# Experimental Flow Performance Investigation of Francis Turbines from Model to Prototype
## 2 Methodology and Experimental Measurements Techniques
### Similarity Theory between the Francis Turbine Model and Prototype  
The model turbine used for experimental measurements was scaled to 1/17.2 of the prototype size. The model turbine for the present investigation included the following components: a spiral case, 16 stay vanes, 24 guide vanes, a model runner with 15 long \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Instruments** & **Purpose** & **Range** & **Uncertainty** \\ \hline BOI Measurement and Testing & Frequency & 0–10 MHz & \(\pm\)0.08 \\ Pressure Calibration & Pressure & 0.1–6 Mpa & \(\pm\)0.005 \\ Emerson Instrument & Temperature sensor & 0–100 \({}^{\circ}\)C & \(\pm\)0.04 \\ Emerson Instrument & Barometer & 0–600 KPa & \(\pm\)0.5\% \\ Load cell & Generator torque & 0–1600 N-m & \(\pm\)0.05 \\ Load cell & Friction torque & 0–22 N-m & \(\pm\)0.0025 \\ China Electronics CKCW36 & Inlet pressure & 0–400 KPa & \(\pm\)0.0019 \\ China Electronics CKCW36 & Outlet pressure & 0–400 KPa & \(\pm\)0.001 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Descriptions of the instruments used in the experiments. Figure 2: Pressure sensor installation in the Francis turbine model. blades and 15 splitter blades, and a draft tube. The model runner had an inlet diameter of 0.5318 m and an outlet diameter of 0.35143 m. The flow rate and the rotational speed for the model turbine were 352 liters per second and 740 rpm, respectively. Figure 3 illustrates the dimensions of the Francis turbine model. Figure 3 presents a comprehensive visualization of the major components of the Francis turbine, highlighting their importance in converting hydraulic energy to mechanical energy and optimizing turbine performance. The spiral case evenly distributes water flow to the runner blades by gradually reducing the cross-sectional area using dimensions \(L_{1}\), \(L_{2}\), and \(D_{1}\) to maintain a constant water velocity, crucial for maximizing energy conversion efficiency. The primary function of the stay vanes and guide vanes is to direct water flow onto the runner with minimal energy loss. The high H is indicative of their structural role in maintaining the shape and stability of the spiral case and water flow directions. The diagram depicts the runner's cross-sectional view, with dimensions \(D_{1}\), \(D_{2}\), and \(H_{1}\) representing the inlet diameter, outlet diameter, and height, respectively. These dimensions are critical for optimizing turbine efficiency and power output. The draft tube is a conduit that gradually expands to recover the kinetic energy of water and convert it into pressure energy. Dimensions \(D_{1}\), \(D_{2}\), \(H_{1}\), \(H_{2}\), \(H_{3}\), \(H_{4}\), \(L_{1}\), and \(L_{2}\) represent critical factors for minimizing energy loss and vortex formation. The experiment was conducted in accordance with international standards (IEC 60193) to ensure the stability and accuracy of measurements related to output, hydraulic efficiency, cavitation effects, and pressure pulsation behavior. The concepts of geometric and kinematic similarity, along with the fundamental dimensions and conversion parameters, are expressed in the following equations:  
\[\text{Rotation speed (rpm)}:\frac{n_{p}D_{p}}{\sqrt{H_{p}}}=\frac{n_{m}D_{m}}{ \sqrt{H_{m}}} \tag{4}\]  
\[\text{Flow rate (m}^{3}/\text{s}):\frac{Q_{p}}{D_{p}^{2}\sqrt{H_{p}}}=\frac{Q_ {m}}{D_{m}^{2}\sqrt{H_{m}}} \tag{5}\]  
\[\text{Power (W)}:\frac{P_{p}}{D_{p}^{3}\sqrt{H_{p}^{3}}}=\frac{P_{m}}{D_{m}^{3} \sqrt{H_{m}^{3}}} \tag{6}\]  
where \(n_{p}\), \(D_{p}\), \(Q_{p}\), \(H_{p}\), and \(P_{p}\) are the rotational speed, runner diameter, flow rates, proto-type head, and power output for the prototype turbine, respectively. Figure 3: Dimensions and design of the Francis turbine model.