Numerical Simulation of Jumping-Droplet Condensation | Langmuir

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Numerical Simulation of Jumping-Droplet Condensation Patrick Birbarah, Shreyas Chavan, and Nenad Miljkovic Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.9b01253 • Publication Date (Web): 12 Jul 2019 Downloaded from pubs.acs.org on July 18, 2019

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Numerical Simulation of Jumping-Droplet Condensation

Patrick Birbarah1, Shreyas Chavan1, and Nenad Miljkovic1,2,3,4,* 1 Department

2 Department

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

of Electrical and Computer Engineering, University of Illinois, Urbana, IL, 61801, USA

3 Materials 4

Research Laboratory, University of Illinois, Urbana, Illinois 61801, USA

International Institute for Carbon Neutral Energy Research (WPI-I2CNER), Kyushu University, 744 Moto-oka, Nishi-ku, Fukuoka 819-0395, Japan

* Corresponding Author E-mail: [email protected] Address: 105 S. Mathews Avenue, Mechanical Engineering Laboratory, Room 2136, Urbana, IL 61801, USA

1 ACS Paragon Plus Environment

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Abstract Jumping-droplet condensation has been shown to enhance the heat transfer performance ( ≈ 100%) when compared to dropwise condensation by reducing the time-averaged droplet size ( ≈ 10 μm) on the condensing surface. Here, we develop a rigorous, three dimensional numerical simulation of jumping-droplet condensation in order to compute the steady-state time-averaged droplet size distribution. In order to characterize the criteria for achieving steady-state, we use maximum radii (𝑅max) tracking on the surface, showing that 𝑅max settles to an average in time once steady-state is reached. The effects of the minimum jumping radius (0.1µm – 10 µm), maximum jumping radius, apparent advancing contact angle (150° – 175°), and droplet growth rate were investigated. We provide a numerical fit for the jumping droplet condensation size distribution with an overall correlation coefficient greater than 0.995. The heat transfer performance was evaluated as a function of contact angle and hydrophobic coating thickness, showing excellent agreement with prior experimentally measured values. Our simulations uncovered that droplet size mismatch during coalescence has the potential to impede the achievement of steady state, and describe a new flooding mechanism for jumping droplet condensation. Our work not only develops a unified numerical model for jumping droplet condensation that is extendable to a plethora of other conditions, it demonstrates design criteria for non-wetting surface manufacture for enhanced jumping-droplet condensation heat transfer.

Keywords: Jumping droplet, distribution, droplet size, condensation, simulation, heat transfer


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Introduction Dropwise condensation has been widely studied since its discovery in 1930 1. Dropwise condensation on non-wetting surfaces has the potential to enhance heat transfer performance by up to 2000% when compared to filmwise condensation on wetting surfaces 2-5. A more recent study revealed that dropwise condensation on suitably-designed low-adhesion superhydrophobic surfaces has the potential to achieve even higher heat transfer coefficients, approaching 100% higher than observed on hydrophobic dropwise condensing substrates

6-7.

The heat transfer

enhancement stems from the coalescence-induced jumping of condensate droplets from the surface 8-9. Termed “jumping-droplet condensation”, the coalescence-induced removal of condensate leads

to a lower average droplet size residing on the surface, and hence lower vapor-to-surface thermal resistance10-12. For thin hydrophobic coatings (< 1 μm, 𝑘 ≈ 0.2 W/m·K), smaller droplets (100 nm to 10 μm in radii) contribute more to the condensation heat transfer coefficient since their conduction thermal resistance is lower as compared to larger droplets (>10 μm to 1mm in radii). However, the overall heat flux through the condensing surface is a weighted average of the heat fluxes through individual droplets, with the weight being the covered area per size range. Hence, in order to characterize the overall surface heat transfer for dropwise and jumping-droplet condensation, the droplet size distribution must be known. For dropwise condensation on hydrophobic surfaces, the steady state distribution of droplet sizes has been studied by Rose et al.

13

for the case of droplets growing mainly by coalescence.

The droplet distribution has been verified experimentally and numerically

14.

While the Rose

distribution was originally derived through empirical observations, it has also been derived separately through fractal theory 15. For droplets smaller than the coalescence radius, Abu Orabi devised the analytical population balance model to predict the size distribution function 16-17. The


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derivation relied on the Rose distribution for the boundary conditions at the coalescence radius. However, the population balance theory has not been verified experimentally or computationally. As for the distribution provided by Rose, although valid for classical dropwise condensation, it cannot be applied to, and does not reflect the physics governing, jumping-droplet condensation. Specifically: 1) sweeping does not take place, as droplet jumping is the main mechanism for droplet removal. 2) For larger apparent contact angles that are observed on superhydrophobic surfaces (𝜃 > 150°), small droplets can grow in the geometric shadow of larger droplets, a process not possible on hydrophobic surfaces (𝜃 ≈ 90°). And 3) the main growth mechanism of droplets is via direct condensation with only a fraction of growth occurring via coalescence. Prior works have attempted to derive an analytical expression for the droplet size distribution for jumping-droplet condensation 9. In addition to the uncertainty related to the population balance theory on which these derivations relied, three main limitations exist: 1) the assumption that droplets are removed exactly when their radius reaches half the coalescence radius defined by the average spacing between nucleation sites. Although the coalescence radius assumption is exact for non-random nucleation site distributions (e.g. square lattices, close-packed arrays, etc…), the effective departure radius for randomly distributed droplets differs from that of well-ordered distributions. 2) For jumping droplet condensation, characterized by droplets having large apparent contact angles (𝜃 > 150°), small droplets can grow in the geometric shadow of larger droplets, a process not captured by past analytical solutions. 3) Droplet size mismatch is an important parameter that determines whether droplets jump or coalesce and remain on the condensing surface, a physical phenomenon currently not tractable with analytical solutions. Here, we provide a comprehensive analysis of droplet size distributions during jumpingdroplet condensation. We start by rigorously examining the criteria for achieving steady-state


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condensation. After achieving steady-state, the droplet size distribution was obtained for several nucleation site densities. The effect of number of droplets simulated, droplet growth rate, apparent advancing contact angle, minimum jumping radius, and maximum jumping radius was studied. A numerical correlation for the droplet distribution function is provided and the overall heat transfer on the condensing surface was quantified. Furthermore, we studied the effect of droplet size mismatch during coalescence on the achievement of steady state and present a previously unidentified mechanism for flooding. This study develops a unified numerical model for jumping droplet condensation that is extendable to a plethora of other condensing conditions.


