## 3 Results and Discussion
### Influence of Cavitation on Flow Characteristics  
A comparative analysis of four different numbers of teeth is conducted to analyze the influence of the number of teeth flow characteristics and performance under operating conditions with an inlet and outlet pressure difference of 5 MPa. These graphs, shown in Figure 8, depict in detail the influence of cavitation on the flow characteristics for gear pumps with six to nine teeth over a cycle. Figure 8 shows the curves of the outlet flow Q under cavitated and non-cavitated conditions for gear pumps with six to nine teeth at a speed of 10,000 RPM. A curve of the gear pump after stable operation is selected for analysis, and the results show that the outlet flow presents obvious periodicity, while the influence of cavitation on the pump outlet flow varies significantly with the number of teeth. Among them, the smallest reduction in flow due to cavitation is achieved with the nine-teeth gear pump, only 0.165 L/min, while the outlet flow of the six-teeth, seven-teeth, and eight-teeth gear pump is reduced by 0.992 L/min, 0.369 L/min, and 0.245 L/min, respectively. It can be seen that as the number of teeth increases, the output flow of the \begin{table}
\begin{tabular}{c c c c c c} \hline
**Gas Volume Fraction** & **7000 RPM** & **8000 RPM** & **9000 RPM** & **10,000 RPM** & **11,000 RPM** \\ \hline
6 & 1.888\% & 2.484\% & 3.311\% & 8.541\% & \\
7 & 1.886\% & 2.443\% & 3.117\% & 5.036\% & \\
8 & 1.884\% & 2.425\% & 3.033\% & 3.853\% & 8.394\% \\
9 & 1.935\% & 2.485\% & 3.123\% & 4.014\% & 6.665\% \\ Average value & 1.898\% & 2.459\% & 3.146\% & & \\ \hline \end{tabular}
\end{table} 
Table 2: Gas volume fraction table corresponding to different speeds. Figure 8: Influence of cavitation on outlet flow. gear pump is affected by cavitation to a decreasing degree. The average outlet flow of six- to nine-teeth gear pumps is 11.18 L/min, 10.66 L/min, 9.89 L/min, and 9.31 L/min, respectively, and the volumetric efficiencies are 82.69%, 87.09%, 88.55%, and 90.73%. Figure 9 shows the relationship between the outlet flows as well as the volumetric efficiency and speed for gear pumps with six to nine teeth, with each data point corresponding to the outlet flow and volumetric efficiency at that speed. According to the flow calculation formula for external gear pumps, it can be concluded that there is a positive relationship between the pump outlet flow and speed under fixed displacement. This means that as the speed rises, so does the output. However, this linear relationship is not unlimited and fails once the speed exceeds a certain threshold. This is because too high a speed may result in the gear pump not being able to efficiently draw in the required oil and ensure that the tooth cavities are filled with oil, which can cause problems such as cavitation, vibration, and noise, as well as reduce the pump's volumetric efficiency; this can impair the gear pump's normal operation. Therefore, when the speed increases to the critical point where the volumetric efficiency begins to decrease, the gear pump is considered to have reached its critical speed. From Figure 9, the gear pump outlet flow shows an increasing distribution with the increase in speed, while the volumetric efficiency shows an increasing and then decreasing trend with the speed. The speed is increased from 7000 RPM to 11,000 RPM; the export flow curve of the six-teeth gear pump changes most significantly; the slope of the flow curve slowly slows down, corresponding to the volumetric efficiency of 81.57%, 83.69%, 85.41%, 82.69%, and 78.02%, in order. At 9000 RPM, the volumetric efficiency reaches its highest  
Figure 9: Relationship between outlet flow and volumetric efficiency and speed under different numbers of teeth. (**a**) Z = 6; (**b**) Z = 7; (**c**) Z = 8; (**d**) Z = 9. The left y-axis represents the export flow and the right y-axis represents the volumetric efficiency.