An application of the generalized least squares method to the analysis of the heat transfer process with supplementary data

Most theoretical approaches for analysing heat transfer processes yield a unique solution from a specified set of governing equations, boundary and initial conditions and thermophysical properties of fluids and materials. This deterministic approach does not include uncertainties connected with the inaccuracy of directly measured variables and model simplifications. This study presents the idea of evaluating the most probable value obtained in a theoretical solution and a measure of its uncertainty. The proposed methodology - the Generalized Least Squares (GLS) method - allows for including additional data, which are commonly used for validation purposes, in the mathematical model as the supplementary variables. Theoretical considerations are then illustrated by applying the proposed methodology to the steady-state heat conduction process. On the basis of a formal mathematical model with the implementation of the GLS method, the computer program was prepared and applied to an analysis of several different cases, which demonstrate that the GLS methodology can be adopted for both: the process of planning experiments and the analysis of overdetermined problems. The advantages of the proposed method ensure obtaining optimal solutions to the problems of finding the proper position of the probe in the experiment design process, the determination of the empirical parameters and calculating the temperature distribution. The presented results proved that the proposed method is useful in verifying the incorrectly defined models as well as in identifying faulty measurement devices. The analysis pointed out that the experimental inaccuracy can be reduced and the most probable values of all unknown variables can be calculated.