## 3 Results and Discussion
### Analysis of the Orthogonal Experiment Results
#### 3.1.1 Range Analysis of the Erosion Depth  
Table 4 shows the range analysis of the erosion depth. In the table, \(K_{ij}\) is the sum of the test values corresponding to the \(j\) level of the \(i\) factor, and \(\overline{K_{ij}}\) is the average value of \(K_{ij}\). An example is as follows,  
\[K_{V1}\ =\ 1.444\ +\ 1.130\ +\ 0.730\ =\ 3.304\]  
\[\overline{K_{V1}}=\ \frac{K_{V1}}{3}\ =\ 1.101\]  
\(R_{i}\) is the range, which reflects the variation amplitude of the evaluation index. The greater the variation amplitude, the more significant the influence of the factors on the evaluation index. The value of \(R_{i}\) is calculated by subtracting the minimum \(\overline{K_{i}}\) from the maximum \(\overline{K_{i}}\), with the following expression,  
\[R_{i}=\overline{K_{imax}}-\overline{K_{imin}} \tag{3}\]  
For example, the range of the jet velocity (Factor A) is calculated as follows,  
\[R_{V}=\overline{K_{Vmax}}-\overline{K_{Vmin}}\ =\ 3.524\]  
As shown in Table 4, \(R_{V}>R_{L}>R_{D}\). Therefore, as for the erosion depth, the jet velocity is the most significant factor, followed by the standoff distance and then the nozzle diameter. Meanwhile, the erosion depth of the seventh test in Table 3 is the largest at 4.976 cm. Therefore, the work parameter combination of the seventh test, namely, \(V3L1D3\), is optimal, and the corresponding jet velocity, standoff distance and nozzle diameter are 150 m\(\cdot\)s\({}^{-1}\), 0.5 cm and 2.5 mm, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Levels**} & \multicolumn{3}{c}{**Factors**} \\ \cline{2-4}  & \(V\) **(m/s)** & \(L\) **(cm)** & \(D\) **(mm)** \\ \hline
1 & 50 & 0.5 & 1.5 \\
2 & 100 & 1.0 & 2.0 \\
3 & 150 & 1.5 & 2.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Factors and levels of the orthogonal experimental design. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Test Number**} & \multicolumn{3}{c}{**Factors**} & \multicolumn{3}{c}{**Evaluation Indexes**} \\ \cline{2-5}  & \(V\) **(m/s)** & \(L\) **(cm)** & \(D\) **(mm)** & **Empty** & **Erosion** & **Erosion** \\  & & & & & **Depth, (cm)** & **Volume, (cm\({}^{3}\))** \\ \hline
1 & 1 & 1 & 1 & 1 & 1.444 & 0.35949 \\
2 & 1 & 2 & 2 & 2 & 1.130 & 0.36834 \\
3 & 1 & 3 & 3 & 3 & 0.730 & 0.35621 \\
4 & 2 & 1 & 2 & 3 & 3.315 & 1.88403 \\
5 & 2 & 2 & 3 & 1 & 2.892 & 2.98578 \\
6 & 2 & 3 & 1 & 2 & 2.480 & 0.84000 \\
7 & 3 & 1 & 3 & 2 & 4.976 & 6.03290 \\
8 & 3 & 2 & 1 & 3 & 4.656 & 2.20350 \\
9 & 3 & 3 & 2 & 1 & 4.244 & 3.03904 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of the orthogonal experiment. To verify the appropriateness of the selected factors and levels for the design of the orthogonal experiment and to check whether a better design scheme exists, the factors and levels of the orthogonal experiment were used as the abscissa. Meanwhile, \(\overline{K_{ij}}\) was used as the ordinate to draw a trend chart of the erosion depth variation of HBS. Figure 5 shows the trend chart. From Figure 5, the erosion depth of HBS reaches the maximum when the jet velocity, standoff distance, and nozzle diameter are 150 m\(\cdot\)s\({}^{-1}\), 0.5 cm and 2.0 mm, respectively, and the corresponding work parameter combination is \(V3L1D2\). However, the parameter combination of \(V3L1D2\) did not exist in the design of the orthogonal experiment. Thus, additional numerical simulation is needed to determine the erosion depth of the \(V3L1D2\). Through numerical computation, the erosion depth of 4.998 cm is obtained, which is larger than that of 4.976 cm in the seventh tests of Table 3. As a consequence, it can be concluded that, with respects to erosion depth, the \(V3L1D2\) is the optimal work parameter combination, which is better than the work parameter combination of \(V3L1D3\). Additionally, Figure 5 shows three main points. First, the erosion depth of HBS increases rapidly as the jet velocity increases. Second, an increase in standoff distance corresponds to a decrease in erosion depth.