A Bayesian approach for the estimation of the thermal diffusivity of aerodynamically levitated solid metals at high temperatures

Abstract

The flash method has been the standard technique for the measurement of thermal diffusivity since it was first developed by Parker and co-workers. Several modifications of the original method have been proposed in the literature, to deal with different physical situations and materials. In this work, we extend the front-face flash method for the simultaneous estimation of the thermal diffusivity and thermal conductivity of aerodynamically levitated spherical solid metal samples at high temperatures, with laser excitation and contactless temperature measurements taken with a pyrometer. The mathematical model associated to the physical problem is formulated as an axisymmetric heat conduction problem, with heat losses by radiation and convection. Since the measurements of a pyrometer are in fact radiative fluxes, the transfer function of the pyrometer is integrated in the mathematical model. A sensitivity analysis is performed and reveals that the thermal diffusivity and the thermal conductivity of the sample can be simultaneously estimated, provided that the other model parameters are known. The solution of the inverse problem is obtained with algorithms within the Bayesian framework of statistics, by using a heat conduction reduced model with linear boundary conditions, for which an analytical solution is developed for fast computations. The Bayesian Approximation Error Model is used to compensate for errors between a complete heat conduction model with nonlinear boundary conditions and a reduced heat conduction model with linear boundary conditions. Moreover, errors related to the inaccuracy and the uncertainties of all model parameters are accounted for in the inverse analysis. Simulated transient flux measurements are assumed available from a multi-spectral pyrometer and are used for the inverse problem solution.