Three-Dimensional Transient Heat Conduction in a Solid Block with Fully Implemented Convective Boundary Conditions

Abstract

Accurate prediction of transient cooling in three-dimensional solid bodies is essential for thermal processing and electronic cooling applications. This study develops a physically consistent 3D transient heat conduction model that explicitly estimates convective heat transfer on all external boundaries, including faces, edges, and corners, using an explicit finite-difference scheme on a uniform Cartesian grid. Unlike conventional models that simplify convective boundary conditions to planar surfaces, this approach implements directionally consistent Robin conditions to allow multi-directional heat loss at boundary intersections. The model was applied to an aluminum block cooling from 180°C to an ambient temperature of 25°C. Results reveal that that enhanced cooling at edges and corners, exposed to convection in two and three orthogonal directions, respectively, results in significant local temperature variations. A grid independence study comparing spatial step sizes from 4 mm to 2 mm confirmed that the numerical solution is robust, with a relative difference in center cooling time of less than 2% between the two finest grids. Quantitative analysis indicates that neglecting these edge and corner effects can result in a significant underestimation of local heat loss and inaccuracies in total cooling time predictions. This framework provides a transparent numerical approach for analyzing transient heat transfer where surface-to-volume ratios and localized cooling are critical.