# Flow-Induced Strength Analysis of Large Francis Turbine Under Extended Load Range
## 4 Results and Discussions of Flow-Induced Strength Analysis
### Flow-Induced Strength Analysis of the Francis Runner  
In order to evaluate the strength of the Francis runner under three different operating conditions, the flow-induced strength analysis of the runner needs to be performed by combining the above CFD simulations with finite element analysis (FEA). The data of the pressure distributions on the runner at 33.3%, 66.6%, and 100% loads of rated power can be exported from the CFD analysis performed in the previous section and mapped on the finite element of the Francis runner as surface loads. High-quality tetrahedral elements are used to create the Francis runner's finite element model (Figure 12). The Francis turbine runner blades' geometrical curvature varies significantly, particularly at the fillets, which are likely to become locations of concentrated stress. To accurately record the stress values, the mesh at the runner blades' fillets is improved and refined. The Francis runner is subjected to the boundary conditions, which include the fixed restrictions, rotating speed, and gravity acceleration. Figure 12: Finite element model of the Francis turbine for the flow-induced stress analysis. The Francis runner in this study is made of stainless steel, and the material parameters of the runner are summarized in Table 4 and include a density of 7700 kg m\({}^{-3}\), a Young's modulus of 2.1 \(\times\) 10\({}^{5}\) MPa, a Poisson's ratio of 0.3, and a yield stress of 550 MPa. This value of yield stress ensures that the Francis runner can withstand operational loads under normal and transient conditions without permanent deformation. The pressure files across the entire flow passage of the Francis turbine unit at three different loads can be exported from the CFD analysis discussed in the previous section. This CFD-derived pressure data can then be accurately mapped onto the nodes of the finite element model to facilitate the subsequent flow-induced structural stress analysis (Figure 13). As shown in Figure 14, the maximum stresses of the runner at 33.3%, 66.6%, and 100% loads of rated power are located in the fillets between the blade trailing edge and the crown. When the Francis runner is running stably at a rated speed, its load is determined by the hydraulic torque acting on the runner. As the load of the Francis unit increases, the opening of the unit's guide vanes increases accordingly; therefore, the water flow entering the runner flow passage and the runner hydraulic torque increases. Eventually, the value of the runner hydraulic torque is equal to the value of the counter load torque on the turbine shaft so that a new torque equilibrium is formed in the Francis turbine unit. As shown in Figure 15, the maximum stress of the runner gradually increases with the power from 33.3% to 100% of the rated power load. As the hydraulic torque on each runner blade increases with the load, the flow-induced stress of the runner blades also increases accordingly. The maximum stress values of the runner at the 33.3% rated power load and the 66.6% rated power load are 0.45 and 0.74 times the stress at the 100% rated power load, which is around 1/3 of the yield stress of runner material. Therefore, the runner is safe during the extended operation range from 33.3% to 100% of the rated power load. As the service life of the Francis turbine increases, the performance of the materials used in the turbine runner may degrade, especially for materials that are susceptible to cavitation erosion or corrosion. To ensure the safe and efficient operation of the units, it is still recommended to strengthen the inspection of the runners to prevent problems before they occur. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline Density & 7700 & kg m\({}^{-3}\) \\ Young’s modulus & 2.1 \(\times\) 10\({}^{5}\) & MPa \\ Poisson’s ratio & 0.3 & - \\ Yield stress & 550 & MPa \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of the hydro-generator unit.