline close to the membrane is remarkable, and in the permeate side its magnitude is smallest among those of all profiles. This is due to a concentration polarization that decreases the flux of the more permeable gas. Fig. 8 is the schematic representation of the concentration polarization. xb and xm are the mole fraction of H2 in the feed side and at the membrane, respectively. y is the mole fraction of H2 on the permeate side. Considering this representation, a modulus of concentration polarization, M, can be defined as [6]: M=
y − xb y − xm
(4)
M is always less than one when concentration polarization takes place. The M calculated from the CFD result is shown in Fig. 9 with the experimental result [6]. In the CFD calculation, xb and xm is a function of slice position. Thus, the average values of them at each three sub-cells are used for
Fig. 8. Schematic figure of concentration polarization.
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lation compared well with that reported experimentally. This suggests the CFD simulator can be used to design a membrane module involving prediction of selectivity and cut.
Acknowledgement This work has been supported by NEDO as a part of the R&D Project for High Efficiency Hydrogen Production/Separation System using Ceramic Membrane promoted by METI, Japan.
References Fig. 9. Comparison of the modulus of concentration polarization obtained by CFD and experimentally.
the estimation of M in Fig. 9. In the experiment, M slightly increased with increasing F and the trend for M corresponds with that obtained by CFD. Although the slight discrepancy between M obtained by CFD and experimentally is observed when F is small, the CFD result is in good agreement with the experimental result. This suggests the validity of the CFD simulation for evaluation of concentration polarization in a membrane module for gas separation.
4. Conclusion A CFD simulator for simulation of gas permeation through membranes was developed and tested for the H2 /CO mixture gas permeation in the bi-channel and tube geometry models of membrane modules. The CFD calculation results for the bi-channel geometry model agree well with the results from the PF model when P is smaller than 10−7 mol m−2 s−1 Pa−1 , whereas there is a discrepancy when P and θ are larger than 10−6 mol m−2 s−1 Pa−1 and 0.1, respectively. The gas pressure distribution analysis suggests that this is because of the concentration polarization at the membrane surface. The CFD calculation of H2 /CO permeation through the membranes was performed using the tube geometry model with the same conditions as the experiment of Haraya et al. [6]. The concentration polarization observed in CFD simu-
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