Applications of Markov Chains in Particulate Process Engineering: A Review

Processes involving particles, are known to exhibit extremely unpredictable behaviour, mainly due to the mesoscopic nature of granular media. Understanding particulate processes, not only for intellectual satisfaction, but also for process design and operation, basically requires a systems approach in modelling. Because they combine simplicity and flexibility, the stochastic models based on the Markov chain theory are very valuable mathematical tools to this respect. However, they are still largely ignored by the whole core of chemical engineering researchers. This motivates the existence of this review paper, in which we examine the three traditional issues: mixing and transport, separation and transformation.

[1]  L. T. Fan,et al.  Stochastic diffusion model of non-ideal mixing in a horizontal drum mixer , 1979 .

[2]  I. M. Leichkis Filtration on precoat filters as a Markov process , 1993 .

[3]  L. T. Fan,et al.  A stochastic model of the unsteady state age distribution in a flow system , 1979 .

[4]  L. Fan,et al.  NUMERICAL AND EXPERIMENTAL SIMULATION STUDIES ON THE MIXING OF PARTICULATE SOLIDS AND THE SYNTHESIS OF A MIXING SYSTEM , 1978 .

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

[6]  Alex C. Hoffmann,et al.  Stochastic Models for Transport in a Fluidized Bed , 1999, SIAM J. Appl. Math..

[7]  L. Fan,et al.  Stochastic modelling of chemical engineering systems. Application of the generalized master equation to the bubble population in a bubbling fluidized bed , 1987 .

[8]  Bhaskar Gupta,et al.  A Finite‐State Markov Chain Model for Axial Mixing of Solids in a Fluidized Bed , 1990 .

[9]  Julio M. Ottino,et al.  Computational studies of granular mixing , 2000 .

[10]  L. T. Fan,et al.  Birth-death modeling of deep bed filtration: sectional analysis , 1985 .

[11]  Hu Liu,et al.  Markovian inventory policy with application to the paper industry , 2002 .

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

[13]  Biao Huang,et al.  Estimation of Markov parameters and time-delay/interactor matrix , 2000 .

[14]  Ka Ming Ng,et al.  Design and development of solids processes - a process systems engineering perspective , 2002 .

[15]  Masayuki Horio,et al.  Numerical analysis of particle mixing characteristics in a single helical ribbon agitator using DEM simulation , 2000 .

[16]  L. Fan,et al.  Modeling of complex chemical reactions in a continuous-flow reactor: A Markov chain approach , 1988 .

[17]  Liang-Shih Fan,et al.  A stochastic model for particle disintegration—I: Grinding mechanism , 1981 .

[18]  Mónica Alonso,et al.  Stochastic modeling of particle coating , 2001 .

[19]  Harold E. Hoelscher,et al.  A stochastic model of packed‐bed mixing and mass transfer , 1971 .

[20]  K. Ceylan,et al.  Stochastical modeling of the granule size distribution in the agglomeration processes of powdered materials , 2001 .

[21]  J. Raghuraman,et al.  A markov chain model for residence time and contact time distributions in packed beds , 1975 .

[22]  János Gyenis,et al.  New stochastic modelling of mixing in process operations , 1999 .

[23]  L. Fan,et al.  Analysis of deep bed filtration data: Modeling as a birth-death process , 1985 .

[24]  L. T. Fan,et al.  The mixing of solid particles in a motionless mixer—a stochastic approach , 1972 .

[25]  Yu Guangsuo,et al.  Experimental studying and stochastic modeling of residence time distribution in jet-entrained gasifier , 2002 .

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

[27]  Vadim Mizonov,et al.  Algorithme de construction de modèles markoviens multidimensionnels pour le mélange des poudres , 2001 .

[28]  L. Fan,et al.  Mixing of randomly moving particles in liquid—solid fluidized beds , 1985 .

[29]  H. Berthiaux,et al.  A markov chain model to describe the residence time distribution in a stirred bead mill , 2001 .

[30]  L. T. Fan,et al.  Stochastic analysis of attrition — a general cell model , 1989 .

[31]  L. T. Fan,et al.  Stochastic modeling of segregation in a motionless mixer , 1977 .

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

[33]  J. Johnson,et al.  Polydisperse particulate solids mixing and segregation: nonstationary Markov chains. , 1977, Journal of pharmaceutical sciences.

