Second-Order Model for Transient Heat Conduction in a Sphere

Abstract

T HE use of simple mathematical formulations to solve the heat conduction problems is of great interest for engineering calculations. Though the traditional analytical models are widely used because of their numerous industrial applications [1–4], they sometimes do not provide satisfaction for large values of Biot number. In addition, these models do not predict the temperature inside the solid (at the center, for example). In the case of spherical geometries, the analytical solution is expressed as an infinite series which necessitates the numerical solution of a characteristic equation for each value of theBiot number [5]. Moreover, this analytical method sometimes requires the calculation of several hundred terms in the series in order to reach the expected accuracy [6]. A perturbation method has been used in [7] to develop first-order models which can be viewed as improved lumped models for the slab, infinite cylinder, and spherical geometries. A second-order model has been examined in [8] for the slab and in [9] for the cylinder. The aim of this paper is to extend the ideas of [7–9] to the case of a sphere subjected to convection to a surrounding fluid. It is shown that this model gives accurate enough results even for an infinite Biot number. The validity of the model is discussed and the region of the sphere where it can be used is determined.