The Statistical Mechanical Microscopic Population Balance Formulation

3 The Statistical Mechanical Microscopic Population Balance Formulation 

The statistical description of multiphase flow is developed based on the Boltzmann theory of gases [37, 121, 93, 11, 94, 58, 61]. The fundamental variable is the particle distribution function with an appropriate choice of internal coordinates relevant for the particular problem in question. Most of the multiphase flow modeling work performed so far has focused on isothermal, non-reactive mono-disperse mixtures. However, in chemical reactor engineering the industrial interest lies in multiphase systems that include multiple particle t3q)es and reactive flow mixtures, with their associated effects of mixing, segregation and heat transfer. 

Defining a single distribution function, p(x, r, c, t)dxdrdc, as the probable number of particles with internal coordinates in the range dx. about x, located in the spatial range dr about the position r, with a velocity range dc about c. 

The single distribution function /(x, r, t)dxdr thus denotes the probable number of particles within the internal coordinate space in the range dx about x, in the external (spatial) range dr about r at time t. is the mean velocity of all particles of properties x at a location r at time t. The velocity independent birth and death terms are defined by  

To close the population balance problem, models are required for the growth, birth and death kernels. In the kinetic theory context, as distinct from the continuum mechanical approach, the continuum closure may be considered macroscopic in a similar manner as in the granular theory treating macroscopic particle properties. 

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