Local Continuum Mechanical Population Balance Equation

2 Local Continuum Mechanical Population Balance Equation 

In the previous subsection it was mentioned that the formulations of the population balance equation based on the continuum mechanical approach might be split into two categories, the macroscopic- and the local instantaneous population balance equation formulations. The local instantaneous approach considers a continuum mechanical representation of a particle density function. This population balance equation is formulated from scratch on the continuum scales using generalized versions of the Leibnitz- and Gauss theorems. 

In this section the population balance modeling approach established by Randolph [105], Randolph and Larson [106], and Himmelblau and Bischoff [37] is outlined. Ramkrishna [103, 104] employed the same continuum mechanical concepts. In this modeling approach the population balance model is considered a concept for describing the evolution of populations of countable entities like bubbles, 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 [39,105], In the terminology of Hulburt and Katz [39], 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. 

If dVx and dVr denote infinitesimal volumes in property space and physical space respectively located at (x, r), then the number of particles in dVjdVr is given by 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 Af(r, t), is given by  

For an arbitrary combined material volume element constituting a combined subvolume Vsvit) 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 [37, 103-106]. 

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In the following sections four alternative approaches for deriving population balance equations are outlined. The four types of PBEs comprise a macroscopic PBE, a local instantaneous PBE, a microscopic PBE, and a PBE on the moment form. Two of these population balance forms are formulated in accordance with the conventional continuum mechanical theory. 

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