Population balance modeling of non-linear effects in milling processes

Specific breakage rate (selection) and breakage distribution functions are used to describe the birth and death terms in population balance models (PBMs) for milling processes. Traditional PBMs for milling processes are inherently linear because the breakage rate is assumed first-order. The specific breakage rate is independent of the population density while it depends on particle size and possibly on time. Even though the linear theory has been applied with some success to the modeling, optimization, and design of various mills in the last 50 years, many researchers have indicated its restrictions and subjected it to serious criticism. In this paper, we first categorize the experimentally observed deviations from the linear theory and suggest the multi-particle interactions as the origin of these deviations. To account for the peculiar non-linear effects, a phenomenological theory has been proposed via multiplicative decomposition of the specific breakage into a size-dependent apparent breakage rate and a population density dependent functional. The proposed theory recovers the first-order breakage kinetics in the limit, yet it is sufficiently general to explain all experimentally observed deviations. Numerical simulations of a batch milling process have demonstrated the potential of the non-linear theory.