The extended Graetz problem with piecewise constant wall temperature for pipe and channel flows

Abstract

Axial heat conduction effects within the fluid can be important for duct flows if the Prandtl number is relatively low (liquid metals). In addition, axial heat conduction effects within the flow might also be important, if the heating zone is relatively short in length. The present paper shows an entirely analytical solution to the extended Graetz problem with piecewise constant wall temperature boundary conditions. The solution is based on a selfadjoint formalism resulting from a decomposition of the convective diffusion equation into a pair of first order partial differential equations. The obtained analytical solution is as simple to compute as the one without axial heat conduction. The analytical results are compared to available numerical calculations and good agreement is found.

Influence of Design Parameters on Hydrodynamics and Heat Transfer of a Modularized Millireactor

“…It is known that heat conduction in the fluid occurs at low Pe-numbers [21] and plays a significant role especially for T w = const. [22].…”

Steady-periodic heating in parallel-plate microchannel flow with participating walls

“…Several researchers included thermal participation of the channel walls which is strongest near the entrance region [5,6] and can influence the overall performance of heat exchangers involving microchannel flow passages [7]. A heated region confined to a small region along the wall was studied by Weigand [8,9], who found that a small heated region extends the impact of axial conduction to higher Peclet numbers. There have been several thermal-entrance studies of microchannels with other velocity distributions, including Hartmann flow [10,11], porous-saturated flow [12], and slip flow [13].…”

The effect of wall conduction for the extended Graetz problem for laminar and turbulent channel flows

“…Their investigation showed that the normally used assumption that the eddy diffusivity in axial and normal direction is the same is correct for the range of parameters under investigation. The effect of a piecewise changing wall heat flux or wall temperature has been analytically studied by Weigand et al [10] and Weigand and Lauffer [11]. Despite the large amount of work, which has been done on the extended Graetz problem for laminar and turbulent duct flows, very little investigations are known which study analytically the effect of conduction in the wall on the heat transfer in a duct.…”