# Study on the Population Balance Dynamics Simulation of Grinding under Impact Crushing
###### Abstract  
During the grinding process, the crushing of minerals mainly depends on the impact action of the grinding medium. Based on the JK drop weight test data of quartz and pyrrhotite and the research results of their impact crushing characteristic parameters, this paper calculates the specific crushing energy (\(E_{\alpha}\)) of mineral samples subjected to impact in a ball mill using the grinding medium motion theory and then calculates the cumulative particle size distribution under screening under any mesh size using the JK drop weight test method. On this basis, the breakage distribution function of mineral samples is calculated, and a selection function is obtained based on grinding experiments. Finally, using Matlab programming and function-fitting mathematical methods, as well as a particle size population balance dynamics simulation of grinding, the particle size distribution characteristics of the grinding products of the two mineral samples in the mill that are only subjected to impact action are calculated. The results show that the selection function of quartz and pyrrhotite decreases overall with the prolongation of the grinding time, and the selection function of the coarse particle size changes more significantly than that of the fine particle size. At the same time, the selection function decreases with the decrease in feed particle size, and the smaller the feed particle size, the lower the probability of impact crushing. The \(E_{\alpha}\) values of quartz and pyrrhotite at each particle level in the mill are different, and the degree of mineral crushing is closely related to the impact energy, feed particle size, and mineral properties. g 20  
physical and chemical attributes [11; 12]. Such complexity poses significant challenges to the efficiency of the grinding separation processes and impacts the economic and technical performance of the concentration plant [13; 14]. Operating conditions such as grinding concentration, grinding time, filling rate, and rotational speed can also significantly affect the grinding process [15; 16; 17]. Therefore, optimizing grinding operations, delving into grinding theory, enhancing the efficiency of grinding operations, and cutting down on grinding costs are pivotal in slashing production expenses and improving the efficiency of resource recycling and usage [18; 19; 20; 21]. Grinding behavior is a process in which numerous solid particles continuously rupture under the continuous or periodic external force of the grinding medium. The breakage results of solid particles are closely related to various factors such as the strength of solid particles to resist damage, manner and magnitude of external forces, direction of the application of forces, time of application, particle flow velocity, and particle size distribution of the particle groups. Therefore, grinding behavior and the grinding process are the most fundamental, critical, and intricate scientific challenges in formulating a comprehensive grinding theory. With the progress of modern scientific and technological research methods and means, the means and capabilities of grinding research have been strengthened, further enriching the analytical theory of grinding. The concept of population balance dynamics simulations of grinding is proposed based on statistical changes in terms of quantity particles in the grinding process. Many scholars have carried out a lot of work to improve models for different grinding environments [22; 23; 24; 25; 26; 27; 28; 29; 30]. [PERSON] devised a performance model for impact crushers, and their simulation outcomes align closely with experimental data, indicating that an increase in rotor speed leads to a finer product size distribution, assuming the feed rate and particle size remain constant [31]. [PERSON] conducted an overview of the various applications of the JK size-dependent breakage model [32].