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Advective flow of non-homogeneous permeable sphere in an electrical field
Colloids and Surfaces A: Physicochem. Eng. Aspects 402 (2012) 168–171
Contents lists available at SciVerse ScienceDirect
Colloids and Surfaces A: Physicochemical and Engineering Aspects journal homepage: www.elsevier.com/locate/colsurfa
Short communication
Advective flow of non-homogeneous permeable sphere in an electrical field Xiao-Peng Zhang a , Zhen Yang a , Duu-Jong Lee b,c,∗ , Yuan-Yuan Duan a a
Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Thermal Engineering, Tsinghua University, Beijing 100084, China Department of Chemical Engineering, National Taiwan University of Science and Technology, Taipei 106, Taiwan c Department of Chemical Engineering, National Taiwan University, Taipei 106, Taiwan b
a r t i c l e
i n f o
Article history: Received 20 January 2012 Received in revised form 14 March 2012 Accepted 16 March 2012 Available online 5 April 2012 Keywords: Electroosmosis flow Porous Drag force Radially varying permeability
a b s t r a c t Advective flow in permeable spheres of radially varying permeability in an electrical filed is studied. A numerical model is developed to elucidate the effect of sphere structure, including the volume-average permeability and the primary particle size, on flow and drag force of the sphere. The radially varying sphere would have stronger internal electroosmotic flow relative to its homogeneous counterpart and increased primary particles size would enhance intra-sphere electroosmotic flow. © 2012 Elsevier B.V. All rights reserved.
1. Introduction
2. Materials and methods
Migration of charged sphere in an electrical field occurs in numerous applications [1]. Permeability is a key parameter to a porous sphere when moving in a fluid [2,3]. Naturally occurred permeable spheres, such as wastewater sludge flocs, have complex structures [4–6] and locally changed permeability, resulting in intricate intra-floc flows [7–10]. Simplified models which assume uniform permeability [11–13] and radially varying permeability [14–18] are extensively used to study permeable sphere flow. In our previous paper [3], flows for permeable spheres with uniform permeability in an electrical field were theoretically investigated. Experimental and numerical investigations showed that structures of numerous colloidal aggregates are nonhomogeneous [19–21]. An aggregate formed through a growth processes in which smaller particles are added to the exterior of a “seed” aggregate can have permeability varied in its radial location [22–26]. The flow models considering radially varying permeability were developed [27–31]. In this work, flow and drag force of permeable spheres with radially varying permeability moving in an infinite Newtonian fluid under external electric filed were numerically evaluated.
The domain of problem statement was shown in Fig. S1 in Supporting Materials and the governing equations and the solution logics were listed in Supporting Materials.
∗ Corresponding author at: Department of Chemical Engineering, National Taiwan University of Science and Technology, Taipei 106, Taiwan. Tel.: +886 2 33663028. E-mail address: [email protected] (D.-J. Lee). 0927-7757/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2012.03.042
3. Results and discussion 3.1. Flow fields-purely hydraulic flow For purely hydraulic flow ( = 0, Fig. 1), increasing the value of ˇ results in a decrease in the hydraulic permeability of the sphere, thereby reducing the intra-sphere flow. At a large ˇ (e.g. ˇ = 24), the sphere becomes less permeable and the incoming flow detour more around the sphere, unlike that at a small ˇ (e.g. ˇ = 3) with easy advective flow. In a homogeneous sphere, the low-velocity region is more uniformly distributed compared to the radially varying spheres, as shown by a comparison between the panels in Fig. 1a. Results in Fig. 1 are the comparison basis in this study. 3.2. Flow fields-electroosmotic and hydraulic flow At = 1, the external electrical filed would cause an electroosmotic flow in the incoming flow direction inside the sphere. In the homogeneous sphere (left panels of Fig. S2 in Supporting Materials), the eletroosmotic flow accelerates incoming flow passing through the sphere. The sphere is seemingly “invisible” to the incoming flow and therefore experiencing no drag force from the fluid [1].
