## 6 Conclusions  
In this paper, the characteristics of the internal flow field of the vortex pump are deeply studied by the numerical simulation method and compared with the experimental data to verify the accuracy of the simulation results. On this basis, the impeller structure of the vortex pump is optimized using the orthogonal design method, and the overall performance of the optimized vortex pump has been significantly improved. In summary, the following conclusions can be formulated:  
1. The internal fluid velocity of the vortex pump is closely related to the speed of the pump. As the rotational speed increases, the fluid velocity gradually increases, especially in the outer edge area of the impeller. At the same time, the fluid pressure also increases, which promotes the formation of higher pressure at the outlet of the pump, thereby achieving high head. 2. The flow field structure internal to the vortex pump is extremely complex, and the fluid generates longitudinal vortices in the blade root area, which are subsequently transformed into transverse vortices. The fluid in the flow channel forms circular longitudinal vortices, and the continuous formation and shedding of these vortices provide continuous energy output for the vortex pump, which improves the performance of the pump. 3. The system performance of the vortex pump is optimized by applying the orthogonal design method. The range analysis method further determines the primary factors affecting pump performance, which are the number of blades, impeller width, impeller diameter, and blade groove radius. Through the optimization, the optimal combination of impeller structural parameters is determined as impeller diameter \(D_{2}\) of 45.7 mm, blade groove radius \(r\) of 14.9 mm, blade number \(Z\) of 32, and impeller \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Index** & **Before Optimization** & **After Optimization** & **Improvement Rate** \\ \hline \(H\) & 7.64 m & 9.09 m & 18.90\% \\ \hline \(\eta\) & 11.79\% & 13.16\% & 11.60\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Performance changes of the vortex pump before and after optimization. Figure 14: Comparison results before and after optimization of vortex pump: (**a**) head of vortex pump and (**b**) efficiency of vortex pump. width \(b\) of 5.15 mm. The optimization of these parameters significantly improved the overall performance of the vortex pump. Optimizing the performance of vortex pumps can significantly improve energy efficiency and reliability, extend service life, and reduce operating costs. In industrial applications, it can better meet the needs for small flow and high head in the chemical, aviation, marine, fire protection, agricultural irrigation, and building water supply industries, providing more efficient and reliable fluid transport solutions to enhance overall system performance. It should be emphasized that in this paper, the interactions between the influencing factors are ignored during the orthogonal design process. Therefore, this needs to be studied in depth in future studies. Conceptualization, J.Y. and D.C.; methodology, J.Y., X.L. and D.C.; software, J.Y. ; validation, J.Y. and X.L. ; formal analysis, J.Y. and X.L. ; investigation, J.Y., X.L., D.C. and J.J.; resources, J.Y. ; data curation, J.Y. ; writing--original draft preparation, J.Y. and D.C.; writing--review and editing, J.Y. and D.C.; visualization, J.Y. and D.C.; supervision, X.L. and M.Z. ; project administration, X.L., W.G. and L.H. ; funding acquisition, X.L., W.G. and L.H. All authors have read and agreed to the published version of the manuscript. This research was funded by the Natural Science Basic Research Project of Shaanxi Province, China (No. S2024-JC-YB-0014) and the Hydraulic Technology Key Laboratory of Shaanxi Province Open Fund Research Project (No. YYJS2022 KF09 and No. YYJS2022 KF11). Not applicable. The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.