Simulating Heat Transfer During Transient Dropwise Condensation on

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Simulating Heat Transfer During Transient Dropwise Condensation on a Low Thermal Conductivity Substrate Ashley Marie Macner, Susan Daniel, and Paul H. Steen Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.9b01231 • Publication Date (Web): 05 Aug 2019 Downloaded from pubs.acs.org on August 12, 2019

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Simulating Heat Transfer During Transient Dropwise Condensation on a Low Thermal Conductivity Substrate Ashley M. Macner, Susan Daniel*, Paul H. Steen* Robert Frederick Smith School of Chemical & Biomolecular Engineering, 120 Olin Hall, Cornell University, Ithaca, NY, 14853 *Author to whom correspondence should be addressed: Dr. Susan Daniel Email: [email protected] Telephone: 607-255-4675 Dr. Paul Steen Email: [email protected] Telephone: 607-255-4749

KEYWORDS: Dropwise condensation, low thermal conductivity substrate, drop-size distributions The instantaneous heat transfer performance of a surface is dictated by the number and the sizes of drops on the surface. While performance averaged over longer times is of interest from a technology standpoint, accurate simulation of the transient state is important in condenser design, because the maximum heat rejection of the surface occurs in this range. Steady-state dropwise

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condensation can be thought of as a collection of transient dropwise condensation cycles occurring in parallel. Traditional simulation of dropwise condensation has focused on making comparisons with experimental drop-size distributions at later times, after the process has reached statistical stationarity where the heat transfer is lower. Understanding how to model and simulate transient dropwise condensation where a maximum in heat transfer occurs, will help us design higher heat rejecting surfaces. Additionally, a constant temperature difference between the steam and the surface below the drop is assumed. While often valid, there are some cases where this isn’t valid, and measuring the drop growth rate is required. We report a way to simulate transient dropwise condensation using a measured population averaged drop growth rate. The simulation reasonably predicts the time evolution of the number density of drops, fractional coverage, normalized condensate volume, and median drop radius for pendant mode dropwise condensation experiments on a cooled, horizontal, dodecyltrichlorosilane coated, glass surface. It was also found that assuming a constant temperature difference grossly under predicts the heat transfer. Modification of the single drop heat transfer model to include substrate conduction and a thermal boundary layer shows that in the limit of low thermal conductivity, the drop growth rate is constant for large drops. Additionally, comparison between experiments and simulation shows that condensation might initialize by nucleation onto fixed sites and then transitions to random nucleation as more sites become activated and more favorable. Understanding how a substrate’s thermal properties affect the progression of dropwise condensation is important in determining the removal performance of the surface. With the commercialization of 3D printing, it’s possible to fabricate low cost, lightweight, plastic substrates with physical texturing for condensation applications where mass and cost savings are critical.

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INTRODUCTION There are several applications covering many length scales where efficient heat transfer is important: heat pipes1, the cooling of microelectronics2, the cooling of nuclear power reactors, and temperature control on board spacecraft3. In all of these systems, latent heat is released into the substrate as the hot vapor condenses onto a subcooled surface. However, the inability to maintain a steady dropwise condensation state has prevented its widespread use on an industrial scale. One of the main barriers is the design of cost-effective, long-lasting surfaces whose coatings do not degrade or foul4-6. The degradation of surface treatments over time typically leads to the transition from dropwise to filmwise condensation. However, a brief review by Ma et al. of research in China on surface enhancements to promote dropwise condensation indicates that some progress has been made7. Most notable, is the surface by Zhao et al. that has been integrated into a condenser in an industrial heating system in China8-10. Condensation on a surface can occur either as a film or as a population of drops. The latter method is preferred because the heat transfer coefficient is often much higher4,11. In filmwise condensation the film acts like an insulator, providing resistance to heat transfer from the steam to the condensing surface. Large drops in dropwise condensation behave the same way. Thick films of liquid are costly from an energy management perspective. A contact-angle of 90° or higher is needed to promote dropwise condensation7,12. Since most traditional condensing surfaces are not naturally hydrophobic, they are either chemically coated13 and/or physically textured14,15 to make them non-wetting by creating a low surface energy solid. However, simple promotion of a surface from wetting to non-wetting is not enough to prevent filmwise condensation. The drops must be removed from the surface either by gravity, a surface energy gradient, or some other means. Gradients remove drops from the surface by creating a force imbalance between the leading and

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trailing edges of a drop that can cause large enough drops to move16,17. The most common gradients are thermal (thermocapillary)18-20, electrostatic potential (electrocapillary)21,22, surface modified (e.g., chemical16,23-28 and/or physical texturing29-31), and periodic asymmetric oscillation32,33 of a substrate. For chemical and physical gradients, a comprehensive review on fabrication methods is provided by Genzer et al.34. The instantaneous heat transfer performance of a surface is dictated by the number and sizes of drops on the surface at any instant in time. Despite advances in computer hardware, simulation of dropwise condensation still remains a computational challenge because of the large range of length and time scales involved and also the number of drops. The length scales in the problem span nearly six orders of magnitude from the smallest thermodynamically stable drops ℴ

[10 ―9𝑚] to the departure size ℴ[10 ―3𝑚]. Additionally, the density of nucleation sites is often cited as being as high35 as 1011 sites/cm2. The highest nucleation site density that has been simulated36 for a large condenser area (~10 cm2) is 107 sites/cm2. The computational expense arises from having to do pairwise comparisons in checking for drop overlap. The expense can be somewhat mitigated by partitioning the surface into a grid, where the drops are evenly distributed amongst the squares. Drop overlap with drops within the same square and within adjacent squares can then be checked37. This greatly reduces the number of comparisons that need to be made. A further complication arises with choosing the time step, especially when simulating for longer times such as required for statistical steady-state. A fairly small time step, whose size depends on the growth rate, is needed to avoid complicated pattern overlaps. Traditional simulation of dropwise condensation has always either assumed a power law growth rate38-40 for isolated drops or a growth rate calculated from a steady-state conduction model with an assumed temperature difference of a few Kelvin between the steam and the surface below the drop (to be called the

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classical model going forward)36,41-44. To grow the drops in the simulation, we have instead chosen to measure a population averaged drop growth rate in situ during a dropwise condensation experiment because the classical model grossly underpredicts the heat transfer for the current experimental setup. Additionally, the simulation is a meso-scale simulation. Instead of using a high nucleation site density with small nucleating drops (𝑎𝑛=1 μm), a low nucleation site density (𝑛~5000 drops) with large nucleating drops (𝑎𝑛=20 μm) is used. This enables input of the initial condition at 𝑡0 from experiment after fitting the initial drop-size distribution. To date, no growth rate model exists for low thermal conductivity surfaces. With the commercialization of 3D printing, the possibility exists for mass production of low cost, light weight plastic condensing surfaces. 3D printing offers more degrees of freedom in surface design that cannot be achieved by current photolithographic methods. Such “designer” surfaces are potentially the condensing surfaces of the future. Hence, it is important to understand what the dominating resistances are to heat transfer for this class of surfaces and how the existing single drop heat transfer model can be modified and incorporated into the simulation. The current work simulates dropwise condensation on the underside of a horizontal hydrophobic surface where a population averaged drop growth rate measured from experiment is used to grow the drops. Modification of the classical single drop heat transfer model to include substrate conduction and a thermal boundary layer shows that in the limit of low thermal conductivity, the drop growth rate is constant for large drops. For our surface and experimental setup, the dominating terms are substrate conduction and thermal boundary layer convection on the coolant side of the condensing surface. By comparison, in the limit of infinite thermal conductivity using the classical heat transfer model, the growth rate is inversely proportional to the drop size, because of the dominance of conduction through the drop. Two different nucleation schemes, fixed site and random

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nucleation, are used in the simulations. Simulations are also reported for the classical heat transfer model with an assumed temperature difference and the experimentally measured drop growth rate. Comparison is made with experiment for the time evolution of number density, fractional coverage, condensate volume, and median radius for a coolant temperature of 1°C. Additionally, comparison between simulation and experiment for the time evolution of the drop-size and drop volume distributions is made at several instants throughout the transient process. Finally, the overall heat transfer coefficient is estimated and compared with simulation. While measurement of the drop growth rate in-situ was developed for the current experimental setup, the method is applicable for all surfaces where the drop shapes can be clearly visualized. Direct measurement of the drop growth rate is more general, and does not require any complicated equipment or intrusive measurement.

