Estimation of Non-homogeneous Thermal Conductivity using Fourier Heat Equation Considering Uncertainty and Error Propagation

Abstract

The present work develops an estimator for thermal conductivity using a simple experiment implemented in a simulated solid metallic bar. The bar is sectioned in a finite number of segments, lately called nodes, and a discretization of the Fourier heat equation is applied in each node to generate a timed-spaced model of the temperature behavior along the bar. Considering only one-dimensional heat flow, an algorithm based on the temperature measured in each node generates the calculus of the estimated thermal conductivity for every segment of the bar. The calculations of thermal conductivity depends on previous values, such as temperature measurements and adjacent segments thermal conductivity, leading to an error propagation. The analysis of uncertainty related to this values is used to establish a range of values for thermal conductivity estimation. Using the proposed technique allows to calculate thermal conductivity in real time and add to the results a uncertainty estimation for thermal conductivity, providing a more complete information about the measurement procedure. Knowing the uncertainty allows to indicate, in statistical terms, the dispersion of the actual values for thermal conductivity, since the values calculated may vary from real, a higher uncertainty implies a lest reliable calculation according to statistics.