The Microscopic Continuum Mechanical Population Balance Formulation

2 The Microscopic Continuum Mechanical Population Balance Formulation 

In this section the population balance modeling approach established by Randolph [95], Randolph and Larson [96], Himmelblau and Bischoff [35], and Ramkrishna [93, 94] is outlined. The population balance model is considered a concept for describing the evolution of populations of countable entities like bubble, drops and particles. In particular, in multiphase reactive flow the dispersed phase is treated as a population of particles distributed not only in physical space (i.e., in the ambient continuous phase) but also in an abstract property space [37, 95]. In the terminology of Hulburt and Katz [37], one refers to the spatial coordinates as external coordinates and the property coordinates as internal coordinates. The joint space of internal and external coordinates is referred to as the particle phase space. In this case the quantity of basic interest is a density function like the average number of particles per unit volume of the particle state space. The population balance may thus be considered an equation for the number density and regarded as a number balance for particles of a particular state. 

A model describing the evolution of this density function must consider the various ways in which the particles of a specific state can either form or disappear from the system. The change of the density function with respect to 

If dVx and dVr denote infinitesimal volumes in property space and physical space respectively located at (x, r), then the number of particles in dVxdVr is given by f x,r,t)dVxdVr. The local (average) number density in physical space, that is, the total number of particles per unit volume of physical space, denoted N[r,t), is given by  

For an arbitrary combined material volume element constituting a combined sub-volume Vsv[t) of the particle phase space the integral formulation of the population balance states that the only way in which the number of particles can change is by birth and death processes [95, 96, 35, 93, 94[. The system balance is thus written on the form  

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