Population Balance Modeling of Environment Dependent Breakage: Role of Granular Viscosity, Density and Compaction. Model Formulation and Similarity Analysis

In this work, we introduce mechanistic considerations in nonlinear milling of particulate material. We show that the mobility of the powder can have a significant effect on the breakage frequency. During grinding, the change in particle size leads to changes in the mobility of the powder which in turn alters the breakage rate. The mechanistic modeling, being able to accommodate different responses of powder mobility for wet and dry grinding, is able to distinguish naturally between dry and wet grinding, an attribute not possessed by other nonlinear models. It is shown that the rich variety of phenomena observed in grinding mills emanates from the interplay between opposing factors that influence the breakage environment in a mill. Such factors include viscosity, density, and compaction of the powder. For dry grinding, the change in powder porosity and viscosity leads to a variety of nonlinear effects. For wet grinding the viscosity and density of the slurry impart a nonmonotonic variation in the breakage rate with particle size. A population balance equation (PBE) for environment dependent breakage has been formulated by considering the foregoing effects. Similarity analysis of the model equation shows the existence of self-similar behavior even for the environment dependent breakage. Experimental investigation of such self-similarity provides an effective diagnostic process for validating model framework. Furthermore, when combined with the methodology of solving inverse problems (e.g., Sathyagal, A. N.; Ramkrishna, D. Chem. Eng. Sci. 1996, 51, 1377), an effective approach becomes available for model identification.