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Simulation Variables The variables considered for our simulation are summarized in Tables 1 and 2. Table 1. Simulation variables, nomenclature, and values of constants. Name Total covered area Area of 3D coalescences Constant relating coalescing droplets sizes Time step

Symbol (value) 𝐴c 𝐴coa,3D 𝐶g

d𝑡

Latent heat of vaporization of water

ℎfg ≈ 2300 kJ/kg

Combined interfacial and droplet heat transfer coefficient

ℎi + drop

Thermal conductivity of liquid water

𝑘w ≈ 0.6 W/(m·K)

Number of nucleation sites Number of 3D coalescences Overall condensation heat flux

𝑁0 𝑁coa,3D 𝑞′′

Droplet Radius

𝑅

Specific gas constant

𝑅g

Minimum jumping droplet radius Time Saturation temperature Surface tension (liquid-vapor)

𝑅jump,min 𝑡 𝑇sat 𝛾 = 72 mN/m

Name Area of 2D coalescences

Symbol (value) 𝐴coa,2D

Total projected area

𝐴p

Constant expressing the ratio 𝑅max/s

𝐶m

Droplet size distribution density function Liquid vapor interfacial heat transfer coefficient18 Thermal conductivity of hydrophobic coating Side length of condensing (square) surface Number of 2D coalescences

𝑓

ℎi ≈ 0.4 MW/ (m2 ∙ K) 18

𝑘coat ≈ 0.2 W/(m·K)

𝐿 𝑁coa,2D

Number of jumps

𝑁j

Droplet heat transfer

𝑞d

Average radius of large coalescing droplets Maximum jumping droplet radius Maximum droplet radius on the surface Temperature of condensing surface Condensation accommodation coefficient Thickness of hydrophobic coating


𝑅coalesce 𝑅jump,max 𝑅max 𝑇s 𝛼 𝛿

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Apparent advancing contact angle

𝜃

Water vapor density

𝜌v

Thermal resistance

𝜓

Droplet conduction thermal resistance

𝜓drop

Liquid water density

𝜌l ≈ 998 kg/m3

Standard deviation (population) of the radius of large coalescing droplets Coating thermal resistance Interfacial thermal resistance

𝜎R,coalesce

𝜓coat 𝜓i

Table 2. Equation definitions of key simulation variables. Name

Symbol (Formula)

Number density of nucleation sites

𝑁s =

Average spacing between random nucleation sites 19

𝑠=

Coalescence radius

𝑁0 𝐿2 1 4𝑁s

𝜎s =

(1-a) (1-b)

𝑠 2

(1-c)

(4 ― 𝜋) 4𝜋𝑁s

(1-d)

𝑅c =

Standard deviation for site spacing 19

Equation number

Droplet radius of the largest droplet present on the surface

𝑅min = 𝜌lℎfg∆𝑇 ≈ 10 nm

(1-e)

Condensation temperature potential

∆𝑇 = 𝑇𝑠𝑎𝑡 ― 𝑇s

(1-f)

Droplet coalescence mismatch parameter

𝑚=

Biot number

Bi =

Nusselt number

Nu =

2𝛾𝑇sat

𝜎R,coalesce 𝑅coalesce ℎi 𝑅sin 𝜃𝑘w ℎi + drop 𝑘w𝑅sin 𝜃


(1-g) (1-h) (1-i)

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Individual Droplet Growth Rate The growth rate of a single droplet condensing on a non-wetting surface is modeled with a thermal resistance approach 16. The thermal resistances (𝜓) include the liquid-vapor interfacial resistance (eq. 2), droplet conduction resistance (eq. 3), and hydrophobic coating resistance (eq. 4), along with the temperature drop due to droplet curvature:

𝜓i =

1 ℎi2𝜋𝑅2(1 ― cos 𝜃)

𝜓drop = 𝜓coat =

,

𝜃 , 4𝜋𝑅𝑘wsin 𝜃 𝛿

𝑘coat𝜋𝑅2sin2 𝜃

(2)

(3)

.

(4)

Figure 1 depicts the thermal resistance breakdown. For small droplets (𝑅 < 100 nm), the coating thermal resistance dominates, whereas at larger length scales (𝑅 > 1 mm), the droplet conduction thermal resistance dominates. The coating thickness (𝛿) is an important parameter that affects the thermal resistance circuit and is chosen here to be 1 μm, with a thermal conductivity of 0.2 W/(m·K) 16. The temperature drop due to droplet curvature is only important for very small droplets (𝑅 ~ 𝑅min < 100 nm) and is accounted for in the droplet growth rate calculation. The resistance of the micro-air pockets between droplets and the condensing surface for a particular surface architecture (feature shape, height, spacing and solid fraction) was not considered here. However, the coating thickness has been used as a variable parameter that reflects the total thermal resistance between the droplet and the surface, including any porous effects.


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Figure 1. Thermal resistance (𝜓𝜋𝑅2) of a droplet having radius, 𝑅, condensing on a hydrophobic surface as a function of the apparent advancing contact angle, 𝜃. The thermal resistances depicted are the (a) droplet conduction resistance (𝜓𝑑𝑟𝑜𝑝𝜋𝑅2, eq. 3), and (b) hydrophobic coating resistance (𝜓𝑐𝑜𝑎𝑡𝜋𝑅2, eq. 4) with the liquid-vapor interface resistance shown (𝜓𝑖𝜋𝑅2, eq.2).

