A rational function approximation to the effectiveness factor for multi-particle interactions in dense-phase dry milling

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. First, computer generated effectiveness factor was approximated by a rational function. Then, evolution of the particle size distribution during batch dry milling of quartz in a tumbling ball mill was fitted with various forms of the effectiveness factor in the context of linear and non-linear PBMs. Goodness-of-fit and statistical significance of the parameters estimated were evaluated to discriminate various forms of the effectiveness factor. The statistical analysis suggests that a rational function 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 allowed us to describe the dense-phase dry milling data accurately. 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.