Self-Similarity in Systems with Growth

Instances of self-similarity in the presence of particle growth have been relatively rare in the literature. However, the author has discussed the possibility of self-similar behavior in the dynamics of microbial populations (Ramkrishna, 1994 Ramkrishna and Schell, 1999). We present a simple example to illustrate the broad ideas from the foregoing development. 

We consider a population of cells distributed according to cell mass x with growth rate JT(x). The cells further undergo binary division at the rate b(x) into daughter cells whose mass distribution is given by P x y) where y represents the mass of the mother cell and x is the mass of either daughter cell. The population balance equation may be written as 

The condition on the exponents a and P appears to be overly restrictive but one to which we shall return presently. On solving the differential equation for /z, and using the relationship (5.1.3) to set gh = we obtain 

We now return to the conditions in (5.4.5), which were necessary for self-similarity. It was observed that the last condition on the exponents a and P is overly restrictive. In the absence of this equality, however, the time-dependence of the third term in the left-hand side of (5.4.4) is incompatible with its time-independent right-hand side. In this case, consistency with self-similarity can come about only by the diminishing of the term relative to the other terms in the left-hand side of (5.4.4). Thus, if h decreases with time (which occurs when is positive) then self-similarity can occur only when the exponents a and P satisfy the inequality a — — 1 0. This inequality is reversed if h increases with time. Self- 

11 Equality constraints were used by the author in a multi-dimensional setting (Ramkrishna, 1994). 

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