Analysis of Laminar Slip-Flow Thermal Transport in Microchannels With Transverse Rib and Cavity Structured Superhydrophobic Walls at Constant Heat Flux

Abstract

This paper presents an analytical investigation of the thermal transport in a parallel-plate channel comprised of superhydrophobic walls. An analytical solution is obtained for the thermally developing state, however, it is the condition far downstream from the entrance where the temperature field exhibits repeating periodic streamwise variation that is of primary interest here. The superhydrophobic walls considered in this paper exhibit alternating microribs and cavities positioned perpendicular to the flow direction and the transport scenario analyzed is that of constant wall heat flux through the rib surfaces with negligible thermal transport through the vapor cavity interface. Axial conduction is neglected in the analysis and the problem is one of Graetz flow with apparent slip-flow and periodicity of constant heating. Closed form solutions for the local Nusselt number and wall temperature are presented and are in the form of infinite series expansions. Previously, it has been shown that significant reductions in the overall frictional pressure drop can be expected relative to the classical smooth channel laminar flow. The present results reveal that the overall thermal transport is markedly influenced by the relative cavity region (cavity fraction), the relative rib/cavity module width, and the flow Peclet number. The following conclusions can be made regarding thermal transport for a constant heat flux channel exhibiting the superhydrophobic surfaces considered: (1) Increases in the cavity fraction lead to decreases in the average Nusselt number; (2) Increasing the relative rib/cavity module length yields a decrease in the average Nusselt number; and (3) as the Peclet number increases the average Nusselt number increases. For all parameters explored, the limiting upper bound on the fully developed average Nusselt number corresponds to the limiting case scenario of classical laminar flow through a smooth-walled channel with constant heat flux.

Apparent Temperature Jump and Thermal Transport in Channels With Streamwise Rib and Cavity Featured Superhydrophobic Walls at Constant Heat Flux

“…10 are results at ^" = 0.1 from Ref. [30] where the transverse rib scenario was explored. For this scenario, the influence of axial conduction was neglected.…”

“…However, in this study, periodicity of the heating was neglected although it is a critical feature of the thermal transport dynamics. Two studies by our group have explored heat transfer in laminar duct fiows with SH walls exhibiting transverse ribs maintained at constant temperature [23] and at constant heat flux [30]. The results from these works showed that the average convective heat transfer coefficient, h, is smaller for SH surfaces than for the non-SH surfaces.…”

“…This scenario differs from that considered in Refs. [23,30] since the rib/cavity direction is parallel, rather than transverse, to the flow direction. A consequence of this change in rib/cavity orientation is a significant change to the overall thermal transport physics and this thermal transpon scenario has never been considered previously in the literature.…”

Thermal transport in laminar flow over superhydrophobic surfaces, utilizing an effective medium approach

“…2), with ridge height d, with air (of thermal conductivity k a = 0.028 W/mK) filling the interstices between the ridges, and assuming a flat air meniscus. 18,22 A linear air fraction was defined through φ a = l a /L (where l a is the width of the air pocket and L is the combined width of the solid rib of length l s and the air pocket, i.e., L = l s + l a ). The substrate is of thickness D and thermal conductivity k s .…”

The Graetz–Nusselt problem extended to continuum flows with finite slip

“…Distinct results are expected for systems concerning effective slip (Maynes, Webb & Davies 2008;Maynes et al 2012;Enright et al 2013). As such, the Graetz-Nusselt problem for heterogeneously slippery surfaces deserves to be studied separately.…”