MATERIALS AND METHODS Substrate Preparation, Treatment, and Characterization. A circular polished glass substrate (Technical Glass Products) was chemically treated by vapor deposition of dodecyltrichlorosilane (dodecyl) to deliver static (𝜃𝑠), advancing (𝜃𝑎), and receding (𝜃𝑟) contact-angles of 98°, 108°, and 85°, respectively. For the ice bath experiments as described earlier45 by Macner et al., all contactangles were calculated using an in-house Matlab routine that minimized the least squares difference between the experimentally and numerically determined drop profiles. Using a side view image of the drop, the experimental profile was found by using a grayscale thresholding routine. The numerical profile was found by numerically integrating the Young-Laplace equation over the shape of the drop. The Nelder-Mead optimization routine was used to navigate the

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parameter space and find the optimal set of parameters to minimize the least squares difference between the two profiles46. Condensation Chamber. A custom designed chamber was used for the condensation experiments (cf. Figure 1). The chamber is cylindrical in shape and hollow in the center to allow for visualization of the condensation process. The chamber is broken up into three sections. Steam enters at the bottom of the chamber (section 3) and a combination of buoyancy and forced convection brings the steam into contact with the underside of the condensing surface. The steam was generated by boiling water in an Erlenmeyer flask using a hot plate. The coolant flowed through section 2 and made direct contact with the top-side of the condensing surface. Another circular polished glass substrate separates sections 1 and 2. Drops were imaged from above and illuminated from below using collimated light. Further details on the experimental setup can be found45 in Macner et al. The coolant was circulated at a volumetric flow rate of 9.6 L/min (LPM) with a circulator pump (Fisher Scientific, Model 73). The coolant was tap water maintained at a temperature of 1°C by excess ice.

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Figure 1. Experimental setup. Condensation chamber (sections 1, 2, and 3) is imaged from above and illuminated from below. A hot plate (left) boils deionized water to produce steam (dots). Condensate buildup drains out through the condensate vent and into a beaker. Chamber sections 1 and 2 are separated by a window, coolant flows through section 2, a downward facing treated surface (cross-hatch) separates sections 2 and 3, and the steam enters into section 3. A diffuser piece at the steam inlet evenly distributes the steam. EXPERIMENTAL METHODS The coolant was circulated 1-2 minutes before the steam entered the chamber, in order to minimize premature condensate buildup on the surface prior to the start of the experiment, while balancing the time required to cool down the surface to the prescribed coolant temperature. For a 1/16" thick substrate with a specific heat capacity of 670 J/(kg*°C) (T=20°C), a density of 2200 kg/m3, and a thermal conductivity of 1.4 W/(m*°C) (T=20°C), the time needed to reach the coolant temperature, can be estimated by the conduction time scale, 𝑡𝑐 =

𝑙2𝜌𝑠𝑐𝑝,𝑠 𝑘𝑠

≈ 2.7𝑠. The estimated time

scale is much smaller than the time allotted for the surface to reach the prescribed coolant temperature. More details on the experimental methods can be found45 in Macner et al. SINGLE DROP HEAT TRANSFER Figure 2 is a definition sketch of heat transfer through a single drop that has condensed onto a chemically treated substrate. The total temperature difference between the steam and the coolant is ∆𝑇𝑠𝑎𝑡 ∞ =

𝑞𝑑 2𝜋𝑎2ℎ𝑖(1 ― cos 𝜃𝑠)

+

2𝑇𝑠𝑎𝑡𝜎 ℎ𝑓𝑔𝑎𝜌𝑙

+

𝑞𝑑𝜃𝑠 4𝜋𝑎𝑘𝑐sin 𝜃𝑠

+

𝑞𝑑𝑙 𝑘𝑠𝜋𝑅

+ 2

𝑞𝑑 ℎ𝐵𝐿𝜋𝑅2

(1)

where the terms describe the temperature difference from the vapor to the liquid-gas interface as the vapor molecules condense, from the curvature of the interface, conduction through the condensate, conduction through the substrate, and convection through the thermal boundary layer on the coolant side of the substrate, respectively. The boundary layer heat transfer coefficient was estimated using the Gnielinksi correlation, which is valid for a large Reynolds number range4,47. 8


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Figure 2. (left) Definition sketch of thermal resistances. (right) Parameter, variable, and dimensionless group definitions. Resistances are color coded to match the curves in Figure 3. Using the dimensionless groups of Figure 2, Equation (1) can be non-dimensionalized ∆𝑇 =

𝑓(𝜃𝑠) 𝑑𝑎

(

sin2 𝜃𝑠

sin3 𝜃𝑠 𝑑𝑡 2(1 ― cos 𝜃𝑠)

+

𝑁𝑢1𝑎𝜃𝑠 4

+ 𝑁𝑢2 +

ℎ𝑖

)

ℎ𝐵𝐿

+

sin 𝜃𝑠 𝑎

(2)

The terms in Equation (2) are plotted for the thermophysical parameters listed in Table S.1 in the Supporting Information to determine the dominating terms. From Figure 3 for 𝑅 ≤ 1.8 mm (most of the drops are smaller than 2 mm), substrate conduction and thermal boundary layer convection dominate for a glass substrate. The interfacial and the curvature resistances are negligible for both types of surfaces for the observed drop size range. Retaining only the substrate conduction and thermal boundary layer convection terms in Equation (2) and unscaling gives the final expression 9


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𝑎=

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(𝑇𝑠𝑎𝑡 ― 𝑇∞)𝑅2𝑐 sin2 𝜃𝑠

[

𝐶𝑓(𝜃𝑠)𝑡𝑐ℎ𝑖

𝑙 1 + 𝑘𝑠 ℎ𝐵𝐿

]

(3)

which shows that the growth rate is independent of the drop size. This result is in agreement with that of Leach et al., obtained by geometric arguments37. By comparison, the classical model is 𝑎=

4(𝑇𝑠𝑎𝑡 ― 𝑇𝑠)𝑘𝑐sin 𝜃𝑠 𝜃𝑠𝑓(𝜃𝑠)𝑎ℎ𝑓𝑔𝜌𝑙

(4)

where (𝑇𝑠𝑎𝑡 ― 𝑇𝑠) is generally assumed to be on the order of a few Kelvin.

