Simulation of sedimentation and fluidization of polydisperse suspensions via a Markov model

Abstract The particle-based approach to sedimentation is extended to include velocity fluctuations that result in hydrodynamic diffusion. The vector process describing the joint values of position and velocity is Markov. Thus, no integration of velocity is required. Height–velocity “skeletons” for each particle are generated from a bivariate-normal distribution with means, variances, and covariance that depend on three parameters. For each particle, there is a unique region in which the vector of species concentrations determines that particle's parameters and hence its Markov process, but the concentrations in that region depend on the Markov processes of neighboring particles. Though only discrete values of height and velocity are generated, the model ensures that sample paths and particle velocities are continuous. Furthermore, steady-state velocities are normally distributed and velocity autocorrelations decay exponentially. Published experimental results indicate that both are excellent approximations. For polydisperse suspensions, the Markov model is much simpler than the standard hydrodynamic-diffusion model and represents the actual process much better. We simulate the sedimentation and fluidization of polydisperse suspensions and study the effects of two additional parameters: variance and autocorrelation decay rate of particle velocities.

[1]  Raimund Bürger,et al.  On mathematical models and numerical simulation of the fluidization of polydisperse suspensions , 2005 .

[2]  D. Drew,et al.  Theory of Multicomponent Fluids , 1998 .

[3]  R. A. Ford,et al.  Simulation of Sedimentation of Polydisperse Suspensions: A Particle-Based Approach , 2005 .

[4]  G. J. Kynch A theory of sedimentation , 1952 .

[5]  Yannick Peysson,et al.  Velocity fluctuations in a bidisperse sedimenting suspension , 1999 .

[6]  Fernando Porté-Agel,et al.  Moore's Law and Numerical Modeling , 2002 .

[7]  Raimund Bürger,et al.  Central schemes and systems of conservation laws with discontinuous coefficients modeling gravity separation of polydisperse suspensions , 2004 .

[8]  E. Hinch,et al.  Spreading fronts and fluctuations in sedimentation , 2003 .

[9]  D. K. Pickard,et al.  Experimental implications of a Markov model for sedimentation , 1979 .

[10]  Tadatoshi Chiba,et al.  A comprehensive interpretation of solid layer inversion in liquid fluidised beds , 1982 .

[11]  R. Jackson,et al.  The nature of aggregative and particulate fluidization , 1964 .

[12]  W. Ramirez,et al.  Dynamic model of multi‐species segregation and dispersion in liquid fluidized beds , 2005 .

[13]  J. C. Luke Decay of velocity fluctuations in a stably stratified suspension , 2000 .

[14]  A three-parameter markov model for sedimentation II. Simulation of transit times and comparison with experimental results , 1987 .

[15]  George M. Homsy,et al.  HINDERED SETTLING AND HYDRODYNAMIC DISPERSION IN QUIESCENT SEDIMENTING SUSPENSIONS , 1988 .

[16]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[17]  Raimund Bürger,et al.  Model Equations and Instability Regions for the Sedimentation of Polydisperse Suspensions of Spheres , 2002 .

[18]  R. Bürger,et al.  Sedimentation and Thickening : Phenomenological Foundation and Mathematical Theory , 1999 .

[19]  E. M. Tory,et al.  Sedimentation of Small Particles in a Viscous Fluid , 1996 .

[20]  Manuel Laso Stochastic dynamic approach to transport phenomena , 1994 .

[21]  M. T. Kamel,et al.  Mean velocities in polydisperse suspensions , 1997 .

[22]  Robert H. Davis,et al.  Spreading of the interface at the top of a slightly polydisperse sedimenting suspension , 1988, Journal of Fluid Mechanics.

[23]  Kai Höfler,et al.  Simulation and modeling of mono- and bidisperse suspensions , 2000 .

[24]  E. J. Hinch,et al.  Particle velocity fluctuations and hydrodynamic self-diffusion of sedimenting non-Brownian spheres , 1995 .

