A Piecewise Linear-in-Time Approximation of the Surface Heat Flux for Solving Forward and Inverse Heat Conduction Problems

An approximate method based on superposition of exact analytical solutions is given for solving direct and inverse transient linear heat conduction problems. The starting point is to model the time-varying surface heat flux as piecewise linear (connected line segments). Solutions constructed using the piecewise linear assumption employ both constant and linear-in-time heat flux-based “building blocks,” where the constant basic building block is typical of a piecewise constant profile approximation (stair step variation) of the surface heat flux. The above superposition is relevant in investigation of inverse heat conduction problems (IHCPs) as it gives a convenient expression for the temperature in terms of the unknown heat flux components. Also, its sequential-in-time nature indicates that one segment after another can be estimated, starting with the earliest times, and then moving successively to larger times. Some examples of comparison between the proposed piecewise linear profile approximation and the well-established piecewise constant one are given for transient power variation of the surface heat flux when solving direct problems. The combined effect of the accuracy of the basic building block solutions and the number of time steps chosen when discretizing the surface heat flux is also analyzed.