A five-parameter Markov model for simulating the paths of sedimenting particles

Applied Mathematical Modelling

Volumn 31, Issue 10, 2007, Pages 2080-2094

  Bargieł, Monika ^a   Tory, Elmer M ^b  

^a AGH UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Poland)

^b MOUNT ALLISON UNIVERSITY   (Canada)

Author keywords

Chaotic systems; Markov model; Sedimentation; Simulation; Stochastic differential equations

Indexed keywords

CHAOTIC SYSTEMS; COMPUTER SIMULATION; DIFFERENTIAL EQUATIONS; EQUATIONS OF MOTION; INTERPOLATION; MATHEMATICAL MODELS; SEDIMENTATION; SEDIMENTS; STATISTICAL METHODS;

CREEPING-MOTION EQUATIONS; MARKOV MODEL; SEDIMENTING PARTICLES;

MARKOV PROCESSES;

CHAOTIC SYSTEMS; COMPUTER SIMULATION; DIFFERENTIAL EQUATIONS; EQUATIONS OF MOTION; INTERPOLATION; MARKOV PROCESSES; MATHEMATICAL MODELS; SEDIMENTATION; SEDIMENTS; STATISTICAL METHODS;

EID: 34247240455     PISSN: 0307904X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apm.2006.08.023     Document Type: Article

References (55)

