# Study on the Population Balance Dynamics Simulation of Grinding under Impact Crushing
## 3 Results and Discussion
#### 3.1.4 Study on Simulation Results of the Population Balance Dynamics of Grinding for Quartz  
Based on the obtained breakage distribution function \(b_{ij}\) and selection function \(S_{i}\) of the quartz, combined with the Reid solution of the grinding overall balance equation [26], the yield of each product particle size generated by crushing the quartz under impact is calculated using MATLAB2018a programming. The variation in product yield with time for the quartz with the feed particle size of 3.35 + 2.36 mm is shown in Figure 3.  
It can be seen from Figure 3 that the quartz with the feed particle size of \(-3.35+2.36\) mm is crushed to products of various particle sizes after impact grinding. The yield of particle sizes of \(-2.36+1.7\) mm, \(-1.7+1.18\) mm, and \(-1.18+0.85\) mm was higher, with the highest  
Figure 3: The variation in product yield with time for quartz with the feed particle size of 3.35 + 2.36 mm: (**a**) +0.425 mm particle size, (**b**) \(-0.425\) mm particle size. Figure 2: The relationship curve between the \(S_{i}\) and time of quartz: (**a**) \(-3.35+2.36\) mm, (**b**) \(-2.36+1.7\) mm, (**c**) \(-1.7+1.18\) mm. yield of 23.09%, 15.5%, and 9.61%, respectively. The maximum yield observed for the remaining particle sizes is 4.25%. As depicted in Figure 3, when the ordinate range is large, the yield curve of the \(-0.85+0.6\) mm particle size is very smooth. Upon magnification of the ordinate scale, as illustrated in Figure 3b, the yield of each particle size at the 8 min mark remains below 2%, and the yield curve is smoother. This indicates a notably low rate of fine particle production under impact. Using the same method, we calculated the yield \(\mathit{Wi}(t)\) of the quartz with particle sizes of \(-2.36+1.7\) mm and \(-1.7+1.18\) mm broken into each particle size under impact action. The results are shown in Figures 4 and 5, respectively. It can be seen from Figure 4 that with the feed particle size of \(-2.36+1.7\) mm, the quartz is also crushed to products of various particle sizes after impact grinding. The highest yield of \(-1.7+1.18\) mm and \(-1.18+0.85\) mm particle sizes was 30.92% and 10.74%, respectively. The highest yield of other particle sizes is 5.86%. For the same product particle size, the yield of the \(-2.36+1.7\) mm feed is higher than that of the \(-3.35+2.36\) mm feed, and the generation rate is also faster. From the comparison between Figures 3b and 4b, it can be seen that the generation rate curve of the \(-0.425\) mm particle sizes generated by impact grinding with a feed particle size of \(-2.36+1.7\) mm has a significantly different trend compared to that of a feed particle size of \(-3.35+2.36\) mm. This is because when the feed particle size decreases, the impact energy of the quartz becomes larger than the crushing energy, so the rate of crushing to generate various particle sizes is faster. As can be seen from Figure 5, compared to Figures 3 and 4, the yield of each fine particle size broken from the quartz is the highest with the feed particle size of \(-1.7+1.18\) mm. Figure 4: The variation in product yield with time for quartz with the feed particle size of \(-2.36+1.7\) mm: (**a**) +0.425 mm particle size, (**b**) \(-0.425\) mm particle size. Figure 5: The variation in product yield with time for quartz with the feed particle size of \(-1.7+1.18\) mm: (**a**) +0.425 mm particle size, (**b**) \(-0.425\) mm particle size. The highest yields of the \(-1.18+0.85\) mm and \(-0.85+0.6\) mm particle sizes were 26.12% and 11.17%, respectively, while the highest yield of the other particle sizes were 9.14%. For the same product particle size, the yield of the \(-1.7+1.18\) mm feed is higher than that of the \(-2.36+1.7\) mm and \(-3.35+2.36\) mm feeds. From a comparison between Figures 3b, 4b and 5b, it can be seen that the generation rate curve of the \(-1.7+1.18\) mm feed particle sizes broken into \(-0.425\) mm particle sizes has the most significantly different trend.