Similarity Analysis of Population Balance Equations

The application of similar advanced distribution functions in the context of population balance analysis of polymerization processes is familiar in reaction engineering [40, 97]. However, the microscopic balance equations used for this purpose are normally averaged over the whole reactor volume so that simplified macroscopic (global) reactor analysis of the chemical process behavior is generally performed [35]. 

Since we are interested in dynamic analysis starting from some initial conditions which can be arbitrary, it is clear that one cannot associate a self-similar solution from the very beginning of the process except for an initial condition that happens to be compatible with the self-similar solution. Thus, the question of a self-similar solution basically arises when the system has evolved away from the initial condition. There is thus a sense of independence of the self-similar solution from the initial condition. This independence may, however, apply only for a class of initial conditions outside of which no self-similar solution may be attained. Questions in regard to the conditions under which a self-similar solution exists for a population balance equation, and the class of initial conditions for which the solution can approach such a self-similar solution, are indeed mathematically very deep and cannot be answered within the scope of this treatment. On the other hand, numerical solutions can be examined for their approach to self-similarity. 

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