Upper critical dimension for aggregation processes.

Aggregation kinetics are often described by the population balance (or Smoluchowski coagulation) equation. The population balance equation is a mean-field equation for an aggregation process. An upper critical dimension d, calculated for a given aggregation frequency has been used by other investigators to characterize the validity of this mean-field equation. It is shown that the upper critical dimension is also related to the singularity of the self-similar spectrum. This paper demonstrates that the usefulness of the upper critical dimension for determining the validity of the population balance equation is lost when the aggregation frequency is unknown. The validity of the equation can be inferred from the similarity or scaling distribution and the evolution of the average particle size in such cases.