Mathematical Modeling of High-Energy Shaker Mill Process with Lumped Parameter Approach for One-Dimensional Oscillatory Ball Motion with Collisional Heat Generation

This study presents an advanced mathematical model for the high-energy shaker mill process, incorporating thermal interactions among the milling ball, shaker mill vial, and the air contained within. Unlike previous models focusing solely on the ball’s temperature, this research emphasizes the heat produced by impacts and the thermal exchange among all three components. Incorporating these thermal interactions allows the model to provide a more comprehensive depiction of the energy dynamics within the system, leading to more precise predictions of temperature changes. Utilizing a lumped parameter method, the study simplifies complex airflow dynamics and non-uniform temperature distributions in the milling system, enabling efficient numerical analysis. Hamilton’s equations are extended to include supplementary state variables that account for the internal energies of both the vial and the air, in addition to the thermomechanical state variables of the ball. High-energy milling techniques are essential in mechanochemical synthesis and various industrial applications, where the optimization of heat transfer and energy efficiency is crucial. Numerical simulations computed using the Bogacki–Shampine integration algorithm significantly align with experimental data, confirming the model’s accuracy. This comprehensive framework enhances understanding of heat transfer in one-dimensional ball motion, optimizing milling parameters for better performance. The mathematical model facilitates the computation of heat production due to collisions, based on operational parameters like shaking frequency and amplitude, thereby allowing for the anticipation of chemical reaction activation potential in mechanochemistry.