Least-squares adjustment of mathematical model of heat and mass transfer processes during solidification of binary alloy
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Motto: “Observations are useless until they have been interpreted. . . . The analysis of experimental data forms a critical stage in every scientific inquire—a stage which can be responsible for most of the failures and fallacies of the past.” (from E.B. Wilson; An Introduction to Scientific Research) I. INTRODUCTION
CONVENTIONAL methods of mathematical modeling of heat and mass transfer processes are commonly based on schemes leading to unique solutions. For example, solving initial-boundary value problems, the mathematical model consists of the governing equations and initial and boundary conditions. In the case of a linear form of the mathematical model, the uniqueness of the solution can be proved mathematically for different kinds of boundary conditions and the problems are well posed. In the classical approach, each variable or physical property (thermal conductivity, thermal emissivity, and diffusivity coefficient in mass transfer) and model parameters (heat transfer coefficient and mass transfer coefficient), describing the thermodynamic state of the system, are treated as exactly known (zero errors) and simplifications in the model are usually neglected. Simplifications can be of different kinds. For example, a linear system of differential equations is ZYGMUNT KOLENDA, Professor and Chairman, and JANUSZ DONIZAK, Adjunct Professor, are with the Department of Theoretical Metallurgy and Metallurgical Engineering, St. Staszic University of Mining and Metallurgy, 30059 Cracow, Poland. JOSE C. ESCOBEDO BOCARDO, Researcher, is with the Research and Advanced Studies Center, 25000 Saltillo, Coahuila, Mexico. Manuscript submitted October 31, 1997. METALLURGICAL AND MATERIALS TRANSACTIONS B
solved instead of a nonlinear system, the convective heat transfer mechanism is neglected in the liquid region (when solidification process is considered), or one- or two-dimensional problems are solved instead of three-dimensional problems. In such a case, the real state of the physical system will be different from the mathematical model solution mainly because of model simplifications, measurement errors of the directly measured variables (observations), and errors of evaluation of the values of physical properties and model parameters. Although the solution obtained in this way uniquely describes the behavior of the system, the degree of its accuracy is usually difficult to establish. If, however, the mathematical model contains more information (called here “supplementary data”) than necessary for a unique solution, it becomes possible to check its accuracy and, what is equally significant, to determine the influence of the measurement errors, model simplification, and errors of model parameters on the accuracy of the general model. Statements such as “The analytical results agree satisfactorily with the corresponding experimental data,” “the experimental results agree well with the theoretical results,” or “comparison with the data shows reasonable agreement between the present theory and experimental results” can often b
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