# Study on the Population Balance Dynamics Simulation of Grinding under Impact Crushing
## 2 Materials and Methods
### Methods  
#### 2.2.1 JK Drop Weight Test [35]  
The JK drop weight testing equipment used in this article is a falling weight testing machine developed by the JK Mineral Research Center (JKMRC) of the University of Queensland, Brisbane, QLD, Australia. Quartz and pyrrhotite were screened into five particle sizes: \(-63\) + 53 mm, \(-45\) + 37.5 mm, \(-31.5\) + 26.5 mm, \(-22.4\) + 19 mm, and \(-16\) + 13.2 mm, with representative particles chosen from each category of particle size for testing. The number of particles corresponding to the five particle sizes is 30, 45, 90, 90, and 90. Then minerals were divided into three equal parts for each particle size, totaling 15 parts. According to the screening analysis of impact crushing products and the calculation method provided by JKMRC, the respective impact crushing parameters A and b for the two mineral types can be determined individually. Based on particle size analysis of quartz and pyrrhotite impact crushing products, particle size characteristic curves can be plotted. Using the Origin function to fit the regression analysis, the DoseResp particle size distribution model for mineral impact crushing products can be obtained, as shown in Equation (1) [36]. Therefore, the cumulative particle size distribution under sieve for any desired particle size can be calculated. \[y=A_{1}+\frac{A_{2}-A_{1}}{1+10^{(logx_{0}-x)\cdot p}} \tag{1}\]  
where \(x\) is the sieving particle size (mm), and \(y\) is the corresponding cumulative weight undersize percentage (%), \(A_{1}\) and \(A_{2}\) are the upper and lower asymptotes of the particle size characteristic curve (%), \(logx_{0}\) is the particle size at (\(A_{1}+A_{2}\))/2(mm), and \(p\) is the absolute value of the maximum slope on the particle size curve (%/mm). Utilizing the established DoseResp particle size distribution model, the cumulative particle size distribution under sieve for any specific mesh size from the products of \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Component** & **SiO2** & **Fe2O3** & **MgO** & **Al2O3** & **S** & **CaO** \\ \hline Content/\% & 99.15 & 0.49 & 0.12 & 0.081 & 0.065 & 0.062 \\ \hline
**Component** & **Mn** & **Cr** & **Ni** & **Cu** & **Zn** & **Others** \\ \hline Content/\% & 0.0059 & 0.0047 & 0.0038 & 0.0035 & 0.0022 & 0.0119 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Chemical components of quartz. \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Component** & **SiO2** & **CaO** & **Fe** & **Zn** & **S** & **Pb** \\ \hline Content/\% & 29.41 & 0.18 & 37.76 & 0.94 & 25.94 & 0.18 \\ \hline
**Component** & **Al2O3** & **As** & **K2O** & **Pb** & **Others** & \\ \hline Content/\% & 1.85 & 3.34 & 0.20 & 0.18 & 0.02 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Chemical components of pyrrhotite. 15 crushing tests on individual minerals can be calculated. Generally, a particle size within the crushed product that is smaller than one tenth of the original feed particle size is designated as the characteristic particle size. The cumulative particle size distribution under sieve corresponding to this characteristic particle size reflects the degree of crushing of the mineral and is labeled \(t_{10}\). Similarly, in the subsequent description, \(t_{xx}\) is used to represent the cumulative mass percentage of particles in the crushing product whose particle size is less than 1/xx of the feed particle size. Utilizing the 15 datasets of \(t_{10}\) values and \(E_{cs}\), a scatter diagram illustrating the correlation between \(t_{10}\) and \(E_{cs}\) can be constructed. Subsequent fitting analysis of this \(t_{10}\)-\(E_{cs}\) relationship allows for the determination of the mineral's impact crushing characteristic parameters A and b, as represented in Equation (2). \[t_{10}=A\Big{(}1-e^{-b-E_{cs}}\Big{)} \tag{2}\]  
The \(t_{10}\)-\(E_{cs}\) correlation was initially proposed by [PERSON] [37] and subsequently validated by [PERSON] et al.