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Research Papers: Forced Convection

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

[+] Author and Article Information
D. Maynes

e-mail: maynes@byu.edu

V. Solovjov

Department of Mechanical Engineering,
Brigham Young University,
Provo, UT 84602

Contributed by the Heat Transfer Division of ASME for publication in the Journal of Heat Transfer. Manuscript received August 10, 2011; final manuscript received August 13, 2012; published online December 26, 2012. Assoc. Editor: Kenneth Goodson.

J. Heat Transfer 135(2), 021701 (Dec 26, 2012) (10 pages) Paper No: HT-11-1393; doi: 10.1115/1.4007429 History: Received August 10, 2011; Revised August 13, 2012

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.

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Figures

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Fig. 1

Scanning electron microscope image of a superhydrophobic surface patterned with alternating ribs and cavities with nominal dimensions of wr = 12 μm and wc = 28 μm, with a cavity depth of 20 μm

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Fig. 2

Qualitative representation of the velocity distribution at y = H/2 = η along a single rib and cavity. The insert illustrates the apparent slip velocity and slip length at a superhydrophobic surface.

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Fig. 3

Schematic illustration of liquid flowing through a superhydrophobic channel with alternating ribs and cavities aligned perpendicular to the flow direction with constant heat flux through the ribs

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Fig. 4

Variation of the coefficients a and b with Us in the relation λm = a + bm, appropriate for m > 10

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Fig. 5

Variation of the coefficients c and d in the relation GmFm(1)=cλm-d, appropriate for m  > 10

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Fig. 6

Nu versus X in the thermal entry region of a parallel-plate channel for Fc = 0, 0.5, and 0.9. Pe = 102 and Wm = 0.25 for all scenarios.

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Fig. 7

Variation in Nu (top) and θw-θm (bottom) with χ for Fc = 0.9, Wm = 0.1, and Pe ranging from 1 to 104

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Fig. 8

Variation in Nu (top) and θw-θm (bottom) with χ for Fc = 0.9, Pe = 102, and Wm ranging from 0.003 to 1

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Fig. 9

Variation in Nu (top) and θw-θm (bottom) with χ for Wm = 0.1, Pe = 102, and Fc ranging from 0.2 to 0.98

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Fig. 10

Average Nusselt numbers on the rib, Nu¯r, and for the entire rib and cavity module, Nu¯, as a function of the solid fraction (1 − Fc) at four values of Wm and for Pe = 102

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Fig. 11

Rib and cavity module averaged Nusselt number as a function of Pe for five cavity fractions and Wm = 0.1

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Fig. 12

Rib and cavity module averaged Nusselt number (left axis) as a function of solid fraction (1 − Fc) for Pe = 10, 102, 103, and 104 and Wm = 0.1. Also shown (right axis) is the normalized characteristic slip length (λ/w) for a surface at Pe < 700.

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Fig. 13

Comparison of Nu¯ as a function of Pe for the present scenario (denoted H), with numerical results from Ref. [22] where a constant wall temperature (denoted T) boundary condition was specified at Fc = 0.5 and 0.98 for Wm = 0.25

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