Hydrodynamic cell model: General formulation and comparative analysis of different approaches

Canadian Journal of Chemical Engineering

Volumn 85, Issue 5, 2007, Pages 701-725

  Zholkovskiy, Emiliy K ^a   Shilov, Vladimir N ^a   Masliyah, Jacob H ^b   Bondarenko, Mykola P ^a  

^a INSTITUTE OF BIOCOLLOID CHEMISTRY   (Ukraine)

^b UNIVERSITY OF ALBERTA   (Canada)

Author keywords

Cunningham; Diaphragm; Electroosmotic flow and sedimentation; Happel; Hydraulic resistivity; Hydrodynamic flow; Kuwabara cell model; Stokes equation

Indexed keywords

DIAPHRAGMS; ELECTROOSMOSIS; MATHEMATICAL MODELS; SEDIMENTATION;

CUNNINGHAM; HYDRAULIC RESISTIVITY; HYDRODYNAMIC FLOW; KUWABARA CELL MODEL; STOKES EQUATION;

HYDRODYNAMICS;

DIAPHRAGMS; ELECTROOSMOSIS; HYDRODYNAMICS; MATHEMATICAL MODELS; SEDIMENTATION;

EID: 36048971645     PISSN: 00084034     EISSN: None     Source Type: Journal    
DOI: 10.1002/cjce.5450850517     Document Type: Article

References (68)