1. Bustos M.C., Concha F., Bürger R., and Tory E.M. Sedimentation and Thickening. Phenomenological Foundation and Mathematical Theory (1999), Kluwer Academic, Dordrecht
2. Drew D.A., and Passman S. Theory of Multicomponent Fluids (1999), Springer-Verlag, New York
3. K. Höfler, Simulation and modeling of mono and bidisperse suspensions, Ph.D. thesis, U. Stuttgart, July 2000, 144 pp. .
4. Bargieł M., and Tory E.M. Simulation of sedimentation and fluidization of polydisperse suspensions via a Markov model. Chem. Eng. Sci. 61 (2006) 5575-5589
5. Gösele W., and Wambsganss R. Thickening process in gravity thickeners. Ger. Chem. Eng. 6 (1983) 351-355
6. Ham J.M., and Homsy G.M. Hindered settling and hydrodynamic dispersion in quiescent sedimenting suspensions. Int. J. Multiphase Flow 14 (1988) 533-546
7. Nicolai H., Herzhaft B., Hinch E.J., Oger L., and Guazzelli E. Particle velocity fluctuations and hydrodynamic self-diffusion of sedimenting non-Brownian spheres. Phys. Fluids 7 (1995) 2-23
8. Tory E.M. Stochastic sedimentation and hydrodynamic diffusion. Chem. Eng. J. 80 (2000) 81-89
9. Kim S., and Karrila S.J. Microhydrodynamics, Principles and Selected Applications (1991), Butterworth-Heineman, Boston
10. Mazur P., and Van Saarloos W. Many-sphere hydrodynamic interactions and mobilities in a suspension. Physica A 115 (1982) 21-57
11. Ladd A.J.C. Dynamical simulations of sedimenting spheres. Phys. Fluids A 5 (1993) 299-310
12. Ladd A.J.C. Numerical simulations of particulate suspensions via a discretized Boltzmann equation, Part 1. Theoretical foundation. J. Fluid Mech. 271 (1994) 285-309
13. Ladd A.J.C. Sedimentation of homogeneous suspensions of non-Brownian spheres. Phys. Fluids 9 (1997) 491-499
14. Voller V.R., and Porté-Agel F. Moore's law and numerical modeling. J. Comput. Phys. 197 (2002) 698-703
15. Laso M. Stochastic dynamic approach to transport phenomena. AIChE J. 40 (1994) 1297-1311
16. Kloeden P.E., and Platen E. Numerical Solution of Stochastic Differential Equations (1992), Springer-Verlag, Berlin
17. Karlin S., and Taylor H.M. A Second Course in Stochastic Processes (1981), Academic Press, San Diego
18. Pickard D.K., Tory E.M., and Tuckman B.A. A three-parameter Markov model for sedimentation II. Simulation of transit times and comparison with experimental results. Powder Technol. 49 (1987) 227-240
19. Ganatos P., Pfeffer R., and Weinbaum S. A numerical-solution technique for three-dimensional Stokes flows with application to the motion of strongly interacting spheres in a plane. J. Fluid Mech. 84 (1978) 79-111
20. Kamel M.T., and Tory E.M. Sedimentation of clusters of identical spheres I. Comparison of methods for computing velocities. Powder Technol. 59 (1989) 227-248 Erratum
21. Kamel M.T., and Tory E.M. Sedimentation of clusters of identical spheres I. Comparison of methods for computing velocities. Powder Technol. 94 (1997) 266
22. Beenakker C.W.J., van Saarloos W., and Mazur P. Many-sphere hydrodynamic interactions III. The influence of a plane wall. Physica A 127 (1984) 451-472
23. Beenakker C.W.J., and Mazur P. Many-sphere hydrodynamic interactions IV. Wall-effects inside a spherical container. Physica A 131 (1985) 311-328
24. Tory E.M., Bargieł M., and Honeycutt R.L. A three-parameter Markov model for sedimentation III. A stochastic Runge-Kutta method for computing first-passage times. Powder Technol. 80 (1994) 133-146
25. Jánosi I.M., Tél T., Wolf D.E., and Gallas J.A.C. Chaotic particle dynamics in viscous flows: the three-particle Stokeslet problem. Phys. Rev. E 56 (1997) 2858-2868
26. Durlofsky L., Brady J.F., and Bossis G. Dynamic simulation of hydrodynamically interacting particles. J. Fluid Mech. 180 (1987) 21-49
27. Tory E.M., and Pickard D.K. Sedimentation of slurries: the long and winding road to chaos. In: Small R.D. (Ed). Proc. Atlantic Math Days 1983 (1984), University of New Brunswick, Fredericton 84-96
28. Tory E.M. A stochastic model for the slow sedimentation of small particles in a viscous fluid. In: Rahman M. (Ed). Ocean Waves Mechanics, Computational Fluid Dynamics and Mathematical Modelling (1990), Computational Mechanics Publications, Boston 671-685
29. Tory E.M., and Pickard D.K. A three-parameter Markov model for sedimentation. Can. J. Chem. Eng. 55 (1977) 655-665
30. Tory E.M., Bargieł M., and Kamel M.T. The distribution of velocities of a concentric sphere in a dispersion of spheres sedimenting in a spherical container. Powder Technol. 74 (1993) 159-169 Erratum
31. Tory E.M., Bargieł M., and Kamel M.T. The distribution of velocities of a concentric sphere in a dispersion of spheres sedimenting in a spherical container. Powder Technol. 94 (1997) 267
32. Caflisch R.E., and Luke H.C. Variance in the sedimentation speed of a suspension. Phys. Fluids 28 (1985) 759-760
33. K. Höfler, Räumliche Simulation von Zweiphasenflüssen, Diplomarbeit, Inst. für Computeranwendungen, U. Stuttgart, September 1997.
34. da Cunha F.R., Rosa O.S., and de Sousa A.J. Do we understand hydrodynamic dispersion in sedimenting suspensions?. J. Braz. Soc. Mech. Sci. 23 4 (2001) 379-399. Available from the web: http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0100-7386200 1000400001&lng=en&nrm=iso
35. Larsen L.J., and Marx M.L. An Introduction to Mathematical Statistics and Its Applications. third ed. (2001), Prentice Hall, Upper Saddle River, NJ
36. Bürger R., and Tory E.M. On upper rarefaction waves in batch settling. Powder Technol. 108 (2000) 74-87
37. E.M. Tory, D.K. Pickard, Original data from laboratory notebook, 1967.
38. Bürger R., Karlsen K.V., Tory E.M., and Wendland W.L. Model equations and instability regions for the sedimentation of equal spheres. Z. Angew. Math. Mech. 82 (2002) 699-722
39. J. Verhoeven, Prediction of batch settling behaviour, B. Chem. Eng. thesis, McMaster U., April, 1963.
40. Bargieł M., Ford R.A., and Tory E.M. Simulation of sedimentation of polydisperse suspensions: a particle-based approach. AIChE J. 51 (2005) 2457-2468
41. Bargieł M., and Tory E.M. An extension of the particle-based approach to simulating the sedimentation of polydisperse suspensions. Int. J. Miner. Process. 79 (2006) 235-252
42. Batchelor G.K., and Janse van Rensburg R.W. Structure formation in bidisperse sedimentation. J. Fluid Mech. 119 (1986) 379-407
43. Berres S., Bürger R., Karlsen K.H., and Tory E.M. Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression. SIAM J. Appl. Math. 64 (2003) 41-80
44. Weiland R.H., Fessas Y.P., and Ramarao B.V. On instabilities arising during sedimentation of two-component mixtures of solids. J. Fluid Mech. 142 (1984) 383-389
45. Law H.-S., Masliyah J.H., MacTaggart R.S., and Nandakumar K. Gravity separation of bidisperse suspensions: light and heavy particle species. Chem. Eng. Sci. 42 (1987) 1527-1538
46. Biesheuvel P.M., Verweij H., and Breedveld V. Evaluation of instability criterion for bidisperse sedimentation. AIChE J. 47 (2001) 45-52
47. M. Bargieł, E.M. Tory, Stability of tridisperse suspensions, Comput. Visualization Sci., in press.
48. Mucha P.J., Tee S.-Y., Weitz D.A., Shraiman B.I., and Brenner M.P. A model for velocity fluctuations in sedimentation. J. Fluid Mech. 501 (2004) 71-104
49. Peysson Y., and Guazzelli E. Velocity fluctuations in a bidisperse sedimenting suspension. Phys. Fluids 11 (1999) 1953-1955
50. Pickard D.K., and Tory E.M. A Markov model for sedimentation: fundamental issues and insights. In: MacNeill I.E., and Umphrey G.J. (Eds). Advances in the Statistical Sciences. Stochastic Hydrology vol. 4 (1987), D. Reidell, Dordrecht 1-25
51. Bargieł M., and Tory E.M. Stochastic dynamic solution of nonlinear differential equations for transport phenomena. AIChE J. 42 (1996) 889-891
52. Happel J., and Brenner H. Low Reynolds Number Hydrodynamics (1986), Martinus Nijhoff Publishers, Dordrecht, The Netherlands
53. Hesse C.H., and Ramos E. Dynamic simulation of a stochastic model for particle sedimentation in fluids. Appl. Math Modell. 18 (1994) 437-445
54. Herzhaft B., Guazzelli E., Mackaplow M.B., and Shaqfeh E.S.G. Experimental investigation of the sedimentation of a dilute fiber suspension. Phys. Rev. Lett. 77 (1996) 290-293
55. Herzhaft B., and Guazzelli E. Experimental study of the sedimentation of dilute and semi-dilute suspensions of fibres. J. Fluid Mech. 384 (1999) 133-158