Numerical formulation of relationship between optimized runner blade angle and specific speed in a francis turbine

Francis turbine is the most widely acceptable hydraulic machine due to its higher efficiency. The establishment of the link between the optimized blade angle and specific speed can provide a turbine model with increased efficiency. In this study, a numerical study was conducted to derive the correlation between runner blade angles and the specific speed ($N_S$). The blade angles at the inlet and outlet of the hub and shroud spans were selected as the design variables. For the numerical calculations, Reynolds-averaged Navier-Stokes equations were modelled with the Shear Stress Transport turbulence model to predict the hydraulic performances. The Grid Convergence Index technique was applied to ensure grid accuracy. Response surface methodology was used as an optimization technique to derive the optimal model with higher efficiency. Based on the optimized blade angles, the efficiencies are improved by 1.12 % and 1.42 % at $N_S=150$ and 270 respectively with a constant power output of 30 MW. For both $N_S$, the internal flow characteristics were improved with a reduction of 24 % in the circumferential velocity at $N_S=150$ and 43 % in the low-speed zone area at $N_S=270$. The correlation was established to formulate a basis for providing a runner's blade angle value close to the optimal point for a wider range of $N_S$ of the turbine.