An Analytical quasi 2D steady-state Francis turbine model using first principles

Abstract

Accurate models are required for obtaining accurate simulation results. High fidelity numerical models exist, but for the simulations of large hydropower conduit systems one-dimensional (1D) models are still required. For the operation and characteristics of hydraulic machinery the state variables in 1D pipe computation are not necessarily providing sufficient level of detail on what happens inside the machine to accurately predict component and system behaviour. This work is looking into the physics of the spatial distribution of the flow at the outlet of a Francis turbine runner in order to include 2D effects into a 1D analysis. The key finding is a differential equation describing the distribution of the relative velocity W between the flow and the runner outlet, which enables numerical integrating for finding the distribution of W itself, and subsequently the circumferential and meridional components needed to execute the integrals turning the 2D results into functions of the 1D state variables. The presented work is reflecting condition at the best efficiency point, but ongoing work is expanding this to map the entire region of a turbine runner. The approach should be very relevant for future implementations into digitalization schemes such as digital twins