# Study on the Population Balance Dynamics Simulation of Grinding under Impact Crushing
## 3 Results and Discussion
#### 3.1.3 Calculation of the Selection Function  
Based on the batch-grinding test data, according to Section 2.2.5, the selection function \(S_{i}\) for each particle size of quartz with a feed particle size of \(3.35+2.36\) mm, \(-2.36+1.7\) mm, and \(-1.7+1.18\) mm is calculated. The relationship curve between the selection function and time is shown in Figure 2.  
According to Figure 2, the selection function of each particle size of quartz generally shows a downward trend with time. The selection function of the coarse particle size is higher than that of the fine particle size. This is because the larger the particle size, the greater the chance of contact with the steel ball medium and the higher the probability of being broken. In addition, the selection functions of the three feed particle sizes were significantly higher than those of the other product particle sizes. This is because after the feed particle size enters the mill, it is in full contact with the steel ball medium and has the highest probability of being crushed by impact, so the selection function is the highest. Except for the feed particle size, other product particle sizes need to increase with the grinding time before they can be further broken by steel balls, so their selection function is significantly lower than the feed particle size. It can also be seen from Figure 2 that the selection function exhibits a decline with the reduction in feed particle size. This is because the smaller the particle size, the greater the energy required to be broken. And the smaller the particle size, the less chance of contact with the steel ball, so the lower the probability of being broken.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Particle Size/mm**} & \multicolumn{10}{c}{**Nominal Particle Size/mm**} \\ \cline{2-13}  & **2.8118** & **2.003** & **1.4163** & **1.0015** & **0.7141** & **0.5050** & **0.3571** & **0.2522** & **0.1783** & **0.1261** & **0.0892** & **0.0630** & **0.0449** \\ \hline \(-3.35+2.36\) & 0.0001 & & & & & & & & & & & \\ \(-2.36+1.7\) & 0.0012 & 0.0000 & & & & & & & & & & \\ \(-1.7+1.18\) & 0.0096 & 0.0012 & 0.0001 & & & & & & & & & \\ \(-1.18+0.85\) & 0.0858 & 0.0096 & 0.0012 & 0.0000 & & & & & & & & \\ \(-0.85+0.60\) & 0.0010 & 0.0858 & 0.0096 & 0.0012 & 0.0002 & & & & & & & \\ \(-0.6+0.425\) & 0.0657 & 0.0010 & 0.0858 & 0.0096 & 0.0012 & 0.0001 & & & & & & \\ \(-0.425+0.3\) & 0.1063 & 0.0657 & 0.0010 & 0.0858 & 0.0096 & 0.0012 & 0.0000 & & & & & \\ \(-0.3+0.212\) & 0.1299 & 0.1063 & 0.0657 & 0.0010 & 0.0858 & 0.0096 & 0.0012 & 0.0000 & & & & \\ \(-0.212+0.15\) & 0.1202 & 0.1299 & 0.1063 & 0.0657 & 0.0010 & 0.0858 & 0.0096 & 0.0012 & 0.0002 & & & \\ \(-0.15+0.106\) & 0.1007 & 0.1202 & 0.1299 & 0.1063 & 0.0657 & 0.0010 & 0.0858 & 0.0096 & 0.0012 & 0.0000 & & & \\ \(-0.106+0.075\) & 0.0835 & 0.1007 & 0.1202 & 0.1299 & 0.1063 & 0.0657 & 0.0010 & 0.0858 & 0.0096 & 0.0012 & 0.0000 & & \\ \(-0.075+0.053\) & 0.0540 & 0.0835 & 0.1007 & 0.1202 & 0.1299 & 0.1063 & 0.0657 & 0.0010 & 0.0858 & 0.0096 & 0.0012 & 0.0001 & \\ \(-0.053+0.038\) & 0.0427 & 0.0540 & 0.0835 & 0.1007 & 0.1202 & 0.1299 & 0.1063 & 0.0657 & 0.0010 & 0.0858 & 0.0096 & 0.0012 & 0.0000 \\ \(-0.038+0.013\) & 0.0600 & 0.0427 & 0.0540 & 0.0835 & 0.1007 & 0.1202 & 0.1299 & 0.1063 & 0.0657 & 0.0010 & 0.0858 & 0.0096 & 0.0012 & 0.0000 \\ \hline \hline \end{tabular}
\end{table}
Table 6: The \(b_{ij}\) matrix of quartz under impact crushing.