The overall heat transfer rate through a droplet can be determined from the vapor-to-surface temperature difference (∆𝑇) and the sum of the thermal resistances, which has been shown previously to be 16: ∆𝑇𝜋𝑅2(1 ― 𝑅min 𝑅) 𝑞d, Kim =

(

)

𝛿

𝑅𝜃 1 + + 𝑘coatsin2 𝜃 4𝑘wsin 𝜃 2ℎi(1 ― cos 𝜃)

. (5)

The heat transfer through a droplet can also be calculated from the enthalpy dissipated by the condensing vapor: d𝑅 . d𝑡

𝑞d = 𝜌lℎfg𝜋𝑅2(1 ― cos 𝜃)2(2 + cos 𝜃)

(6)

In order to determine the interfacial heat transfer coefficient, we used the kinetic theory formulation 9, 18, 20:


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𝜌vℎfg2 2𝛼 1 ℎi = , 2 ― 𝛼 2𝜋𝑅g𝑇s 𝑇s

(7)

where 𝛼 = 1 is the condensation accommodation coefficient 9, 16. At ambient conditions in a pure saturated water vapor, ℎi ≈ 0.4 MW/(m2·K) 18, and is held constant throughout the simulations. Recent simulations have suggested that the isotherm model provided in the aforementioned analysis for droplet conduction resistance (eq. 3) is inaccurate and results in an infinite heat flux at the contact line

21.

A more recent work proposed a modified correlation for the interfacial

resistance and the droplet conduction thermal resistance. The non-dimensional correlation represents the Nusselt number (Nu, eq. 1-i) as a function of Biot number (Bi, eq. 1-h) and the apparent advancing contact angle (𝜃). The modified individual droplet heat transfer becomes: ∆𝑇𝜋𝑅2(1 ― 𝑅min 𝑅) 𝑞d, Chavan =

(

𝛿

𝜋𝑅 + 𝑘coatsin2 𝜃 Nu(𝜃,𝐵𝑖)𝑘wsin 𝜃

)

. (8)

In order to determine the individual droplet growth rate, eq. (5) or eq. (8) were equated with eq. (6). The parameters used in the model were ∆𝑇 = 10 K, 𝑅min = 10 nm, 𝛿 = 1 μm, 𝑘w = 0.6 W/m·K, ℎi = 0.4 MW/m2·K, 𝜌l = 1000 kg/m3, ℎfg = 2.3 MJ/kg, 𝑘coat = 0.2 W/m·K. The apparent advancing contact angle was 𝜃 = 175°, unless explicitly noted. Figure 2 depicts the two independent droplet growth rate calculations. While the approaches agree with each other for small droplets (𝑅 < 10 μm), they differ for larger droplets (𝑅 > 10 μm). For our current study, eq. (8) was used to calculate the individual growth rate during simulations.


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Figure 2. Individual droplet growth rate (𝑑𝑅/𝑑𝑡) as a function of droplet radius 𝑅. The heat transfer through the droplet was calculated via eq. (5) for the thermal resistance model (Kim & Kim,16) and via eq. (8) for the more recent model (Chavan, 21). Model parameters: ∆𝑇 = 10 K, 𝑅min = 10 nm, 𝛿 = 1 μm, 𝑘w = 0.6 W/m·K, ℎi = 0.4 MW/m2·K, 𝜌l = 1000 kg/m3, ℎfg = 2.3 MJ/kg, 𝑘coat = 0.2 W/m·K, and 𝜃 = 175°.


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The Simulation The simulation relies on the following main assumptions: 1. Condensation in a pure vapor environment devoid of any non-condensable gases. 2. Droplet nucleation sites are randomly distributed on the surface, and are fixed throughout the simulation. 3. Droplets jump upon coalescence with any neighboring droplet. This assumption is investigated in a later section in our study where the effect of coalescing droplets size mismatch is included. 4. Droplet jumping is observed for a range of droplet sizes bound by the minimum jumping radius 𝑅jump,min and the maximum jumping radius 𝑅jump,max. 5. Droplets do not return to the condensing surface after jumping. 6. Droplets are small enough that no gravitational shedding is present. To verify this assumption, computation of the Bond number revealed (Bo = ∆𝜌𝑔𝑅2/𝛾 < 0.01 for 𝑅 = 100 µm, the largest droplet size considered in our simulations. 7. The condensing surface is simulated as a flat surface with droplets taking on an apparent advancing contact angle throughout the simulation. 8. The superhydrophobic surface does not degrade over time. i.e. the apparent advancing contact angle of water on the surface is constant throughout the simulation. The simulation begins by randomly distributing the number of condensate nucleation sites, 𝑁0, on the surface (Figure 3). The nucleation sites are defined as a randomly assigned array of coordinates in the frame of the condensing surface. Next, we initiated a droplet on each nucleation site with a size equivalent to the critical nucleation radius as specified by eq. (1-e). Prior to iterating in time, the time step was chosen such that the product of the maximum growth rate and the time step


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resulted in a change of radius equivalent to a tenth of the average spacing between droplets, or a fifth of the average coalescence radius. The finite time step guaranteed adequate granularity to access the direct droplet growth mode prior to coalescence. After initiating time iteration, coalescence checks were made for each iteration such that clusters of droplets that intersect were captured. From the intersecting droplets, the radius of the resulting coalesced droplet was calculated via conservation of mass. If the calculated radius (post-coalescence) resided between the minimum jumping radius (𝑅jump,min = 1 μm) and the maximum jumping radius (𝑅jump,max = 100 μm) 22 then jumping occurred and the coalescing droplets were removed from the surface. At each iteration, the number of jumping events (𝑁j) was recorded. The next step in each iteration was the growth of droplets and droplet re-nucleation. For re-nucleation, we checked whether the initial nucleation sites were covered by existing droplets. If a nucleation site was “free” (i.e. not covered), then a droplet with radius 𝑅min was seeded at that location. The seeding step marked the end of each iteration. The algorithm was repeated until the number of jumping events had reached a specified value. Figure 3 shows a schematic breakdown of the algorithms used in the simulation. Figure 4 shows representative top-view snapshots of the surface at multiples of the time step, 𝑑𝑡. The outlined regions mark jumping events that occurred due to coalescence on the superhydrophobic surface at consecutives time steps. It is important to note, although Figure 4 shows a top view two-dimensional projection, the numerical model was implemented in threedimensions. Figure 5 shows representative three-dimensional isometric snapshots during condensation on the superhydrophobic surface with a higher magnification than Figure 4 (see Video S1). The simulation Matlab codes are supplied in the Supporting Information.


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Figure 3. Flow chart of the numerical algorithm used to implement the jumping droplet simulation.


