10th International Conference on Boiling and Condensation Heat Transfer

12th-15th March 2018 in Nagasaki, Japan

www.icbcht2018.org

DROPLET SIZE DISTRIBUTIONS ON VERTICAL AND HORIZONTAL

SUPERHYDROPHOBIC SURFACES DURING JUMPING-DROPLET

CONDENSATION

Patrick Birbarah1, Chengpu Li1 and Nenad Miljkovic1,2,*

1 Department of Mechanical Science and Engineering, University of Illinois, Urbana, Illinois 61801, USA

2 International Institute for Carbon Neutral Energy Research (WPI-I2CNER), Kyushu University, 744 Moto-

oka, Nishi-ku, Fukuoka 819-0395, Japan

ABSTRACT

Water vapor condensation governs the efficiency of a number of important industrial processes. Jumping-droplet

condensation of water has been shown to have a 10X heat transfer enhancement compared to state-of-the-art

filmwise condensation due to the removal of condensate at much smaller length scales (~1 μm) than what is

capable with gravitational shedding (~1 mm). In order to model heat transfer performance during jumping-droplet

condensation, individual droplet heat transfer models and droplet size distributions are needed. Although heat

transfer through a condensate droplet is relatively well understood, jumping-droplet size distributions are lacking.

In this study, we develop a full numerical simulation of jumping droplet condensation on vertically and

horizontally oriented superhydrophobic surfaces. We start by simulating hydrophobic surfaces with a contact angle

of 95° in order to compare our results with the well-known distribution for hemispherical droplets undergoing

dropwise condensation [1]. Figure 1 shows time lapse images of the simulated dropwise condensation on a

hydrophobic vertical surface with a random nucleation site density of 104 sites/cm2.

Figure 1 : Droplet distribution on a vertical surface (1cm x 1cm) with a contact angle of 95°. The simulation

screenshots are taken (a) 20 ms, (b) 80 ms, (c) 100 ms and (d) 500 ms after the onset of condensation

(heterogeneous nucleation). Droplets nucleate in a spatially random fashion with a nucleation site density of 104

sites/cm2. Droplets are considered to nucleate with a nucleation radius of 10 nm and shed from the surface once

they reach radii of 500 µm.

In order to model the individual droplet growth, we utilize the a recent droplet grow model [2], with a surface

temperature of 15°C and water saturation temperature of 24°C, neglecting the effects of roughness features for

hydrophobic surfaces. The correlations for the droplet growth rate were fitted in a form of d𝑅/d𝑡 = 𝐴/𝑅𝐵 where

𝑅 is the droplet radius, 𝑡 is time, and 𝐴 = 0.015 and 𝐵 = 0.3 are the fitted parameters. The initial nucleation

radius was assumed to be 10 nm in accordance with heterogeneous nucleation theory on hydrophobic substrates

[3]. The droplets are assumed to fall due to gravitational force after reaching a radius of 0.5 mm. The growth and

coalescence are achieved sequentially through the algorithm, with the falling droplets merging with stationary

droplets their way down. Time steps of 10 ms are considered ion order to allow for sufficient growth resolution

without inducing substantial computational cost. Figure 2 shows the computed droplet distribution function 𝑓

[drops/cm3] as obtained from two simulations (with a density of 103 and 104 nucleation sites/ cm2) along with the

Rose distribution, 𝑓ROSE =1

3𝜋𝑅2𝑟0(𝑅

𝑟0)

−2/3 with 𝑟0 = 0.45 mm being the maiimum radius of a generation of

droplets (falling radius).

Figure 2 : Time averaged droplet distribution density function 𝑓 during dropwise condensation on a vertical

surface. The average was taken for a simulated real condensation time of 1 minute. Droplets having radii 𝑅

nucleate in a spatially random fashion (103-104 sites/cm2) for a 9°C supersaturation, with an advancing contact

angle of 95 degrees. The simulation is compared to the distribution presented by Rose [1] We observe that the

nucleation site density is an important factor in the convergence of the distribution functions to the Rose

distribution. The decreased slope for smaller droplets in both distributions is due to droplet growth dominated

distribution (average spacing of 300 µm for 103 sites/cm2 and 100 µm for 104 sites/cm2) as opposed to

coalescence dominated distribution as valid for the Rose distribution.

The effects of nucleation density, superhydrophobicity of the surface, supersaturation, and heat flui are coupled

to determine the transient and steady state droplet size distribution, frequency of droplet departure, and overall

heat transfer. Importantly, localized effects including localized droplets beneath larger spherical droplets are

accounted for, which are not observable in optical microscopy studies. While comparing randomized nucleation

with controlled nucleation, we consider droplet return to the surface and predict flooding behavior for different

surfaces and heat fluies. This work not only theoretically and eiperimentally develops the steady-state droplet

size distribution of jumping droplet surfaces, it elucidates the importance of localized effects during the

condensation process that cannot be observed with eiperimental imaging techniques currently employed.

NOMENCLATURE R Droplet radius [m] A, B Fitting coefficients t Time [s] r0 Maiimum droplet radius on surface [m] f Droplet number density [drops/m3]

REFERENCES [1] Rose, J.W. and Glicksma.Lr, Dropwise Condensation - Distribution of Drop Sizes. International Journal

of Heat and Mass Transfer, 1973. 16(2): p. 411-425.

[2] Chavan, S., et al., Heat Transfer through a Condensate Droplet on Hydrophobic and Nanostructured

Superhydrophobic Surfaces. Langmuir, 2016. 32(31): p. 7774-7787.

[3] Carey, V.P., Liquid-vapor phase-change phenomena : an introduction to the thermophysics of

vaporization and condensation processes in heat transfer equipment. 2nd ed. 2008, New York: Taylor and

Francis. iiii, 742 p.