Tamkang Journal of Science and Engineering, Vol. 4, No. 2, pp. 127-132 (2001)
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Effect of Pore Morphology on Fluid Flow through Track-Etched Polycarbonate Membrane Kuo-Lun Tung, Ya-Ling Chang and Ching-Jung Chuang Department of Chemical Engineering Chung Yuan University Chung-Li 320, Taiwan E-mail:
[email protected]
Abstract The effect of pore morphology on fluid flow through track-etched polycarbonate membrane was studied numerically by using the fluid flow software FLUENT. The model of membrane structure is a 3D bi-periodic porous surface with circular holes of finite thickness. Both of the effects of constriction and expansion of inner pore geometries on fluid flow are taken into consideration in numerical calculation, and are compared with that of capillary pore. The flow pattern and resistance to flow in the interstics are obtained as results of the numerical solution. Results show that the inner construction of the filter pores has a significant influence on the flow pattern in the interstics and downstream and deposition of particles at the beginning stage of cake filtration. Results also show that neglecting of 10% expansion of inner pore diameter will result in 30% of over-estimated of pressure drop as compared to capillary pore model for fluid flow through track-etched polycarbonate membrane. Key Words: Polycarbonate Membrane, Flow Field, Flow Resistance, Pore Morphology
1. Introduction The track-etched membranes as shown in Figure 1 are typically regarded to be regular porous media with cylindrical pores of uniform size and length [4, 11]. Due to these unique characteristics, these membranes have been used as model pores in fundamental studies of membrane transport phenomena [5-7, 10, 15]. While some researchers reported that there could be dramatic differences in hydraulic permeability and clogging resistance between cylindrical model pore and actual pore due to the irregularities such as irregular pore shape, multiple pores and existence of pore size distribution [8-9,13, 16-17]. The geometry of the pores is highly dependent on the etching conditions [4]. The polymerization or γ -irradiation grafting can be initiated in these pores by the highly localized depositing energy along the ion path.
Figure 1. The SEM picture of polycarbonate membrane. (Millipore, Isopore®) Most of the track-etched membranes are prepared by irradiation of thin plastic sheets with fission fragments generated from neutron induced of uranium [18]. The morphology and orientation of the pores depends critically on the collimation of the irradiation process. This is likely to vary among
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Kuo-Lun Tung, Ya-Ling Chang and Ching-Jung Chuang
manufacturers. Inadequate collimation during irradiation process may cause some of pores to deviate as much as 30o from the perpendicular [8]. Multiple pores due to overlapping of the pores and inhomogeneous pore distribution may arise due to high pore density and non-uniform target thickness [12]. The shape of pores formed in the thin plastic sheets is affected by the conditions of chemical etching used for enlarging the latent track [3,13]. The irregularities of irregular pore shape; multiple pores and existence of pore size distribution could cause dramatic differences in hydraulic permeability and clogging resistance between cylindrical model pore and actual pore. In this study, the effect of pore morphology on fluid flow through track-etched polycarbonate membrane was studied numerically by using the fluid flow software FLUENT. The model of membrane structure is a 3D bi-periodic porous surface with circular holes of finite thickness. Both of the effects of constriction and expansion of inner pore geometries on fluid flow are taken into consideration in numerical calculation, and are compared with that of capillary pore.
All space variables and flow quantities are normalized with respect to the equivalent pore diameter and approach velocity far from the inlet of the filter pore.
(a) (b) Figure 2. Pore model of polycarbonate membrane for calculation.
2. Theoretical Model The model of membrane structure is a 3D bi-periodic flat plate porous surface with circular holes of finite thickness as shown in Figure 2. However, the cross-sectional view of track-etched polycarbonate membrane as illustrated in Figure 3 depicts that the flow channels of filter pore are not straight thoroughly. Since at the beginning of the etching, two cones were formed at both of the entrance and exit points of the pore. As the etching progress, pores of double conical geometry were formed. Thus, the constriction tube model with sinusoidal wall variation as indicated in Figure 4 could more realistically describe the geometry of flow channel in filter pore. The flow of fluid in the domain of this study is assumed to be laminar and steady state. And the flow field can be obtained by solving the equation of the continuity and the momentum balance equations of the system with appropriate boundary conditions. The dimensionless equation of continuity and equation of motion for the system can be obtained as (I) Equation of continuity
∇v * = 0 ,
(1)
(II) Momentum equation
v * ∇v * = −∇p * +
1 2 * 1 * ∇v + g . (2) Re Fr
Figure 3. The morphology of cross-section of polycarbonate membrane.
Figure 4. Inner pore structure model with sinusoidal wall variation. The boundary conditions for equations (1) and (2) in axisymmetrical cylindrical coordinate are as follows, and the system is shown in Figure 5. (I) Upstream inlet and downstream outlet: The upstream and downstream velocity profile far from the inlet and outlet of the filter pore model are uniform. Both the inlet and outlet of the calculation domain are set to 5(l − d ) from the filter pore model:
Effect of Pore Morphology on Fluid Flow through Track-Etched Polycarbonate Membrane
u *z = u z / u ∞ = 1, v r* = 0.
