Incorporating engineering intuition for parameter estimation in thermal sciences

Abstract

This paper proposes a new method of incorporating priors based on engineering intuition for solving inverse problems. The thesis of this paper is that if an asymptote can be found to a problem in applied sciences or engineering, estimation of parameters can be first done for this asymptotic variant, which in principle should be simpler, since one or more parameters of the original problem may vanish for the asymptotic variant. Even so, by solving the inverse problem associated with the asymptotic variant, estimates of key parameters of the full problem can be obtained. This information can then be quantitatively incorporated as priors in the estimation of parameters for the full version of the problem which we call as prior generation through asymptotic variant. The goal is to see if this methodology will significantly reduce the uncertainties in the resulting estimates. To demonstrate this methodology, the classic problem of unsteady heat transfer from a one dimensional fin is chosen. The inverse problem is posed as the simultaneous estimation of the temperature dependent transfer coefficient (h θ ) and the thermal diffusivity (α) of the fin material, given experimentally measured temperature–time histories at various locations along the fin. The asymptotic variant θ (x,t) is the steady state problem where the influence of thermal diffusivity vanishes. Using surrogate temperature data generated from assumed values of h θ , first the asymptotic variant of the problem is solved using the Markov Chain Monte Carlo method in a Bayesian framework to generate an estimate of h θ . The estimate of h θ is then used as an informative prior for solving the inverse problem of determining h θ and α from θ (x,t), and the effect of prior is quantitatively assessed by performing estimation with and without the prior. Finally, for purposes of validation, in-house experiments have been done where θ (x,t) is generated using liquid crystal thermography and these data are used to estimate h θ and α. A comparison of experimentally measured temperatures with those that are simulated by using estimated values of (h θ , α) to solve the governing equation to the problem is also done.