Advanced modelling for energy transport in solids at both micro and macro dimensions, utilizing relaxation concepts to analyze transient states

This study derives a generalized energy equation for solids with arbitrary geometries, including anisotropic materials and curvilinear or moving coordinate systems. The derived equation incorporates enthalpy fluxes under significant motion, allowing for broad applicability. Specific energy equations for Cartesian, cylindrical polar, and spherical systems are presented and applied to case studies. Additional coordinate systems, such as bipolar, elliptic cylindrical, and prolate spheroidal, can also be accommodated. The analysis further incorporates non-Fourier heat conduction, when Fourier's law fails to accurately describe thermal behaviour. The proposed formulations eliminate the necessity for case-specific derivations, thereby streamlining analysis for practical designs. A case study examines the thermal responses in straight and cylindrical fins, taking into account convection, radiation, and internal heat generation, through corrected Fourier and non-Fourier analyses. Results demonstrate the superior performance of the non-Fourier model, highlighting the significance of relaxation time and variable energy sources. The propagation of thermal energy oscillation depends not only on the relaxation time factor but also on the temperature-dependent volumetric energy source and surface treatment. This study also examines thermal propagation in solids without heat loss through lateral surfaces, as observed in the case study of a non-Fourier system, which results in higher oscillations of heat propagation.