A Numerical Method for Solving the Inverse Problem of Heat Transfer and Its Application to Determination of the Thermal Conductivity
The usual practice in the measurement of the thermal conductivity is as follows: a sample with specified shape is employed in a given experimental apparatus; as the foundation of the experimental measurement, the thermal conductivity is calculated by a formula based on the assumption that the differential equation of heat conduction is a linear one, i.e. the thermal conductivity is constant. In this paper, an approach is employed in which a numerical method for solving the inverse problem of heat conduction is used to determine the thermal conductivity. With this approach, the thermal conductivity can be determined from the data obtained directly from an object in its natural situation instead of from the data obtained by a sample in an experimental apparatus. In order to show the applicability of the suggested approach, the thermal conductivity is measured first for a sample of stainless steel by using a method of direct electric heating within the temperature region from 200° to 800°C. The thermal conductivity and its dependence on temperature are calculated by using Fourier’s law. The thermal conductivity and its temperature dependence are then calculated by using a finite-difference method and a finite-element method for solving the inverse problem of heat conduction. All these experimental and calculated results are compared with the curve recommended by TPRC. The agreement is good (within ±5%).
The approach is also employed to determine the thermal conductivity of engineering material. By means of the temperature history measured directly at several points within a 4.5 ton vermicular-graphite-cast-iron ingot mold, the thermal conductivity and its temperature dependence within the region 0°–800°C are calculated by using the finite-difference method for solving the inverse problem of nonlinear heat conduction. The calculated values of the thermal conductivity compared favorably with the experimental ones obtained by use of a sample of the same material as the ingot mold and of the direct electric heating method.