Figure 3. Comparison of dimensionless terms appearing in Equation (2) for 𝑎=5.4 μm/s, 𝜃𝑠=98°, and 78 𝜇𝑚 ≤ 𝑅 ≤ 2 𝑚𝑚. COMPUTATIONAL METHODS

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Condensation. For simplicity, the drops are assumed to be hemispherical (i.e., spherical radius a is equivalent to footprint radius R), the shape distortion due to gravity is assumed to be negligible (i.e., zero Bond number), and the substrates are assumed to be defect free (i.e., ∆𝜃 = 𝜃𝑎 ― 𝜃𝑟 = 0). The last assumption implies that there is no resistance to contact-line retraction after a coalescence event. Additionally, contact-line retraction is assumed to be instantaneous. We are interested in condensing drops on the underside of a cooled surface, called pendant mode dropwise condensation. Figure 4 is a flow diagram showing the cycle of dropwise condensation for fixed site nucleation.

Figure 4. Example schematic of dropwise condensation cycle for fixed site nucleation. Black points denote fixed nucleation sites (in this example, 3). The number of fixed sites in the simulation is the same as the number of drops at t=0, as prescribed by the experiment. The simulation initializes by randomly nucleating n drops of radius 𝑅𝑚𝑖𝑛 from a gamma distribution characterized by shape parameter A and scale parameter B (see Matlab help file) onto a condenser surface area, 𝐴𝑠, of size 1 cm2. Nucleation can be accomplished by either nucleating

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onto sites fixed for the duration of the simulation or anywhere (randomly) in the interstitial space between drops. For each finite time step ∆𝑡, each drop grows according to

𝑎𝑚 + 1 = where

𝑑𝑎 𝑑𝑡

𝑑𝑎 ∆𝑡 + 𝑎𝑚 𝑤ℎ𝑒𝑟𝑒 𝑚 = 1,2,…𝑛𝑡 𝑑𝑡

(5)

= 𝑎, an input parameter. As mentioned above, we will use a measured isolated drop

growth rate between coalescence events for 𝑎. The time step is limited by

0 < ∆𝑡 ≤

0.2𝑎𝑚 𝑎

(6)

since no more than a 20% increase (arbitrarily chosen) from the drop size at the previous step (𝑎𝑚) is desired. If the time step is too large, then the radius of the nucleating drop will grow by an order of magnitude (i.e., 𝑎𝑚= 1 μm to 𝑎𝑚 + 1= 10 μm) in a single time step. Additionally, large time steps (e.g., Δt = 1 s) resulted in complicated overlap patterns that usually involved four or more drops. For these types of scenarios, it is nearly impossible to devise an algorithm that can reduce all possible cases to a series of pairwise events. To avoid these two issues, a relatively small time step (e.g., Δt = 0.05 s) was used. After growing the drops, they are checked for overlap by doing a pairwise comparison between drop i and drop j to see if the center-to-center distance dij is less than the sum of the radii 𝑑𝑖𝑗 = (𝑥𝑖 ― 𝑥𝑗)2 + (𝑦𝑖 ― 𝑦𝑗)2 ≤ 𝑎𝑖 + 𝑎𝑗 𝑤ℎ𝑒𝑟𝑒 𝑖 ≠ 𝑗

(7)

where (𝑥𝑖,𝑦𝑖) and (𝑥𝑗,𝑦𝑗) are the center coordinate pairs. For drops that do overlap, coalescence is assumed to be instantaneous. This is a reasonable assumption for smaller drops 𝑎 ≤ 300 μm (determined from contact-line motion in experimental images). The center-of-mass (COM) of the new coalesced drop is a mass-weighted average of the two unmerged ‘parent’ drops

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(𝑥′,𝑦′) =

(

𝑥𝑖𝑉𝑖 + 𝑥𝑗𝑉𝑗 𝑦𝑖𝑉𝑖 + 𝑦𝑗𝑉𝑗 , 𝑉𝑖 + 𝑉𝑗 𝑉𝑖 + 𝑉𝑗

)

(8)

where 𝑉𝑖 and 𝑉𝑗 are the volumes of the unmerged parent drops. After a coalescence event, subsequent resulting coalescences are checked for. In the event there is simultaneous overlap between two drops and a third shared drop, then the two drops that overlap the most are chosen to coalesce, since they would have intersected each other first. Once all of the coalescence events have been taken care of, then nucleation of new drops occurs. Drops can be either nucleated onto fixed sites or randomly in the interstitial space between drops. In the lab, a nucleation site is usually a defect on the surface such as a scratch, a pit, a piece of dust, etc. The defects could also be SAM layer defects such as islands of molecules. These defects are fixed in space for all time. However, not all of the sites may be active48. When running a fixed nucleation site simulation, the centers of the sites are specified to be the centers of the randomly distributed initial distribution of drops. The sites are checked for coverage after a coalescence event. If a drop can be nucleated onto a site without overlapping an existing drop, then it is nucleated. Otherwise, the site is left empty. In contrast, random nucleation involves trying to randomly nucleate drops in the interstitial space between drops. Random nucleation captures the activation/deactivation of nucleation sites over the course of an experiment49. Coalescence induced sweeping of the surface could lead to smaller net water deposits in or around surface defects that activate the site for nucleation. Additional nucleation sites could also be created if a receding contact-line retracts non-uniformly and pinches off tiny drops at its perimeter. Another method used by Leach et al.37 involves nucleating only in areas near drops that had just coalesced. Leach et al. concluded that there was no difference among the different nucleation methods. However, we noticed that there were some slight differences in our simulations that will be discussed in the

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results and discussion section. Random nucleation can be thought of as a constant reassignment of the nucleation sites each time a drop is nucleated. For random nucleation, a maximum attempt counter 𝑛𝑎 is used to set the maximum allowable attempts for nucleation. For example, if the counter is set to a value of 15, then the nucleation loop has 15 tries to randomly place a new drop of radius 𝑅𝑚𝑖𝑛 subject to the constraint that it cannot overlap an existing drop. If this is not accomplished, then the program aborts the loop and continues with execution of the rest of the routine. The final loop prior to moving on to the next time step involves checking for drop departure by gravity. The final departure radius is taken from experiment (𝑎𝑑~3.0 mm). To compare with experiment, simulations were only carried out when there was no drop detachment. This is the transient part of dropwise condensation. Once the drops begin to depart from the surface by gravity, a steady-state is reached as nucleation begins on the fresh surface revealed by departing drops. Here, the fractional coverage becomes constant and the drop-size distribution reaches statistical stationarity. Figure S.1 in the Supporting Information shows a more detailed flow chart for the condensation simulation. The metrics used for comparison between the experiment and the simulation are the time evolution of the number density of drops, the fractional coverage, the condensate volume, and the median drop size. The number density of drops N is just the number of drops per unit of condenser surface area above a certain size threshold, 𝑎𝑐. The fractional coverage ε is the percentage of condenser surface area 𝐴𝑠 covered by drops

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𝑛

𝜀=

𝜋sin2 𝜃𝑠∑𝑖 = 1𝑎2𝑖

(9)

𝐴𝑠

where the geometric relationship relating the spherical cap radius to the footprint radius, 𝑅 = 𝑎 sin 𝜃𝑠, has been used. The condensate volume is the total volume of liquid condensed onto the surface 𝑛

𝑉𝑐𝑜𝑛𝑑 =

𝜋(2 ― 3cos 𝜃𝑠 + cos3 𝜃𝑠)∑𝑖 = 1𝑎3𝑖 3𝐴𝑠

(10)