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(3)
(II) Symmetric planes :
∂u *z = 0, v r* = 0 , * ∂n
(4)
(III) Pore inner surface: The pore inner surface boundary conditions are no-slip and impermeable. (5) v r* = u *z = 0 . The body-fitted coordinate system was adopted to generate the grids of various pores as shown in Figures 6, 7 and 8 for flow field calculation. FLUENT (Version 5.3), a commercially available CFD software package [1, 2], was used to calculate the flow field in the filter pore. The SIMPLE (semi-implicity method for pressure-linked equations) algorithm [17] with the power law difference scheme and single direction sweep solution method were used in this study. The result corresponding to each control volume is an algebraic equation containing unknown values of the dependent variable at the central grid and its immediate neighboring grids. The algebraic equations corresponding to all control volumes in the calculation domain are then solved iteratively to obtain discrete values of the dependent variable. The sum of normalized residuals of all variables converge to less than 1 × 10 −3 within 1500-2000 iterations.
(a) expanded pore (b) constricted pore (c) capillary pore
Figure 6. Body-fitted coordinate grids for various pores.
(a)
(b)
(c)
(d)
Figure 7. Body-fitted coordinate grids for various pores of (a) Capillary, (b) Constricted10%, (c) Constricted-20% and (d) Constricted-30%.
Figure 5. Flow system and BCs.
(a)
(b)
(c)
(d)
Figure 8. Body-fitted coordinate grids for various pores of (a) capillary, (b) expansion-10%, (c) expansion-20% and (d) expansion- 30%.
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Kuo-Lun Tung, Ya-Ling Chang and Ching-Jung Chuang
3. Results and Discussion The flow pattern and resistance to flow in the pores are obtained as results of the numerical solution. Figure 9 depicts the streamlines and velocity vector profiles for fluid flow through various pores. The membrane pore diameter is 2 µm with an opening ratio of 50% and a thickness of 1 µm. The percentages of constriction and expansion of inner pore diameter as shown in Figures 9(a) and 9(b), respectively, are 5%. The flow field is altered considerably due to the irregularity of the wall, especially in the neighborhood of the filter pore. As indicted in the figures of streamline pattern, the fluid flow through constriction pore is smoother than that through expansion pore. The velocity vectors and streamlines in Figure 9 also shows that the highly tortuous walls of inner pore geometry noticeably affect the flow pattern in the interstice and the streamlines downstream. Figure 10 illustrates the pressure contours of the four different pore geometries of various inner constriction diameters. The membrane pore diameter is 3 µm with an opening ratio of 15% and a thickness of 20 µm. The specifications represent the commercial available track-etched polycarbonate membrane of Isopore® manufactured by Millipore Co. The colorized contours evidently illustrate that increase of the constriction percentage results in the increase of pressure. The results depicted in Figure 11 indicate that as the inlet velocity increases, there is a corresponding increase in pressure drop. The constriction pore show a higher values of pressure drop than that of expansion pore due to the constriction of fluid flow path. The deviations of pressure drop values of constriction and expansion pores from that of cylindrical pore are depicted in Figure 12. Results show that neglecting of 10% expansion of inner pore diameter will result in 30% of over estimation of pressure drop as compared to that of capillary pore model for fluid flow through track-etched polycarbonate membrane; and neglecting of 30% constriction of inner pore diameter will result in ca. 150% of under estimation of pressure drop as compared to that of capillary pore model. Furthermore, all of the simulated values of pressure drop through irregular pores are larges than experimental value of Isopore® membrane manufactured by Millipore Co. The discrepancy between experimental and simulated results is due to the neglecting the existence of the irregularities of multiple pores and pore size distribution as it can be observed in Figure 1. The irregularities of irregular pore shape; multiple pores and existence
(a) expansion pore
(b) constricted pore
(c) capillary pore Figure 9. Stream lines and velocity vector profile for fluid flow through various pores, Re=0.1
Effect of Pore Morphology on Fluid Flow through Track-Etched Polycarbonate Membrane
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of pore size distribution could cause dramatic differences in hydraulic permeability and clogging resistance between cylindrical model pore and actual pore. Effects of multiple pores and existence of pore size distribution are under investigation in our further studies. capillary
0.6
constriction-10% constriction-20%
0.5
constriction-30%
∆P x 10-6 (Pa )
expansion-10% expansion-20%
0.4
expansion-30%
0.3
0.2
0.1 0.00
0.01
0.02
0.03
0.04
v(m/sec)
(a)
(b)
Figure 11. Pressure drop vs. superficial velocity for fluid flow through various pores. 1.6
σ
0.8
0.0
constriction-10%
-0.8
constriction-20% constriction-30% expansion-10% expansion-20% expansion-30%
-1.6 0.00
0.01
0.02
0.03
0.04
v(m/sec)
Figure 12. Deviation of Pressure drop of constriction and expansion pores from cylindrical pore.
