Generalized population balance equation nucleation

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. 

Analytical solutions of the self-preserving distribution do exist for some coalescence kernels, and such behavior is sometimes seen in practice (see Fig. 40). For most practical applications, numerical solutions to the population balance are necessary. Several numerical solution techniques have been proposed. It is usual to break the size range into discrete intervals and then solve the series of ordinary differential equations that result. A geometric discretization reduces the number of size intervals (and equations) that are required. Litster, Smit and Hounslow (1995) give a general discretized population balance for nucleation, growth and coalescence. Figure 41 illustrates the evolution of the size distribution for coalescence alone, based on the kernel of Ennis Adetayo (1994). 

In general, both nucleation and crystal growth depend on supersaturation and to lesser extent temperature and magma characteristics. Such data must therefore be collected to gain maximum benefit from the population balance approach (Jones and Mullin, 1974 Jones, 1974). Further simplifications to the describing equations are also possible, however (as follows). 

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