[25]  Monika Bargieł,et al.  An extension of the particle-based approach to simulating the sedimentation of polydisperse suspensions , 2006 .

[26]  J. Happel,et al.  Low Reynolds number hydrodynamics , 1965 .

[27]  Robert H. Davis Hydrodynamic diffusion of suspended particles: a symposium , 1996, Journal of Fluid Mechanics.

[28]  B. Koglin Statistische Verteilung der Sedimentationsgeschwindigkeit in niedrig konzentrierten Suspensionen , 1971 .

[29]  S. Lee,et al.  Combined effect of sedimentation velocity fluctuation and self-sharpening on interface broadening , 1992 .

[30]  D. K. Pickard,et al.  A three‐parameter markov model for sedimentation , 1977 .

[31]  Jacob H. Masliyah,et al.  ]Hindered settling in a multi-species particle system , 1979 .

[32]  G. Batchelor,et al.  Sedimentation in a dilute polydisperse system of interacting spheres. Part 1. General theory , 1982, Journal of Fluid Mechanics.

[33]  R. Bürger,et al.  Central difference solutions of the kinematic model of settling of polydisperse suspensions and three-dimensional particle-scale simulations , 2001 .

[34]  Sangtae Kim,et al.  Microhydrodynamics: Principles and Selected Applications , 1991 .

[35]  M. J. Lockett,et al.  Sedimentation of Binary Particle Mixtures , 1979 .

[36]  M. T. Kamel,et al.  Sedimentation is container-size dependent , 1992 .

[37]  D. K. Pickard,et al.  A Markov Model for Sedimentation: Fundamental Issues and Insights , 1987 .

[38]  S. Karlin,et al.  A second course in stochastic processes , 1981 .

[39]  Monika Bargieł,et al.  A three-parameter Markov model for sedimentation III. A stochastic Runge—Kutta method for computing first-passage times , 1994 .

[40]  D. K. Pickard,et al.  Extensions and refinements of a Markov model for sedimentation , 1982 .

[41]  Prediction of complete mixing of liquid-fluidized binary solid particles , 1986 .

[42]  John R. Grace,et al.  Steady state dispersion of mono-size, binary and multi-size particles in a liquid fluidized bed classifier , 2002 .

[43]  Elisabeth Guazzelli,et al.  Evolution of particle-velocity correlations in sedimentation , 2001 .

[44]  D. K. Pickard,et al.  A Markov model for sedimentation , 1977 .

[45]  E. M. Tory,et al.  Stochastic sedimentation and hydrodynamic diffusion , 2000 .

[46]  P. Mucha,et al.  A model for velocity fluctuations in sedimentation , 2004, Journal of Fluid Mechanics.

[47]  R. Larsen,et al.  An introduction to mathematical statistics and its applications (2nd edition) , by R. J. Larsen and M. L. Marx. Pp 630. £17·95. 1987. ISBN 13-487166-9 (Prentice-Hall) , 1987, The Mathematical Gazette.

[48]  A. Ladd Sedimentation of homogeneous suspensions of non-Brownian spheres , 1997 .

[49]  R. A. Ford,et al.  Simulation of sedimentation of bidisperse suspensions , 2004 .

[50]  P. T. Shannon,et al.  Batch and Continuous Thickening. Basic Theory. Solids Flux for Rigid Spheres , 1963 .

[51]  G. Gee,et al.  Particle-size Analysis , 2018, SSSA Book Series.

[52]  Krishnaswamy Nandakumar,et al.  Gravity separation of bidisperse suspensions: Light and heavy particle species , 1987 .

[53]  Russel E. Caflisch,et al.  Variance in the sedimentation speed of a suspension , 1985 .

[54]  Raimund Bürger,et al.  On upper rarefaction waves in batch settling , 2000 .

[55]  Raimund Bürger,et al.  Strongly Degenerate Parabolic-Hyperbolic Systems Modeling Polydisperse Sedimentation with Compression , 2003, SIAM J. Appl. Math..