Estimating a pressure dependent thermal conductivity coefficient with applications in food technology

Abstract

ABSTRACTIn this paper, we introduce a method to estimate a pressure dependent thermal conductivity coefficient arising in a heat diffusion model with applications in food technology. The strong smoothing effect of the corresponding direct problem renders the inverse problem under consideration severely ill-posed. Thus specially tailored methods are necessary in order to obtain a stable solution. Consequently, we model the uncertainty of the conductivity coefficient as a hierarchical Gaussian Markov random field (GMRF) restricted to uniqueness conditions for the solution of the inverse problem established in Fraguela et al. [A uniqueness result for the identification of a time-dependent diffusion coefficient. Inverse Probl. 2013;29(12):125009]. Furthermore, we propose a Single Variable Exchange Metropolis–Hastings algorithm (SVEMH) to sample the corresponding conditional probability distribution of the conductivity coefficient given observations of the temperature. Sensitivity analysis of the direct problem suggests that large integration times are necessary to identify the thermal conductivity coefficient. Numerical evidence indicates that a signal to noise ratio of roughly suffices to reliably retrieve the thermal conductivity coefficient.