Statistical Microscopic Population Balance Formulation

A statistical description of multiphase flow might be developed based on an analogy to the Boltzmann theory of gases [11, 39, 60,63, 66, 91, 125,135]. 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 reactive flow systems that include poly-dispersed mixtures of multiple particle types, with their associated effects of mixing, segregation and heat and mass transfer. 

Integrating over the whole particle velocity space to eliminate the particle velocity dependence, one obtains the following equation  

The single distribution function/(x, r, t)dxdv thus denotes the probable number of particles within the internal coordinate space in the range dx about x, in the external (spatial) range dv about r at time t. r 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  

For non-reactive, isothermal, particle mixtures it is customary to assume that all the relevant internal variables can be calculated in some way from the particle volume or diameter. It is natural to use the particle diameter as inner coordinate for the population balance analysis because this choice of inner coordinate coincides with the classical kinetic theory of gases. This inner coordinate is especially useful describing solid particle and fluid particle dispersions. A statistical description of dispersed multiphase flow can then be obtained by means of a distribution function fdp dp, r, t), defined so that the number of particles with diameters in the range dp and dp+d dp), located in a volume dr around a spatial location r at time t is f dp, r, t)d(dp)dr. In this case the population balance (9.118) is usually expressed as  

To close the population balance problem, models are required for the growth, birth and death kernels. It is required that these kernels are consistent with the inner coordinate used. The coalescence and breakage kernels presented in this chapter are expressed in terms of the particle diameter. 

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The Statistical Mechanical Microscopic Population Balance Formulation... 

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