Inverse Solution to Steady Two-Dimensional Heat Conduction

This paper presents an analytical method for solving an inverse problem of steady, two-dimensional, heat conduction in a solid plate with space-variable temperature and heat flux on one of its boundaries. The method doesn’t need the knowledge of the thermal conditions on the other plate boundaries. Explicit closed-form expressions are derived for calculating the two-dimensional distributions of the temperature and heat flux in the plate for arbitrary over-specified boundary temperature and heat flux profiles. The special problem case of isothermal boundary with a specified heat flux profile, and that of insulated boundary with a specified temperature profile are considered realistic cases for this problem. Validity and exactness of the proposed method is proved via test problems with known exact solutions. Sensitivity of the current technique to errors in the input boundary conditions is examined. The proposed method is considered of a theoretical value for the heat conduction area.