1. Barnea, E. and J. Mizrahi, "A Generalized Approach to the Fluid Dynamics of Particulate Systems: Part 1. General Correlation for Fluidization and Sedimentation in Solid Multiparticle Systems," Chem. Eng. J. 5, 171-189 (1973).
2. Batchelor, G. K., "Sedimentation in a Dilute Dispersion of Spheres," J. Fluid Mech. 52, 245-268 (1972).
3. Batchelor, G. K., "Transport Properties of Two-Phase Materials with Random Structure," Ann. Rev. Fluid Mech. 6, 227-255 (1974).
4. Batchelor, G. K., "Sedimentation in a Dilute Polydisperse System of Interacting Spheres Part 1. General Theory," J. Fluid Mech. 119, 379-408 (1982).
5. Batchelor, G. K. and C. S. Wen, "Sedimentation in a Dilute Polydisperse System of Interacting Spheres. Part 2. Numerical Results," J. Fluid Mech., 124, 495-528 (1982).
6. Carrique, F., F. J. Arroyo and A. V. Delgado, "Electrokinetics of Concentrated Suspensions of Spherical Colloidal Particles: Effect of a Dynamic Stern Layer on Electrophoresis and DC Conductivity," J. Colloid Interface Sci. 243, 351-361 (2001a).
7. Carrique, F., F. J. Arroyo and A. V. Delgado, "Sedimentation Velocity and Potential in a Concentrated Colloidal Suspension. Effect of a Dynamic Stern Layer," Colloids Surf. A: Physicochem. Eng. Aspects 195 157-169 (2001b).
8. Carrique, F., F. J. Arroyo and A. V. Delgado, "Electrokinetics of Concentrated Suspensions of Spherical Colloidal Particles with Surface Conductance, Arbitrary Zeta Potential, and Double-Layer Thickness in Static Electric Fields," J. Colloid Interface Sci. 252, 126-137, (2002).
9. Chen, S. H, S. L. Lin, C. Kung and Y. M. Nian, "Concentration Effects on Sedimentation Velocity of Monodispersed Aerosol Particles," J. Chem. Eng. Japan 32, 635-644 (1999).
10. Cheng, H. and G. Papanicolaou, "Flow Past Periodic Arrays of Spheres at Low Reynolds Number," J. Fluid Mech. 335, 189-212 (1997).
11. Cunningham, E., "On the Steady State Fall of Spherical Particles through Fluid Medium," Proc. Roy. Soc. (London) A83, 357-365 (1910).
12. Dassios, G., M. Hadjinicolaou, F. A. Coutelieris and A. C. Payatakes, "Stokes Flow in Spheroidal Particle-in-Cell Models with Happel and Kuwabara Boundary Conditions," Int. J. Eng. Sci. 33, 1465-1490 (1995).
13. Datta, S. and S. Deo, "Stokes Flow with Slip and Kuwabara Boundary Conditions," Proc. Indian Acad. Sci. (Math. Sci.), 112, 463-475 (2002).
14. Davis, R. H. and A. Acrivos, "Sedimentation of Noncolloidal Particles at Low Reynolds Numbers," Ann. Rev. Fluid Mech. 17, 91-118 (1985).
15. Ding, J. M. and H. J. Keh, "The Electrophoretic Mobility and Electric Conductivity of a Concentrated Suspension of Colloidal Spheres with Arbitrary Double-Layer Thickness," J. Colloid Interface Sci. 236, 180-193 (2001).
16. Dukhin A. S., H. Ohshima, V. N. Shilov and P. J. Goetz, "Electroacoustics for Concentrated Dispersions," Langmuir 15, 3445-3451 (1999a).
17. Dukhin A. S., V. N. Shilov and Y. B. Borkovskaya, "Dynamic Electrophoretic Mobility in Concentrated Dispersed Systems. Cell Model," Langmuir 15, 3452-3457 (1999b).
18. Dukhin A. S., V. N. Shilov, H. J. Ohshima and P. J. Goetz, "Electroacoustic Phenomena in Concentrated Dispersions: New Theory and CVI Experiment," Langmuir 15, 6692-6706 (1999c).
19. Dukhin, A. S. and P. J. Goetz, "Ultrasound for Characterizing Colloids," in "Studies in Interfacial Science," D. Möbius and R. Miller, Eds., Elsevier (2002), p. 372
20. Epstein, N. and J. H. Masliyah, "Creeping Flow through Clusters of Spheroids and Elliptical Cylinders," Loughborough Univ. Technol. Chem. Eng. J. 3, 169-175 (1972).
21. Filippov, A. N., S. I. Vasin and V. M. Starov, "Mathematical Modeling of the Hydrodynamic Permeability of a Membrane Built Up from Porous Particles with a Permeable Shell," Colloids Surf. A 282-283, 272-278 (2006).