X.-P. Zhang et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 402 (2012) 168–171
169
Fig. 1. Streamlines and velocity magnitude around a sphere at = 0: left – homogenous; middle – radially varying model of dp /df = 0.01; right – radially varying model of dp /df = 0.1. (a) ˇ = 3 and (b) ˇ = 24.
The flow directions for radially varying sphere (middle and right panels in Fig. S2) are distorted due to change in permeability: flow is enhanced in the outer layer regions of the sphere as well as in its vicinity region while diminished at the sphere center. Increasing ˇ value extends high-velocity region around the sphere, as shown by a comparison between the right panels of Fig. S2a and b. At = 2, the internal electroosmotic flow surpasses the incoming flow, making the sphere as a pump, sucking the incoming flow and accelerating it inside the sphere. The streamlines hence are concentrated into the sphere, as shown in Fig. 2. As ˇ is increased, the pumping effect of sphere becomes more significant. For the radially varying spheres with identical volume-average permeability (at the same ˇ), increasing primary particle diameter dp would reduce sphere’s permeability at core region but increase that at rim region, resulting in an expansion of lowvelocity region at the sphere center and an expanding high-velocity region near the sphere surface (comparing cases in Figs. 1 and 2 and S2). At = −1, the internal electroosmotic flow is against the incoming flow, forming a vortex pair around the sphere (Fig. 3). In radially varying spheres, high-velocity region appears in the rim region, especially at large ˇ, showing that electroosmotic flow in the rim region (more permeable) is stronger than in the core region (less permeable). Note that is a ratio of electroosmotic flow to incoming flow. The electroosmotic flow has the same strength in Fig. 3a–c due to the same . The sphere is relatively permeable at ˇ = 3 (Fig. 3a) so the incoming flow easily flows into the sphere and significantly suppresses the electroosmotic flow in the opposite direction, reducing flow velocity magnitudes inside the sphere. The sphere is less permeable at ˇ = 24 (Fig. 3c) and the incoming flow becomes weaker in the sphere so the electroosmotic flow is less attenuated by the incoming flow and its magnitude is larger compared to that at ˇ = 3. Increasing the primary particle diameter dp causes an expansion of low-velocity region in the sphere center and of high-velocity
region around the sphere, as shown by the changes from dp /df = 0.01 to dp /df = 0.1 in Fig. 3.
3.3. Drag force coefficient ˝ Since fluid can flow easily through permeable spheres, the corresponding ˝ value at = 0 (without electrical field) is less than unity. At = 0, ˝ → 1 at ˇ → ∞, corresponding to purely hydraulic solid sphere flow. At all s, ˝ → 0 at ˇ → 0, a self-evident result. At > 0, the internal electroosmotic force accelerates the incoming fluid, thereby reducing ˝. At > 1, the electroosmotic flow in the incoming flow direction becomes strong so does the reacting force on the sphere in the opposite direction; the reacting force changes the overall drag force direction from along to against the incoming flow and the value of ˝ accordingly changes form positive to negative (˝ < 0). The ˝ values at = −1 are about double to those at = 0, owing to retardation of incoming fluid by sphere-pumping stream. At = 0 (no electroosmotic flow), the drag force seems weakly dependent on the sphere structure, since the three ˝ lines almost overlap in Fig. 4. It seems that in the radially varying spheres the blockage of flow in the dense center region is largely compensated by the easy flow in the loose rim region, resulting in the overall flow drag force being close to that of the homogeneous sphere although the flow streamlines through the spheres are different, as shown in Fig. 1. At either > 0 or < 0, the radially varying sphere would experience a greater drag force (in the absolute value of ˝) compared with the homogeneous sphere. This observation is attributable to that electroosmotic flow would be enhanced in a loose structure and the radially varying sphere has larger volume in the loose rim region than the homogeneous sphere. Additionally, increase in primary particle size from dp /df = 0.01 to dp /df = 0.1 increases drag force (Fig. 4). This occurrence is attributable to the fact that radially varying sphere comprised by large primary
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X.-P. Zhang et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 402 (2012) 168–171
Fig. 2. Streamlines and velocity magnitude around a sphere at = 2: left – homogenous; middle – radially varying model of dp /df = 0.01; right – radially varying model of dp /df = 0.1. (a) ˇ = 3 and (b) ˇ = 24.
