Laminar forced convection slip-flow in a micro-annulus between two concentric cylinders

Abstract

Forced convection heat transfer in hydrodynamically and thermally fully developed flows of viscous dissipating gases in annular microducts between two concentric micro cylinders is analyzed analytically. The viscous dissipation effect, the velocity slip and the temperature jump at the wall are taken into consideration. Two different cases of the thermal boundary conditions are considered: uniform heat flux at the outer wall and adiabatic inner wall (Case A) and uniform heat flux at the inner wall and adiabatic outer wall (Case B). Solutions for the velocity and temperature distributions and the Nusselt number are obtained for different values of the aspect ratio, the Knudsen number and the Brinkman number. The analytical results obtained are compared with those available in the literature and an excellent agreement is observed.

Introduction

Fluid flow and heat transfer at microscale have attracted an important research interest in recent years due to the rapid growth of novel techniques applied in MEMS (microelectromechanical systems) and biomedical applications such as drug delivery, DNA sequencing, and bio-MEMS. Readers are referred to see recent excellent reviews related to transport phenomena in microchannels by Ho and Tai [1], Palm [2], Sobhan and Garimella [3], Obot [4] Rostami et al. [5], [6], Gad-el-Hak [7], Guo and Li [8], [9], Morini [10].

It is experimentally shown that fluid flow and heat transfer at microscale differ greatly from those at macroscale, especially in terms of wall friction and heat transfer performance, with generally inconsistent and contradictory experimental results published. Therefore, there is a need and a good potential for theoretical investigations. At macroscale, classical conservation equations are successfully coupled with the corresponding wall boundary conditions, usual no-slip for the hydrodynamic boundary condition and no-temperature-jump for the thermal boundary condition. These two boundary conditions are valid only if the fluid flow adjacent to the surface is in thermal equilibrium. However, they are not valid for gas flow at microscale. For this case, the gas no longer reaches the velocity or the temperature of the surface and therefore a slip condition for the velocity and a jump condition for the temperature should be adopted.

Barron et al. [11], [12] extended the Graetz problem to slip-flow and developed simplified relationships to describe the effect of slip-flow on the convection heat transfer coefficient. Ameel et al. [13] analytically treated the problem of laminar gas flow in microtubes with a constant heat flux boundary condition at the wall assuming a slip-flow hydrodynamic condition and a temperature jump thermal condition at the wall. They disclosed that the fully developed Nusselt number decreased with Knudsen number. In recent studies, Aydın and Avcı [14], [15], [16], [17] theoretically investigated the steady, laminar forced convective heat transfer of a Newtonian fluid in a microtube and microduct between two parallel plates including the velocity slip and the temperature jump at the wall for the fully developed flow case [14], [15] and for the thermally developing flow [16], [17].

An example of micro heat exchangers is the one for which the hot and cold fluids move in the same or opposite directions in a concentric microtube. Fabrication of such microexchangers is now possible due to recent innovative microfabrication techniques such as etching/lithography/deposition and silicon micromachining. Such a geometry can also be found in cooling of high power resistive magnets, compact fission reactor cores, fusion reactor blankets, advanced space thermal management systems, manufacturing and material processing operations and high-density multi-chip modules in supercomputers and other modular electronics [18]. The objective of the present study is to theoretically investigate the gas flow in a concentric annular microduct representing a micro heat exchanger for the hydrodynamically and thermally fully developed case. The effect of the Knudsen number, the aspect ratio of the annular geometry and the Brinkman number on the temperature profile and, in the following, the Nusselt number are determined for two different configurations of the thermal boundary conditions.