Governing Equations for Polydispersed Multiphase Systems

To describe polydispersed multiphase systems the Boltzmann equation can be extended by including the dependency of the internal property coordinates such as the particle size and shape in the definition of the distribution function. In this way a statistical balance formulation can be obtained by means of a distribution function on the form p, r,v, c,t)d dr dv dc, defined as the probable number of particles with internal properties in the range about with a velocity range in property space dv about v, located in the spatial range dr about the position r, with a velocity range dc about c, at time t. The particular Boltzman type of equation is given by  

The main challenge in formulating these equations is related to the definition of the collision operator. So far this approach has been restricted to the formulation of the population balance equation. That is, in most cases a general transport equation which is complemented with postulated source term formulations for the particle behavior is used. Randolph [80] and Randolph and Larson [81] used this approach deriving a microscopic population balance equation for the purpose of describing the behavior of particulate systems. Ramkrishna [79] provides further details on this approach considering also fluid particle systems. 

The population balance modeling framework is further discussed in chap 9. 

Aris R (1962) Vectors, Tensors, and the Basic Eqnations of Flnid Mechanics. Dover, Inc., New York 

Arnold VI (1989) Mathematical Methods of Classical Mechanics. Second Edition, Springer-Verlag, New York 

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