An Inverse Method of Designing the Cooling Passages of Turbine Blades Based on the Heat Adjoint Equation

A method of solution of the inverse problem in heat conduction is presented. The method, based on an adjoint optimization procedure, is applied to the design of the pattern of cooling passages inside turbine blades. For blade coating technologies, the general case of a non-homogeneous solid material is considered. The numerical solution of both the temperature field and of the adjoint problem is based on a finite element method. The new formulation of the adjoint thermal problem is deduced for three different parametric representation of the internal cooling passages. This allows the designer to select the most adequate blade parametrization, going from blades with circular coolant passages to modern multi-holed hollow blades. The mathematical method, the adjoint problem solution and the enforcement of geometric constraints are explained and the procedure is validated against theoretical, experimental data and numerical solution available in open literature.