22. Frazer, R. A., "On the Motion of Circular Cylinders in a Viscous Fluid," Philosophical Transactions of the Royal Society of London, A 225, 93-130 (1926).
23. Gal-Or, B., "On Motion of Bubbles and Drops," Can. J. Chem. Eng. 48, 526-530, (1970).
24. Garside, J. and M. R. Al-Dibouni, "Velocity-Voidage Relationship for Fluidization and Sedimentation in Solid-Liquid Systems," Ind. Eng. Chem. Process Des. Dev. 16, 206-214 (1977).
25. Happel, J., "Viscosity of Suspensions of Uniform Spheres," J. Appl. Phys 28, 1288-1292 (1957).
26. Happel, J., "Viscous Flow in Multiparticle Systems: Slow Motion of Fluids Relative to Beds of Spherical Particles," AICHE J. 4, 197-201 (1958).
27. Happel, J. and H. Brenner, "Low Reynolds Number Hydrodynamics, with Special Applications to Particulate Media," Prentice-Hall: Englewood Cliffs, NJ (1965).
28. Hasimoto, H., "On the Fundamental Solutions of the Stokes Equations and Their Application to Viscous Flow Past a Cubic Array of Spheres," J. Fluid Mech. 5, 317-328 (1959).
29. Kvashnin, A. G., "Cell Model of Suspension of Spherical Particles," Fluid Dynamics 14, 598-602 (1979).
30. Kuwabara, S., "The Forces Experienced by Randomly Distributed Parallel Circular Cylinders or Spheres in a Viscous Flow at Small Reynolds Numbers," J. Phys. Soc., Japan 14, 527-532 (1959).
31. Leclair, B. P. and A. E. Hamielec, "Viscous Flow through Particle Assemblages at Intermediate Reynolds Numbers - A Cell Model for Transport in Bubble Swarms," Can. J. Chem. Eng. 49, 713-720 (1971).
32. Lee, K. W., L. D. Reed and J. A. Gieseke, "Pressure Drop across Packed Beds in the Low Knudsen Number Regime," J. Aerosol Sci. 9, 557-565 (1978).
33. Lee, E., F. Y. Yen and J. P. Hsu, "Dynamic Electrophoretic Mobility of Concentrated Spherical Dispersions," J. Phys. Chem. B 105, 7239-7245 (2001).
34. Lee, E., C. H. Fu and J. P. Hsu, "Dynamic Electrophoretic Mobility of a Concentrated Dispersion of Particles with a Charge-Regulated Surface at Arbitrary Potential," J. Colloid Interface Sci. 250, 327-336 (2002).
35. Levine, S. and G. H. Neale, "The Prediction of Electrokinetic Phenomena within Multiparticle Systems. I. Electrophoresis and Electroosmosis," J. Colloid Interface Sci. 47, 520-529 (1974).
36. Levine, S. and G. H. Neale and N. J. Epstein, "The Prediction of Electrokinetic Phenomena within Multiparticle Systems: II. Sedimentation Potential," J. Colloid Interface Sci. 57, 424-437 (1976).
37. Lynch, E. D. and E. Herbolzheimer, "Formation of Microscale Structure in Sedimenting Suspensions," Bull. Am. Phys. Soc. 28, 1365 (1983).
38. Masliyah, J. H., "Viscous Flow across Banks of Circular and Elliptic Cylinders: Momentum and Heat Transfer," Can. J. Chem. Eng. 51, 550-555 (1973).
39. Masliyah, J. H. and T. G. M. van de Ven, "A Two Dimensional Model of the Flow of Ordered Suspensions of Rods," Int. J. Multiphase Flow 12, 791-806 (1986).
40. Masliyah, J. H. and S. Bhattacharjee, "Electrokinetic and Colloid Transport Phenomena," John Wiley and Sons, Inc., Hoboken, NJ (2006) p. 707.
41. Maxey, M. R. and B. K. Patel, "Localized Force Representations for Particles Sedimenting in Stokes Flow," Int. J. Multiphase Flow 27, 1603-1626 (2001).
42. Maxwell J. C., "A Treatise of Electricity and Magnetism," 1, Clarendon Press, Oxford, 2nd ed., 435 (1881).
43. Mehta, G. D. and T. F. Morse, "Flow through Charged Membranes," J. Chem. Phys. 63, 1878-1889 (1975).
44. Neale, G. H. and W. K. Nader, "Prediction of Transport Processes within Porous Media: Creeping Flow Relative to a Fixed Swam of Spherical Particles," AICJE J. 20, 530-538 (1974).
45. Neale, G. H. and J. H. Masliyah, "Flow Perpendicular to Mats of Randomly Arranged Cylindrical Fibers (Importance of Cell Models)," AICHE J. 21, 805-807 (1975).