Fig. 3. Streamline and velocity magnitude around sphere at = −1: left – homogenous; middle–radially varying model of dp /df = 0.01; right – radially varying model of dp /df = 0.1. (a) ˇ = 3, (b) ˇ = 6, and (c) ˇ = 24.
X.-P. Zhang et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 402 (2012) 168–171
Fig. 4. Drag force coefficient ˝ (H–homogeneous sphere, F1 – radially varying sphere of dp /df = 0.01; F2 – radially varying sphere of dp /df = 0.1).
particles would have a looser outer layer compared to that of smaller primary particles. Acknowledgement This research is currently supported by the National Natural Science Foundation of China (Contract No. 51176090). Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.colsurfa.2012.03.042. References [1] S. Bhattacharyya, P.P. Gopmandal, Migration of a charged sphere at an arbitrary velocity in an axial electric field, Colloids Surf. A 390 (2011) 86–94. [2] R.M. Wu, D.J. Lee, Hydrodynamic drag on non-spherical floc and free-settling tests, Water Res. 35 (2001) 3226–3234. [3] Z.D. Yang, J. Lee, T. Liu, Advective flow of permeable sphere in an electric field, J. Colloid Interface Sci. 344 (2010) 214–220. [4] R.M. Wu, D.J. Lee, T.D. Waite, J. Guan, Multilevel structure of sludge floc, J. Colloid Interface Sci. 252 (2002) 383–392. [5] Z. Yang, X.F. Peng, C.P. Chu, D.J. Lee, Image processing and geometrical parameters extracted from sliced image of porous biomaterial, Adv. Powder Technol. 18 (2007) 187–213.
171
[6] Z. Yang, D.J. Lee, A.S. Mujumdar, X.F. Peng, A. Su, C. Hsu, Probing heterogeneous structure of aggregates, Drying Technol. 26 (2008) 1018–1023. [7] Z. Yang, X.F. Peng, D.J. Lee, S. Ay, Advective flow in spherical floc, J. Colloid Interface Sci. 308 (2007) 451–459. [8] J.P. Hsu, C.H. Chou, S.J. Tseng, R.M. Wu, 3D simulations of hydrodynamic drag on a nonhomogeneously structured permeable sphere and advective flow thereof, J. Colloid Interface Sci. 336 (2009) 850–856. [9] R.M. Wu, M.H. Lin, H.Y. Lin, R.Y. Hsu, 3D simulations of hydrodynamic drag forces on two porous spheres moving along their centerline, J. Colloid Interface Sci. 301 (2006) 227–235. [10] J.P. Hsu, C.H. Chou, C.C. Kuo, S. Tseng, R.M. Wu, Electrophoresis of an arbitrarily oriented toroid in an unbounded electrolyte solution, Colloids Surf. B 82 (2011) 505–512. [11] H.C. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Appl. Sci. Res. A 1 (1947) 27–34. [12] H.C. Brinkman, Problems of fluid flow through swarms of particles and through macromolecules in solution, Research (London) 2 (1949) 190. [13] M. Vanni, Creeping flow over spherical permeable aggregates, Chem. Eng. Sci. 55 (2000) 685–698. [14] A.S. Kim, R.R. Yuan, Hydrodynamics of an ideal aggregate with quadratically increasing permeability, Colloid Interface Sci. 285 (2005) 627–633. [15] S. Chellam, M.R. Wiesner, Fluid-mechanics and fractal aggregates, Water Res. 27 (1993) 1943–1946. [16] S. Veerapaneni, M.R. Wiesner, Hydrodynamics of fractal aggregates with radialy varying permeability, J. Colloid Interface Sci. 177 (1996) 45–57. [17] J.P. Hsu, Y.H. Hsieh, Boundary effect on the drag force on a nonhomogeneous floc, J. Colloid Interface Sci. 264 (2003) 517–525. [18] R.M. Wu, D.J. Lee, Hydrodynamic drag force exerted on a moving floc and its implication to free-settling tests, Water Res. 32 (1998) 760–768. [19] N. Tambo, Y. Watanabe, Physical characteristics of flocs. 1: floc density function and aluminum floc, Water Res. 13 (1979) 409–419. [20] R.C. Klimpel, R. Hogg, Effects of flocculation conditions on agglomerate structure, J. Colloid Interface Sci. 113 (1986) 121–131. [21] D.H. Li, J. Ganczarczyk, Fractal geometry of particle aggregates generated in water and wastewater treatment processes, Environ. Sci. Technol. 