A Systematic Approach for Quantifying the Multi-Particle Interactions in Dry Milling: Rational Function Type Effectiveness Factor

Non-linear population balance models (PBMs) have recently been shown to be superior to their linear counterparts for quantifying the breakage kinetics in dense-phase dry milling processes, where mechanical multi-particle interactions are significant. At the particle ensemble scale, particles of different sizes interact causing a population dependence of the specific breakage rate function. Non-linear PBMs explicitly account for these multi-particle interactions through a population-dependent functional called the effectiveness factor. Due to the inherent difficulty and complexity of proposing and selecting an appropriate form of the effectiveness factor, a general and systematic approach is desired to assess breakage kinetics in an unbiased manner. In this study, toward addressing the need for a general form of the effectiveness factor, we propose a rational function approximation. Computer generated data on the effectiveness factor and evolution of the particle size distribution during batch dry milling of quartz in a tumbling ball mill were fitted. Goodness-of-fit and statistical significance of the parameters estimated were evaluated to discriminate various forms of the effectiveness factor. It is concluded that a rational function approximation may replace specific forms of the effectiveness factor for a less restrictive analysis of the multi-particle interactions. Additionally, fitting a non-linear PBM using the rational function approximation to the effectiveness factor was able to accurately describe the dense-phase dry milling data. Accordingly, this study enables experimenters to quantify the impact of the multi-particle interactions in a robust, systematic, and unbiased manner via the rational function approximation to the effectiveness factor within the context of the non-linear PBM.