Inverse estimation of front surface temperature of a locally heated plate with temperature-dependent conductivity via Kirchhoff transformation

• Solved heat conduction problem with temperature dependent thermal conductivity.
• Formulated an iterative procedure to use the discretized sensor data.
• Presented comparison with results obtained using Conjugate Gradient Method.

In this paper, by Kirchhoff transformation of the temperature variable, the temperature dependence of thermal conductivity is eliminated, thereby simplifying the 3-dimensional heat conduction equation. Through Hadamard Factorization Theorem, transfer function relating the front and back surface temperature as infinite product of polynomial is established. The inverse Laplace transform of the polynomial provide the relationship for every mode in the time domain. The front surface temperature is revealed through iterative time domain operations from the data on the back surface. Seven points for smoothing and third order polynomial in derivative calculation were used in Savitzky–Golay (S–G) method. The comparison between direct solution, Conjugate Gradient Method (CGM) and DCT/Laplace transform solutions are given. Root Mean Square (RMS) of the errors at different time steps for DCT/Laplace solution and CGM method are also presented.