Derivation of Analytical Formulas for Local Thermal Nonequilibrium Parabolic Flows in Porous Media

This study presents an analytical investigation of local thermal nonequilibrium (L.T.N.E.) parabolic flow problems in fluid-saturated porous media, focusing on two fundamental configurations: mixed convective thermal boundary layer flow over a vertical flat plate and thermally developing flow in channels and tubes. Practical analytical formulas were derived to support engineering applications. For the isothermal vertical flat plate, results show that while the local Nusselt number under L.T.N.E. Conditions can exceed that of the local thermal equilibrium (L.T.E.) case near the leading edge, the average Nusselt number remains consistently lower, emphasizing the necessity of the L.T.N.E. model for accurate heat transfer predictions. As for the thermally developing flows in conduits, a region map was constructed to classify the thermal development stages - developing, transition, and nearly fully developed - in conduits under constant heat flux. The effects of the controlling parameters, namely, the inverse of Biot number and thermal conductivity ratio on thermal development were also systematically characterized through derived analytical relationships.