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Figure 4. Time-lapse top-view images of simulated jumping droplet condensation. Black represents liquid condensate while white represents the superhydrophobic surface. Snapshots are taken at different multiples of the time step (𝑑𝑡 = 0.3 ms for this simulation). Areas outlined in dotted red boxes reveal droplet jumping. Simulation parameters: 𝑁0 = 5000 sites, 𝐿 = 2.2 mm, 𝜃 = 175°.


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Figure 5. Three dimensional time-lapse images of jumping droplet condensation simulation showing (a) small droplets shortly after the start of the simulation (5𝑑𝑡), and (b) larger droplets at 25𝑑𝑡. Images shown in (c) and (d) were taken within 2 time steps from each other at steady state (>200𝑑𝑡). Areas outlined in the red solid square depict jumping droplets that disappear between (c) and (d). Simulation parameters: 𝑁0 = 100, 𝐿 = 50 μm, 𝜃 = 170°, and 𝑑𝑡 = 0.3 ms. For a isometric view of the simulation, see Video S1 of the Supporting Information.


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Steady State Criterion We hypothesize that a way of identifying steady-state during jumping-droplet condensation is by checking the maximum droplet radius on the surface. If the maximum radius reaches a constant value that is bounded, we know that droplets on average are not growing anymore. An alternate, and much more time consuming method to check whether steady-state has been reached is to check if the droplet distribution is changing over time, by calculating a standard cumulative error. The latter method, however, lacks physical insight compared to the former, hence we tracked the maximum droplet radius on the surface as an indicator of achieving steady state, and double checked the result with the obtained distribution. Figure 6 depicts the maximum droplet size (𝑅max) obtained as an ensemble average over 50 simulations as a function of the number of jumping events (𝑁j) that have occurred on the condensing surface (1 mm2 with 𝑁0 = 1,000 sites). The simulation used the droplet growth rate formulation developed in the previous section with 𝜃 = 175°, and droplet mismatch not considered (i.e. all coalescing droplets within the correct size range jump). The cumulative average and the right-half (RH) cumulative average were plotted as metrics for convergence. The RH cumulative average was calculated from half the data points taken from the right side as opposed to the whole dataset, both in cumulative terms. It was observed, as expected, that the RH cumulative average converged faster than the cumulative average since the starting set of data (left half) was dispatched during the calculation. The number of jumping events to steady state, however, varies when the nucleation site density changes and a check for the achievement of steady state is performed for each case.


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Figure 6. Steady state convergence of the maximum droplet size 𝑅𝑚𝑎𝑥, taken as an ensemble average over 50 simulations, as a function of the (a) number of jumping events (𝑁𝑗) that have occurred on the condensing surface having dimensions 1 mm2 with 1,000 nucleation sites, 𝜃 = 175°. The cumulative and right half (RH) cumulative time averages are plotted as metrics for convergence. The RH cumulative average was calculated from half the data points taken from the right side as opposed to the whole dataset, both in cumulative terms. The average spacing, 𝑠, between nucleation sites was calculated from Eq. 1-b.


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Effect of Number of Simulated Sites The initial number of droplets simulated (number of nucleation sites), 𝑁0, was varied for a constant number density (changing condensing area), in order to determine its effect on the jumping droplet distribution. We observed that increasing 𝑁0 from 1,000 to 10,000 increased the maximum droplet radius on the surface. Figure 7 shows the normalized maximum droplet radius 𝑅max as a function of the normalized number of jumping events (representing the time axis). For a nucleation site density of 𝑁s = 103 sites/mm2, the steady state average of 𝑅max increased by 19% as 𝑁0 is increased from 1,000 to 5,000 nucleation sites. Furthermore, the steady state average of 𝑅max increased by 35% as 𝑁0 was increased from 1,000 to 10,000 sites. The increase in 𝑅max was justifiable since 𝑅max is correlated with the maximum spacing between two nearest droplets as defined by the number of nucleation sites. As the sample size was increased, a higher probability for deviation from the mean occurred. However, the distribution of nucleation site spacing becomes more narrow at higher number densities (as shown by the decreasing standard deviation in eq. 1-d) 19. Nevertheless, the effect of the number of sites simulated on the overall droplet size distribution was shown to be negligible even at lower nucleation densities (𝑁s = 103 sites/mm2).


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Figure 7. Normalized maximum droplet radius 𝑅𝑚𝑎𝑥/𝑠 as a function of time, as represented by the number of jumping events normalized by the number of nucleation sites simulated, 𝑁𝑗/𝑁0. The curves are plotted for two different number densities (𝑁𝑠 = 103 sites/mm2), with variation of the number of simulated sites (𝑁0 = 1,000, 5,000, and 10,000). For both nucleation densities, 𝑅𝑚𝑎𝑥 remained below four times the average spacing, 𝑠, with an increasing value correlated with the increase in sample size.


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Droplet Size Distribution Function After determining the criteria for achieving steady-state jumping-droplet condensation, we performed simulations with 𝑁0 = 1,000 sites. We simulated jumping droplet condensation to study the droplet size distribution for a variety of conditions. Effect of Nucleation Site Density Figure 8 shows the droplet size distribution (𝑓) as a function of droplet radius for varying nucleation site densities (𝑁s). To achieve a more physical comparison, we non-dimensionalized the curves by normalizing the droplet radius (𝑅) by the characteristic length scale (𝑠, eq. 1-b, Table 2) and the distribution function (𝑓) by 𝑁s/𝑠. Figure 8 shows that the droplet population has a fractal-like behavior. While the curves for 𝑁s = 104 and 105 sites/mm2 show similar behavior in terms of their decay characteristics, distributions with larger nucleation densities increasingly deviated from the norm (Figure 8, inset). For 𝑁s = 104 and 105 sites/mm2, the left side of the distribution function (𝑅/𝑠 < 0.1) asymptoted to a constant value. However, for 𝑁s = 106 and 107 sites/mm2, the asymptotic behavior was not attained since the coalescence radius (𝑠) approached the critical nucleation radius (𝑅min). The tail of the distribution also changed at higher droplet densities due to the minimum jumping radius criteria. To gain more insight into the distribution, we plotted the normalized maximum radius (𝑅max/𝑠) as a function of the number of jumping events (𝑁j) (Figure 9). We observed that increasing the nucleation site density from 104 to 106 sites/mm2 slightly increased the coalescence based growth rate (slope of the straight line) 13. Furthermore, the increase in nucleation site density tended to decrease 𝑅max/𝑠, as expected, due to the decreasing standard deviation of site spacing (eq. 1-d). The highest nucleation density (107 sites/mm2) revealed a different behavior that will be explored in the next section.


