## 3 CFD Analysis of the Francis Turbine Unit
Table 3 lists the operating parameters of the load conditions analyzed based on field measurements, and the net head of these four conditions is 54 m. The CFD-simulated torque of the Francis turbine under the four loading conditions is relatively close to the measured results, and the deviations of the simulation from the measurement at 25%, 50%, 75%, and 100% loads are 1.6%, 1.3%, 1.2%, and 1.1%, respectively, which proves the reliability of the simulation. Figure 4 presents the normalized pressure distribution in the flow domain of the Francis turbine unit under four different load conditions. As shown in Figure 4, the water flow of the Francis turbine unit enters the spiral case with high pressure under different loads, and the stay vanes help direct the flow smoothly towards the guide vanes. The guide vanes can be adjusted to control the direction and flow rate of the water entering the runner under different load conditions. By changing the guide vane opening (GVO), the flow direction in the guide vane passage can be better aligned with the flow direction of the runner, to minimize the energy loss and improve the efficiency of the hydraulic turbine. The runner is the key rotating component of the Francis \begin{table}
\begin{tabular}{c c c} \hline \hline
**Fluid Region** & **Element Number \(\times\)10\({}^{6}\)** & **Element Type** \\ \hline Spiral case & 0.8 & Tetrahedral \\ Stay vane & 0.35 & Hexahedral \\ Guide vane & 4.4 & Tetrahedral \\ Crown and band chamber & 0.42 & Tetrahedral and hexahedral \\ Runner & 4.6 & Tetrahedral \\ Draft tube & 0.63 & Tetrahedral \\ Total & 11.2 & Tetrahedral and hexahedral \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the hydro-generator unit. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Load Condition** & **Power Output (\%)** & **Guide Vane Opening (\%)** & **Flow Discharge (\%)** \\ \hline
1 & 25 & 47 & 27 \\
2 & 50 & 66 & 64 \\
3 & 75 & 83 & 87 \\
4 & 100 & 100 & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The analyzed load cases of the prototype Francis turbine unit. turbine for energy conversion. As water flows drive the runner to rotate, the kinetic and potential energies of the water are converted into the mechanical energy of the rotating shaft system, and then the rotating generator cuts the magnetic lines and generates electricity. The draft tube is located at the outlet of the runner and ensures a smooth discharge of water downstream. It also helps to recover the kinetic energy of the water by gradually enlarging its cross-sectional area, reducing the flow velocity, and increasing the pressure of the outgoing water. In the global view, the pressure in the turbine flow passage decreases from the spiral case to the draft tube under four different working conditions, but the local pressure distribution of the Francis turbine unit under different loads is different (Figures 5-7). Figure 5 shows the normalized pressure distribution in the crown chamber of the Francis turbine under different loads. Under the 25% load, the GVO is small, the water flow mainly enters the runner flow channel, and the pressure gradient in the crown chamber changes significantly, with a high value at the outer side close to the guide vanes and a low value at the inner side close to the turbine shaft. With the increase in GVO and power, the flow in the crown chamber gradually develops, and the pressure difference from the outside to the inside decreases. Figures 6 and 7 demonstrate the normalized local pressure distribution of the crown chamber and around guide vanes for the Francis turbine runner under different loads. For the low-head Francis turbine runner, the diameter of the band is larger than the diameter of the crown. It can be seen that at 25% load, the pressure value at the junction between the leading edges of the blades and the band is higher than the pressure in other areas of the runner.