Estimation of a Thermal Conductivity in a Stationary Heat Transfer Problem with a Solid-Solid Interface

TL;DR

This paper presents a method to identify the unknown thermal conductivity of a material in a stationary heat transfer problem involving a solid-solid interface, using noisy flow measurements and analyzing the estimation error bounds.

Contribution

It introduces a novel inverse problem approach for estimating unknown material properties in a 1D heat transfer model with interface conditions, including error bounds and elasticity analysis.

Findings

Derived necessary and sufficient conditions for conductivity estimation.

Provided bounds for the estimation error.

Numerical examples demonstrate the method's effectiveness.

Abstract

An inverse problem for a stationary heat transfer process is studied for a totally isolated bar on its lateral surface, made up of two consecutive sections of different, isotropic and homogeneous materials, perfectly assembly, where one of the materials, that is unreachable and unknown, has to be identified. The length of the bar is assumed to be much greater that the diameter so that a 1D heat transfer process is considered. A constant temperature is assumed at the end of the unknown part of the rod while the other end is let free for convection. We propose a procedure to identify the unknown material of the bar based on a noisy flow measurement at the opposite end. Necessary and sufficient conditions are derived together with a bound for the estimation error. Moreover, elasticity analysis is performed to study the influence of the data in the conductivity estimation and numerical examples are included to illustrate the proposed ideas and show the estimation performance.