Optimization the shape of the Francis hydraulic turbine

Department of Electrical Equipment Kazan National Research Technical University named after A.N. Tupolev, Kazan, Russia

Abstract

The article presents one of the possible methods for optimizing the blade shape of a Francis radial-axial hydraulic turbine. The method for optimizing the shape of the turbine blade is based on the criterion of the maximum mechanical moment developed by the turbine. The blade shape optimization operation is conventionally divided into two stages. At the first stage of optimization of the blade shape, the analytical expression of the moment developed by the turbine is presented in a Taylor series by variable parameters − the coordinates of the vertices of the characteristic polyhedron of the median surface of the turbine blade. Adding the boundary conditions in the formulation of the optimization problem in the form of equalities − the contact of the median surface of the turbine blade with the turbine hub and rim, as well as conditions in the form of inequalities − the concavity of the greater part the median surface allows to reduce the problem of optimizing the blade shape to a standard linear programming problem. It is proposed to carry out 50−60 similar operations with small steps in the variables − the coordinates of the vertices of the characteristic polyhedron. Thus, it is necessary to move into the zone of optimal values of the coordinates of the vertices of the characteristic polyhedron of the median surface of the blade. At the second stage, it is proposed to continue the search for the optimal values of the coordinates of the vertices of the characteristic polyhedron of the median surface of the blade, applying for this purpose one of the most effective algorithms of genetic optimization.