Analytical and numerical methodology for the temperature distribution on surfaces exposed to a radiant heat source

Abstract

In this work, we investigate the effect of a small, high-temperature radiant source on a surface located at a significant distance. Starting from a partial differential equation with Neumann boundary conditions, we develop a nonlinear mathematical model for the heat transfer, aiming to determine the temperature distribution on the surface of a thin, initially flat rectangular plate, which serves as the initial case study. The solution is obtained through numerical methods. The proposed equation accounts for radiative heat transfer incident on the upper surface of the plate from the source, as well as radiation emitted from both the upper and lower surfaces of the plate. The inherent non-linearity is dealt with using a weak variational formulation, applying Newton's method, which is discretized using finite elements achieving high accuracy (tolerance < 10-6). Convergence of the numerical solution is demonstrated, and results are validated by upper and lower bound estimates. A priori bounds for the solution are established, ensuring the avoidance of inconsistent numerical approximations. Results reveal that temperature gradients exhibit a sharp peak localized beneath the source, with maximum temperatures decreasing from 10 K to 0.1 K as source height increases from 0.1 m to 10.0 m. The proposed numerical method demonstrated fast convergence and robustness of the proposed scheme, while maintaining physical consistency. The numerical results show close agreement with both the analytical bounds and the Method of Manufactured Solutions benchmark. This study provides good insights into how small, high-temperature radiant bodies influence the temperature distribution in distant surfaces. Moreover, the mathematical approach adopted here lays the groundwork for applications to other geometries and sophisticated nonlinear heat transfer problems.