Analysis of nonlinear batch grinding in stirred media mills using self-similarity solution

Generally, particle breakage rate is considered to be independent of the grinding environment, and hence, the system is referred to as a linear time-invariant grinding system with first-order grinding kinetics. However, time-dependent breakage rate exists and perhaps, is more critical for fine grinding of particles. The time-dependent breakage rate also introduces nonlinearity in the grinding phenomena. In the present work, a self-similarity based approach is described to model the evolution of fine particle size distributions in a batch stirred media milling with an emphasis on the nonlinear breakage rate function by considering the breakage rate to be a function of the grind time. The present approach yields analytical expressions for cumulative weight percent finer distributions for the continuous-size continuous-time population balance equation. The breakage parameters in the analytical solution can be estimated for a given system from any three measured size distributions that show self-similarity and these parameters can be used to predict distributions evolving at higher grind times. Several sets of published data of stirred media milling are employed to validate the model.