1

Supporting Information

Heat Transfer through a Condensate Droplet on

Hydrophobic and Nanostructured Superhydrophobic

Surfaces

Shreyas Chavan,1 Hyeongyun Cha,

1,2 Daniel Orejon,

2 Kashif Nawaz,

3 Nitish

Singla,1 Yip Fun Yeung,

1 Deokgeun Park,

1 Dong Hoon Kang,

1 Yujin Chang,

1

Yasuyuki Takata,2 and Nenad Miljkovic

1,2,*

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

61801, USA

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

University, 744 Moto-oka, Nishi-ku, Fukuoka 819-0395, Japan

3Heat Transfer Center of Excellence, Johnson Controls, Norman, 73069, USA

*Corresponding Author E-mail: nmiljkov@illinois.edu

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S1. ANSYS Model

To study individual droplet heat transfer, a 2D axisymmetric numerical model based on

the finite element method was used to solve the heat equation through a single droplet. The

ANSYS steady-state thermal analysis module was used to model the droplet, with quadrilateral

meshing as shown in Figure S1 with 10,000 nodes. The ratio of maximum element size to the

droplet radius was set at 0.015. Mesh refinement was used at the liquid-vapor interface and

solid-liquid interface (ratio of element size to the droplet radius ≈ 0.008). Mesh at the three

phase contact line was further refined (ratio of element size to the droplet radius ≈ 0.004) to

resolve the large temperature gradients present there. To reduce computation time, a 2D

axisymmetric model was simulated.

ba

3

FIGURE S1. Quadrilateral mesh used to model droplet heat transfer with ratio of maximum element size

to droplet radius of (a) 0.06, and (b) 0.015. The differing ratios were used to prove grid independence and

convergence of the numerical solution

The ratio of maximum element size to droplet radius used in all the simulations was

0.015. Ratios smaller than 0.015 resulted in < 1% change in heat transfer, as shown in Figure

S2.

FIGURE S2. Individual droplet heat transfer, 𝑄, as a function of mesh size normalized by the

droplet radius for 𝜃𝑎 = 140˚, 𝐵𝑖 = 100, 𝑅 = 0.243 mm and 𝛼 = 0.04

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S2. Non Dimensional Analysis

Consider the system of a condensing water droplet on a superhydrophobic surface as

shown in Figure S3. Using Buckingham Pi theorem,1 we can reduce the number of explicit

physical variables to a reduced number of dimensionless parameters.

Figure S3. Schematic showing the simulation domain and the relevant dimensional parameters

during droplet condensation on a superhydrophobic surface. Please see the Nomenclature section

at the end of the Supplemental for symbol definitions.

The number of physical variables defining the system are 𝑚 = 7 [𝑅, 𝑇𝑠𝑎𝑡, 𝑇𝑠, 𝑞″, 𝑘𝑤, ℎ𝑖,

𝜃𝑎]. Meanwhile, the number of physical dimensions, 𝑛 = 4 [𝐿, 𝑀, 𝜃, 𝑇]. Therefore, the number

of dimensionless groups which define the system can be represented by Π = 𝑚 – 𝑛 = 3. Since θa

is already dimensionless, we define it as our first dimensionless group, Π1 = 𝜃𝑎.

For the second dimensionless group, we use 𝑅𝑏, 𝑘𝑤, ℎ𝑖, to non-dimensionalize the problem.

ℎi Wm-2

K-1

[M θ-1

T-3

]

𝑘w Wm-1

K-1

[L M θ-1

T-3

]

𝑅b m [L]

5

Using the Buckingham Π theorem, we get,

[𝑀θ−1𝑇−3][𝐿𝑀θ−1𝑇−3]𝑎[𝐿]𝑏 = 1

Solving, we obtain 𝑎 = −1, 𝑏 = 1, therefore:

Π2 = 𝐵𝑖 =ℎ𝑖𝑅𝑏

𝑘𝑤

For the third dimensionless group, we use 𝑞″, 𝑅𝑏, 𝑘𝑤, 𝛥𝑇 to non-dimensionalize:

𝑘w W/mK [L-1

M θ-1

T-3

]

𝑞″ W/m2 [M

T

-3]

𝑅b m [L]

𝛥𝑇 K [θ]

Using the Buckingham Π theorem, we get,

[𝑀𝑇−3][𝐿]𝑎[θ]𝑏[𝐿−1𝑀θ−1𝑇−3]𝑐 = 1

Solving, we obtain 𝑎 = −1, 𝑏 = −1, 𝑐 = −1

Π3 = 𝑁𝑢 =𝑞″

𝑘𝑤𝑅𝑏Δ𝑇

Using Buckingham Π theorem, we can write

Π3 = 𝑓(Π2, Π1)

Thus,

𝑁𝑢 = 𝑓(𝐵𝑖, 𝜃𝑎)

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S3. Condensation Heat Transfer Model

To study the overall steady-state condensation heat flux, we combined the simulation

results with droplet distribution theory to account for the fraction of droplets on the surface of a

given radius 𝑅 for the surfaces undergoing shedding and jumping. For small droplets (𝑅 ≤ 𝑅e),

the size distribution 𝑛(𝑅) is determined by:2

𝑛(𝑅) =1

3𝜋𝑅e3�̂�

(𝑅e

�̂�)

−23 𝑅(𝑅e − 𝑅min)

𝑅 − 𝑅min

𝐴2𝑅 + 𝐴3

𝐴2𝑅e + 𝐴3exp(𝐵1 + 𝐵2) , (S4)

Where, where �̂� is the average maximum droplet radius (departure radius), 𝑅e is the

radius when droplets growing by direct vapor addition begin to merge and grow by droplet

coalescence, 𝑅min is the critical nucleation radius for condensing droplets (≈10 nm for water).

For large droplets (𝑅 ≥ 𝑅e) growing due to coalescence, the droplet distribution 𝑁(𝑅) is

determined from:3

𝑁(𝑅) =1

3𝜋𝑅e2�̂�

(𝑅e

�̂�)

−23 (S5)

The variables 𝐴1, 𝐴2, 𝐴3, 𝐵1, 𝐵2 are constants associated with droplet sweeping, defined as:4

𝐴1 =∆𝑇

ℎ𝑓𝑔𝜌𝑤(1 − cos 𝜃)2(2 + cos 𝜃) (S6)

𝐴2 =𝜃

4𝑘𝑤 sin 𝜃 (S7)

𝐴3 =1

2ℎ𝑖(1 − cos 𝜃)+

1

𝑘𝐻𝐶 sin2 𝜃[

𝑘𝑝𝜙

𝛿𝐻𝐶𝑘𝑝 + ℎ𝑘𝐻𝐶+

𝑘𝑝(1 − 𝜙)

𝛿𝐻𝐶𝑘𝑤 + ℎ𝑘𝐻𝐶]

−1

(S8)

𝐵1 =𝐴2

𝜏𝐴1[𝑅𝑒

2 − 𝑅2

2+ 𝑅𝑚𝑖𝑛(𝑅𝑒 − 𝑅) − 𝑅𝑚𝑖𝑛

2 ln (𝑅 − 𝑅𝑚𝑖𝑛

𝑅𝑒 − 𝑅𝑚𝑖𝑛)] (S9)

𝐵2 =𝐴3

𝜏𝐴1[𝑅𝑒 − 𝑅 − 𝑅𝑚𝑖𝑛 ln (

𝑅 − 𝑅𝑚𝑖𝑛

𝑅𝑒 − 𝑅𝑚𝑖𝑛)] (S10)

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𝜏 =3𝑅𝑒

2(𝐴2𝑅𝑒 + 𝐴3)2

𝐴1(11𝐴2𝑅𝑒2 − 14𝑅𝑒𝑅𝑚𝑖𝑛 + 8𝐴3𝑅𝑒 − 11𝐴3𝑅𝑚𝑖𝑛)

(S11)