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Figure 8. Non-dimensional droplet size distribution (𝑓𝑠/𝑁𝑠) as a function of the normalized droplet radius (𝑅/𝑠). Inset: Dimensional droplet size distribution functions (𝑓) as a function of droplet radius (𝑅) prior to non-dimensionalization. Error bars arise due to the random error (standard deviation) obtained by repeating the simulation 50 times and have been made partially transparent for clarity.

Figure 9. Normalized maximum droplet radius (𝑅𝑚𝑎𝑥/𝑠) as a function of the number of jumping events (𝑁𝑗) for different nucleation site densities (𝑁𝑠). The number of simulated sites was set to 𝑁0 = 1000 droplets for all simulations. 22 ACS Paragon Plus Environment

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Effect of Minimum Jumping Droplet Radius The minimum jumping droplet radius (𝑅jump,min) was varied in order to study how it effects the droplet size distribution. Physically, 𝑅jump,min has been shown to depend on the droplet-surface adhesion, manifested macroscopically by the contact angle hysteresis on the condensing surface. Experimentally, the minimum jumping droplet departure radius for water has been shown to be as low as 𝑅jump,min = 375 nm, and as high as 𝑅jump,min = 100 µm23 . Theory reveals that inertialcapillary forces govern droplet coalescence even at nanometric length scales 23-24. It is important to note that as droplet radii approach 300 nm, jumping ceases to happen after coalescence due to viscous effects starting to dominate inertial-capillary forces in the coalescing bridge 8. For 𝑁s = 105 sites/mm2 with 𝑁0 = 1000 sites, we ran simulations for 𝑅jump,min = 0.1, 1, 3, 6 and 10 μm, with an average droplet spacing 𝑠 = 1.6 μm. Figure 10 shows the jumping droplet distribution as a function of droplet radius for varying 𝑅jump,min. The results show that the right side of the distribution flattens for large 𝑅jump,min (> 6 μm) similarly to the curve for large number density (𝑁s=107/mm2), indicating that curve flattening was due to the average spacing between droplets becoming small compared to the minimum jumping radius. For 𝑁0 = 1000 sites, we observed that the maximum radius of droplets residing on the surface at steady-state reached approximately three times the average spacing between droplets ( ≈ 3𝑠, Figure 7). However if the maximum droplet radius exceeded the jumping radius, then the maximum radius on the surface became the jumping radius, as given by the equation: 𝑅max = min (𝐶m·𝑠 , 𝐶g·𝑅jump,min) ,


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where 𝐶m is a constant depending on the number of droplet simulated. For 𝑁0 = 1000, 𝐶m ≈ 3. However, for 𝑁0 = 104, 𝐶m ≈ 3.9 (Figure 7) and can increase further at a decreasing rate for higher 𝑁0. The increase of 𝐶m as a function of 𝑁0 tempered at higher droplet densities. The factor 𝐶g (0 < 𝐶g < 1) represents the ratio of pre-to-post coalescence radius and depends on the average size mismatch between coalescing droplets. The factor 𝐶g can be obtained from the conservation of volume at a constant condensate density. For 𝑛 equally sized coalescing droplets, 𝐶g = 1 𝑛1/3, ( ≈ 0.8 for 𝑛 = 2). For the cases simulated here, 𝐶g ≈ 1 showing that droplets close to the minimum jumping radius merged with smaller droplets on average. Hence, the distribution flattening can be explained by the fact that all droplets were well beyond their natural maximum spacing and would jump as soon as they reach the jumping radius. The curve flattening was observed to initiate at 𝑅 ≈ 2𝑠, increasing as the difference between 𝐶m·𝑠 and 𝑅jump,min increased, as seen from the cases of 𝑅jump,min = 3 and 6 μm. The case of 𝑅jump,min = 3 μm shows a transient regime where the 𝑅max approaches the value of 3 μm expressed by 𝑅jump,min. A clear cut-off in the tail of the droplet distribution appears for 𝑅max = 6 μm and 𝑅max = 10 μm.

Effect of Maximum Jumping Droplet Radius The radius at which droplets coalesce and jump can exhibit a range of values for the same surface2526.

One way to modify 𝑅jump,max is by microstructuring the surface to pin droplets between pre-

designed structures such as pillars. In our simulations, for values where 𝑅jump,max was higher than the maximum droplet radius on the surface at steady state, changing 𝑅jump,max had no effect on the droplet distribution. If, however, 𝑅jump,max was lowered to a value below the maximum radius on


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the surface (𝐶m·𝑠), then steady state was not reached as droplets existed on the surface that would not jump.

Figure 10. Droplet size distribution as a function of droplet radius for varying minimum jumping radii (𝑅𝑗𝑢𝑚𝑝,𝑚𝑖𝑛). As 𝑅𝑗𝑢𝑚𝑝,𝑚𝑖𝑛 increased from 0.1, 1, 3, 6 to 10 μm, the distribution tail flattened. The flattening behavior explained the trends observed in Figures 8 and 9 for large droplet densities. Error bars have been made partially transparent for clarity.

Effect of Contact Angle The apparent contact angle affects both the droplet growth rate (Eqns. 6, 8), and droplet coalescence due to the 3D geometry of droplets in the superhydrophobic state. Figure 11 shows the distribution function for apparent advancing contact angles of 𝜃 = 150°,160° and 175° while keeping all other parameters constant (𝑁𝑠 = 105 sites/mm2, 𝑁0 = 1000 sites). It is important to note, 25 ACS Paragon Plus Environment

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although the apparent advancing contact angle is used here for simulations due to droplets residing in the advancing state during growth 27, the apparent receding contact angle is of equal importance as it is a macroscopic measure of whether jumping occurs (adhesion)23,

28.