Because the condenser surface area being viewed can vary in size between the different trials, it is necessary to normalize the condensate volume. The median drop size is determined by first calculating the cumulative distribution function of drop sizes

𝐹𝑋(𝑎) =

∑ 𝑃(𝑋 = 𝑎 ) 𝑘

(11)

𝑎𝑘 ≤ 𝑎

and then interpolating to find the drop radius a where 𝐹(𝑎) = 0.5. In Equation (11), X is a discrete random variable consisting of all the drop-sizes, P is the probability, 𝑎𝑘 are the values of the dropsizes in X, and F is the probability that X will have a value less than or equation to a. Drop Departure. Because we are interested in pendant mode dropwise condensation on a horizontal surface, drop departure is complicated. Unlike experiments involving an inclined plane or a gradient surface where the drop is completely removed from the surface, a piece of a departing drop typically remains on the underside of the horizontal surface after pinch off (cf. Figure 5). Measurement of the drop departure from experiment is not a static measurement. The drop contactline is often retracting for a long period of time prior to detachment of a drop. Oftentimes, or so it appears, drop departure is triggered by a coalescence event. After a coalescence event, there is retraction of the contact-line in order to minimize the surface free energy of the drop. Because the 15


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contact-line is moving inwards, the drop has some momentum for dripping. And because no two coalescence events are the same, a universal threshold for drop departure seems unrealistic. It is also not clear where to measure the radius of the drop as it is departing. Additionally, the departure radius depends on the contact-angle (or the chemical treatment of the surface). In the simulation, once a drop reaches the departure size, it is treated as being completely removed from the surface. In the absence of accurate model detachment, the simulation is thought to be faithful only when there is no drop detachment.

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Figure 5. (top left) Coalescence event involving multiple drops on a dodecyl treated surface (𝜃𝑠 =98°). (top right) Formation of a large drop from coalescence of several drops. (bottom left) Drop contact-line retraction (white arcs on surface near drop perimeter) prior to pinch off. (bottom right) Drop departure from surface (blurry spot). A smaller version of the original drop remains on the surface after pinch off. Isolated Drop Growth. It is believed that a ‘small’ drop (𝑎 < 25 μm) grows primarily by surface adsorbed water molecules diffusing to the drop perimeter whereas a ‘large’ drop (𝑎 > 25 μm) grows primarily by condensation onto the liquid-gas interface37. For the current experiments, only the latter isolated drop growth mechanism is applicable, since we are limited optically to drops with a footprint radius 𝑅 > 78 μm. Additional drop growth comes from coalescence with neighboring drops. Whereas isolated drop growth is continuous in time, growth by coalescence is a discrete event, happening almost instantaneously (with the exception of large drops; to be discussed). Hence, if the radius of a single drop were tracked in time, the function would resemble a staircase. For each step, 𝑎 increases monotonically in time because of the volume increase due to condensation onto the drop interface. The discontinuous jump in 𝑎 corresponds to an instantaneous coalescence event (cf. Figure 6). For medium-sized drops, (300 μm ≤ 𝑎 ≤ 800 μm), there is a period just after coalescence when the contact-line retracts to minimize the surface energy of the drop in its quest to reach the lowest equilibrium state. During this period, any growth due to condensation is hidden. If the drop is able to avoid coalescing with its neighbors, it will then grow to a size where the decrease in 𝑎 due to contact-line retraction is balanced by an increase in 𝑎 as a result of condensation (𝑎𝑛𝑒𝑡 = 0). After a long enough period of time, the contact-line pins very near (or perhaps even at) the lowest energy state and the drop begins to grow only by condensation until it coalesces again. For bigger drops (𝑎 ≥ 1000 μm), the time between coalescence events is

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short enough that only retraction of the contact-line is observed (𝑎𝑛𝑒𝑡 < 0). Consequently, growth rates can only be accurately obtained for smaller drops where contact-line retraction is minimal. However, an exception can be made for medium and large drops if they experience uninterrupted growth over a long enough time (typically a few seconds).

Figure 6. Experimental growth rate of a single drop. Steps correspond to isolated growth by condensation onto the liquid-gas interface, discontinuous jumps correspond to growth by coalescence. Contact-line retraction after a coalescence event (red arrows) masks growth by condensation for larger sized drops. To estimate the isolated drop growth rate, either a population average or a time-average of a single drop can be used. There are several simulation algorithm complications with the latter method. The first being that a set of rules to determine when a drop has coalesced is difficult to define. The most straightforward rule is to assume that the radius of the new coalesced drop overlaps the radius of the old drop. This is certainly true for most coalescence events. However, if the drop contact-line is locally mobile, there could be some shape deformation after coalescence

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that triggers subsequent coalescence events and these could translate the drop a great distance. In this case, the radius of the new coalesced drop would not overlap the radius of the original uncoalesced drop, making it hard to keep track. Additionally, taking a time average of a single drop means calculating the growth rate from small and large drops alike. As mentioned above, the growth rate of a large drop can be complicated because of simultaneous contact-line retraction as the drop relaxes. This limits the number of steps that can be used in the time-average of a single drop. For these reasons, the average growth rate will be calculated using a population average. A routine was written in Matlab to track all drops (i=1,…n) on the condenser surface from a time 𝑡0 to a final time 𝑡𝑓,𝑖, where the final time is when the drop first coalesces with another drop. The final time depends on the local spatial distribution of drops. In other words, a drop of radius 50 μm at position (𝑥1,𝑦1) and another drop of the same radius at position (𝑥2,𝑦2) may not coalesce with a neighboring drop at the same time, because the spatial distribution of drops local to each drop is different. Once the final time is reached for a drop, it is no longer tracked. The program continues until all of the drops have coalesced and are no longer being tracked. Drops tracked for less than ~1.7 s (50 time steps at 30 fps) are not retained for the growth rate average because of the scatter in determining the drop radius. The growth rate is determined by fitting a linear trend to a plot of 𝑎 versus time, 𝑡. The radius assigned to the growth rate is calculated as a simple average 𝑎(𝑡0) + 𝑎(𝑡𝑓) 2

. The tracking process is then restarted at a later time a few seconds after the termination

of the previous tracking session to determine how the growth rate depends on time for the same drop radius. For the current set of experiments, an average growth rate of 𝑎 = 5.4 μm/s was measured. Calculation of the predicted growth rate from Equation (3) yields 𝑎 = 4.9 μm/s, which is in agreement with the experimentally measured value.