46. Ohshima, H., "Electrophoretic Mobility of Spherical Colloidal Particles in Concentrated Suspensions," J. Colloid Interface Sci. 188, 481-485 (1997a).
47. Ohshima, H., "Dynamic Electrophoretic Mobility of Spherical Colloidal Particles in Concentrated Suspensions," J. Colloid Interface Sci. 195, 137-148 (1997b).
48. Ohshima, H., "Sedimentation Potential of a Concentrated Suspension of Spherical Colloidal Particles," J. Colloid Interface Sci. 208, 295-301 (1998).
49. Ohshima, H., "Sedimentation Potential and Velocity in a Concentrated Suspension of Soft Colloidal Particles," J. Colloid Interface Sci. 229, 140-147 (2000a).
50. Ohshima, H., "Cell Model Calculation for Electrokinetic Phenomena in Concentrated Suspensions: An Onsager Relation between Sedimentation Potential and Electrophoretic Mobility," Adv. Colloid Interface Sci. 88, 1-18 (2000b).
51. Richardson, J. F. and W. N. Zaki, "Sedimentation and Fluidization: Part 1," Trans. Inst. Chem. Eng. 32, 35-53 (1954).
52. Ruiz-Reina, E., F. Carrique, F. J. Rubio-Hernandez, A. I. Gomez-Merino and P. Garcia-Sanchez, "Electroviscous Effect of Moderately Concentrated Colloidal Suspensions," J. Phys. Chem. B 107, 9528-9534 (2003).
53. Rubio-Hernandez, F. J., F. Carrique and E. Ruiz-Reina, "The Primary Electroviscous Effect in Colloidal Suspensions," Adv. in Colloid and Interface Sci. 107, 51-60 (2004).
54. Sangani, A. S. and A. Acrivos, "Slow Flow through a Periodic Array of Spheres," Int. J. Multiphase Flow 8, 343-360, (1982).
55. Shilov, V. N., N. I. Zharkikh and Y. B. Borkovskaya, "Theory of Non-Equilibrium Electro-Surface Phenomena in Concentrated Disperse Systems: 1. Application of Non-Equilibrium Thermodynamics to Cell Model of Concentrated Dispersions," Colloid J. U.S.S.R. (English translation) 43, 434-438 (1981).
56. Shilov, V. N., Y. B. Borkovskaya and A. S. Dukhin, "Electroacoustic Theory for Concentrated Colloids with Overlapped DLs at Arbitrary Ka: I. Application to Nanocolloids and Nonaqueous Colloids," J. Colloid Interface Sci. 277, 347-358 (2004).
57. Simha, R., "A Treatment of Viscosity in Concentrated Suspensions," J. Appl. Phys 23, 1020-1024 (1952).
58. Smith, T. N., "The Differential Sedimentation of Particles of Various Species," Trans. Inst. Chem. Eng. 44, T153-T177 (1966).
59. Smith, T. N., "The Sedimentation of Particles Having Dispersion of Sizes," Trans. Inst. Chem. Eng. 45, T311-313 (1967).
60. Smoluchowski, M., "Contribution a la theorie de l'endosmose electroque et de quelques phenomenes correlatifs," Bull. Int. Académie des Sciences de Cracovie 8, 183-200 (1903).
61. Sparrow, E. M. and A. L. Loeffler, "Longitudinal Laminar Flow between Cylinders Arranged in Regular Array," AICHE J. 5, 325-330 (1959).
62. Uchida, S., "Slow Viscous Flow through a Mass of Particles," Rep. Inst. Sci. Tech. Univ. Tokyo 3, 97-100 (1949).
63. Yaron, J. and B. Gal-Or, "Relative Velocities and Pressure Drop in Clouds of Drops, Bubbles or Solid Particles," AICHE J. 17, 1064-1074 (1971).
64. Yu, C. P. and T. T. Soong, "A Random Cell Model for Pressure Drop in Fibrous Filters," J. Appl. Mech.-Trans. ASME 42, 301-304 (1975).
65. Zharkikh, N. I. and V. N. Shilov, "Theory of Collective Electrophoresis of Spherical Particles in the Henry Approximation," Colloid J. U.S.S.R. (English translation) 43, 865-870 (1981).
66. Zholkovskiy, E. K., O. B. Adeyinka and J. H. Masliyah, "Spherical Cell Approach for the Effective Viscosity of Suspensions," J. Phys. Chem. B. 110, 19726-19734 (2006).
67. Ziemer, W. and B. Foerester, "A New Cell Model Filtration Theory," J. Aerosol Sci. 27, Supplement 1, S605-S606 (1996).
68. Zick, A. A. and G. M. Homsy, "Stokes Flow through Periodic Arrays of Spheres," J. Fluid Mech. 115, 13-26 (1982).