On the Temperature Distribution in a Flat Plate Induced by an External Thermal Radiant Source

Abstract

This work presents a comprehensive numerical and analytical analysis of steady-state temperature distribution in flat plates subjected to external thermal radiation sources. The mathematical model incorporates nonlinear radiative heat transfer governed by the Stefan-Boltzmann law, including both radiative emission from plate surfaces and heating from compact external sources. A robust finite element formulation with Newton’s method linearization is developed to handle the inherent nonlinearity of the fourth-power temperature dependence. The variational weak form is discretized using bi-quadratic finite elements, and the linearized system is solved through an iterative Newton-Raphson scheme. Analytical bounds are established to ensure solution validity and numerical stability. The methodology is validated through manufactured solutions and applied to practical scenarios relevant to aerospace applications, where flat surfaces are exposed to intense radiative heating. Results demonstrate sharp temperature localization beneath radiation sources at close distances, with gradual smoothing as source height increases. The numerical framework shows excellent convergence properties and physical consistency, providing valuable insights for thermal management in high-temperature engineering applications.