Synergy between the solution based on the Transversal Method Of Lines and the Leveque solution in the platform of the Graetz/Nusselt problem

Abstract

The aim of the present study is to generate an approximate, semi–analytical solution of the Graetz/Nusselt problem for the demanding upstream sub-region of the tube employing the Transversal Method Of Lines (TMOL). In the heat convection literature, the approximate, semi–analytical solution of the Graetz/Nusselt problem in the upstream sub-region of the tube that is available refers to the traditional physics-based Lévêque problem that possesses certain idealizations. TMOL is a mathematics-based transformation that discretizes the partial derivative in the axial direction of the two–dimensional energy equation in cylindrical coordinates, while leaving the partial derivatives in the radial direction continuous. As a consequence, the TMOL procedure provides an adjoint one–dimensional energy equation in the radial variable linked to a transversal line placed in the cross-section of the tube. The resulting ordinary differential equation of second order with variable coefficients and inhomogeneous is named the confluent hypergeometric differential equation. This equation can be readily solved in exact, analytical form with a symbolic algebra code in terms of the Kummer function of first kind. This gives way to the approximate, analytical temperature profile in the upstream sub-region of the tube near the entrance. From here, the obtained approximate, analytical mean-bulk temperature profile based on TMOL gently overestimates the exact, analytical Graetz mean-bulk temperature profile, whereas the traditional approximate, analytical mean-bulk temperature profile rendered by Lévêque gently underestimates the exact, analytical Graetz mean-bulk temperature profile. Overall, the two dissimilar methodologies share reasonable quality levels in the specific upstream sub-region of the tube, X ≤ 0.01.