Models for transient conduction in a flat plate subjected to a variable heat flux Ali Grine * , Jean Yves Desmons, Souad Harmand Laboratoire de Me ´ canique et Energe ´ tique, Universite ´ de Valenciennes et du Hainaut Cambre ´sis, Le Mont Houy F59 313, Valenciennes Cedex 9, France Received 27 September 2005; accepted 16 June 2006 Available online 18 September 2006 Abstract This work presents analytical models allowing to identify the transient temperature distribution in a flat plate. The plate is exposed to a convective heat transfer on a face and to a heat flux on the other one. The heating flux is Heaviside (crenel type) and is maintained during a t 1 time. The heating phase is followed by a relaxation one. The theoretical method is original because it uses Green’s functions method to determine the analytical solutions of the heat propagation equation in the plate during the heating and relaxation phases. These analytical solutions allow to identify the temperature distribution as well as wall heat flux versus time. The results of our work can be useful at different levels: during the identification of parameters (such as the thermal conductivity or the thermal diffusivity of a plate), during the identification of the boundary conditions (like the heating flux or the convection coefficient) in industrial processes using this kind of systems, or even with educational intents for teaching transient conduction. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Convective heat transfer; Transient state; Analytical method; Flat plate; Identification of parameters; Green’s functions 1. Introduction This work presents analytical models describing the tem- perature distribution in a flat plate exposed to a heat flux convective boundary conditions (Fig. 1). The development of these models is based on the Green’s functions theory applied to the resolution of the heat transfer equation [1–4]. The Green’s function method, well known in many fields of physics where it definitively becomes essential, remains however not widely used in thermal science. Its main advantage is in its ability to synthesize the thermal behavior of complex solid structure featuring compound boundary conditions. This advantage can a priori appear purely formal but in reality it is not, because this formula- tion supposes physical assumptions concerning the bound- ary conditions which are less restrictive than the methods usually used. The results of the developed models are plotted as tem- peratures and wall heat flux versus the Biot and Fourier numbers [5]. This dimensionless form allows very easy usage of the results and allows the latter to be accessible in many fields of research or industry. The models devel- oped can be useful and be used as a basis to validate numerical simulations, to identify parameters such as the thermophysical properties of materials [6] or to identify the convective heat transfer coefficient or the heat flux in a given process [7]. The originality of the method used to develop these models can also be useful, from a teaching point of view, to enlarge and diversify the traditional techniques usually employed to teach transient modes in thermal conduction [8]. 2. Problem presentation A flat plate made of a homogeneous material with a ther- mal conductivity k is exposed to a wall heating flux density 1359-4311/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2006.06.013 * Corresponding author. Tel.: +33 3 27 51 19 80; fax: +33 3 27 51 19 61. E-mail address: ali.grine@univ-valenciennes.fr (A. Grine). www.elsevier.com/locate/apthermeng Applied Thermal Engineering 27 (2007) 492–500