On the consequences of non-first-order breakage kinetics in comminution processes: Absence of self-similar size spectra

Comminution or size reduction is a commonly used unit operation in a variety of industries including pharmaceuticals. In quantitative analysis of comminution at the process length scale, population balance modeling has been used with an assumption that the breakage rate of particles is first-order and that the population balance model is linear. For certain forms of the specific breakage rate and breakage distribution functions, a similarity solution to the linear population balance exists, and self-similarity plots have been used to show its validity in batch comminution processes. On the other hand, ample experimental batch milling data in the literature indicate the inadequacy of the linear theory. In this study, a theoretical investigation has been conducted within the framework of a recently proposed non-linear theory. The non-linear theory takes into account phenomenologically the non-first-order effects resulting from mechanical multi-particle interactions. Specifically, the cushioning action of small particles on the coarser ones is considered. It is found that self-similarity does not exist in the presence of the non-first-order breakage kinetics. The present study suggests the use of self-similarity plots for an approximate assessment of the impact of non-first-order kinetics on the shape of the cumulative particle size distribution.