Semi-Analytical Solution of the Graetz Problem With Uniform Wall Heat Flux Utilizing the Transversal Method of Lines

Abstract

This study addresses the second Graetz problem with prescribed wall heat flux employing the transversal method of lines (TMOL), which deviates significantly from the traditional mathematical procedures employed in the past. The wall heat flux is customarily provided by electrical, radiative or solar heating in engineering applications. The TMOL transforms the governing two-dimensional energy equation with temperature-invariant thermo-physical properties into a sequence of adjoint ordinary differential equations of second order with the radial variable as the independent variable. The singular feature in those equations is the embedded axial variable interval. For the implementation of TMOL, a special computational domain consists in a condensed set of transversal lines displayed in the cross section of the tube. An approximate, semi-analytical temperature distribution is obtained with the solution of the first adjoint ordinary differential equation of second order, which is expressed in terms of the Kummer function of first kind M(a,b,c). From here, the approximate, semi-analytical wall and center temperature distributions exhibit excellent quality because the two compare favorably with the exact, analytical wall and center temperature distributions given by the classical Graetz infinite series. As a beneficial consequence, usage of the second adjoint ordinary differential equation of second order having more complex structure becomes unnecessary.