On the Similarity of Austin Model and Kotake-Kanda Model and Implications for Tumbling Ball Mill Scale-Up

The aim of this theoretical investigation is to seek any similarities between the Austin model and the Kotake-Kanda (KK) model for the specific breakage rate function in the population balance model (PBM) used for tumbling ball milling and assess feasibility of the KK model for scale-up. For both models, the limiting behavior for small particle size-to-ball size ratio and the extremum behavior for a given ball size are described by "power-law." Motivated by this similarity, specific breakage rate data were generated using the Austin model parameters obtained from the lab -scale ball milling of coal and fitted by the KK model successfully. Then, using the Austin's scale-up methodology, the specific breakage rate was scaled-up numerically for various mill diameter scale-up ratios and ball sizes of 30-49 mm and coal particle sizes of 0.0106-30 mm. PBM simulations suggest that the KK model predicts identical evolution of the particle size distribution to that by the Austin model prior to scale-up. Upon scale-up, the differences are relatively small. Hence, modification of the exponents in the Austin's scale-up methodology is not warranted for scale-up with the KK model. Overall, this study has established the similarity of both models for simulation and scale-up.