Least-squares adjustment of mathematical model of heat and mass transfer processes during solidification of binary alloys

Classical boundary and initial-boundary value problems in heat and mass transfer are generally formulated in a mathematically unique way. Boundary and initial conditions together with physical properties of the thermodynamic system are treated as exactly known. The influence of different kinds of mathematical model simplifications on the accuracy of solution and reliability of the model are not usually analyzed. The problems become more complicated when inverse ill-posed initial-boundary problems are considered. The widely used procedure of model validation is based on direct comparison of analytical or numerical solution, unique in a mathematical sense, with measurement results. The main feature of the method presented in this article is that all experimental results are included into the mathematical model. Thus, because of the inevitable errors of measurements, the system of model equations becomes internally contradicted as the number of unknown variables is less than the number of equations. In consequence, basic laws of energy and mass conservation are not satisfied. To adjust the experimental data to the mathematical model, an orthogonal least-squares method is proposed. Special attention has been paid to the coupling of experimental data with the nucleation and grain growth models formulated by Rappaz and co-workers. Theoretical considerations are illustrated with experimental data for an Al-Si alloy.