Our simulations

incorporate the contact angle hysteresis through the specification of 𝑅𝑗𝑢𝑚𝑝,𝑚𝑖𝑛, i.e. the lower 𝑅𝑗𝑢𝑚𝑝,𝑚𝑖𝑛, the lower the contact angle hysteresis. The inset of Figure 11 depicts the maximum droplet radius for the simulated scenarios, showing that the distribution is negligibly altered at the tail end of the distribution (𝑅 > 2 μm, 𝑠 = 1.6 μm). The results show that the maximum droplet radius residing on the surface increases from 3.8 μm to 4.2 μm ( ≈ 10%) when the apparent advancing contact angle increases from 150° to 175°. The increase in maximum radius can be explained by the increased coalescence beneath large droplets at elevated contact angles29. Small droplets coalescing with large droplets “feed” the large droplets without any subsequent jumping (large size mismatch, Figure 5-d). However, this shadowed coalescence phenomenon does not perturb the distribution significantly. The lack of distribution change is observable in Figure 12 where the normalized covered surface area (𝐴𝑐/𝐿2), projected surface area (𝐴𝑝/𝐿2) and their ratio are plotted as a function of the apparent advancing contact angle. The values (𝐴𝑐/𝐿2) and (𝐴𝑝/𝐿2) are calculated by integrating the distribution function times the base area and projected area respectively. The ratio 𝐴𝑝/𝐴𝑐 follows the square of the ratio of the droplet radius to base radius (1/ 𝑠𝑖𝑛 𝜃), a consequence of the invariance of the distribution with apparent advancing contact angle. Although the apparent advancing contact angle directly affects 3D coalescence effects, it also plays a large effect on individual droplet growth rate.


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Figure 11. Condensate droplet distribution (𝑓) as a function or droplet radius (𝑅) for varying apparent advancing contact angles (𝜃). Inset: maximum droplet radius (𝑅𝑚𝑎𝑥) as a function of time as expressed by the number of droplet jumping events (𝑁𝑗) on the surface. Simulation parameters: 𝑁0 = 1000 sites, 𝑁𝑠 = 105 sites/mm2. Larger apparent advancing contact angles cause a slight increase in the droplet size for 𝑅 > 2 μm due to increased coalescence with smaller droplets within their geometric shadow. Error bars have been made partially transparent for clarity.


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Figure 12. Normalized covered surface area (𝐴𝑐/𝐿) , projected surface area (𝐴𝑝/𝐿) and their ratio as a function of apparent advancing contact angle (𝜃). The values are calculated by integrating the distribution function (𝑓) times the base and projected areas, respectively. The Ratio 𝐴𝑝/𝐴𝑐 follows the square of the ratio of the droplet radius to the base radius (𝑅/𝑅𝑠𝑖𝑛 𝜃), which is a consequence of the invariance of the distribution with the apparent advancing contact angle.

Effect of Droplet Growth Rate Although the magnitude of the growth rate varies as a function of apparent advancing contact angle (eq. 8), the previous section showed that the contact angle did not significantly affect the distribution function. To reconcile the effect of individual droplet growth rate on the distribution function, we conducted additional simulations with differing growth rate functions. Although the growth rate of a condensing droplet in pure saturated vapor is well defined, additional thermal and mass transfer resistances can change the experimentally observed growth rate of individual 28 ACS Paragon Plus Environment

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droplets

21, 30-31.

We perform simulations for different growth rate functions with respect to the

droplet radius R. The growth rate obtained from the droplet heat transfer analysis was fitted with a quadratic curve (log 𝑓 ~ (log 𝑅)2) that captures the shape closely, a linear curve (log 𝑓 ~ (log 𝑅)) and a constant value (Figure 13). A different hydrophobic coating thickness (𝛿 = 1 nm), which acted to change the growth rate by more than an order of magnitude, was also investigated. Despite the growth rate changes, the distribution did not alter in any considerable way. The results are in agreement with past studies of dropwise condensation on smooth surfaces which have shown the droplet distribution to be invariant with surface heat flux or droplet growth rate dynamics 32-33.

Figure 13. Condensate droplet size distribution (𝑓) as a function of droplet radius (𝑅) for varying individual droplet growth rates (𝑑𝑅/𝑑𝑡). Quadratic (log f ~ (log R)2), linear (log f ~ (log R)) and constant growth curves were simulated. A different coating thickness (𝛿 = 1 nm) was also investigated. Error bars have been made partially transparent for clarity. 29 ACS Paragon Plus Environment

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Suggested Droplet Size Distribution Fit Based on the aforementioned results, we suggest the following curve fit for the non-dimensional droplet size distribution during jumping droplet condensation (Figure 14) for realistic number densities (𝑁s > 105 mm-2)4, according to the assumptions laid out in the “Simulation” section, with an overall correlation coefficient 𝑟2 > 0.995.

𝑌=

{

―0.25𝑋4 ― 1.37𝑋2 ― 2.88𝑋2 ― 2.84𝑋 ― 0.61 ―3.18𝑋2 ― 3.67𝑋 ― 0.73

where 𝑋 = log10 (𝑅/𝑠), and 𝑌 = log10 (𝑓𝑠/𝑁s).


|

𝑋 ≤ 0.15 𝑋 > 0.15 ,

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Two-Dimensional (2D) versus Three-dimensional (3D) Coalescence The high apparent contact angles (𝜃>150°) observed on superhydrophobic surfaces create a 3D aspect for the coalescence that is not present on hydrophobic surfaces (𝜃 ≈ 90°) 29. We highlight this effect by counting, at steady state, the number of coalescence events that would have been counted if the simulation were 2D (𝑁coa,2D) and the number of coalescence events that are effectively happening in 3D and cannot be determined from a top view analysis, (𝑁coa,3D). The 2D comparison is particularly important as experimental observation of droplet nucleation and growth has typically been done with top-view optical microscopy, which is currently not capable of obtaining 3D information. Figure 14 shows the ratio of the 2D to 3D coalescence count (𝑁coa,2D/ 𝑁coa,3D) as a function of nucleation density for the growth rate corresponding to 𝛿 = 1 μm, 𝜃 = 175°, and 𝐿 = 1 mm. The inset in Figure 14 reveals the criteria for 2D versus 3D coalescence and depicts each case for 𝜃 ≈ 160°. The importance of 3D coalescence is highlighted further as we plot the ratio of the base areas associated with the 3D and 2D coalescences respectively, 𝐴coa,3D/ 𝐴coa,2D. The results reveal the paramount importance of 3D coalescence as 𝑁coa,3D/𝑁coa,2D reaches values of ≈ 10% and 𝐴coa,3D/𝐴coa,2D ≈ 5% for nucleation site densities 𝑁s > 104 mm-2, which results in a large error if the simulation was performed in 2D instead of 3D. Furthermore, the results show the utility of conducting fully 3D numerical simulations as conducted here in order to obtain droplet distribution dynamics not obtainable using experimentation alone. In fact, the jumping-droplet condensation simulations shown here are capable of capturing droplet distribution physics not previously capable of being characterized with experiments alone.


