Short Communication A second order model for transient heat conduction in a slab with convective boundary conditions H. Sadat * Laboratoire d’Etudes Thermiques, UMR 6608, Ecole Supe ´ rieure d’Inge ´nieurs de Poitiers, 40 Avenue du Recteur Pineau, Poitiers Cedex, France Received 30 January 2005; accepted 13 October 2005 Available online 28 November 2005 Abstract In this paper we present a second order model for transient heat conduction in a slab obtained by using a perturbation method. We show that this simple model is accurate even for high values of the Biot number in a region surrounding the center of the slab. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Heat conduction; Lumped model; Step response 1. Introduction It is of interest for engineering calculations to pro- pose simple mathematical formulations to deal with heat conduction problems. In the present work, we treat the unsteady heat conduction in a slab with convective boundary conditions. Typical applications may be found in the thermal study of building envelopes, metal- lurgical processes which involve heat treatment of steel strips and in many other industries. The well-established classical lumping procedure is extensively used [1–5]. However it is known to be inaccurate for great values of the Biot number which measures the competition between heat conduction in the slab and the convective heat exchanged with the surrounding fluid. Further it gives only the temperature of the surface of the consid- ered medium as it has been shown in [6], where we have given a first order model for the slab, the infinite cylin- der and the sphere. In the following sections, we extend the mathematical developments given in [6], to the sec- ond order, in the case of a slab. It will be shown that this leads to a simple and accurate model even for very high values of the Biot number in a region surrounding the center of the slab. The lower bounds of this region are given. Finally, a computational example for an infinite Biot number demonstrates the accuracy of the method. 2. Problem formulation We consider unsteady heat conduction in a slab of thickness L, initially at a uniform temperature T i . At t = 0, the solid material is exposed to a fluid at a variable temperature f(t) with a heat transfer coefficient h. It is assumed that the thermophysical properties are con- stant. By using the dimensionless parameters defined by h ¼ T  T i T 1  T i X ¼ x L Fo ¼ at L 2 B ¼ hL k where T 1 is a reference temperature which will be the fluid constant temperature, the mathematical formula- tion of the problem is given by oh oFo ¼ o 2 h oX 2 ð1Þ 1359-4311/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2005.10.013 * Tel.: +33 549 453 543; fax: +33 549 453 545. E-mail address: hamou.sadat@univ-poitiers.fr www.elsevier.com/locate/apthermeng Applied Thermal Engineering 26 (2006) 962–965