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 tra nsfer 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 ma thematical model simplifications on the accuracy of solution and reliabilit y of the model are not usually analyzed. The problems become more complicat ed when inverse ill-posed initial-boundary problems are considered. The wid ely used procedure of model validation is based on direct comparison of ana lytical or numerical solution, unique in a mathematical sense, with measure ment results. The main feature of the method presented in this article is t hat all experimental results are included into the mathematical model. Thus , because of the inevitable errors of measurements, the system of model equ ations becomes internally contradicted as the number of unknown variables i s less than the number of equations. In consequence, basic laws of energy a nd mass conservation are not satisfied. To adjust the experimental data to the mathematical model, an orthogonal least-squares method is proposed. Spe cial attention has been paid to the coupling of experimental data with the nucleation and grain growth models formulated by Rappaz and co-workers. The oretical considerations are illustrated with experimental data for an Al-Si alloy.