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Figure 14. Ratio of the number of 3D coalescence events (𝑁𝑐𝑜𝑎,3𝐷), and the corresponding base area of coalescing droplets (𝐴𝑐𝑜𝑎,3𝐷), to the number of 2D coalescence events (𝑁𝑐𝑜𝑎,2𝐷) and its corresponding base area (𝐴𝑐𝑜𝑎,2𝐷) at steady-state as a function of the nucleation site density (𝑁𝑠). The inset depicts schematic representations of both 2D and 3D coalescence for 3 adjacent droplets. The error bars arise from the ten repeated measurements on snapshots taken at steady state.


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Heat Transfer The main purpose of quantifying the droplet distribution function was to predict the overall (average) condensation heat transfer on the surface. The overall heat flux 𝑞" can be obtained by:

𝑞" =

∫

∞

𝑞d𝑓𝑑𝑅 .

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𝑅min

The value of the overall heat flux was plotted as a function of apparent advancing contact angle for four different coating thicknesses (𝛿 = 1 µm, 100 nm, 10 nm, and 1 nm) with a nucleation site density 𝑁s = 105 mm-2 (Figure 15). The results reveal a diminishing overall heat flux as a function of increasing apparent advancing contact angle due to the increased droplet conduction thermal resistance for more spherical droplet geometries. Numerically, the results make sense as the droplet distribution is not changed significantly with increasing apparent advancing contact angle (Figure 12), while the droplet base area is decreased. The simulations reveal that the overall heat flux can reach values approaching 80 W/cm2 for thin coatings (𝛿 = 1 nm) with condensation heat transfer coefficient of 80 kW/m2K (∆𝑇 = 10 K) which is consistent with previous experimentally measure values 6. To gain a better understanding of physical mechanisms governing jumping droplet condensation heat transfer, the cumulative heat flux fraction (𝑞′′cumul 𝑞′′) was calculated as a function of droplet radius 𝑅, for 𝑁s = 105 mm-2 and 𝜃 = 175° (Figure 17, right axis). Figure 16 (left axis) shows the droplet distribution function (𝑓), heat transfer per droplet (𝑞d), heat transfer per droplet size (𝑓𝑞d), and total area per droplet size (𝑓𝑅2) as a function of condensing droplet radius. All variables were normalized by their maximum over the plotted range for ease of visualization. The results show that the biggest contributors to the overall heat flux are droplets in the size range 800 nm < 𝑅 < 1.2 µm. The peak observed for the heat transfer per droplet size 33 ACS Paragon Plus Environment

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matches well with the peak in total area covered by this size range, indicating that the area covered by droplets residing in this size range (800 nm < 𝑅 < 1.2 µm) is the largest amongst other size ranges, hence contributing most to the overall heat transfer.

Figure 15. (a) Overall condensation heat flux (𝑞") as a function of apparent advancing contact angle (𝜃) for varying hydrophobic coating thickness for 𝑁𝑠 = 105 sites/mm2. To compare the simulation results to analytical theory, a previously developed analytical model21 was used to calculate the heat flux. (b) Droplet size distribution comparison between this work and previous analytical expressions.


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Figure 16. Jumping droplet heat transfer parameters as a function of individual droplet radius (𝑅). The left axis plots the labeled normalized heat transfer variables. The variables have been normalized by their maximum over the plotted range and represent: 1) the droplet size distribution 𝑓, 2) the total heat transfer per droplet 𝑞𝑑, 3) the total heat transfer per droplet radius, 𝑞𝑑𝑓, and 4) the total area covered by droplets of finite size, 𝑓𝑅2. On the right axis (blue), the cumulative fraction of the total heat flux (𝑞′′𝑐𝑢𝑚𝑢𝑙/𝑞′′) is plotted on a linear scale. The cumulative plot is the integral of the heat transfer per droplet size, 𝑞𝑑𝑓.


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Effect of Droplet Size Mismatch The droplet size distributions obtained thus far we calculated for the case where droplet jumping is not constrained by the relative size of coalescing droplets. In a real experiments, however, coalescing droplets do not jump regardless of their respective sizes 25, 34-43. Droplets having large size mismatch (𝑅1/𝑅2 > 1.5) may not jump, partly due to loss of momentum transfer in the direction normal to the surface. For this reason, we included an additional constraint for the jumping condition that allowed only droplets of comparable sizes to jump. The mismatch factor 𝑚 (Eq. 1-g) was introduced representing the ratio of the standard deviation of large droplets with respect to their average. Large droplets were defined as droplets that exceed 50% of the area-based average radius of the coalescing droplets, meaning their surface energy is comparable to the average surface energy of coalescing droplets. This allows for very small droplets to be neglected for the jumping condition since they would be entrained with the large droplets. If the large droplets are comparable in size (𝑚 < 0.2, as has been observed experimentally for two droplet coalescence), then jumping was set to occur in the simulation. If only a single large droplets was present, then jumping did not occur. Figure 17 shows the simulation results for jumping droplet condensation with the size constraint considered. It was observed that the constraint led to the maximum radius on the surface 𝑅max constantly growing (Figure 17-a) causing eventual flooding of the condensing surface. The simulation is stopped before the droplets become large enough (≈ 0.5 mm in radius) to be shed away with gravity44-45 (Figure 17-b, see Video S2 of the Supporting Information.). Although Figure 17 shows one particular simulation with set initial conditions, the observation of flowing with droplet mismatch considered was always present for all initial conditions simulated.


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Figure 17. (a) Maximum droplet radius 𝑅𝑚𝑎𝑥 as a function of the number of jumps 𝑁𝑗 with mismatch constraint for jumping droplets. 𝑁0 = 1000, 𝑁𝑠 = 105 mm-2. (b) Close-up time lapse image of the simulation (𝑡 = 60 ms) showing a large droplet growing indefinitely on the surface leading to flooding on the surface. See Video S2 of the Supporting Information.


