Self-Similarity in Systems with Breakage and Aggregation Processes

3 SELF-SIMILARITY IN SYSTEMS WITH BREAKAGE AND AGGREGATION PROCESSES 

A somewhat difTerent form of scaling behavior is observable in systems in which both aggregation and breakage processes occur together. We are concerned here with systems in which the two processes occur such that they can be described by a population balance equation featuring an additive combination of the breakage and aggregation rates. Thus we may write the equation 

Following Meakin and Ernst (1988), we assume that the transient period well before the attainment of steady state is aggregation dominated. Suppose we now allow the breakage parameter k to vary (continuously) from one system to another among a (continuous) collection of systems characterized by k. (The system in which no breakage occurs belongs to this family of systems with k = 0.) Then we shall be concerned with the number density / (x, t k) of particles of size x at instant t in system /c, which satisfies the population balance given by 

A similarity analysis of the foregoing equation with the form 

Vigil and ZilF (1989) dispense with the assumption of binary breakage in their analysis of self-similarity but appear to assume a constant mean number of fragments independently of the size of the fragmenting particle. 

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Self-Similarity in Systems with Breakage and Aggregation Processes... 

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