# Experimental Flow Performance Investigation of Francis Turbines from Model to Prototype
## 3 Experimental Results and Discussions  
### Performance Observation through Hill Chart and Torque  
The torque output of a [PERSON] turbine is influenced by various factors, including the flow rate \(Q\) (m\({}^{3}\)/s), head \(h\) (m), blade geometry (inlet angle \(\beta_{1}\) and outlet angle \(\beta_{2}\)), rotational speed \(\omega\), tangential velocities (\(V_{t1}\) and \(V_{t2}\)), blade inlet radius \(r_{1}\), blade outlet radius \(r_{2}\), and absolute velocity \(\Delta V\). Blade design plays a crucial role in optimizing turbine performance and efficient torque generation. Figure 4 illustrates the velocity triangles at the inlet and outlet of the blades and the angles required to calculate the torque for the model [PERSON] turbine [28]. Equation (11) is used to determine the torque for different load conditions. \[T_{(N\times m)}=\Delta\;V.\frac{r_{2}.\tan(\;\beta\;2)-r_{1}.\tan(\;\beta\;1)} {\cos(\;\beta\;2)-\cos(\;\beta\;1)} \tag{11}\]Figure 5a illustrates the hydraulic efficiency and output torque across various operating loads, with different guide vane opening angles at the design head. At a guide vane opening angle of 3.06\({}^{\circ}\) and 50% load, the Francis model exhibited the minimum torque, with a unit speed (\(n_{11}\)) of 54.51 rpm and a unit discharge (\(Q_{11}\)) of 140.38 L/s. As the load increased and the guide vane angle changed, the torque also increased, reaching the optimal condition. At partial load conditions from 70% to 80%, the efficiency improved slightly from 92% to 93.10% as the torque increased. These variations were attributed to enhanced flow direction through the guide vanes and the absolute velocity at the inlet angle of the blades. The model Francis turbine operated at 100% load with a maximum guide vane opening of 8.72\({}^{\circ}\) and a flow rate of 392.48 L/s, achieving a unit speed (\(n_{11}\)) of 64.26 rpm. Under 100% load, the turbine attained maximum torque and the best efficiency of 93.66%. The ability of the turbine to operate effectively under varying load conditions was demonstrated by the increasing load, changing flow rates, and guide vane openings. Figure 4: Velocity triangle of a Francis turbine. Figure 5: _Cont._  
Figure 5b illustrates the estimated performance of the prototype Francis turbine and validates its design through model testing. The maximum efficiency, calculated at 95.03%, was achieved at 100% full load with a guide vane opening of 8.72\({}^{\circ}\) and a flow rate of 15.4 m\({}^{3}\)/s and 50% load; the efficiency was calculated to be 91.2% with a minimum guide vane opening of 3.06\({}^{\circ}\). Table 2 provides detailed information on other partial load conditions and the efficiency estimation from the model to the prototype Francis turbine. Figure 6 presents a hill chart demonstrating the design and performance validation of the model and prototype performance along with the determination of the best efficiency point (BEP) during model testing under various operating conditions of the turbine. The hill chart provides detailed insights into the variation of the turbine as the load increased from 50% to 60%. In this load range, both unit discharge (Q\({}_{11}\)) and unit speed (\(n_{11}\)) exhibited rapid increases and guide vane opening from 3.32\({}^{\circ}\) to 4.25\({}^{\circ}\), showcasing that the turbine's performance also rapidly increased. Within the 70% to 90% load range, there was a notable escalation in the flow rates factor (Q\({}_{11}\)) and the opening angle of the guide vanes (\(\alpha\)) as they approached their maximum opening angle, resulting in an efficiency of 92.5%. The various load conditions are commonly encountered within the turbine's operational range, making it crucial to analyze the correlation between guide vane opening, flow rates, speed, and head to achieve the BEP. At 100% load, the hill chart demonstrates that the model and prototype achieved their respective best efficiencies of 93.66% and 95.03%. The study conducted by [PERSON] et al. [29] investigated the efficiency of the Francis model turbine.