Formulation of a non-linear framework for population balance modeling of batch grinding: Beyond first-order kinetics

Population balance models (PBMs) for batch grinding are based on the concepts of specific breakage rate and breakage distribution. In the traditional PBMs, the breakage rate is assumed first-order, thus neglecting the effects of the temporally evolving material properties and multi-particle interactions. As an attempt to explain some of the above effects, a time-dependent specific breakage rate was introduced in the literature. The time-variant PBMs are inadequate to explain the multi-particle interactions explicitly and thoroughly. In this paper, we formulate a non-linear population balance framework to explain the non-first-order breakage rates that originate from multi-particle interactions. Based on this framework, four size-discrete non-linear models with varying complexity have been derived. A simple non-linear model with non-uniform kinetics assumption, Model B, was used to simulate the slowing-down phenomenon commonly observed in dry grinding processes. Not only does the model explain the effects of the fines accumulation on the specific breakage rate of the coarse, but also it is capable of predicting the significant influence of the initial population density. Identification of the proposed models, i.e., the solution of the inverse problem is also discussed.