Population balance microscopic

The accumulation term is the chstnge in the population with time, dij /dt. The input and the output terms are considered together. For the input—output terms, we have one term due to flow and anoBier 

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The microscopic population balance is obtained by accounting for all the particles in a differential volume AV = Ajc Ay Az as shown in Figure... 

For the net generation terms we have birth, B, minus death, D. Combining all these terms we have the microscopic population balance ... 

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 formulations of the population balance equation based on the continuum mechanical approach can be split into two categories, the macroscopic- and the microscopic population balance equation formulations. The macroscopic approach consists in describing the evolution in time and space of several groups or classes of the dispersed phase properties. The microscopic approach considers a continuum representation of a particle density function. 

The fundamental and thus more general microscopic population balance equation is formulated from scratch on the continuum scales using generalized versions of the Leibnitz- and Gauss theorems. 

The coalescence terms in the average microscopic population balance can then be expressed in terms of the local effective swept volume rate and the coalescence probability variables ... 

The Statistical Mechanical Microscopic Population Balance Formulation... 

For bubbly flows most of the early papers either adopted a macroscopic population balance approach with an inherent discrete discretization scheme as described earlier, or rather semi-empirical transport equations for the contact area and/or the particle diameter. Actually, very few consistent source term closures exist for the microscopic population balance formulation. The existing models are usually solved using discrete semi-integral techniques, as will be outlined in the next sub-section. 

General 3D microscopic population balance equations, including several distributed properties, e.g. size and moisture content, can be derived using the same balance method. 

This relation expresses that not all collisions lead to coalescence. The modeling of the coalescence processes thus means to find adequate physical expressions for hc d d, Y) and pc d d, Y). Kamp et al [39], among others, suggested that microscopic closures can be formulated in line with the macroscopic population balance approach, thus we may define ... 

The application of similar advanced distribution functions in the context of population balance analysis of polymerization processes is familiar in reaction engineering [40, 97]. However, the microscopic balance equations used for this purpose are normally averaged over the whole reactor volume so that simplified macroscopic (global) reactor analysis of the chemical process behavior is generally performed [35]. 

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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