Introducing a proper definition of the Nusselt number for fluid flow in a pipe partially filled with porous media

In the present study the validity of two conventional Nusselt number definitions were investigated using analytical and numerical methods for convection heat transfer in a pipe partially filled with porous media. The first definition is denoted as Nu_1 (x)=(2R(∂T/∂r)_(r=R))⁄((T_w-T_m (x))) and the second one follows: Nu_2 (x)=(2Rq_cond^'')⁄(k_ref (T_w-T_m (x))). The Nusselt number resulted from these two definitions was investigated analytically in a pipe for different porous configurations. The results show that the calculated Nusselt numbers using these two definitions, are different in porous media boundary arrangement. In the first definition, the heat transferred to the fluid flowing thorough the porous media is not considered, so the Nusselt number which is calculated via this definition cannot demonstrate the physics of heat transfer phenomenon properly. The boundary arrangement of porous in a pipe with turbulent flow is simulated numerically and the Nusselt number was calculated by the two definitions. The calculated Nusselt from the first definition shows that the Nusselt number increases as the heat conduction coefficient of porous grows which is not a proper expression of physics of this problem. So, the first definition of the Nusselt number is not proper for porous boundary arrangement in a pipe. However, with investigating of the second definition, it is seen that with increasing the porous heat conduction coefficient, the Nusselt number increases which this result is physically valid; therefore the second definition is more appropriate for the porous media boundary arrangement.