Generalized population balance equation

The general population balance equation requires numerical methods for its solution and several have been proposed (e.g. Gelbard and Seinfeld, 1978 Hounslow, 1990a,b Hounslow etai, 1988, 1990), of which more later. Fortunately, however, some analytic solutions for simplified cases also exist. 

In principle, given expressions for the crystallization kinetics and solubility of the system, equation 9.1 can be solved (along with its auxiliary equations -Chapter 3) to predict the performance of continuous crystallizers, at either steady- or unsteady-state (Chapter 7). As is evident, however, the general population balance equations are complex and thus numerical methods are required for their general solution. Nevertheless, some useful analytic solutions for design purposes are available for particular cases. 

Note that the RANS formulation used in (B.44) and (B.45) can easily be extended to the LES, as outlined in Section 5.10. Moreover, by following the same steps as outlined above, DQMOM can be used with the joint velocity, composition PDF transport equation. Finally, the reader can observe that the same methodology is applicable to more general distribution functions than probability density functions. Indeed, DQMOM can be applied to general population balance equations such as those used to describe multi-phase flows. 

A number of numerical approaches have been applied to the solution of the general population balance equation ... 

This book provides a consistent treatment of these issues that is based on a general theoretical framework. This, in turn, stems from the generalized population-balance equation (GPBE), which includes as special cases all the other governing equations previously mentioned (e.g. PBE and BE). After discussing how this equation originates, the different computational models for its numerical solution are presented. The book is structured as follows. 

Substituting into the general population balance equation yields for the micro-distributed case... 

A general population balance equation for bubbles located at position vector with a bubble volume Vj, at time t, can be written as (Chen et al., 2005)... 

The Reynolds transport theorem is a convenient device to derive conservation equations in continuum mechanics. Toward derivation of the general population balance equation, we envisage the application of this theorem to the deforming particle space continuum defined in the previous section. We assume that particles are embedded on this continuum at every point such that the distribution of particles is described by the continuous density function / (x, r, t). Let i//(x, r) be an extensive property associated with a single particle located at (x, r). 

One-dimensional population balance models for both batch and continuous systems are described in this section as special cases of the generalized population balance model stated in General Population Balance Equations. ... 

As is evident, however, the general population balance equations are complex and thus numerical methods are required for their general solution. Nevertheless, some useful analytic solutions are available for particular cases. 

Equation (3.14) is thus consistent with the general population balance (equation 3.8) when B = D = 0 i.e. nucleation occurs at zero size and both the birth and death terms due to agglomeration and breakage are neglected, and the feed is crystal-free. 

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