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RESULTS AND DISCUSSION The results from a parametric study will be presented to quantify the effect of the most important simulation parameter, the growth rate, on the time evolution of the number-density, the fractional coverage, the condensate volume, and the median radius. The effects of other parameters, including the nucleating radius (random nucleation), the threshold radius (random nucleation), the number of nucleation attempts (random nucleation), and the nucleation site density (fixed site nucleation) on these metrics can be found in Figures S.2-S.5 in the Supporting Information. Next, a comparison of the simulation results with the experiments for a coolant temperature of 1°C will be discussed for both the classical heat transfer model and a measured constant growth rate. Finally, an overall heat transfer coefficient will be calculated. Validation. Simulation validation was completed by comparing the fractional coverage results with the literature for the cases of nucleation and no nucleation. The meso-scale simulation converges to the appropriate limits of ~55% coverage without nucleation49 and ~76% coverage with nucleation36. The former corresponds to the jamming limit of equal sized disks on a planar surface50. For a distribution of disk sizes with a small standard deviation (due to coarsening by coalescence), this limit is approximate. A higher packing fraction can only be achieved with nucleation of small drops to fill the interstitial space between drops. Parametric Study. A parametric study was completed to assess the sensitivity of the simulation to various input parameters. For the same initial density and number of nucleation attempts, a larger growth rate leads to a faster transition between regions I and II (cf. Figure 7; top left). Region I spans from t = 0 to t = tm, where tm is the instant where the number density is maximum. Here, the drops come into view as they nucleate and coalesce on a small length scale to produce drops of increasing size. Region II spans from tm to ts, where ts corresponds to the start of the sawtooth 20


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region, region III. Here, there is a net decrease in the total number of drops because the number of coalescence events exceeds the number of nucleation events. A more detailed description of the different regimes can be found45 in Macner et al. For a faster growth rate, the drops reach the maximum packing fraction, or smallest neighbor distance, at an earlier time than for a slower growth rate. This has the advantage of clearing the surface earlier to make room for more nucleation of small drops. Over time, this leads to more condensed volume (cf. Figure 7; bottom left) and a higher fractional coverage (cf. Figure 7; top right).

Figure 7. Effects of varying the growth rate, 𝑎, on number density (top left), fractional coverage (top right), normalized condensate volume (bottom left), median radius (bottom right) for 5.4 μm/s (yellow), 4 μm/s (gray), 3.5 μm/s (light blue), and 1.5 μm/s (orange). Simulation parameters: 𝑛

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=4943 drops, 𝑅𝑚𝑖𝑛=20 μm, ∆𝑡=0.05 s, 𝑡𝑓=100 s, 𝑅𝑐=90 μm, 𝐴𝑠=1 cm2, 𝑅𝑚𝑎𝑥=3 mm, 𝐴=19.9, 𝐵 =2.6×10-6, 𝑛𝑎=300 attempts, random nucleation. In the case of random nucleation, other parameters that were varied included the nucleating radius, the threshold radius, and the number of nucleation attempts. In the case of fixed site nucleation, the nucleation site density was also varied. The results of these parameter variations can be found in the Figures S.2-S.5 in the Supporting Information. To summarize, the condensate volume curves are most sensitive to the growth rate. Hence, accurate determination of the growth rate is important in obtaining agreement between the simulation and the experiments. The isolated growth rate essentially controls the rate of volume growth by tuning the time scale between successive coalescence events for a drop. Simulation vs. Experiments. In addition to determining the isolated drop growth rate 𝑎, it is necessary to determine the number of drops per unit condenser surface at 𝑡0, and the distribution of drops at 𝑡0. The value of 𝑡0 corresponds to the time when the number density reaches a value above a user-defined threshold, such as 70 drops/cm2. The value is arbitrarily chosen and serves as a shifting factor that enables comparison between data sets and simulations. The shape parameters from fitting a gamma distribution to the initial drop distribution at t0 are plugged into the ‘gamrnd’ command in Matlab to generate the initial distribution of drops on the surface in the simulation. Figures 8 and 9 compare the experimental results with simulation (random nucleation sites) for a measured growth rate and for the classical heat transfer model, respectively. The number density and the fractional coverage plots are nearly identical for the two cases. While the median radius is better predicted using the classical heat transfer model, the condensate volume is better predicted using the measured growth rate. The classical heat transfer model underpredicts the 22


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condensate volume because it underestimates the growth rate of the larger drops, which contribute significantly to the fractional coverage and the condensate volume at later times (cf. Figure 11).

Figure 8. Constant 𝒂, random nucleation. Comparison between experiments (symbols) and simulation (black curve) for number density (top left), fractional coverage (top right), normalized condensate volume (bottom left), and median radius (bottom right). (), (), (), and () denote first, second, third, and fourth trials, respectively. Drops were nucleated randomly. Simulation parameters: 𝑛=4943 drops, 𝑅𝑚𝑖𝑛=20 μm, 𝑎=5.4 μm/s, ∆𝑡=0.05 s, 𝑡𝑓=100 s, 𝑅𝑐=95 μm, 𝐴𝑠=1 cm2, 𝑅𝑚𝑎𝑥=3 mm, 𝐴=19.9, 𝐵=2.6×10-6, 𝑛𝑎=300 attempts. Fractional coverage is scaled by difference in drop overhang between experiment and simulation because of different contact-angles.

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Figure 9. Constant ∆𝑻, random nucleation. Comparison between experiments (symbols) and simulation (black curve) for number density (top left), fractional coverage (top right), normalized condensate volume (bottom left), and median radius (bottom right). (), (), (), and () denote first, second, third, and fourth trials, respectively. Drops were nucleated randomly. Simulation parameters: 𝑛=4943 drops, 𝑅𝑚𝑖𝑛=20 μm, ∆𝑇=1 K, ∆𝑡=0.05 s, 𝑡𝑓=100 s, 𝑅𝑐=95 μm, 𝐴𝑠=1 cm2, 𝑅𝑚𝑎𝑥 =3 mm, 𝐴=19.9, 𝐵=2.6×10-6, 𝑛𝑎=300 attempts. Fractional coverage is scaled by difference in drop overhang between experiment and simulation because of different contact-angles. Simulation results for fixed nucleation sites and a constant growth rate are shown for the ice bath experiments in Figure 10. Comparing with the results in Figure 8, the fixed site nucleation simulation fits the number density data better in regions I and II (at earlier times) while the random nucleation simulation fits better in region III (at later times). The fixed site nucleation simulation

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fits the fractional coverage data better for all regions. However, in the case of the condensate volume, the fixed site nucleation fits better at earlier times whereas the random nucleation simulation fits better at later times. For the median radius, the disagreement between the experiments and the simulation with fixed nucleation sites starts at the beginning of the sawtooth regime. The simulation predicts much larger drops. Hence, the fixed site nucleation simulation fits the data better at earlier times whereas the random nucleation simulation fits the data better at later times. This suggests that condensation initializes by nucleation of vapor onto fixed nucleation sites and then transitions to random nucleation as more sites become activated and more favorable. Nucleation sites that arise from surface defects, such as cracks and pits, are more energetically favorable when there is liquid that wets and/or fills the defect48. The water can get deposited by either nucleation, or by drops sweeping over the cavity. In region III, there are several large-scale coalescence events occurring that create large drops that sweep large areas of the surface. Hence, the activation of large numbers of nucleation sites at later times could be the result of large sweeping events. The results for fixed nucleation sites using the classical heat transfer model are shown in Figure S.6 in the Supplementary Information.