4. Conclusions (c)
(d)
Figure 10. Pressure contours for fluid flow through various pores.
The effect of pore morphology on fluid flow through track-etched polycarbonate membrane was studied numerically by using the fluid flow software FLUENT. The model of membrane structure is a 3D bi-periodic porous surface with
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Kuo-Lun Tung, Ya-Ling Chang and Ching-Jung Chuang
circular holes of finite thickness. Both of the effects of constriction and expansion of inner pore geometries on fluid flow are taken into consideration in numerical calculation, and are compared with that of capillary pore. The flow pattern and resistance to flow in the interstics are obtained as results of the numerical solution. Results show that the inner construction of the filter pores has a significant influence on the flow pattern in the interstics and downstream and deposition of particles at the beginning stage of cake filtration. Results also show that neglecting of 10% expansion of inner pore diameter will result in 30% of over estimation of pressure drop as compared to capillary pore model for fluid flow through track-etched polycarbonate membrane. The irregularities of irregular pore shape; multiple pores and existence of pore size distribution could cause dramatic differences in hydraulic permeability and clogging resistance between cylindrical model pore and actual pore.
Acknowledgement The authors wish to express their sincere gratitude to the National Science Council of the Republic of China for its financial support. References [1] Anon., "FLUENT User's Guide," Ver. 5.3, Fluent Inc., Lebanon, NH, June (2000). [2] Anon., "PreBFC User's Guide," Ver. 5.3, Fluent Inc., Lebanon, NH, June (2000). [3] Durrani, S.A. and Bull, R.K., Solid State Nuclear Track Detectors: Principles, Methods and Application, Pergamon Press, Oxford, 1985. [4] Flescher, R.L., Price, P.B., Walker, R.M. Nuclear Tracks in Solids: Principles and Application, University of California, Berkeley, 1975. [5] Frey, J. M. and Schmitz, P. “Particle Transport and Capture at the Membrane Surface in Cross-flow Microfiltration,” Chem. Eng. Sci., Vol.55, pp.4053-4065 (2000). [6] Grzywna, Z.J., Siwy, Z. and Bashford, C.L., “Non-linear theory for ionic transport through track-etched nuclear membranes,” J. Membrane Sci., Vol.121, pp.261-269 (1996). [7] Kawakatsu, T., Nakajima, M., Nakao, S. and Kimura, S. “Three-dimensional Simulation of Random Packing and Pore Blocking Phenomena during Microfiltration,” Desalination, Vol.101, pp.203-209 (1995). [8] Kim, K.J., Stevens, P.V., and Fane, A.G.,
“Porosity Dependence of Pore Entry Shape In Track-etched Membranes by Image Analysis,” J. Membrane Sci., Vol.93, pp.79-90 (1994). [9] Kim, K.J. and Stevens, P.V., “Hydraulic and Surface Characteristics of Membrane with Parallel Cylindrical Pores,” J. Membrane Sci., Vol.123, pp.303-314 (1997). [10] Long, T.D., Jacobs, D.L., and Anderson, J.L., “Configurational Effects on Membrane Rejection,” J. Membrane Sci., Vol.9, pp.13-27 (1981). [11] Lonsdale, H.K. “The Growth of Membrane Technology,” J. Membrane Sci., Vol.10, pp.81-181 (1982). [12] Pandey, A.K., Sharma, R.C., Kalsi, P.C. and Iyer, R.H., “Measurement of Alpha to Fission Branching Ratios of Heavy Actinides by Sequential Etching of Alpha and Fission Tracks in CR-39,” Nucl. Instr. Methods Phys. Res. B, Vol.82, pp.151-163 (1993). [13] Pandey, A.K., Gautam, M.M., Shukla, J.P. and, Iyer, R.H. “Effect of Pore Characteristics on Carrier-facilitated Transport of Am(III) across Track-etched Membranes,” J. Membrane Sci., Vol.190, pp.9-20 (2001). [14] Patankar, S. V., Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York (1980). [15] Schmitz, P. and Prat, M., “3-D Laminar Stationary Flow over a Porous Surface with Suction: Description at Pore Level,” AIChE J., Vol.41(10), pp.2212-2226 (1995). [16] Tung, K. L., “Study on the Mechanism of Particle Collection and Particle Packing in the Neighborhood of a Circular Hole,” Private Report, NTU, Taipei, Taiwan (1994). [17] Urase, T., Yamamoto, K. and Ohgaki, S. “Effect of Pore structure of Membranes and Module Configuration of Virus Retention,” J. Membrane Sci., Vol.115 pp.21-33 (1996). [18] Yamazaki, I.M., Pateron, R. and Geraldo, L.P., “A New Generation of Track-etched Membranes for Microfiltration and Ultrafiltration. Part I. Preparation and characterisation,” J. Membrane Sci., Vol.118, pp.239-245 (1996).
Accepted: Jun. 29, 2001
