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Research Papers

Isoflux Nusselt Number and Slip Length Formulae for Superhydrophobic Microchannels

[+] Author and Article Information
Ryan Enright

Stokes Institute,
University of Limerick,
Limerick, Ireland
e-mail: ryan.enright@alcatel-lucent.com

Marc Hodes

Department of Mechanical Engineering,
Tufts University,
Medford, MA 02155

Todd Salamon

Bell Labs,
Alcatel-Lucent,
600 Mountain Avenue,
Murray Hill, NJ 07974

Yuri Muzychka

Faculty of Engineering and Applied Science,
Memorial University of Newfoundland,
St. John's, NL A1B 3X5

1Present address: Thermal Management Research Group, Efficient Energy Transfer (ηET) Dept., Bell Labs Ireland, Alcatel-Lucent Ireland, Blanchardstown Business & Technology Park, Blanchardstown, Snugborough Road, Dublin 15, Ireland.

Contributed by the Heat Transfer Division of ASME for publication in the Journal of Heat Transfer. Manuscript received September 10, 2012; final manuscript received May 8, 2013; published online October 17, 2013. Assoc. Editor: Jose L. Lage.

J. Heat Transfer 136(1), 012402 (Oct 17, 2013) (9 pages) Paper No: HT-12-1493; doi: 10.1115/1.4024837 History: Received September 10, 2012; Revised May 08, 2013

We analytically and numerically consider the hydrodynamic and thermal transport behavior of fully developed laminar flow through a superhydrophobic (SH) parallel-plate channel. Hydrodynamic slip length, thermal slip length and heat flux are prescribed at each surface. We first develop a general expression for the Nusselt number valid for asymmetric velocity profiles. Next, we demonstrate that, in the limit of Stokes flow near the surface and an adiabatic and shear-free liquid–gas interface, both thermal and hydrodynamic slip lengths can be found by redefining existing solutions for conduction spreading resistances. Expressions for the thermal slip length for pillar and ridge surface topographies are determined. Comparison of fundamental half-space solutions for the Laplace and Stokes equations facilitate the development of expressions for hydrodynamic slip length over pillar-structured surfaces based on existing solutions for the conduction spreading resistance from an isothermal source. Numerical validation is performed and an analysis of the idealized thermal transport behavior suggests conditions under which superhydrophobic microchannels may enhance heat transfer.

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Figures

Grahic Jump Location
Fig. 1

Geometry of a parallel-plate channel where H is the channel height. Both the channel width and length are considered much larger than H. Each of the surfaces exhibits arbitrary hydrodynamic and thermal slip.

Grahic Jump Location
Fig. 2

Isometric (a) and plan (b) views of the solution domain. The origin of the coordinate system is defined at the center of the pillar tip. The composite interface at y = 0 is composed of a liquid–solid interface, Bs, subjected to no-slip and constant heat flux. The liquid–gas interface, Bi, was specified as shear-free and adiabatic. The domain extended to a symmetric boundary, Bsymy=H/2, at the channel center-line. Flow imposed by a pressure gradient entered the domain from the left (x-direction) through Bperx=-l/2 with a specified mean temperature and exited to the right through Bperx=l/2 in a periodic manner. Bsymz=0 and Bsymz=l/2 represent symmetry boundaries spaced l/2 apart in the z-direction.

Grahic Jump Location
Fig. 3

Dimensionless temperature profile for φs = 0.01 (a) in the channel and (b) a magnified view near the surface

Grahic Jump Location
Fig. 4

The numerical thermal slip length data (•) are compared to the analytical conduction solution, Eq. (29) (solid curve). The predicted behavior of ridges, Eq. (31), is shown by the dashed curve.

Grahic Jump Location
Fig. 5

Comparison of our numerical hydrodynamic slip length data (▲) to the asymptotic expression of Davis and Lauga [21], b/l=(3π/16)φs-1/2-0.421 (dotted–dashed curve), Eq. (39) (dashed curve), and Eq. (41) (solid curve) as a function of (a) φs-1/2 and (b) φs. The inset of (a) shows the ratio of the slip length predicted by Davis and Lauga [21] (dashed curve) and Eq. (37) (solid curve) to Eq. (41). The numerical data of Ng and Wang [23] (open circles/squares corresponding to circular/square pillars) and Ybert et al. [14] (filled squares corresponding to square pillars) are also shown in (b).

Grahic Jump Location
Fig. 6

(a) Relationship between the thermal and hydrodynamic slip lengths. The numerical data (▲) are compared to the solid line which corresponds to the analytical prediction for pillars. The dashed and dashed–dotted curves correspond to the analytical prediction for ridges aligned parallel and transverse to the channel flow, respectively. (b) Nusselt number Nu and (c) the thermohydraulic measure Φ/Φc plotted as a function of b˜ for pillars (b˜t≈1.5b˜) (solid curve), parallel ridges (b˜t≈1.05b˜) (dashed curve) and transverse ridges (b˜t≈2.1b˜) (dotted–dashed curve) in the Stokes flow limit. Also shown is the limit of b˜t=0 (blue dashed curve) and a range of 0≤b˜t≤2.5b* in increments of 0.25b˜t (light dotted curves).

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