EFFECTS OF LONGITUDINAL HEAT-CONDUCTION OF A VERTICAL THIN-PLATE IN ANATURAL CONVECTIVE COOLING PROCESS

This paper analyzes the cooling process of a vertical thin plate cause d by a free convective flow, taking into account the effects of both l ongitudinal and transversal heat conduction in the plate. Due to the f inite thermal conductivity of the plate, a longitudinal temperature gr adient arises within it, which prevents any similarity solution in the boundary layer, changing the mathematical character of the problem fr om parabolic to elliptic, for large values of the Rayleigh number. The energy balance equations are reduced to a system of three differentia l equations with two parameters: the Prandtl number and a non-dimensio nal plate thermal conductivity alpha. In order to obtain the evolution of the temperature of the plate as a function of time and position, t he coupled balance equations are integrated numerically for several va lues of the parameters, including the cases of very good and poor cond ucting plates. The results obtained, are compared with an asymptotic a nalysis based on the multiple scales technique carried out for the cas e of a very good conducting plate. There is at the beginning a fast tr ansient in non-dimensional time scale of order alpha-1 followed by a s low non-dimensional time scale of order unity, which gives the evoluti on of the cooling process. Good agreement is achieved even for values of the conduction parameter alpha of order unity. The asymptotic solut ion allows us to give closed form analytical solution for the plate te mperature evolution in time and space. The overall thermal energy of t he plate decreases faster for smaller values of alpha.