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Research Papers: Evaporation, Boiling, and Condensation

Mathematical Model for Dropwise Condensation on a Surface With Wettability Gradient

[+] Author and Article Information
Manjinder Singh

Department of Mechanical Engineering,
Indian Institute of Technology Delhi,
Hauz Khas 110016, New Delhi, India

Sasidhar Kondaraju

School of Mechanical Sciences,
Indian Institute of Technology Bhubaneswar,
Bhubaneswar 751013, Orrisa, India

Supreet Singh Bahga

Department of Mechanical Engineering,
Indian Institute of Technology Delhi,
Hauz Khas 110016, New Delhi, India
e-mail: bahga@mech.iitd.ac.in

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received May 5, 2017; final manuscript received January 4, 2018; published online April 6, 2018. Assoc. Editor: Debjyoti Banerjee.

J. Heat Transfer 140(7), 071502 (Apr 06, 2018) (8 pages) Paper No: HT-17-1248; doi: 10.1115/1.4039014 History: Received May 05, 2017; Revised January 04, 2018

We present a mathematical model for dropwise condensation (DWC) heat transfer on a surface with wettability gradient. We adapt well-established population balance model for DWC on inclined surfaces to model DWC on a surface with wettability gradient. In particular, our model takes into account the effect of wettability gradient and energy released during drop coalescence to determine the drop departure size. We validate our model with published experimental data of DWC heat flux and drop size distribution. Based on various experimental studies on drop motion, we also propose a mechanism that explains how the energy released during drop coalescence on a surface with wettability gradient and in a condensation environment aids drop motion. The mechanism correctly explains the shift of center of mass of two coalescing drops on a surface with wettability gradient toward the drop on high wetting region. Using the model, we analyze the effect of wettability gradient on the DWC heat flux. Our model predictions show that the optimal choice of wettability gradient is governed by differential variations in population density and heat transfer through a drop with change in wettability of the surface. We also demonstrate that contact angle at which there is maximum heat transfer through a drop varies with thickness of coating layer leading to change in optimal wettability gradient.

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Figures

Grahic Jump Location
Fig. 1

Schematic showing drop size distribution in DWC. Drop size distribution is divided Into two phases small drops n(r) and large drops N(r). The drop radius re denotes boundary between small and large drops.

Grahic Jump Location
Fig. 2

Schematic illustrating the mechanism of coalescence of two drops on a surface with wettability gradient. P denotes Laplace pressure and rb denotes base radius of the drop. Solid lines show dynamic contact angles at respective drop edges (θd,A,θd,B,θd,D, and θd,E). The dashed lines denote receding (∠b′BA, ∠a′AB, and ∠e′ED) and advancing (∠bBA, ∠aAB, and ∠dDE) contact angle at respective drop edges. (a) Shifting of drop shape from staggered to nearly spherical cap shape on a surface with wettability gradient in the absence of condensation. (b) State of a drop on a surface with wettability gradient before coalescence. (c) Formation of bridge with very small and negative interface radius of curvature at the initiation of coalescence. Larger influx of liquid from the left drop causes contact angle at A to reduce and bridge to grow. When the contact angle at A reduces to the receding value whole of the liquid from left drop flows into the right drop. (d) State of drop just after coalescence. During coalescence, due to out flux of liquid, θd,A becomes smaller than receding contact, which is reflected at the edge E of the merged droplet. While at D, due to influx of liquid θd,D becomes greater than advancing contact angle: (a) drop on a surface with wettability gradient, (b) before coalescence, (c) during coalescence, and (d) after coalescence.

Grahic Jump Location
Fig. 3

Comparison of heat flux predicted by our mathematical model with the experimental results [10,21] as a function degree of subcooling. The calculations are based on nucleation site density, Ns=1×1010m−2, Tsat = 373 K, δ = 100 nm.

Grahic Jump Location
Fig. 4

Comparison of drop size distribution predicted by our mathematical model with the experimental results of Macner et al. [11]. The distributions are plotted for drops with radius, r≥ 80 μm. These calculations are performed for nucleation site density, Ns=1×1010m−2, Tsat = 373 K, δ = 100 nm.

Grahic Jump Location
Fig. 5

Effect of wettability gradient on DWC heat flux and size of drop departure. (a) DWC heat flux at different wettability gradients while keeping the minimum contact angle constant. (b) Variation in size of drop departure and heat transfer through a drop with contact angle for the largest wettability gradient 20 deg–150 deg. These calculations are performed for nucleation site density, Ns=1×1010 m−2, Tsat = 373 K, δ = 100 nm.

Grahic Jump Location
Fig. 6

Effect of thickness of coating layer δ on heat transfer through a drop. These calculations are performed for degree of subcooling ΔT=2 K, r = 5 μm, Tsat = 373 K and conductivity of coating layer Kcoat=2 Wm−1K−1.

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