Aggregation kernel closure

The fundamental derivation of the population balance equation is considered general and not limited to describe gas-liquid dispersions. However, to employ the general population balance framework to model other particulate systems like solid particles and droplets appropriate kernels are required for the particle growth, agglomeration/aggregation/coalescence and breakage processes. Many droplet and solid particle closures are presented elsewhere (e.g., [96, 122, 25, 117, 75, 76, 46]). 

We have shown how this set of differential equations is closed in some simple cases (i.e. for nucleation at zero size, with constant growth rate, and constant aggregation and breakage kernels). In general, when the system is not closed, closure must be sought through a stable numerical method. 

Insofar as the possibility of directly using Eq. (7.104) is concerned, as for univariate cases, the main limitation resides in the difficulty of finding analytical solutions and developing simple and stable closures. Analytical solutions are very rare and exist only for rates of continuous change and for kernels independent of the internal coordinates. Under these simplification hypotheses, for the case of nucleation, growth, diffusion, aggregation, and breakage (with additive internal coordinates), the bivariate PBE becomes... 

A discrete version of the master density equations (7.3.10), without particle growth, has been solved by Bayewitz et al (1974), and later by Williams (1979), to examine the dynamic average particle size distribution in an aggregating system with a constant kernel. When the population is small EN < 50) their predictions reveal significant variations from those predicted by the population balance equation. However, the solution of such master density equations is extremely difficult even for the small populations of interest for nonconstant kernels. It is from this point of view that a suitably closed set of product density equations presents a much better alternative for analysis of such aggregating systems. We take up this issue of closure again in Section 7.4. 

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