23 (1989) 1385–1389. [22] R. Amal, J.A. Raper, T.D. Waite, Fractal structure of hematite aggregates, J. Colloid Interface Sci. 140 (1990) 158–168. [23] S. Stoll, A. Elaissari, E. Pefferkorn, fractal dimensions of latex aggregates–correlation between hydrodynamic radius and cluster size, J. Colloid Interface Sci. 140 (1990) 98–104. [24] B.E. Logan, D.B. Wilkinson, Fractal dimensions and porosity of Zoogloea ramigera and Saccharomyce scerevisae aggregates, Biotechnol. Bioeng. 38 (1991) 389–396. [25] P. Wiltzius, Hydrodynamic behavior of fractal aggregates, Phys. Rev. Lett. 58 (1987) 710–713. [26] X.Y. Li, B.E. Logan, Permeability of fractal aggregates, Water Res. 35 (2001) 3373–3380. [27] X.Y. Li, B.E. Logan, Colision frequencies of fractal aggregates with small particles by differential sedimentation, Environ. Sci. Technol. 31 (1997) 1229–1236. [28] X.Y. Li, B.E. Logan, Collision frequencies between fractal aggregates and small particles in a turbulently sheared fluid, Environ. Sci. Technol. 31 (1997) 1237–1242. [29] B.E. Logan, Environmental Transport Processes, Wiley, New York, 1999, 466–504. [30] D.H. Li, J.J. Ganczarczyk, Flow through activated-sludge flocs, Water Res. 22 (1988) 789–792. [31] D.H. Li, J.J. Ganczarczyk, advective transport in activated-sludge flocs, Water Environ. Res. 64 (1992) 236–240.
Contents lists available at SciVerse ScienceDirect
Colloids and Surfaces A: Physicochemical and Engineering Aspects journal homepage: www.elsevier.com/locate/colsurfa
Short communication
Advective flow of non-homogeneous permeable sphere in an electrical field Xiao-Peng Zhang a , Zhen Yang a , Duu-Jong Lee b,c,∗ , Yuan-Yuan Duan a a
Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Thermal Engineering, Tsinghua University, Beijing 100084, China Department of Chemical Engineering, National Taiwan University of Science and Technology, Taipei 106, Taiwan c Department of Chemical Engineering, National Taiwan University, Taipei 106, Taiwan b
a r t i c l e
i n f o
Article history: Received 20 January 2012 Received in revised form 14 March 2012 Accepted 16 March 2012 Available online 5 April 2012 Keywords: Electroosmosis flow Porous Drag force Radially varying permeability
a b s t r a c t Advective flow in permeable spheres of radially varying permeability in an electrical filed is studied. A numerical model is developed to elucidate the effect of sphere structure, including the volume-average permeability and the primary particle size, on flow and drag force of the sphere. The radially varying sphere would have stronger internal electroosmotic flow relative to its homogeneous counterpart and increased primary particles size would enhance intra-sphere electroosmotic flow. © 2012 Elsevier B.V. All rights reserved.
1. Introduction
2. Materials and methods
Migration of charged sphere in an electrical field occurs in numerous applications [1]. Permeability is a key parameter to a porous sphere when moving in a fluid [2,3]. Naturally occurred permeable spheres, such as wastewater sludge flocs, have complex structures [4–6] and locally changed permeability, resulting in intricate intra-floc flows [7–10]. Simplified models which assume uniform permeability [11–13] and radially varying permeability [14–18] are extensively used to study permeable sphere flow. In our previous paper [3], flows for permeable spheres with uniform permeability in an electrical field were theoretically investigated. Experimental and numerical investigations showed that structures of numerous colloidal aggregates are nonhomogeneous [19–21]. An aggregate formed through a growth processes in which smaller particles are added to the exterior of a “seed” aggregate can have permeability varied in its radial location [22–26]. The flow models considering radially varying permeability were developed [27–31]. In this work, flow and drag force of permeable spheres with radially varying permeability moving in an infinite Newtonian fluid under external electric filed were numerically evaluated.
