Analytical–Numerical Modeling of Transient Heat Flux in a Semi-Infinite Medium Using the Duhamel Integral
Abstract
This article integrates the concepts of general analytical solutions (Model 2) based on the Duhamel integral principle in a semi-infinite body model with digital modelling in the Simulink environment (Model 1) for a thermoelectric battery. The influence of various non-stationary heat loads on the surface temperature of a TEG is analysed. The Duhamel principle allows expressing inhomogeneous boundary conditions of the heat-conduction equation as a superposition of elementary impulses. In Simulink, this integral was implemented numerically through convolution, and the results were compared with analytical solutions. Three classical heat-flux shapes were analysed—polynomially increasing, linearly decreasing, and bell-shaped impulse. A square-pulse heat load was also analysed. The results revealed how the surface temperature’s rise-fall behaviour, peak-time delay, and the relationship between heat diffusion and thermoelectric response depend on the heat-flux shape. The numerical and analytical solutions of the Duhamel integral showed good agreement. The differences were mainly attributed to singularity and discrete step selection. This approach is useful for analysing transient thermal processes in thermoelectric batteries and for systems operating with impulsive or pulsating heat sources. The novelty of this study lies in the combined analytical-numerical implementation of the Duhamel integral in the Simulink environment and its application to transient thermoelectric systems under various non-stationary heat-flux profiles.