In our case, the analysis is valid for smooth hydrophobic surfaces (𝜙 = 1, ℎ = 0, 𝛿𝐻𝐶 ≈ 0)

or nanostructured superhydrophobic surfaces (ℎ ≈ 0, 𝛿𝐻𝐶 ≈ 0), 𝐴3 is defined as:

𝐴3 =1

2ℎ𝑖(1 − cos 𝜃) (S12)

The total surface steady state condensation heat flux (𝑞") is obtained by incorporating the

individual droplet heat transfer rate obtained from simulations, with the droplet size distributions

(Equation (S4) and (S5)):

𝑞" = ∫ 𝑄(𝑅)𝑛(𝑅)𝑑𝑅𝑅e

𝑅min

+ ∫ 𝑄(𝑅)𝑁(𝑅)𝑑𝑅�̂�

𝑅e

(S13)

Note, for symbol definitions, please see the Nomenclature section at the end of the

Supplemental.

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S4. Droplet Growth Studies

To provide insight into the experimental results, capture the growth dynamics related to the

different droplet morphologies, and verify the numerical simulation results, we modeled the

experimental droplet growth behavior with our developed simulations and a the state-of-art

(SoA) analytical model.2 To determine the theoretical growth rate (𝑑𝐷/𝑑𝑡 = 2𝑑𝑅/𝑑𝑡), the

individual droplet heat transfer 𝑄(𝑅, 𝜃) is related to the droplet growth rate by the latent heat of

phase change4

𝑄(𝑅, 𝜃) = �̇�ℎ𝑓𝑔 = 𝜌𝑤ℎ𝑓𝑔

𝑑𝑉

𝑑𝑡=

𝜋

3𝜌𝑤ℎ𝑓𝑔

𝑑

𝑑𝑡[(1 − cos 𝜃)2(2 + cos 𝜃)𝑅3] . (S14)

Differentiating Equation (S14), we obtain explicit term for 𝑑𝑅/𝑑𝑡

𝑄(𝑅, 𝜃) = 𝜋𝜌𝑤ℎ𝑓𝑔𝑅2𝑑𝑅

𝑑𝑡{(1 − cos2 𝜃)2 sin 𝜃

𝑑𝜃

𝑑𝑅𝑅 + (1 − cos 𝜃)2(2 + cos 𝜃)} . (S15)

The individual droplet heat transfer, 𝑄(𝑅, 𝜃) = 𝑓(𝑇s, 𝛼, 𝑃sat), was computed using our

simulation and the SoA analytical model. As shown in Figure S4, by varying the surface-to-

vapor temperature difference ∆𝑇, we can fit for:

1. The mean experimentally measured diameter ⟨𝐷⟩ (black solid line for simulation fitting

(Numerical), brown solid line for fitting SoA analytical model (Analytical mean))

2. The maximum experimentally measured diameter 𝐷max (pink solid line for simulation

fitting (Ansys max), navy blue solid line for fitting SoA analytical model (Analytical

max))

3. The minimum experimentally measured diameter 𝐷min (green solid line for simulation

fitting (Ansys min), purple solid line for fitting SoA analytical model (Analytical min))

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Other than this, Analytical droplet growth (Analytical) is plotted for ∆𝑇 which was obtained by

fitting the numerical solution to the data (Numerical). (i.e. ∆𝑇 (Numerical) = ∆𝑇 (Analytical)).

Figure S4. Time evolution of the average droplet diameter 𝐷 on the (a) superhydrophobic

regions and (b) hydrophobic regions. The surface temperature, 𝑇𝑠 = 5ºC. The values of ∆𝑇 used

for fitting are given in Table S1.

Table S1 provides the values of the surface-to-vapor temperature difference ∆𝑇 used for

fitting the data in Figure 4 for the superhydrophobic CuO surface (𝜃a = 150º for 𝐷 ≤ 7 µm, 𝜃a =

160º for 7 µm< 𝐷 ≤ 14 µm and 𝜃a = 170º for 𝐷 > 14 µm) and hydrophobic Cu surface (𝜃a =

140º), for surface temperatures 𝑇s = 10, 5, and 0ºC.

