On the Teaching of the Lumped Model for Unsteady Heat Conduction: Natural Convection Versus Forced Convection

Within the lumped model platform for unsteady heat conduction, the Biot number criterion between a solid body and a surrounding fluid requires that Bi =       S V k h s < 0.1. Although not clearly stated, this holds true mostly for forced convection where the mean convective coefficient h is affected by the impressed fluid velocity. Conversely, when heat is exchanged by natural convection, the Biot number criterion involves a mean convective coefficient h that depends on the temperature difference between the body and the fluid. Consequently, the above-cited Biot number criterion must be modified to incorporate the variability of the mean convective coefficient h with the temperature difference. This situation gives rise to a new Bimax =       S V k h s max < 0.1 where max h is the maximum mean convective coefficient that, in the case of cooling, happens at the initial temperature Ti and time t = 0. In this paper on engineering education, the exact mean temperature distribution T (t) is deduced for a case study wherein the solid body is a sphere being cooled by natural convection in quiescent air under the premises of the lumped model. A physics-based equivalence of Bimax interweaves the solid thermal conductivity, the fluid thermal conductivity and the extended Grashof number embracing the initial–to–fluid temperature difference.