The domain of problem statement was shown in Fig. S1 in Supporting Materials and the governing equations and the solution logics were listed in Supporting Materials.
∗ Corresponding author at: Department of Chemical Engineering, National Taiwan University of Science and Technology, Taipei 106, Taiwan. Tel.: +886 2 33663028. E-mail address: [email protected] (D.-J. Lee). 0927-7757/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2012.03.042
3. Results and discussion 3.1. Flow fields-purely hydraulic flow For purely hydraulic flow ( = 0, Fig. 1), increasing the value of ˇ results in a decrease in the hydraulic permeability of the sphere, thereby reducing the intra-sphere flow. At a large ˇ (e.g. ˇ = 24), the sphere becomes less permeable and the incoming flow detour more around the sphere, unlike that at a small ˇ (e.g. ˇ = 3) with easy advective flow. In a homogeneous sphere, the low-velocity region is more uniformly distributed compared to the radially varying spheres, as shown by a comparison between the panels in Fig. 1a. Results in Fig. 1 are the comparison basis in this study. 3.2. Flow fields-electroosmotic and hydraulic flow At = 1, the external electrical filed would cause an electroosmotic flow in the incoming flow direction inside the sphere. In the homogeneous sphere (left panels of Fig. S2 in Supporting Materials), the eletroosmotic flow accelerates incoming flow passing through the sphere. The sphere is seemingly “invisible” to the incoming flow and therefore experiencing no drag force from the fluid [1].
X.-P. Zhang et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 402 (2012) 168–171
169
Fig. 1. Streamlines and velocity magnitude around a sphere at = 0: left – homogenous; middle – radially varying model of dp /df = 0.01; right – radially varying model of dp /df = 0.1. (a) ˇ = 3 and (b) ˇ = 24.
The flow directions for radially varying sphere (middle and right panels in Fig. S2) are distorted due to change in permeability: flow is enhanced in the outer layer regions of the sphere as well as in its vicinity region while diminished at the sphere center. Increasing ˇ value extends high-velocity region around the sphere, as shown by a comparison between the right panels of Fig. S2a and b. At = 2, the internal electroosmotic flow surpasses the incoming flow, making the sphere as a pump, sucking the incoming flow and accelerating it inside the sphere. The streamlines hence are concentrated into the sphere, as shown in Fig. 2. As ˇ is increased, the pumping effect of sphere becomes more significant. For the radially varying spheres with identical volume-average permeability (at the same ˇ), increasing primary particle diameter dp would reduce sphere’s permeability at core region but increase that at rim region, resulting in an expansion of lowvelocity region at the sphere center and an expanding high-velocity region near the sphere surface (comparing cases in Figs. 1 and 2 and S2). At = −1, the internal electroosmotic flow is against the incoming flow, forming a vortex pair around the sphere (Fig. 3). In radially varying spheres, high-velocity region appears in the rim region, especially at large ˇ, showing that electroosmotic flow in the rim region (more permeable) is stronger than in the core region (less permeable). Note that is a ratio of electroosmotic flow to incoming flow. The electroosmotic flow has the same strength in Fig. 3a–c due to the same . The sphere is relatively permeable at ˇ = 3 (Fig. 3a) so the incoming flow easily flows into the sphere and significantly suppresses the electroosmotic flow in the opposite direction, reducing flow velocity magnitudes inside the sphere. The sphere is less permeable at ˇ = 24 (Fig. 3c) and the incoming flow becomes weaker in the sphere so the electroosmotic flow is less attenuated by the incoming flow and its magnitude is larger compared to that at ˇ = 3. Increasing the primary particle diameter dp causes an expansion of low-velocity region in the sphere center and of high-velocity
region around the sphere, as shown by the changes from dp /df = 0.01 to dp /df = 0.1 in Fig. 3.