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Discussion The simulations provided in this work from the first complete 3D framework that enables the modeling of condensation heat transfer on superhydrophobic surfaces with jumping-droplets. Prior models 9 suggested that the distribution of droplets below the coalescence radius is governed by the population balance theory, which is shown here to be false. Although a recent numerical simulation

46

obtained the droplet distribution for jumping-droplet condensation on vertical

surfaces, it assumed non-ideal behavior of the jumping droplets (droplets can grow until they become large enough to be swept from the surface). Furthermore, past works did not include 3D phenomena, which are shown here to be critical to understanding droplet distribution dynamics. Our work provides a comprehensive approach to jumping droplet modeling through the investigation of achievement of steady-state conditions for jumping droplet distributions that is capable of addressing 3D coalescence. Furthermore, our simulation generalizes the mismatch dynamics for multiple droplet coalescence and investigates the effects of parameters such as the number of droplets simulated, droplet growth rate, apparent contact angle, minimum and maximum jumping radius, and different nucleation site densities. Interestingly, our simulation framework shows a third possible flooding mechanism on jumping-droplet surfaces pertaining to droplet mismatch. In contrast to nucleation-mediated and progressive flooding6,

47,

which are

nucleation-based and droplet-return based phenomena, respectively, our work (Figure 17) points to a tertiary flooding mechanism simply due to droplet mismatch. This tertiary mechanism is present even for surfaces exhibiting low nucleation density (nucleation mediated flooding is not present) and no droplet return (progressive flooding is not present). Our simulation results beg the question if progressive flooding observed in the past is attributable to a combination of dropletmismatch flooding shown here, and droplet return.


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Although benchmarked here extensively, and used to study canonical jumping-droplet condensation situations, our developed simulation provides a framework for potential simulation of additional phase-change phenomena needing exploration. In the future, it would be prudent to implement contact angle hysteresis and droplet jumping physics from the standpoint of droplet coalescence and the different regimes it involves24,

48-53.

Although currently the model makes

droplets disappear and relies on empirical evidence for jumping criteria (i.e. minimum size, maximum size mismatch), in the future, it would be good to implement a more physics based solver for jumping in one unified code. This would require iteration between a finite element software such as ANSYS and our Matlab/C++ code. Unification of the two codes would enable a better fundamental understanding of the mismatch effects for multiple droplets, which has been lumped as one parameter in this study. Furthermore, it would be good to implement this algorithm for dropwise condensation to validate the previously derived Rose distribution

13

and population balance theory

17

in one

continuous distribution. This validation would require optimization of the algorithm to be able to handle a large size range of droplets. We suggest enhancement of our algorithm through the use of parallelization for the simulation domain. The current code does not consider vapor effects which can significantly be altered due to jumping. It would be useful to use our code to develop a boundary condition for a coupled vaporside code to see the effects in terms of mass transfer enhancement. Droplet jumping has been observed to perturb vapor flow, and this effect is still to be modeled in terms of the hydrodynamic and thermal boundary layer mixing, and potential concentration boundary layer effects for condensation in the presence of non-condensable gases. As a first step, integration of the current simulation framework with convective jumping-droplet condensation simulations presents a first


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step54-55. Coupling of the two simulation frameworks would involve computing the location of droplet coalescence and jumping events from the current simulation, and feeding of the spatial location, size of departing droplets, and their respective velocity, to the convective vapor flow simulation framework. Although assumed in our simulations, the surface structure can exhibit different geometries such as: micro-pillars56, nanotubes57, porous structures58, hierarchical structures41, that may affect the dynamics of droplet interaction. It would be interesting to mimic surface structure geometry in the simulation by manipulating the coordinates of the surface and the nucleation site locations during simulation initialization. Given the relatively small domain size of our developed simulation framework, microscale features approaching 10 µm can be resolved by the simulation. Droplet return to the surface was not considered in this work, which focused on simulation cases such as condensation on vertically oriented surfaces, and on surfaces where droplet return is impeded through a convective fluid flow54-55 or external electric fields59. In the future, it would be interesting to determine the effect of droplet return to the surface, which is expected to change the droplet distribution60-61, and have gradational dependence through the surface orientation62, droplet size, and heat flux59.


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Conclusions In this study, we provide a numerical simulation of jumping-droplet condensation on in order to determine the steady-state time-averaged droplet size distribution on the condensing surface. In order to characterize steady state, we track the maximum radius on the surface. At steady-state, the maximum radius oscillates around an average value in time. The droplet size distribution is obtained in its non-dimensional form, and effects of the minimum jumping radius (0.1µm – 10 µm), maximum jumping radius, contact angle (150° – 175°), and droplet growth rate are provided. The minimum jumping radius flattens the tail of the distribution only if it reaches high enough values (> ≈ 3X average spacing), whereas the maximum jumping radius is not shown to modify the distribution unless it drops below ( ≈ 3X average spacing), where it prohibits the achievement of steady state. The contact angle and growth rate are shown to have negligible effect on the distribution. In this study, we provide a suggested numerical fit for the droplet size distribution with an overall correlation coefficient greater than 0.995. The heat transfer performance is evaluated with the derived distribution and is highly sensitive to contact angle and coating thickness as dictated by the droplet growth rate. The obtained heat fluxes agree well with previous studies of jumping droplet condensation. We found that the size mismatch constraint for droplet jumping can impede the achievement of steady-state and could describe a new flooding mechanism for jumping droplet condensation. Supporting Information   

S.1. Videos: 3D simulations of jumping droplet condensation on a superhydrophobic surface without (video S1) and with (video S2) droplet mismatch affected jumping. S.2. Matlab codes: The matlab codes used for running the simulations are provided in “.m” format. S.3. Matlab code PDF file: source codes are provided as text for ease of access.


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Acknowledgements This work was supported by the National Science Foundation Engineering Research Center for Power Optimization of Electro Thermal Systems (POETS) with cooperative agreement EEC‐1449548. The authors gratefully acknowledge the support of the International Institute for Carbon Neutral Energy Research (WPI-I2CNER), sponsored by the Japanese Ministry of Education, Culture, Sports, Science and Technology.


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Numerical Simulation of Jumping-Droplet Condensation | Langmuir

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