Iterative multi-scale approach for heat conduction with free convection problem in periodic hollow structures

This paper describes a thermal homogenization approach to the application of a multi-scale formulation for steady state heat conduction with free convection problems in a hollow structure. The presented formulation evaluates the effective macroscopic thermal properties of a hollow structure containing fluid within microstructures. This is done by scaling up and down between the microscopic and macroscopic models. The uniqueness of this approach is that it permits the solution of a heat conduction problem on a macroscopic scale, instead of solving heat conduction with free convection problem. The proposed methodology involves an iterative upscaling procedure which solves a coupled convection conduction CFD unit cell problem in each iteration. This reduces the required computational time while maintaining the required level of accuracy. Our method of multi-scale formulation has been verified using two examples of hollow structures containing air within the hollow spaces. The results are in excellent agreement with the reference solutions.