# Calculation Method of the Blasting Throwing Energy and Its Variation Affected by the Burden
## 4 Discuss Test Results
### Casting Velocity Analysis
#### 4.2.2 Throwing Kinetic Energy Distribution Law  
For model tests, the blasting medium is artificially concreted plain concrete, regarded as a homogeneous medium regardless of density. Then, let \(\rho(h)=\rho\); the overall blasting much pile quality is determined by statistics after the explosion, taking the actual statistical result \(M\) as the benchmark. At the same time, it is assumed that the distribution is uniform in the y-direction; the vertical coordinate value y is used instead of the height along the step. The coordinates of the upper and lower masses are denoted by \(y_{\rm max}\) and \(y_{\rm min}\), respectively. Figure 12: Time history curve of the velocity of the center particle. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Number**} & \multirow{2}{*}{ \begin{tabular}{c} **Minimum** \\ **Burden** \\ **W/m** \\ \end{tabular}  } & \multicolumn{4}{c}{**Functional Relationship**} & \multicolumn{2}{c}{**Correlation Parameters**} \\  & & \(y_{0}\) & \(A\) & \(m_{1}\) & \(\mu\) & \(\mathbf{R^{2}}\) \\ \hline
1 & 0.12 & 4.60 & 0.33 & 0.03 & 0.03 & 0.91 \\
2 & 0.14 & 0.93 & 0.59 & 0.03 & 0.02 & 0.99 \\
3 & 0.16 & 3.08 & 0.20 & 0.03 & 0.02 & 0.98 \\
4 & 0. Then, the equation for calculating the kinetic energy of throwing can be transformed by Equation (20), shown as follows. \[E_{v}=\frac{M}{2(y_{max}-y_{min})}\int_{y_{min}}^{y_{max}}v^{2}(y)dy \tag{26}\]  
where \(v(y)\) is obtained by fitting the high-speed photography results of the step test using the numerical difference method for the 20 micro-elements along the step direction in the model test. The specific calculation equation of throwing kinetic energy can be transformed into:  
\[E_{v}=\frac{1}{2}\times\frac{M}{20}\sum_{j=1}^{j=20}\ v_{j}^{2} \tag{27}\]  
By substituting the functional relationship in Table 5 into Equation (27), each model's kinetic energy blasting throwing is calculated, as shown in Table 6.  
According to the specific calculation of the throwing energy in the model test, the energy used to throw the broken rock mass during the explosion shows a significant downward trend with the increasing burden. When the minimum burden is 0.12 m, the throwing energy consumption is 1306.88 J. When the minimum burden increases to 0.20 m, the energy consumption of rock-throwing is 747.49 J, which is reduced by 0.57 times when the minimum burden is 0.12 m. This shows that the change of the minimum burden greatly affects the energy consumed by throwing broken rocks. With the increase of the minimum burden, the breaking time and the breaking range of the rock explosive energy also increase. Therefore, more energy is used to break the rock. The kinetic energy of explosives used for throwing has a significant decrease in the proportion of explosive energy, as shown in Figure 13. By fitting the relationship between the minimum burden and the throwing energy consumption, we obtain the power function relationship between them. \[E_{v}=122.91W^{-1.10},R^{2}=0.95 \tag{28}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Number**} & **Minimum** & **Explusive Energy** & **Throwing Kinetics** & **Throwing Kinetics** \\  & **Burden W/m** & \(E_{t}\)/**J** & \(E_{v}\)/**J** & **Ratio \(\eta_{v}\)/\%** \\ \hline
1 & 0.12 & 8.848 & 1306.88 & 14.77 \\
2 & 0.14 & 8.848 & 1024.29 & 11.58 \\
3 & 0.16 & 8.848 & 985.11 & 11.13 \\
4 & 0.18 & 8.848 & 786.06 & 8.88 \\
5 & 0.20 & 8.848 & 747.49 & 8.45 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Model throwing kinetic energy. Figure 13: Relation of the thrown kinetic energy with the minimum burden. At the same time, the following equation is used to calculate the proportion of energy consumed by broken rock throwing. The calculation results are shown in Table 6.  
\[\eta_{\sigma}=E_{\sigma}/E_{t} \tag{29}\]  
In the model experiment, the proportion of explosive energy used to throw broken rocks is between 8% and 15%. When the charge quantity is the same as the detonated medium, the throwing kinetic energy proportion decreases as the minimum burden increases.