A new front surface heat flux calibration method for a 1-D nonlinear thermal system with a time-varying back boundary condition (Q1700583)

The object of interest is a one-dimensional slab with thermophysical properties depending on the temperature. The heat flux on the front side of the slab is unknown and Robin-type boundary conditions on its rear side depend on the time \(t\). A method is supposed that allows to reconstruct the heat flux based on known time-dependent temperature values \(T(b,t)\) and \(T(w,t)\) in two points \(b\) and \(w\) inside the slab. The method combines several ideas. At first, there is calibration. One should obtain \(T(b,t)\) and \(T(w,t)\) for two known sets of boundary conditions. Then these functions are used in the derivation of an integral equation that includes the unknown heat flux. The second idea is rescaling in time that is used to quasi-linearize the problem. At third, the solution of the derived integral equation involves Tikhonov regularization. The method described is tested on two materials with significantly different thermodynamic properties: stainless steel and carbon composite. While testing, both exact and noisy data are used. As a result, the method is found to be stable and quite exact.