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Figure 10. Constant 𝒂, fixed nucleation. Comparison between experiments (symbols) and simulation (black curve) for number density (top left), fractional coverage (top right), normalized condensate volume (bottom left), and median radius (bottom right). (), (), (), and () denote first, second, third, and fourth trials, respectively. Drops nucleated onto fixed sites. Simulation parameters: 𝑛=4566 drops, 𝑅𝑚𝑖𝑛=20 μm, 𝑎=5.4 μm/s, ∆𝑡=0.05 s, 𝑡𝑓=100 s, 𝑅𝑐=95 μm, 𝐴𝑠=1 cm2, 𝑅𝑚𝑎𝑥=3 mm, 𝐴=19.9, 𝐵=2.6×10-6. Fractional coverage is scaled by difference in drop overhang between experiment and simulation because of different contact-angles. Volume Distribution. To better understand how the growth rate impacts the coarsening of the drop pattern, the evolution of the volume and the drop size distributions are compared in Figure 11 at several different instants. The distributions overlap one another completely up until a little before the maximum number density (B). The divergence is most clearly illustrated by a plot of the median for both the volume and drop size distributions (cf. Figure 12; top). At the divergence point (near B), coarsening by coalescence becomes important. To understand why, one can do the following thought experiment. In the absence of any mass exchange (i.e., condensation of steam onto the drops), volume is conserved. As two smaller drops coalesce, the volume from the histogram bars at smaller drop sizes transfers to those at larger drop sizes. Coalescence shifts

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volume to larger sizes so that at later times, the majority of the condensate volume is contained within a few large drops. Another interesting feature, is that the evolution of the median for the volume distribution has the same linear dependence as the condensate volume curve (cf. Figure 12; top). Hence, the slope of the curve is a constant, indicating again (i.e., in addition to Equation (3)) a constant isolated drop growth rate. Because a large percentage of the total volume on the surface is distributed amongst the larger drops, an under prediction of the growth rate of these largest drops would under predict the total condensate volume.

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Figure 11. Time progression of drop size (blue) and volume (red) distributions. Vertical red dashed lines denoted by a letter correspond to different instants. A comparison between the simulation (black) and the experiment (red) for the median values for the drop size (points) and volume (diamonds) distributions is shown at the top in Figure 12. The median values at time instants A-E correspond to the 50% probability on the cumulative distributions shown at the bottom left and right in Figure 12. Comparison between simulation and experiment was performed using cumulative distributions instead of histograms, because results in the latter case are sensitive to the bin width. Using a cumulative distribution avoids this issue. Immediately evident from Figure 12, is that the agreement is reasonable but gets worse as time goes on. This is likely due to the inability of the simulation to exactly reproduce the coalescence events and predict the exact drop size distribution. For the same total volume and the same number density, there can be multiple populations of drops (solutions), each with a different median. However, at least qualitatively, the bimodality feature in the drop size distribution at instants C, D, and E is well captured by the simulation (i.e., where curve levels off).

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Figure 12. (top) Comparison of the median values for the drop size (points) and volume (diamonds) distributions as a function of time for both simulation (black) and experiments (red), as calculated from a cumulative distribution. Vertical dashed red lines denoted by a letter correspond to different instants. (bottom left) Cumulative distributions of drop sizes at five different instants for experiment (points) and simulation (curves). (bottom right) Cumulative distributions of drop volumes at five different instants for experiment (points) and simulation (curves).

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Overall Heat Transfer Coefficient. The value of the overall heat transfer coefficient gives information about the ability of a surface to remove heat through condensation. The overall heat transfer coefficient can be found from

𝑈=

𝜌𝑙ℎ𝑓𝑔𝑉

(12)

𝐴𝑠(𝑇𝑠𝑎𝑡 ― 𝑇∞)

Equation (12) assumes 1D steady-state heat transfer in the axial direction. Calculated from 𝑊

experiment, the overall heat transfer coefficient was found to be 240 𝑚2 ∙ 𝐾 whereas the simulation 𝑊

predicted 218 𝑚2 ∙ 𝐾, corresponding to a 10% error. The only difference between the two calculations is the value of 𝑉. An under prediction of 𝑈 corresponds to an under prediction of 𝑉. There are several aspects of the experiment that cannot be realistically captured by the current simulation, such as the broad range of drop sizes in the interstitial space between drops, small oscillation of the surface because of the vibrations from the pump, irregularity of the contactangle along the perimeter of a drop because of surface defects and temperature gradients, and possible steam transients. These factors affect the growth and coalescence of drops, which ultimately affect 𝑉. It is speculated that the first factor, the broad range of drop sizes, is likely the most influential. In the experiment, there exists a range of drop sizes from the thermodynamic nucleating radius (nm) up to the departing radius (mm). The simulation is limited to drop sizes ranging from microns to millimeters. It is possible that the smallest drops, which pack themselves in at the perimeter of the large drops, enhance and facilitate more frequent coalescence.

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CONCLUSIONS The heat transfer of a surface at any instant is related to the population of drops present on the surface. It is important to be able to accurately model the time evolution of the drop population in order to be able to predict the heat transfer coefficient. It was found that using a population averaged drop growth rate, instead of the classical model with an assumed temperature difference of a few Kelvin, yielded satisfactory agreement between experiment and simulation for a low thermal conductivity substrate with our setup. From the classical model, larger drops have a larger conduction resistance. A larger resistance leads to slower growth and less coalescence. It was shown that a shift in the drop volume distribution towards larger drops occurs just prior to the maximum number density. Hence, using an inappropriate single drop growth model leads to under prediction of the total amount of liquid condensed and consequently, under prediction of the overall heat transfer coefficient. Simulations for both fixed site and random site nucleation were performed. It was found that the fixed site nucleation simulation fits the data better at earlier times whereas the random site nucleation simulation fits the data better at later times. The reason is possibly due to the activation of nucleation sites as the surface is swept by coalescing drops that leave deposits of liquid behind on any surface defects. By overlaying the drop size and condensate volume distributions at different points in time, it was shown that the medians of each distribution begin to diverge just before the maximum number-density, a time when coarsening of the drop pattern by coalescence first becomes important. Coalescence effectively shifts volume to larger sizes so that at later times, the majority of the condensate volume is contained within a few large drops. Again, emphasizing that accurate

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determination of the growth rate is important in being able to accurately capture the rate of change in condensate volume over time. Using the rate of change in condensate volume, the overall heat transfer coefficient of the surface was found to be 240 W/m2K whereas the simulation predicted 218 W/m2K (about 10% error). The difference likely arises from not being able to better simulate the broad range of drop sizes that exist on the surface. The dense packing of the smallest drops between existing larger drops possibly enhances and facilitates more frequent coalescence. As the resolution of 3D printing improves, the printing of physically textured condensing surfaces with precisely controlled micro- and nano-meter features becomes possible. Printing surfaces is a cheaper, and lighter alternative to photolithographic texturing of metals. In applications where mass savings is critical, such as a heat transfer cycle loop onboard a spacecraft, 3D printing of condensing surfaces is an attractive option. In order to predict the thermal performance of this class of materials, the traditional model for the isolated growth rate of a drop must be modified. In contrast to an infinitely conductive surface where the limiting resistance is conduction through the condensate, here, the limiting resistance is conduction through the substrate. By accurately capturing the growth rate of a single drop, the rate of condensate accumulation can be accurately calculated, and the thermal performance can be successfully predicted. ACKNOWLEDGEMENTS We thank Dr. Peter Ehrhard for help with designing the condensation chamber and the collimated light setup. We also thank Glenn Swan of the Olin Hall machine shop for fabrication support and useful design suggestions. The majority of this work was supported by a NASA Office of the Chief

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Technologist’s Space Technology Research Fellowship Grant NNX11AM81H. Additional support came from NSF Grant CBET-1236582.

