Birth and Death Rates at the Boundary

We begin with revisiting the boundary condition (2.7.12), which represents the crucial boundary condition representing the birth of new particles at the boundary, which subsequently migrate to the interior of the particle state space. If the birth of new particles represented by the boundary condition (2.7.12) occurs at the expense of existing particles, then the right-hand side of the population balance equation (2.7.9) must include a corresponding sink term. 

Boundary conditions of the type (2.7.12) are important in crystallization where secondary nucleation, as pointed out by Randolph and Larson (1988), may be governed by the growth rate of existing particles. For example, consider a well-mixed crystallizer where the number density is only a function of the sole internal coordinate selected as particle size x as represented by a characteristic length, which should satisfy a population balance equation of the type (2.7.6). Randolph and Larson discuss a variety of nucleation mechanisms and conclude that contact nucleation is the most significant form of nucleation. Thus, the mechanical aspects of the crystallizer equipment which provide contact surfaces contribute to increased nucleation rate. When growing crystals, containing adsorbed solute on their surfaces, come into contact with other solid surfaces, nucleation is induced. The boundary condition for the formation of new nuclei in a real crystallizer is therefore considerably more complicated than that implied by (2.7.7). Instead, the boundary condition must read as 

1 By a functional is meant a mapping of a function into a number. The mapping generally depends on the values of the function defined over the entire domain of its arguments. In the present context, we are concerned about a functional, that depends on the entire size distribution at a fixed time. Note that the definition does not extend here to its time history. 

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