# Study on the Population Balance Dynamics Simulation of Grinding under Impact Crushing
## 3 Results and Discussion
### The Population Balance Dynamics Simulation of Grinding under Impact Crushing of PyrrIhotite
#### 3.2.2 Calculation of Breakage Distribution Function  
The calculation process of the breakage distribution function of pyrrhotite is the same as that in Section 3.1.2, and the calculation results are shown in Table 9.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(t_{\text{xx}}\) & **Fitting Equation** & \(R^{2}\) \\ \hline \(t_{1.4}\) & \(t_{1.4}=33.3753+7.12391_{10}-0.3162t_{10}{}^{2}+0.007t_{10}{}^{3}-\) & 0.9496 \\  & \(7.4178\times 10^{-5}t_{10}{}^{4}+3.0379\times 10^{-7}t_{10}{}^{5}\) & \\ \(t_{2.0}\) & \(t_{2.0}=15.8714+4.2643t_{10}-0.0713t_{10}{}^{2}+3.9037\times 10^{-4}t_{10}{}^{3}\) & 0.9771 \\ \(t_{2.8}\) & \(t_{2.8}=-0.7524+3.9874t_{10}-0.0546t_{10}{}^{2}+2.5608\times 10^{-4}t_{10}{}^{3}\) & 0.9939 \\ \(t_{3.9}\) & \(t_{3.9}=-3.8249+2.9503t_{10}-0.0283t_{10}{}^{2}+9.2259\times 10^{-5}t_{10}{}^{3}\) & 0.9982 \\ \(t_{5.6}\) & \(t_{5.6}=-2.5216+1.8573t_{10}-0.0082t_{10}{}^{2}-5.1016\times 10^{-6}t_{10}{}^{3}\) & 0.9995 \\ \(t_{7.9}\) & \(t_{7.9}=-1.2380\times 1.3254t_{10}-0.0028t_{10}{}^{2}\) & 0.9999 \\ \(t_{11.1}\) & \(t_{11.1}=0.5161+0.87991t_{10}+8.9764\times 10^{-4}t_{10}{}^{2}\) & 1 \\ \(t_{15.8}=0.997+0.6198t_{10}+0.0019t_{10}{}^{2}\) & 0.9948 \\ \(t_{22.3}=0.5455+0.6105t_{10}-0.0027t_{10}{}^{2}+4.3588\times 10^{-5}t_{10}{}^{3}\) & 0.9954 \\ \(t_{31.5}=2.8650+0.3002t_{10}+0.0027t_{10}{}^{2}\) & 0.9865 \\ \(t_{44.6}\) & \(t_{44.6}=2.6921+0.2635t_{10}+0.0017t_{10}{}^{2}\) & 0.9689 \\ \(t_{62.6}\) & \(t_{62.6}=2.8678+0.2178t_{10}+0.0016t_{10}{}^{2}\) & 0.9443 \\ \(t_{144.2}=0.2106+0.4458t_{10}-0.0064t_{10}{}^{2}+5.2529\times 10^{-5}t_{10}{}^{3}\) & 0.8515 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Functional relation of \(t_{10}\)-\(t_{xx}\) for pyrrhotite.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Particle** & **Nominal Particle** & **M/kg** & \(E_{\text{cs}}\)/kWh/t & \(t_{10}\)/\% \\ Size/mm & **Size/mm** & **M/kg** & \(E_{\text{cs}}\)/kWh/t & \(t_{10}\)/\% \\ \hline \(-3.35+2.36\) & 2.8118 & \(5.3517\times 10^{-5}\) & 0.8062 & 55.2683 \\ \(-2.36+1.7\) & 2.003 & \(1.9346\times 10^{-5}\) & 2.2302 & 74.3688 \\ \(-1.7+1.18\) & 1.4163 & \(6.8392\times 10^{-6}\) & 6.3085 & 76.5951 \\ \(-1.18+0.85\) & 1.0015 & \(2.4182\times 10^{-6}\) & 17.8417 & 76.5986 \\ \(-0.85+0.60\) & 0.7141 & \(8.7662\times 10^{-7}\) & 49.2166 & 76.5986 \\ \(-0.6+0.425\) & 0.5050 & \(3.1003\times 10^{-7}\) & 139.1602 & 76.5986 \\ \(-0.425+0.3\) & 0.3571 & \(1.0962\times 10^{-7}\) & 393.5678 & 76.5986 \\ \(-0.3+0.212\) & 0.2522 & \(3.8616\times 10^{-8}\) & \(1.1173\times 10^{3}\) & 76.5986 \\ \(-0.212+0.15\) & 0.1783 & \(1.3645\times 10^{-8}\) & \(3.1618\times 10^{3}\) & 76.5986 \\ \(-0.15+0.106\) & 0.1261 & \(4.8271\times 10^{-9}\) & \(8.9381\times 10^{3}\) & 76.5986 \\ \(-0.106+0.075\) & 0.0892 & \(1.7086\times 10^{-9}\) & \(2.5252\times 10^{4}\) & 76.5986 \\ \(-0.075+0.053\) & 0.0630 & \(6.0195\times 10^{-10}\) & \(7.1675\times 10^{4}\) & 76.5986 \\ \(-0.053+0.038\) & 0.0449 & \(2.1791\times 10^{-10}\) & \(1.9799\times 10^{5}\) & 76.5986 \\ \(-0.038+0.013\) & 0.0195 & \(1.7833\times 10^{-11}\) & \(2.4194\times 10^{6}\) & 76.5986 \\ \hline \hline \end{tabular}
\end{table}
Table 8: The calculated results of the \(E_{\text{cs}}\) and \(t_{10}\) values for each particle size of pyrrhotite.