On the Heat-Transfer of Laminar Flow in the Circular Pipe

Abstract

This paper describes a solution for the heat-transfer in a circular pipe of constant temperatur, in which the velocity distribution of fluid is the Poiseuille's flow. The basic equations are the same as that of Graetz or Nusselt, and as a mathematical method we applied the Karman-Pohlhousen's method which is used on the boundary layer problems. We derived an equation from the bsic equations, which correspond to the momentum equation of the bouodary layer problems, and used the conception of thermal boundary layer, in which we assume dthat the temperature distribution varies as the 4th degree equation with respect to the distance from the pipe wall. As a result we obtained the following equations, for the relation of dimensionless mean temperature of fluid θ_m and Graetz number Wc/kL, [numerical formula][numerical formula] Where t_i is an initial temperature of fluid, and t_w is a temperture of pipe wall, t_m is a mean temperature of fluid after passed the pipe length L, Wc/kL is a Graetz number. And δ is a dimensionless thickness of the thermal boundary layer at the pipe end. We can δ between equations (1) and (2), but practically it is more advantageous to draw θ_m-Wc/kL curve by calculate the values of equations (1) and (2) corresponding δ=0.1, 0.2 etc.. This solution will be used over a range between the ranges in which Nusselt-Groeber-Yamagata's solution and Leveque's solution were used respectively. And moreover for the range of Leveque's solution, perhaps this solution will be more exact than the Leveque's solution.