Simultaneous reconstruction of space-dependent heat transfer coefficients and initial temperature

Many complex physical phenomena and engineering systems, e.g., in heat exchanges, reflux condensers, combustion chambers, nuclear vessels, etc., due to the high temperatures/high pressures hostile environment involved, possess certain properties which are inaccessible to measure and therefore their influence/determination using inverse analysis is very important and desirable. In this spirit, the purpose of this paper is to mathematically formulate and analyse a new inverse problem in which given measurements of temperature at two different instants, it is required to obtain the space-dependent heat transfer coefficients (HTCs) and the initial temperature. This simultaneous identification is challenging since it is both nonlinear and ill-posed. The uniqueness of solution is established based on the max–min principle for parabolic equations and the contraction mapping principle for the existence and uniqueness of a fixed point. The novel inverse mathematical model that is proposed offers appropriate scientific guidance to the polymer/heat transfer processing industry as to which data to measure/provide in order to be able to reliably determine the desirable HTCs along with the initial temperature, which is in general unknown. Furthermore, for the reconstruction, the surface HTC is determined separately, whilst the variational formulation is introduced for the simultaneous determination of the domain HTC and the initial temperature. The Fréchet gradient of the minimizing objective functional is derived. The numerical reconstruction process is based on the conjugate gradient method (CGM) regularized by the discrepancy principle. Accurate and stable numerical solutions are obtained even in the presence of noise in the input temperature data. Since noisy data are invented, the study models realistic practical situations in which temperature measurements recorded using sensors or thermocouples are inherently contaminated with random noise.