3.3. Drag force coefficient ˝ Since fluid can flow easily through permeable spheres, the corresponding ˝ value at = 0 (without electrical field) is less than unity. At = 0, ˝ → 1 at ˇ → ∞, corresponding to purely hydraulic solid sphere flow. At all s, ˝ → 0 at ˇ → 0, a self-evident result. At > 0, the internal electroosmotic force accelerates the incoming fluid, thereby reducing ˝. At > 1, the electroosmotic flow in the incoming flow direction becomes strong so does the reacting force on the sphere in the opposite direction; the reacting force changes the overall drag force direction from along to against the incoming flow and the value of ˝ accordingly changes form positive to negative (˝ < 0). The ˝ values at = −1 are about double to those at = 0, owing to retardation of incoming fluid by sphere-pumping stream. At = 0 (no electroosmotic flow), the drag force seems weakly dependent on the sphere structure, since the three ˝ lines almost overlap in Fig. 4. It seems that in the radially varying spheres the blockage of flow in the dense center region is largely compensated by the easy flow in the loose rim region, resulting in the overall flow drag force being close to that of the homogeneous sphere although the flow streamlines through the spheres are different, as shown in Fig. 1. At either > 0 or < 0, the radially varying sphere would experience a greater drag force (in the absolute value of ˝) compared with the homogeneous sphere. This observation is attributable to that electroosmotic flow would be enhanced in a loose structure and the radially varying sphere has larger volume in the loose rim region than the homogeneous sphere. Additionally, increase in primary particle size from dp /df = 0.01 to dp /df = 0.1 increases drag force (Fig. 4). This occurrence is attributable to the fact that radially varying sphere comprised by large primary
170
X.-P. Zhang et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 402 (2012) 168–171
Fig. 2. Streamlines and velocity magnitude around a sphere at = 2: left – homogenous; middle – radially varying model of dp /df = 0.01; right – radially varying model of dp /df = 0.1. (a) ˇ = 3 and (b) ˇ = 24.
Fig. 3. Streamline and velocity magnitude around sphere at = −1: left – homogenous; middle–radially varying model of dp /df = 0.01; right – radially varying model of dp /df = 0.1. (a) ˇ = 3, (b) ˇ = 6, and (c) ˇ = 24.
X.-P. Zhang et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 402 (2012) 168–171
Fig. 4. Drag force coefficient ˝ (H–homogeneous sphere, F1 – radially varying sphere of dp /df = 0.01; F2 – radially varying sphere of dp /df = 0.1).
particles would have a looser outer layer compared to that of smaller primary particles. Acknowledgement This research is currently supported by the National Natural Science Foundation of China (Contract No. 51176090). Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.colsurfa.2012.03.042. References [1] S. Bhattacharyya, P.P. Gopmandal, Migration of a charged sphere at an arbitrary velocity in an axial electric field, Colloids Surf. A 390 (2011) 86–94. [2] R.M. Wu, D.J. Lee, Hydrodynamic drag on non-spherical floc and free-settling tests, Water Res. 35 (2001) 3226–3234. [3] Z.D. Yang, J. Lee, T. Liu, Advective flow of permeable sphere in an electric field, J. Colloid Interface Sci. 344 (2010) 214–220. [4] R.M. Wu, D.J. Lee, T.D. Waite, J. Guan, Multilevel structure of sludge floc, J. Colloid Interface Sci. 252 (2002) 383–392. [5] Z. Yang, X.F. Peng, C.P. Chu, D.J. Lee, Image processing and geometrical parameters extracted from sliced image of porous biomaterial, Adv. Powder Technol. 18 (2007) 187–213.
