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A stochastic model of particle shattering

Published online by Cambridge University Press:  01 July 2016

William S. Griffith ,
Richard J. Kryscio  and
Peter Purdue
William S. Griffith
Affiliation:
University of Kentucky
Richard J. Kryscio
Affiliation:
University of Kentucky
Peter Purdue
Affiliation:
University of Kentucky
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Abstract

Motivated by the problem of solid catalytic particle attrition during chemical reactions we formulate a continuous-time Markov model to describe the shattering of a particle when it is assumed that particles can be classified into a small number of types by size. We then obtain a recursive expression for the joint probability generating function of the count of the different types of particle at time t and derive a system of differential equations for the mean number of particle counts and a system of matrix differential equations for the covariance matrix of the particle counts. Solutions to these differential equations are presented in an important special case.

Keywords

SHATTERING MECHANISMGRINDING MECHANISMGENERALIZED COMPARTMENT MODELPROBABILITY GENERATING FUNCTIONMEAN NUMBER OF PARTICLE COUNTSCOVARIANCE OF PARTICLE COUNTS
Type
Research Article
Information
Advances in Applied Probability , Volume 16 , Issue 1 , March 1984 , pp. 70 - 87
Copyright
Copyright © Applied Probability Trust 1984 

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Footnotes

∗

On leave from Northern Illinois University, DeKalb, Illinois.

∗∗

Research partially supported by NSF Grant No. MCS-81–02215.

References

Capasso, V. and Paveri-Fontana, S. L. (1981) Some results on linear stochastic multicompartment systems. Math. Biosci. 55, 7–26.Google Scholar
Cardenas, ?. and Matis, L. (1974) On the stochastic theory of compartments: solution for m-compartment system with irreversible time-dependent transition probabilities. Bull. Math. Biol. 36, 489–504.Google Scholar
Fan, L. S. and Srivastava, R. C. (1981) A stochastic model for particle disintegration–I. Grinding mechanism. Chem. Eng. Sci. 36, 1091–1096.CrossRefGoogle Scholar
Mode, C. J. (1971) Multitype Branching Processes. American Elsevier, New York.Google Scholar
Purdue, P. (1979) Stochastic compartment models: a review of the mathematical theory with ecological applications. Compartment Analysis of Ecosystem Models, ed. Matis, J. H., Patten, B. C. and White, G. C.. Satellite Programs in Statistical Ecology, Internat. Coop. Pub.-House, Fairland, Md.Google Scholar
Srivastava, R. C. and Fan, L. S. (1982) A stochastic model for particle disintegration–II. Simple shattering mechanism. Technical Report, Department of Statistics, Ohio State University.Google Scholar
Wei, J., Lee, W. and Krambeck, R. F. (1977) Catalyst attrition and deactivation in fluid catalytic cracking system. Chem. Eng. Sci. 32, 1211–1218.Google Scholar