Some refinements concerning the boundary conditions at the macroscopic level

The study of boundary effects initiated in a previous paper is continued. New assumptions regarding the geometrical structure of the boundary surface are introduced. Under these assumptions, it is shown that macroscopic Neumann conditions do not generally affect the determination of the macroscopic field in the case of the transport process considered — heat conduction. For this type of boundary condition, the boundary effect is generally confined within a thin layer near the boundary. When heat sources are taken into account within the porous domain, the result is different. In this case, making use of a Neumann boundary condition, expressed in terms of macroscopic variables, amounts to introducing an extra flux. Under normal circumstances, however, this additional flux is negligible.Roman Letters A cross-sectional area of a unit cell - A _e cross-sectional area of a unit cell at the boundary surface - A _sf interfacial area of the s-f interface contained within the averaging volume - surface area per unit volume (A _sf/ ) - A _sf interfacial area of the s-f interface contained within the macroscopic system - g closure vector - h closure vector - k heat transfer coefficient at the s-f interface - K_eff effective thermal conductivity tensor - _x unit cell length - n unit vector - n_e outwardly directed unit normal vector at the boundary - n_sf outwardly directed unit normal vector for thes-phase at f-s interface - q heat flux density - T ^* macroscopic temperature defined by the macroscopic problem - s closure variable - V volume of the macroscopic system - V boundary surface of the macroscopic domain - V ₁ macroscopic sub-surface of the boundary surface - x local coordinate Greek Letters _s,_f volume fraction - _s, gl_f microscopic thermal conductivities - true microscopic temperature - ^* microscopic temperature corresponding toT ^* - microscopic error temperature - vector defined by Equation (34) - < > spatial average