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Research Papers

Modeling and Optimization of Superhydrophobic Condensation

[+] Author and Article Information
Nenad Miljkovic

Department of Mechanical Engineering,
Massachusetts Institute of Technology,
77 Massachusetts Avenue,
Cambridge, MA 02139

Ryan Enright

Department of Mechanical Engineering,
Massachusetts Institute of Technology,
77 Massachusetts Avenue,
Cambridge, MA 02139;
Stokes Institute,
University of Limerick,
Limerick, Ireland

Evelyn N. Wang

Department of Mechanical Engineering,
Massachusetts Institute of Technology,
77 Massachusetts Avenue,
Cambridge, MA 02139
e-mail: enwang@mit.edu

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the Journal of Heat Transfer. Manuscript received March 31, 2012; final manuscript received September 2, 2012; published online September 23, 2013. Assoc. Editor: Sujoy Kumar Saha.

J. Heat Transfer 135(11), 111004 (Sep 23, 2013) (14 pages) Paper No: HT-12-1145; doi: 10.1115/1.4024597 History: Received March 31, 2012; Revised September 02, 2012

Superhydrophobic micro/nanostructured surfaces for dropwise condensation have recently received significant attention due to their potential to enhance heat transfer performance by shedding water droplets via coalescence-induced droplet jumping at length scales below the capillary length. However, achieving optimal surface designs for such behavior requires capturing the details of transport processes that is currently lacking. While comprehensive models have been developed for flat hydrophobic surfaces, they cannot be directly applied for condensation on micro/nanostructured surfaces due to the dynamic droplet-structure interactions. In this work, we developed a unified model for dropwise condensation on superhydrophobic structured surfaces by incorporating individual droplet heat transfer, size distribution, and wetting morphology. Two droplet size distributions were developed, which are valid for droplets undergoing coalescence-induced droplet jumping, and exhibiting either a constant or variable contact angle droplet growth. Distinct emergent droplet wetting morphologies, Cassie jumping, Cassie nonjumping, or Wenzel, were determined by coupling of the structure geometry with the nucleation density and considering local energy barriers to wetting. The model results suggest a specific range of geometries (0.5–2 μm) allowing for the formation of coalescence-induced jumping droplets with a 190% overall surface heat flux enhancement over conventional flat dropwise condensing surfaces. Subsequently, the effects of four typical self-assembled monolayer promoter coatings on overall heat flux were investigated. Surfaces exhibiting coalescence-induced droplet jumping were not sensitive (<5%) to the coating wetting characteristics (contact angle hysteresis), which was in contrast to surfaces relying on gravitational droplet removal. Furthermore, flat surfaces with low promoter coating contact angle hysteresis (<2 deg) outperformed structured superhydrophobic surfaces when the length scale of the structures was above a certain size (>2 μm). This work provides a unified model for dropwise condensation on micro/nanostructured superhydrophobic surfaces and offers guidelines for the design of structured surfaces to maximize heat transfer. Keywords: superhydrophobic condensation, jumping droplets, droplet coalescence, condensation optimization, environmental scanning electron microscopy; micro/nanoscale water condensation, condensation heat transfer.

FIGURES IN THIS ARTICLE
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Copyright © 2013 by ASME
Topics: Drops , Condensation
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References

Figures

Grahic Jump Location
Fig. 1

Schematics of the modeled structured surface showing (a) side view and (b) top view of the characteristic structure dimensions h, d, and l representing the pillar height, diameter and center-to-center spacing, respectively. Schematics showing the (c) S, (d) PW, and (e) W morphologies.

Grahic Jump Location
Fig. 2

Time-lapse schematics of (a) S, (b) PW, and (c) W droplet morphologies during growth on the structured surface. To the right of the schematics are corresponding environmental scanning electron microscopy (ESEM) images of droplets with the different morphologies on a nanostructured surface (h = 6.1 μm, l = 2 μm, d = 300 nm) [23,24]. Schematics not to scale.

Grahic Jump Location
Fig. 3

(a) Schematic of the droplet on the condensing surface growing in the PW morphology. (b) Droplet thermal resistance diagram showing the liquid–vapor interface (Ri), droplet conduction (Rd), hydrophobic promoter coating (Rhc), pillar (Rp), and gap (Rg) thermal resistances. (c) Thermal resistance network in the droplet and pillar structure. The schematic shows the parallel path of heat flowing through (i) the coating (RHC) followed by the pillar (Rp) and (ii) the liquid bridge (Rg) followed by the coating (RHC). Schematics not to scale.

Grahic Jump Location
Fig. 4

Droplet size distribution for a surface with droplet removal by gravity (flat hydrophobic surfaces) as a function of droplet radius R for various nucleation densities NS. The population density is shown for small droplets (n(R)) with color curves and large coalescing droplets (N(R)) with the black curve. Higher nucleation densities lead to earlier droplet coalescence and smaller coalescence lengths (lc = 2Re). The population of small (R < Re) noninteracting droplets is higher than large (R > Re) droplets because large droplets experience coalescence in addition to being swept off the surface. Model parameters: h = 10 μm, l = 4 μm, d = 300 nm, ΔT = Tsat− Ts = 5 K, θa/θr = 121.6 deg/86.1 deg.

