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

Abstract

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 initialboundary 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.

An Application of Generalized Least Squares Method to an Analysis of Methane/Steam Reforming Process on a Ni/YSZ Catalyst

“…Their researches involved problems connected with theory of coordination of material and energy balance ( 6), ( 7), (8), heat transfer by convection and radiation and the prediction of temperature distribution ( 9), (10). Another concerned issues were investigations related to the carbon dioxide emissivity (11), solidification of binary alloys (12) and modeling of the autocatalytic reactions (13).…”

“…The covariance matrix C VB defined by [12] represents the hyperellipsoid of the (K+N)-dimensional normal distribution. The covariance ellipsoid is the hypersurface in the (K+N)-dimensional space and is always inside the (K+N)-dimensional cuboid defined by the central point P'(u * ,x * ) and the square roots of the diagonal elements of C VB which represent the lengths of the cuboid edges.…”

Efeito da nucleação de grãos nas previsões do modelo multifásico para a solidificação equiaxial.

“…Nos dois casos, nota-se que os resultados estão relativamente próximos. Dessa forma, verifica-se que os resultados apresentados pelo presente trabalho estão de acordo com os obtidos por Kolenda et al (1999). Kolenda et al (1999), através de curvas de resfriamento para a amostra II.…”

“…Os resultados estão bem próximos para a amostra II, porém observa-se uma discrepância significativa na amostra I. Entretanto, deve-se ressaltar que os dois modelos não são completamente iguais, principalmente no que se refere ao modelo de crescimento do envelope, definido pelas equações que determinam a velocidade de crescimento dos mesmos. No modelo deKolenda et al (1999), a velocidade de crescimento dos envelopes foi calculada por: 21 -Evolução do raio médio para dois tipos de amostra, comparando o presente modelo com o modelo deKolenda et al (1999).AsFiguras 5.22 e 5.23 mostram outras comparações entre os dois modelos, através de curvas de resfriamento obtidas para as amostras I e II respectivamente.…”

“…22 -Comparação entre o presente modelo e o modelo deKolenda et al (1999), através de curvas de resfriamento para a amostra I. 23 -Comparação entre o presente modelo e o modelo de…”