Heat transfer by laminar flow in a cylindrical tube

Summary

In a recent paper¹) a method for determining the discrete set of eigenvalues and the associated eigenfunctions of a certain Sturm-Liouville differential system has been developed and it has been pointed out that by this method the problem of heat transfer by laminar flow in cylindrical channels can be worked out withaxial heat conduction, viscous dissipation and any prescribedheat generation included. In the present paper an exact solution of this problem for a circular channel is presented and the first four eigenvalues and the corresponding eigenfunctions are found for Péclet numbers 100 and 1000 (hitherto worked out by assuming thatδ ² T/δx ² ≪δ ² T/δr ² + (1/r)δ T/δr, and also neglecting the viscous dissipation). It is found that the effect of axial heat-conduction is almost negligible for higher Péclet numbers (Pé>100), and hence it can be definitely stated that the approximate solution is good enough under the usual experimental conditions. The heat transfer problem for plug flow is also solved exactly, and it is concluded that the effect of axial heat conduction is negligible forPé>100. The mean mixed temperature and the Nusselt numbers are tabulated and plotted against the Graetz number.