Numerical Solutions for Laminar-Flow Heat Transfer in Circular Tubes

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Abstract
The existing solutions for laminar-flow heat transfer in a circular tube are generally based on the assumption of a fully established velocity profile at the point in the tube where heating begins. For high Prandtl number fluids, such as the viscous liquids, this idealization does not seriously restrict the usefulness of the solutions because the velocity profile is established much more rapidly than the temperature profile. However, for the Prandtl number range near 1.00, which includes the gas range, the velocity and temperature profiles develop at similar rates along the tube, and the assumption of a fully established velocity profile at the tube entrance can, for many applications, lead to a considerable error in predicted performance. In this paper, numerical solutions are presented for a number of heating conditions for the case of a fluid of NPr = 0.7 with both velocity and temperature uniform at the tube entrance. The solution of Langhaar (1) is employed to provide the velocity profiles which are introduced into the energy equation. Experimental data for laminar air flow in circular tubes are presented for two heating conditions, constant wall temperature and constant heat input per unit of tube length. These data correspond closely to the numerical solutions based on the Langhaar velocity profiles, while differing considerably from solutions based on a parabolic profile throughout the tube. All of the solutions and experimental data are presented in the form of either a local or mean Nusselt number as a function of a modified Graetz number, NRNPr/(x/D).

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