Exact Solution of Heat Conduction in a Two-Domain Parallelepiped With an Orthotropic Layer and Its Application to the Estimation of the Three-Dimensional Thermal Conductivity Tensor and Volumetric Heat Capacity of the Orthotropic Layer

The three-dimensional exact solution of heat conduction in a two-layer composite is found applying the method of separation of variables. One layer is orthotropic and the other layer is isotropic. This solution is used to calculate sensitivity coefficients with respect to the thermophysical properties of the orthotropic layer at fourteen thermocouple locations. Numerical experiments are carried out to solve a parameter estimation problem that involves the estimation of the thermal conductivities in the x-, y-, and z-directions, the volumetric heat capacity of the orthotropic layer, the effective thermal conductivity of the isotropic layer, and the heat flux input. The exact solution is used to generate temperature readings at fourteen thermocouple locations. First, the parameter estimation problem is solved using the exact temperatures and a hybrid algorithm to estimate the thermal properties and the heat flux. Second, random noise is added to the exact temperatures and the thermal properties and heat flux are estimated using the same hybrid algorithm. It is found that when using the exact temperatures, the minimized quadratic functional has a value of 2.4×10⁻¹⁶ (°C)² and the estimated properties agree to the ninth decimal place with the “exact” properties.