A theory to describe heat transfer during laminar incompressible flow of a fluid in periodic porous media

Abstract

In this paper, a theory is formulated that describes the heat transfer during the laminar flow of an incompressible fluid within a rigid periodic porous medium. To describe the macroscopic temperature distribution in porous media, the volume average concept is employed to derive the volume averaged energy equation. Previous researchers have used either the one-equation model with the local thermal equilibrium assumption between the various phases or the complex two-equation model to deal with the average temperatures of the solid phase and the fluid phase. Both types of models are derived and applied in a stationary observation frame. In this paper, we propose a generalized one-equation model by relaxing the local thermal equilibrium assumption between the phases and invoking the frame invariant principle. The three new terms arise due to the microscopic transient, convective, and conductive effects. By taking advantage of the objectivity of the generalized model, it is shown that the microscopic temperature distribution in a periodic unit cell results in a periodic heat flux boundary condition. The direct temperature solution is obtained in the periodic unit cell by solving the generalized volume averaged energy equation. A modified effective thermal conductivity tensor is calculated which encompasses the micro-convective effects encountered in porous media. An analytic study of a non-isothermal flow through two parallel thick plates is performed to explore the role of each new term in the generalized model.