171
[6] Z. Yang, D.J. Lee, A.S. Mujumdar, X.F. Peng, A. Su, C. Hsu, Probing heterogeneous structure of aggregates, Drying Technol. 26 (2008) 1018–1023. [7] Z. Yang, X.F. Peng, D.J. Lee, S. Ay, Advective flow in spherical floc, J. Colloid Interface Sci. 308 (2007) 451–459. [8] J.P. Hsu, C.H. Chou, S.J. Tseng, R.M. Wu, 3D simulations of hydrodynamic drag on a nonhomogeneously structured permeable sphere and advective flow thereof, J. Colloid Interface Sci. 336 (2009) 850–856. [9] R.M. Wu, M.H. Lin, H.Y. Lin, R.Y. Hsu, 3D simulations of hydrodynamic drag forces on two porous spheres moving along their centerline, J. Colloid Interface Sci. 301 (2006) 227–235. [10] J.P. Hsu, C.H. Chou, C.C. Kuo, S. Tseng, R.M. Wu, Electrophoresis of an arbitrarily oriented toroid in an unbounded electrolyte solution, Colloids Surf. B 82 (2011) 505–512. [11] H.C. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Appl. Sci. Res. A 1 (1947) 27–34. [12] H.C. Brinkman, Problems of fluid flow through swarms of particles and through macromolecules in solution, Research (London) 2 (1949) 190. [13] M. Vanni, Creeping flow over spherical permeable aggregates, Chem. Eng. Sci. 55 (2000) 685–698. [14] A.S. Kim, R.R. Yuan, Hydrodynamics of an ideal aggregate with quadratically increasing permeability, Colloid Interface Sci. 285 (2005) 627–633. [15] S. Chellam, M.R. Wiesner, Fluid-mechanics and fractal aggregates, Water Res. 27 (1993) 1943–1946. [16] S. Veerapaneni, M.R. Wiesner, Hydrodynamics of fractal aggregates with radialy varying permeability, J. Colloid Interface Sci. 177 (1996) 45–57. [17] J.P. Hsu, Y.H. Hsieh, Boundary effect on the drag force on a nonhomogeneous floc, J. Colloid Interface Sci. 264 (2003) 517–525. [18] R.M. Wu, D.J. Lee, Hydrodynamic drag force exerted on a moving floc and its implication to free-settling tests, Water Res. 32 (1998) 760–768. [19] N. Tambo, Y. Watanabe, Physical characteristics of flocs. 1: floc density function and aluminum floc, Water Res. 13 (1979) 409–419. [20] R.C. Klimpel, R. Hogg, Effects of flocculation conditions on agglomerate structure, J. Colloid Interface Sci. 113 (1986) 121–131. [21] D.H. Li, J. Ganczarczyk, Fractal geometry of particle aggregates generated in water and wastewater treatment processes, Environ. Sci. Technol. 23 (1989) 1385–1389. [22] R. Amal, J.A. Raper, T.D. Waite, Fractal structure of hematite aggregates, J. Colloid Interface Sci. 140 (1990) 158–168. [23] S. Stoll, A. Elaissari, E. Pefferkorn, fractal dimensions of latex aggregates–correlation between hydrodynamic radius and cluster size, J. Colloid Interface Sci. 140 (1990) 98–104. [24] B.E. Logan, D.B. Wilkinson, Fractal dimensions and porosity of Zoogloea ramigera and Saccharomyce scerevisae aggregates, Biotechnol. Bioeng. 38 (1991) 389–396. [25] P. Wiltzius, Hydrodynamic behavior of fractal aggregates, Phys. Rev. Lett. 58 (1987) 710–713. [26] X.Y. Li, B.E. Logan, Permeability of fractal aggregates, Water Res. 35 (2001) 3373–3380. [27] X.Y. Li, B.E. Logan, Colision frequencies of fractal aggregates with small particles by differential sedimentation, Environ. Sci. Technol. 31 (1997) 1229–1236. [28] X.Y. Li, B.E. Logan, Collision frequencies between fractal aggregates and small particles in a turbulently sheared fluid, Environ. Sci. Technol. 31 (1997) 1237–1242. [29] B.E. Logan, Environmental Transport Processes, Wiley, New York, 1999, 466–504. [30] D.H. Li, J.J. Ganczarczyk, Flow through activated-sludge flocs, Water Res. 22 (1988) 789–792. [31] D.H. Li, J.J. Ganczarczyk, advective transport in activated-sludge flocs, Water Environ. Res. 64 (1992) 236–240.