The extended Graetz problem for micro-slit geometries; analytical coupling of rarefaction, axial conduction and viscous dissipation

Abstract

a b s t r a c tIn order to support the recent MEMS and Lab-on-a-chip technologies, we studied heat transport in micro-scale slit channel gas flows. Since the micro convection transport phenomena diverges from conventional macro-scale transport due to rarefaction, axial conduction and viscous heating, an accurate understanding requires a complete coupling of these effects. For such cases, we studied heat transfer in hydrodynamically developed, thermally developing gas flows in micro-slits at various flow conditions. The analytical solution of the energy equation considered both the heat conduction in the axial direction and heat dissipation of viscous forces. Furthermore, updated boundary conditions of velocity slip and temperature jump were applied based on Knudsen number of flow in order to account for the nonequilibrium gas dynamics. Local Nusselt number (Nu) values were calculated as a function of Peclet (Pe), Knudsen (Kn) and Brinkman (Br) numbers which were selected carefully according to possible micro-flow cases. Strong variation of Nu in thermal development length was found to dominate heat transfer behavior of micro-slits with short heating lengths for early slip flow regime. For this instance, influence of axial conduction and viscous dissipation was equally important. On the other hand, high Kn slip flow suppressed the axial conduction while viscous heating in a small surface-gas temperature difference case mostly determined the fully developed Nu and average heat transfer behavior as a function of Kn value.

A review of heat and fluid flow characteristics in microchannel heat sinks

“…Brinkman (Br) number is the ratio of viscous heating over heat conduction in the fluid domain which flows along microchannel, and it is used to determine the effect of viscous dissipation on heat transfer mechanism in MCHS. 110 Kalyoncu and Barisik 71 numerically showed that viscous dissipation increases and decreases heat transfer rate for positive and negative Br cases, respectively. However, they indicated that developed Nu number values are the same for both cases.…”

“…99 An increment in Kn number, Pe number, and hydraulic diameter decreases axial heat conduction rate. 71,[100][101][102] In contrast, the effect of axial heat conduction on heat transfer increases as Re number, thermal conductivity, and thickness of the separating wall increase. 96,102 At the entrance region, the presence of axial heat conduction increases heat transfer rate.…”

“…Continuum assumption cannot be used as Kn number becomes greater than 0.001 and rarefaction effect becomes significant on heat transfer for gaseous flow 67,68 . Researchers 69‐71 indicated that Nu number decreases as the rarefaction effect increases in MCHS. Flow regimes with respect to Kn number and the main solution approaches are listed in Table 3 9,72,73 .…”

“…The effect of axial heat conduction changes with respect to Knudsen number, Peclet number, thermal conductivity, and thickness of the separating wall 99 . An increment in Kn number, Pe number, and hydraulic diameter decreases axial heat conduction rate 71,100‐102 . In contrast, the effect of axial heat conduction on heat transfer increases as Re number, thermal conductivity, and thickness of the separating wall increase 96,102 .…”

Improved lumped analysis of Graetz problems with axial diffusion

“…Even though the classical Graetz problem [1] has been known for over a 100 years, a number of recent investigations related to extended versions of the problem can be found in the literature. Among features that are commonly incorporated as extensions to the classical Graetz formulation, one can mention axial heat diffusion [2][3][4][5][6], different velocity profiles [7], conjugate heat transfer within the surrounding walls [8][9][10][11], slip flow effects [5,6], and heating by viscous dissipation [4,[12][13][14]. These recent examples show that, despite the simplicity of the dynamically developed flows that occur in Graetz formulations, there are many current applications for these problems.…”