Homogenization and two scales convergence of some stationary and non-stationary heat transfer problems, application to gas cooled nuclear rectors.
We are interested in the homogenization of heat transfer in periodic porous media modelling the geometry of a gas cooled nuclear reactor. This geometry is made of a solid media perfora- ted by several long thin parallel cylinders, the diameter of which is of the same order than the period. The heat is transported by conduction in the solid part of the domain and by conduc- tion, convection and radiative transfer in the fluid part (the cylinders). A non-local boundary condition models the radiative heat transfer on the cylinder walls. It is a stationary analysis corresponding to a nominal performance of the reactor core, and also non-stationary corres- ponding to a normal shut-down of the core. To obtain the homogenized problem we first use a formal two-scale asymptotic expansion method. The mathematical justification of our re- sults is based on the notion of two-scale convergence. One feature of this work in dimension 3 is that it combines homogenization with a 3D to 2D asymptotic analysis since the radiative transfer in the limit cell problem is purely two-dimensional. A second feature of this work is the study of this heat transfer when it contains an oscillating thermal source at the mi- croscopic level and a thermal exchange with the perforations. In this context, our numerical analysis shows a non-negligible contribution of the second order corrector which helps us to model the gradients appearing between the source area and the perforations.