ESTIMATION OF SURFACE TEMPERATURES FROM INTERIOR MEASUREMENTS

Abstract

The main objective of the thesis is to estimate the surface temperature of a steel slab by solvinganinverseheatconductionproblem. Thisproblemarisesinapplications, forexample insteelindustry,whereitisofgreatimportancetobeabletocontrolthesurfacetemperature andheatingorcoolingratesduringheattreatmentprocessesinordertoachievegoodquality of the end products. However, in many industrial applications, the surface itself is inac- cessible for direct measurements or locating a measurement device such as a thermocouple on the surface would disturb the measurements so that an incorrect temperature measure- ment is recorded. In this situation, we are restricted to interior measurements, from which we approximate the surface temperature by solving an inverse heat conduction problem in the region between the surface and a measurement point, because this process is strongly influenced by the time dependent temperature and heat-flux close to the surface. In this thesis we formulate the problem as an operator equation Kf = g, where K is an operator that maps the surface temperature f(t) to the interior measured temperature data g(t), and we need to solve for the unknown surface temperature f(t). However, two main complications arise. Firstly, the operator K is non-linear while most efficient regularization methodsaredesignedforsolvinglinearoperatorequations. Secondly, duetorandomnoisein measurements, only noisy version gδ of the exact data g is available in practice, thus solving for f is an ill-posed problem in the sense that the solution does not depend continuously on the data. To address these issues, we present in this thesis an approach of rewriting K as a linear operator equation. Also, ill-posedness is investigated and we implement a regularization approach based on Tikhonov method. Finally, the developed method is applied to a real industrial problem with measured data taken during an industrial steel quenching process. Numerical experiments show that the method works well. We also consider improving the accuracy of the solution by including more measurements, and discuss how making use of the additional data may improve the estimate of the surface temperature as well as improving the stability of the inverse problem.