REFERENCES [1] Putra, N.; Septiadi, W. N.; Rahman, H.; Irwansyah, R. Thermal performance of screen mesh wick heat pipes with nanofluids. Exp. Therm. Fluid Sci. 2012, 40, 10-17. [2] Qu, W.; Mudawar, I. Analysis of three-dimensional heat transfer in micro-channel heat sinks. Int. J. Heat Mass Transfer 2002, 45, 3973-3985. [3] Hill, S.A.; Kostyk, C.; Motil, B.; Notardonato, W.; Rickman, S.; Swanson, T. DRAFT Thermal Management Systems Roadmap: Technology Area 14; National Aeronautics and Space Administration: Washington, DC, November 2010. [4] Incropera, F.P.; DeWitt, D.P.; Bergman, T.L; Lavine, A.S. Fundamentals of Heat and Mass Transfer, 6th ed.; John Wiley & Sons, Inc: Hoboken, NJ, 2007. [5] Vemuri, S.; Kim, K. J.; Wood, B. D.; Govindaraju, S.; Bell, T. W. Long term testing for dropwise condensation using self-assembled monolayer coatings of n-octadecyl mercaptan. Appl. Therm. Eng. 2006, 26, 421-429. [6] Rose, J. W. Dropwise condensation theory and experiment: a review. Proc. Inst. Mech. Eng., Part A 2002, 216, 115-128. [7] Ma, X.; Rose, J. W.; Xu, D.; Lin, J.; Wang, B. Advances in dropwise condensation heat transfer: Chinese research. Chem. Eng. J. 2000, 78, 87-93. 33


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[8] Zhao, Q.; Zhang, D. C.; Zhu, X. B.; Xu, D.; Lin, Z. Q.; Lin, J. F. Industrial application of dropwise condensation. Heat Transfer 1990, Proc. Int. Heat Transfer Conf., 9th 1990, 391-394. [9] Zhao, Q.; Burnside, B. M. Dropwise condensation of steam on ion implanted condenser surfaces. Heat Recovery Syst. CHP 1994, 14, 525-534. [10] Zhao, Q.; Xu, D. The Dropwise Condensation Surface With Corrosion-Resisting Property. Corros. Rev. 1993, 11, 97–104. [11] McCormick, J.L.; Baer, E. On the Mechanism of Heat Transfer in Dropwise Condensation. J. Colloid Sci. 1963, 18, 208-216. [12] Ma, X. H. Dropwise condensation on PTFE coated surfaces and the catastrophe transition mechanisms between dropwise and film condensation. Ph.D. Dissertation, Dalian University of Technology, Dalian, China, 1994. [13] Tanaka, H. A theoretical study of dropwise condensation. J. Heat Transfer 1975, 97, 72-78. [14] Zhong, L.; Xuehu, M.; Sifang, W.; Mingzhe, W.; Xiaonan, L. Effects of surface free energy and nanostructures on dropwise condensation. Chem. Eng. J. 2010, 156, 546-552. [15] Chen, C. H.; Cai, Q.; Tsai, C.; Chen, C. L.; Xiong, G.; Yu, Y.; Ren, Z. Dropwise condensation on superhydrophobic surfaces with two-tier roughness. Appl. Phys. Lett. 2007, 90, 173108. [16] Zhao, H.; Beysens, D. From droplet growth to film growth on a heterogeneous surface: condensation associated with a wettability gradient. Langmuir 1995, 11, 627-634. [17] Daniel, S.; Chaudhury, M. K. Rectified motion of liquid drops on gradient surfaces induced by vibration. Langmuir 2002, 18, 3404-3407. 34


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[18] Farahi, R. H.; Passian, A.; Ferrell, T. L.; Thundat, T. Microfluidic manipulation via Marangoni forces. Appl. Phys. Lett. 2004, 85, 4237-4239. [19] Pratap, V.; Moumen, N.; Subramanian, R. S. Thermocapillary motion of a liquid drop on a horizontal solid surface. Langmuir 2008, 24, 5185-5193. [20] Burns, M. A.; Mastrangelo, C. H.; Sammarco, T. S.; Man, F. P.; Webster, J. R.; Johnsons, B. N.; Foerster, B.; Jones, D.; Fields, Y.; Kaiser, A.R.; Burke, D. T. Microfabricated structures for integrated DNA analysis. Proc. Natl. Acad. Sci. 1996, 93, 5556-5561. [21] Cho, S. K.; Moon, H.; Kim, C. J. Creating, transporting, cutting, and merging liquid droplets by electrowetting-based actuation for digital microfluidic circuits. J. Microelectromech. Syst. 2003, 12, 70-80. [22] Luo, M.; Gupta, R.; Frechette, J. Modulating contact angle hysteresis to direct fluid droplets along a homogenous surface. ACS Appl. Mater. Interfaces 2012, 4, 890-896. [23] Daniel, S.; Chaudhury, M. K.; Chen, J. C. Fast drop movements resulting from the phase change on a gradient surface. Science 2001, 291, 633-636. [24] Chaudhury, M. K.; Whitesides, G. M. How to make water run uphill. Science 1992, 256, 1539-1541. [25] Choi, S. H.; Zhang Newby, B. M. Micrometer-scaled gradient surfaces generated using contact printing of octadecyltrichlorosilane. Langmuir 2003, 19, 7427-7435. [26] Kraus, T.; Stutz, R.; Balmer, T. E.; Schmid, H.; Malaquin, L.; Spencer, N. D.; Wolf, H. Printing chemical gradients. Langmuir 2005, 21, 7796-7804.

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[27] Moumen, N.; Subramanian, R. S.; McLaughlin, J. B. Experiments on the motion of drops on a horizontal solid surface due to a wettability gradient. Langmuir 2006, 22, 2682-2690. [28] Subramanian, R.S. Motion of Drops on Gradient Surfaces. In Soft Matter Gradient Surfaces: Methods and Applications, 1st ed.; Genzer, J., Eds.; John Wiley & Sons, Inc.: Hoboken, NJ, 2012; pp 407-429. [29] Khoo, H. S.; Tseng, F. G. Spontaneous high-speed transport of subnanoliter water droplet on gradient nanotextured surfaces. Appl. Phys. Lett. 2009, 95, 063108. [30] Lv, C.; Hao, P. Driving droplet by scale effect on microstructured hydrophobic surfaces. Langmuir 2012, 28, 16958-16965. [31] Yang, J. T.; Chen, J. C.; Huang, K. J.; Yeh, J. A. Droplet manipulation on a hydrophobic textured surface with roughened patterns. J. Microelectromech. Syst. 2006, 15, 697-707. [32] Mettu, S.; Chaudhury, M. K. Motion of liquid drops on surfaces induced by asymmetric vibration: role of contact angle hysteresis. Langmuir 2011, 27, 10327-10333. [33] Daniel, S.; Chaudhury, M. K.; De Gennes, P. G. Vibration-actuated drop motion on surfaces for batch microfluidic processes. Langmuir 2005, 21, 4240-4248. [34] Genzer, J.; Bhat, R. R. Surface-bound soft matter gradients. Langmuir 2008, 24, 2294-2317. [35] Rose, J. W. Further aspects of dropwise condensation theory. Int. J. Heat Mass Transfer 1976, 19, 1363-1370. [36] Khandekar, S.; Muralidhar, K. Dropwise Condensation on Inclined Textured Surfaces; Springer: New York, NY, 2014 36