[34]  L. T. Fan,et al.  Stochastic modeling of transient residence-time distributions during start-up , 1995 .

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

[36]  K. M. Mehata,et al.  A Markov chain model of a packed bed , 1972 .

[37]  F. Antia,et al.  The effect of stoichiometry on Markov chain models for chemical reaction kinetics , 1985 .

[38]  Henri Berthiaux Analysis of grinding processes by Markov chains , 2000 .

[39]  Henri Berthiaux,et al.  An experimental method and a Markov chain model to describe axial and radial mixing in a hoop mixer , 2002 .

[40]  L. Fan,et al.  A modified coalescence—dispersion model for the axial mixing of segregating particles in a motionless mixer , 1977 .

[41]  V. Zhukov,et al.  The modelling of grinding processes by means of the principle of maximum entropy , 1998 .

[42]  John A. Dodds,et al.  Modeling classifier networks by Markov chains , 1999 .

[43]  L. Fan,et al.  Experimental study of deep bed filtration: A stochastic treatment , 1984 .

[44]  Wooyoung Lee,et al.  Catalyst attrition and deactivation in fluid catalytic cracking system , 1977 .

[45]  L. Fan,et al.  Stochastic analysis of fluctuations in the local void fraction of a gas-solids fluidized bed , 1986 .

[46]  Leonard G. Austin,et al.  Introduction to the mathematical description of grinding as a rate process , 1971 .

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

[48]  L. Fan,et al.  Estimation of the particle diffusivity in a liquid-solids fluidized bed based on a stochastic model , 1982 .

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

[50]  Michael Rubinovitch,et al.  Single-particle approach for analyzing flow systems. Part II: Regional residence times and local flow rates , 1983 .

[51]  L. Fan,et al.  Mixing of large particles in two‐dimensional gas fluidized beds , 1979 .

[52]  L. Fan,et al.  Application of a Discrete Mixing Model to the Study of Mixing of Multicomponent Solid Particles , 1975 .

[53]  L. T. Fan,et al.  Application of the master equation to coalescence and dispersion phenomena , 1988 .

[54]  V. P. Zhukov,et al.  Application of multi-dimensional Markov chains to model kinetics of grinding with internal classification , 2004 .

[55]  L. T. Fan,et al.  Stochastic modeling of controlled-drug release , 1998 .

[56]  L. T. Fan,et al.  Stochastic simulation of residence time distribution curves , 1985 .

[57]  Michael Rubinovitch,et al.  Single‐particle approach for analyzing flow systems. Part I: Visits to flow regions , 1983 .

[58]  Michael Rubinovitch,et al.  A single particle approach for analyzing flow systems. Part III: Multiple fluids , 1985 .

[59]  Prodromos Daoutidis,et al.  Simulation of population dynamics using continuous-time finite-state Markov chains , 2003, Comput. Chem. Eng..

[60]  Bhaskar Gupta,et al.  A multiple state stochastic model for deep‐bed filtration , 1992 .

[61]  Raja Nassar,et al.  Markov chain models of complex chemical reactions in continuous flow reactors , 1983 .

[62]  L. Fan,et al.  Radial particle mixing in gas-solids fluidized beds , 1986 .

[63]  Winfried Pippel,et al.  Utilization of markov chains for simulation of dynamics of chemical systems , 1977 .

[64]  L. Fan,et al.  A stochastic treatment of unimolecular reactions in an unsteady state continuous flow system , 1981 .

[65]  C. Chou,et al.  Kinetics of mass transport in sheared particulate beds: Markov chains , 1978 .

[66]  H. Kropholler,et al.  Solution of a mixing model due to van de Vusse by a simple probability method , 1967 .

[67]  Julio M. Ottino,et al.  Particle dynamics simulation: a hybrid technique applied to granular mixing , 1998 .

[68]  O. Molerus,et al.  Stochastisches Modell der Gleichgewichtssichtung , 1967 .

[69]  Octave Levenspiel,et al.  Modeling in chemical engineering , 2002 .

[70]  Rex B. Thorpe,et al.  Stochastic modelling of the particle residence time distribution in circulating fluidised bed risers , 2002 .

[71]  L. T. Fan,et al.  Modelling and simulation of deep-bed filtration: a stochastic compartmental model , 1986 .