An Approximate Solution for Heat Transfer in the Entrance Region of Laminar Newtonian Pipe Flow

The study of the simultaneously developing pipe flow requires facing nonlinear systems of partial differential equations. In this framework, the aim of this paper is to demonstrate that the integral method can be an effective procedure to obtain analytic-approximate solutions that are easy to handle while allowing the recovery of a satisfactory accordance with the exact solution. To prove the above statement this paper will present a comparison between the approximate solution and the corresponding numerical solution in the entrance region of Newtonian pipe flow. Third-kind thermal boundary conditions are included, while velocity and temperature profiles at the inlet are assumed uniform. Numerical results demonstrate that the proposed approximate solution is quite accurate and readily implemented, both in terms of developing velocity and temperature profiles. Moreover, the expected functional dependence on the main parameters of the problem at hand is retained. As a consequence, the developing Fanning friction coefficient and Nusselt curves are satisfactory and accurate for different thermal boundary conditions at the wall.