Determining the unknown cooling condition and contact heat transfer coefficient during solidification of alloys

A method is developed in this paper to calculate the unknown boundary cooling condition and contact heat transfer coefficient for solidification of alloys. In this method, an objective functional is constructed based on a least squares approach and the difference between measured and estimated temperatures. This functional is minimized to obtain the boundary cooling condition and the contact heat transfer coefficient. The estimated temperatures are calculated by solving the corresponding direct problem. Beck's future time method and a regularization method are used to stabilize the numerical algorithm. A moving finite element method is used to solve the direct problem, where the velocities of the solid/mush and mush/liquid interfaces are primary variables and both interfaces are tracked continuously. In the procedure of solving the direct alloy solidification problem, complete solute mixing with no mass diffusion is assumed and the relationship between the solid fraction and the temperature in the mushy region is known in advance. In the example problems, solidification of binary alloys is studied and Scheil's equation is used to calculate the solid fraction in the mushy region.