Table S1. The values of the surface-to-vapor temperature difference ∆𝑇 used for fitting the

numerical model and SoA analytical model to experimental data.

𝑻𝒔 Surface ∆𝑻𝐍𝐮𝐦𝐞𝐫𝐢𝐜𝐚𝐥 ∆𝑻𝐀𝐧𝐚𝐥𝐲𝐭𝐢𝐜𝐚𝐥

min mean max min mean max

ºC

K K K K K K

0 Cu 0.0015 0.0025 0.006 0.004 0.007 0.02

0 CuO 0.0015 0.0025 0.005 0.004 0.007 0.012

5 Cu 0.001 0.0018 0.0035 0.0035 0.006 0.01

5 CuO 0.001 0.0018 0.003 0.0025 0.005 0.008

10 Cu 0.0005 0.0015 0.001 0.0015 0.003 0.005

10 CuO 0.0006 0.0015 0.003 0.0015 0.004 0.008

a b

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It is important to note that for all experiments, fitting with the same value of ∆𝑇 (for

mean, max, and min) for both superhydrophobic and hydrophobic droplet morphologies yielded

the best numerical model fit to the experimental results. The good agreement is due to our

experimental ability to maintain the same local conditions (supersaturation) for both surfaces via

the bi-philic surface design.

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S5. Individual droplet heat transfer as a function of contact angle for a fixed droplet

volume

To provide a comparison between droplets of the same volume, Figure S5 plots the heat

transfer for an individual droplet versus droplet contact angle for a fixed volume:

Figure S5. Individual droplet heat transfer 𝑄(𝑅, 𝜃) as a function of droplet contact angle 𝜃𝑎.

The volume of water droplet is kept constant (𝑉 = 4.1888 × 10−18 m3) which corresponds to

the volume of a sphere with radius 1 µm. Since the droplet base area decreases with increasing

contact angle, the heat transfer decreases.

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Nomenclature

ℎi Heat transfer coefficient at liquid-vapor interface (m)

ℎfg latent heat of vaporization (J/kg)

�̂� effective maximum droplet radius (m)

𝑅b Droplet base radius (m)

𝑅e droplet coalescence radius (m)

𝑅g specific gas constant (J/mol K)

𝑅min minimum droplet nucleation radius (m)

𝑅t overall droplet thermal resistance (K/W)

𝑇i Temperature at liquid-vapor interface (K)

𝑇s substrate/wall temperature (K)

𝑘HC hydrophobic promoter coating thermal conductivity (W/mK)

𝑘p pillar/substrate thermal conductivity (W/m K)

𝑘w water thermal conductivity (W/m K)

𝛿HC thickness of hydrophobic coating (m)

𝜃𝑎 advancing contact angle (deg)

𝜈g water vapor specific volume (m3/kg)

𝜌w water density (kg/m3)

∆𝑇 = 𝑇𝑠𝑎𝑡 − 𝑇𝑠 surface subcooling temperature (K)

ℎ pillar height (m)

q″ heat flux through the droplet (W/m2)

𝐵𝑖 Biot Number

𝑁 large droplet population density (m-3

)

𝑁𝑢 Nusselt number

𝑅 Radius of the spherical droplet (m)

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𝑛 small droplet population density (m-3

)

𝛼 condensation coefficient

𝜃 contact angle (deg)

𝜏 sweeping period (s)

𝜙 solid fraction

References:

1. White, F. M., Fluid mechanics. 6th ed.; McGraw-Hill: New York, 2008; p xiii, 864 p. ill.

2. Kim, S.; Kim, K. J., Dropwise Condensation Modeling Suitable for Superhydrophobic

Surfaces. J Heat Transf 2011, 133 (8), 081502.

3. Le Fevre, E. J.; Rose, J. W. In A Theory of Heat Transfer by Dropwise Condensation,

Proceedings of the Third International Heat Transfer Conference, Chicago, IL, ASME:

Chicago, IL, 1966; pp 362-375.

4. Miljkovic, N.; Enright, R.; Wang, E. N., Modeling and Optimization of

Superhydrophobic Condensation. J Heat Transf 2013, 135 (11).