Grahic Jump Location
Fig. 5

(a) Droplet population densities for surfaces exhibiting coalescence-induced droplet removal as a function of radius R for a variety of nucleation densities NS with constant contact angle ns(R) (solid lines) and variable contact angle nd(R) (dotted lines). Higher nucleation densities result in earlier droplet coalescence and smaller coalescence lengths (lc = 2Re). Inset: schematic showing coalescence length (lc). (b) Ratio of the dynamic surface heat flux qd (Eq. (46)) to the static surface heat flux qs (Eq. (45)) as a function of droplet coalescence length (lc) and structured surface pillar-to-pillar spacing (l). The shaded region includes the results for the different pillar spacings (0.5 μm < l < 8 μm). The static contact angle droplet model overpredicts the surface heat flux at small departure sizes, which shows the importance of using the dynamic contact model for predicting PW droplet performance. Model parameters: h = 10 μm, l = 4 μm, d = 300 nm, ΔT = 5 K, θa/θr = 121.6 deg/86.1 deg.

Grahic Jump Location
Fig. 6

(a) Condensing droplet apparent contact angle θ as a function of coalescence length (lc/l) and ratio of pillar diameter to center-to-center spacing (d/l). Distinct regions of differing droplet wetting morphologies exist based on the wetting criteria (Sec. 2). For d/l > 0.36, the PW droplet morphology is favored; however, droplet jumping is not possible due to the high solid fraction (φ > 0.1) and high contact line pinning to the surface structure. For lc/l < 2 (not-shown), liquid films and pinned W droplets are formed due to droplet merging within the unit cell of the structure. (b) Condensing droplet departure radius R∧ as a function of coalescence length (lc/l) and ratio of pillar diameter to center-to-center spacing (d/l). Regimes of W droplet formation have higher departure radii than PW droplets due to higher surface adhesion and contact angle hysteresis. Model parameters: h = 10 μm, l = 4 μm, d = 300 nm, ΔT = 5 K, θa/θr = 121.6 deg/86.1 deg, kHC ≈ 0.2 W/mK [28], kP = 150 W/mK, δHC = 1 nm. Insets: emergent droplet morphology schematics for each region.

Grahic Jump Location
Fig. 7

Normalized overall steady-state surface heat flux q″/qmax, as a function of coalescence length (lc/l) and ratio of pillar diameter to center-to-center spacing (d/l) for (a) h = 5 μm, (b) h = 2 μm, and (c) h = 1 μm. Scaling down the surface structure ((a) to (c)) enhances performance due to the reduced micro/nanostructure thermal resistance. Regions favoring PW jumping droplet removal show peak heat fluxes for all three cases ((a) to (c)). qmax was determined from examining the peak heat flux in all three cases, which occurred for the smallest scale structure (c), qmax = 342.12 kW/m2. Model parameters: h/l = 2, ΔT = 5 K, θa/θr = 121.6 deg/86.1 deg, kHC = 0.2 W/mK, kP = 150 W/mK, δHC = 1 nm. Insets: emergent droplet morphology schematics for each region.

Grahic Jump Location
Fig. 8

Structured surface steady-state wetting morphology as a function of the pillar diameter to center-to-center spacing ratio (d/l) and the center-to-center spacing to pillar height ratio (l/h). Scaling down the surface structure (l/h) broadens the d/l regime where PW jumping droplets are observed. Insets: emergent droplet morphology schematics for each region.

Grahic Jump Location
Fig. 9

Structured surface steady-state heat flux q″ as a function of coalescence length lc for four different promoter coatings with (a) coalescence-induced droplet jumping (no sweeping) and (b) gravitational droplet removal (sweeping). The surface heat flux is not sensitive to the promoter coating for surfaces with coalescence-induced droplet departure. Heat flux (q″) is highly dependent on the promoter coating for surfaces relying on gravity for droplet removal due to the strong dependence of droplet/surface adhesion on contact angle hysteresis. Insets: Surface heat flux (q″) as a function of temperature difference (ΔT = TsatTs) for the four different promoter coatings and model parameters: h = 5 μm, l = 2.5 μm, ΔT = 5 K, lc = 7.5 μm, kHC = 0.2 W/mK, kP = 150 W/mK, and δHC = 1 nm.

Grahic Jump Location
Fig. 10

Structured surface steady-state heat flux q″ as a function of the surface inclination angle Θ for SAM coated (SAM1, SAM2, and THIOL) structured surfaces exhibiting coalescence-induced droplet jumping (d/l = 0.3) and gravity based droplet shedding (d/l = 0.4). Jumping surfaces showed little sensitivity to the surface orientation owing to their ability to shed droplets at length scales well below the capillary length (R∧≪1 mm). Surfaces exhibiting gravity based shedding showed a strong dependence on Θ, due to the cosΘ dependence of the gravitational body force acting on the condensing droplets needed to overcome the surface tension force (Eq. (25)). Inset: condensing surface orientation schematic. Model parameters: h = 5 μm, l = 2.5 μm, ΔT = 5 K, lc = 10 μm, kHC = 0.2 W/mK, kP = 150 W/mK, and δHC = 1 nm.

Grahic Jump Location
Fig. 11

Structured (jumping) and flat (gravity shedding) surface steady-state heat flux q″ as a function of intrinsic promoter coating contact angle hysteresis Δθ for three structured surfaces coated with the SAM1 promoter. As Δθ decreases for the flat surface, q″ increases due to the lower droplet adhesion to the surface and lower departure radii (inset). As a result, the flat surfaces begin to show enhanced q″ compared to the structured surfaces. Inset: Droplet departure diameter (R∧), as a function of intrinsic flat surface contact angle hysteresis (Δθ). Model parameters: ΔT = 5 K, lc = 10 μm, kHC = 0.2 W/mK, kP = 150 W/mK, δHC = 1 nm, and SAM1 coating: θa/θr = 121.6 deg/86.1 deg.

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