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[37] Leach, R. N.; Stevens, F.; Langford, S. C.; Dickinson, J. T. Dropwise condensation: experiments and simulations of nucleation and growth of water drops in a cooling system. Langmuir 2006, 22, 8864-8872. [38] Fritter, D.; Knobler, C. M.; Roux, D.; Beysens, D. Computer simulations of the growth of breath figures. J. Stat. Phys. 1988, 52, 1447-1459. [39] Mei, M.; Yu, B.; Zou, M.; Luo, L. A numerical study on growth mechanism of dropwise condensation. Int. J. Heat Mass Transfer 2011, 54, 2004-2013. [40] Briscoe, B. J.; Galvin, K. P. The evolution of a 2D constrained growth system of dropletsbreath figures. J. Phys. D: Appl. Phys. 1990, 23, 422. [41] Kim, S.; Kim, K. J. Dropwise condensation modeling suitable for superhydrophobic surfaces. J. Heat Transfer 2011, 133, 081502. [42] Sikarwar, B. S.; Khandekar, S.; Agrawal, S.; Kumar, S.; Muralidhar, K. Dropwise condensation studies on multiple scales. Heat Transfer Eng. 2012, 33, 301-341. [43] Burnside, B. M.; Hadi, H. A. Digital computer simulation of dropwise condensation from equilibrium droplet to detectable size. Int. J. Heat Mass Transfer 1999, 42, 3137-3146. [44] Sikarwar, B. S.; Battoo, N. K.; Khandekar, S.; Muralidhar, K. Dropwise condensation underneath chemically textured surfaces: simulation and experiments. J. Heat Transfer 2011, 133, 021501. [45] Macner, A. M.; Daniel, S.; Steen, P. H. Condensation on surface energy gradient shifts drop size distribution toward small drops. Langmuir 2014, 30, 1788-1798.

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[46] Cabezas, M. G.; Bateni, A.; Montanero, J. M.; Neumann, A. W. Determination of surface tension and contact angle from the shapes of axisymmetric fluid interfaces without use of apex coordinates. Langmuir 2006, 22, 10053-10060. [47] Gnielinski, V. New equations for heat and mass transfer in turbulent pipe and channel flow. Int. Chem. Eng. 1976, 16, 359-368. [48] McCormick, J. L.; Westwater, J. W. Nucleation sites for dropwise condensation. Chem. Eng. Sci. 1965, 20, 1021-1036. [49] Rose, J. W.; Glicksman, L. R. Dropwise condensation—the distribution of drop sizes. Int. J. Heat Mass Transfer 1973, 16, 411-425. [50] Tanemura, M. On random complete packing by discs. Ann. Inst. Stat. Math. 1979, 31, 351365. [51]

Technical

Glass

Products.

Technical:

Properties.

https://technicalglass.com/technical_properties.html. SUPPORTING INFORMATION Table S.1: List of thermophysical parameters. Figure S.1: Condensation simulation flow chart. Figure S.2. Effects of varying the nucleating radius, 𝑅𝑚𝑖𝑛, on number density, fractional coverage, normalized condensate, and median radius for random nucleation and constant growth rate.

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Figure S.3. Effects of varying the threshold radius, 𝑅𝑐, on number density, fractional coverage, normalized condensate volume, and median radius for random nucleation and constant growth rate. Figure S.4. Effects of varying the number of nucleation attempts, 𝑛𝑎, on number density, fractional coverage, normalized condensate volume, and median radius for random nucleation and constant growth rate.

Figure S.5. Effects of varying the nucleation site density,

𝑛

𝐴𝑠 on number density, fractional

coverage, normalized condensate volume, and median radius for fixed nucleation and constant growth rate. Figure S.6. Constant ∆𝑻, fixed nucleation. Comparison between experiments and simulation for number density, fractional coverage, normalized condensate volume, and median radius.

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Camera

Condensation Chamber 1 Window

Coolant Inlet 2 O-ring 3

Treated Surface

Inverted Petri Dish

Condensate Drain

Hot Plate

Coolant Outlet


Optics

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T∞ Langmuir

Coolant 1 l 23 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57

ks

Ts R

g

θs

σ,ρl ,kc

a


Tsat

Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Name Symbol Units Spherical Cap Radius a m Drop Footprint Radius R m Surface Tension σ N/m Liquid Density kg/m3 𝜌𝑙 Liquid Thermal Conductivity kc W/(m*K) Substrate Thickness l m Substrate Thermal Conductivity ks W/(m*K) Latent Heat of Vaporization hfg kJ/kg Rate of Heat Transfer qd W Interfacial Heat Transfer hi W/(m2*K) Coefficient Thermal BL Heat Transfer W/(m2*K) ℎ𝐵𝐿 Coefficient Critical Threshold Radius m 𝑅𝑐 Saturated Steam Temperature K 𝑇𝑠𝑎𝑡 Coolant Temperature K 𝑇∞ Gravity g m/s2 Dimensionless Groups 𝑡𝑐 𝑎 𝑎 sin 𝜃𝑠 ∆𝑇 𝑡 𝑎̇̃ = 𝑎̇ ; 𝑎̃ = = ; ∆𝑇̃ = ; 𝑡̃ = 𝑎𝑐

𝑎𝑐

𝑁𝑢1 = 𝐶

𝑅𝑐 ℎ𝑖

; 𝑁𝑢2 𝑘𝑐 𝜌𝑙 ℎ𝑓𝑔 𝑅𝑐2

∆𝑇𝑐 = 𝑅 ; 𝑡𝑐 = 𝑐

𝑅𝑐

𝐶ℎ𝑖

=

𝑙ℎ𝑖

𝑡𝑐

𝑘𝑠

;𝐶=

𝑓(𝜃𝑠 ) = 2 − 3𝑐𝑜𝑠(𝜃𝑠 ) +

∆𝑇𝑐

2𝑇𝑠𝑎𝑡 𝜎

ℎ𝑓𝑔 𝜌𝑙 𝑐𝑜𝑠(𝜃𝑠 )3


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Langmuir

Nucleate Vi (xi,yi) Vj (xj,yj)

a

Coalesce

Grow


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Number Density [drops/cm2]

1200 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

1000 800 600 400 200 0 0

20

40

60

80

t [s]


100

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Fractional Coverage [ ]

1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

0.8

0.6

0.4

0.2

0 0

20

40

60

80

t [s]


100

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

Normalized Condensate Volume [m]

−3

2

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1.5

1

0.5

0 0

20

40

60

80

t [s]


100

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Median Radius [mm]

0.3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

0.25

0.2

0.15

0.1

0.05 0

20

40

60

80

t [s]


100

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Number Density [drops/cm2]

1000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

800

600

400

200

0 0

20

40

60

80

t [s]


100

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Fractional Coverage [ ]

1.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

0.8

0.6

0.4

0.2

0 0

20

40

60

80

t [s]


100

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

Normalized Condensate Volume [m]

−3

2

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x 10

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1.5

1

0.5

0 0

20

40

60

80

t [s]


100

−4

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4.5 4 Median Radius [m]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

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x 10

3.5 3 2.5 2 1.5 1 0.5 0

20

40

60

80

t [s]


100

Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

B

A

C

D ACS Paragon Plus Environment

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E

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A


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B


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C


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D


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E


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A

B C

D


E

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A

B

C

D

E

Drop Sizes


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A

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B C D

E


Drop Volumes

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(79 μm)

(394 μm)


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(788 μm)

(2365 μm)

Simulating Heat Transfer During Transient Dropwise Condensation on

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