Two methods for estimation of temperature-dependent thermal conductivity based on constant element approximation

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Two methods for the estimation of temperature-dependent thermal conductivity are developed. The concept of constant element approximation is introduced, which approximates the thermal conductivity dependence on temperature (k-T function) with a step function. A 1-D heat conduction process in a semi-infinite region is considered for the design of the two methods, since an analytical solution describing this process which utilizes the constant element approximation can be found in the literature. The problem concerning the computation of the analytical solution is first solved, and then the analytical solution is applied to develop the two methods. For method I, the surface flux and the movements of the isotherms are recorded. A group of implicit recurrence formulas are established, and the thermal conductivity for each constant element can be determined sequentially in a non-iterative way. For method II, time-varying temperatures at two depths are measured. The thermal conductivities for the constant elements are determined through an optimization process using the Levenberg-Marquardt method. The application of the analytical solution greatly reduces the computational effort spent on the solution process of the inverse problem. Computational examples are presented. The two methods are applied to estimate five types of temperature-dependent thermal conductivities, and the accuracies of the estimated results are discussed. The two methods are proven to be applicable for arbitrary types of k-T function, and a prior knowledge concerning the form of the k-T function is not necessary.