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Research Papers: Evaporation, Boiling, and Condensation

# Directional Passive Condensate Film Drainage on a Horizontal Surface With Periodic Asymmetrical StructuresOPEN ACCESS

[+] Author and Article Information
Shashank Natesh, Eric Truong

Mechanical and Aerospace Engineering, University of California-Davis, One Shields Avenue, Davis, CA 95616

Vinod Narayanan

Mechanical and Aerospace Engineering, University of California-Davis, One Shields Avenue, Davis, CA 95616 e-mail: vnarayanan@ucdavis.edu

Sushil Bhavnani

Department of Mechanical Engineering, Auburn University, Mary Martin Hall, Auburn, AL 36849

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received October 14, 2016; final manuscript received May 8, 2017; published online June 21, 2017. Assoc. Editor: C. A. Dorao.

J. Heat Transfer 139(11), 111507 (Jun 21, 2017) (15 pages) Paper No: HT-16-1662; doi: 10.1115/1.4036708 History: Received October 14, 2016; Revised May 08, 2017

## Abstract

Condensation of a highly wetting fluid on a horizontal surface with asymmetric millimeter-sized ratchets and periodically located film drainage pathways (DPs) in the spanwise direction is characterized. The hypothesis to be tested is whether the geometry would result in a net steady-state preferential drainage of the condensate film. Experiments are performed using PF5060 on a brass surface with ratchets of 3 mm pitch and 75–15 deg asymmetry. Drainage pathways are varied in density as nondimensional drainage pathways per meter depth ranging from 133 to 400. Experiments are performed at varied wall subcooling temperatures from 1 to 10 °C. Results of the asymmetric ratchet are compared against a control test surface with 45–45 deg symmetric ratchets. Both global and film visualization experiments are performed to characterize the differences in condensation between the symmetric and asymmetric surfaces. Global mass collection results indicate that all characterized asymmetric ratchet surfaces exhibit a net directional drainage of condensate while the symmetric control surface exhibited no preferential drainage. Among the asymmetric ratchets, the total mass flux rate increase with decrease in drainage pathway density, while the net mass flux rate increased with pathway density. Visualization of the condensate film was performed to explain the trends in net drainage with subcooling for different drainage pathway densities. For small drainage path density surfaces, a two-dimensional analytical model was developed to further characterize the effect of ratchet angle and Bond number on the net preferential drainage.

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## Introduction

Film condensation plays a critical role in a wide variety of engineering applications, ranging from process heat exchangers and fuel cells, to power generation, heating, ventilation and air-conditioning (HVAC), and refrigeration. Filmwise condensation is prone to condensate buildup that leads to increasing thermal resistance without appropriate drainage. In order to obtain enhanced heat transfer, surfaces that are equally effective at condensation and drainage need to be designed. Most of the surfaces designed for filmwise condensation rely on gravity to draw the condensate from the surface onto drainage channels or ridges; however, this mechanism is nonexistent in horizontal condensation surface or in reduced-gravity. Gregorig [1] noted that when an appropriately contoured surface is chosen, surface tension could be utilized to draw the condensate into drainage channels, wherein gravity is used for drainage. Zener and Lavi [2] presented a number of surface designs along with optimization methods for improving the design of vertical condensing surfaces. They introduced a simple drainage network for a contoured surface that consisted of a system of ridges (drainage wall) and valleys (condensing surface) with valley width four times the ridge width, and the network relied on surface tension to drain the condensing surfaces. Webb [3] extended the optimization analysis of Zener and Lavi [2] to include a broader range of condensing convex surfaces. Three parameters, convex surface length, convex surface radius of curvature at the crest, and angle at the end of the convex surface, were optimized to yield a maximum condensation coefficient based on the projected area of the fluted surface. Shigeki et al. [4] studied the effect of surface tension on the motion of condensate during laminar condensation inside a small trough. They revealed that the suction of liquid flowing into the trough reduced the liquid film thickness and enhanced the local condensation heat transfer. They reported that the heat transfer coefficient decreased with increasing trough width and was at least 3–4 times the Nusselt's prediction. However, the experiment and modeling were performed on a vertical open channel; therefore, the drainage in the depth of the fins was still gravity-controlled. Webb et al. [5] developed a theoretical model to predict the condensation coefficient on horizontal integral-fin tubes designed for surface-tension drainage from the fins. The model showed that a fraction of the tube surface allows bridging of the condensate, and this bridging occurs on the lower side of the tube, whereas surface tension was the dominant drainage force on the integral-fin tubes. However, they reported that the condensation rate on the condensate-bridged zone was negligible. Kedzierski and Webb [6] designed a family of high-performance fin profiles for surface-tension-drained condensation. The trade-offs between condensate retention and number of fins per meter (fpm) had to be considered to estimate the fin thickness, as larger fpm would result in increased heat transfer but would also retain the condensate on the interfin spaces. They suggested that the fin height and fin thickness should be designed based on two parameters: (1) bond number of unity and (2) the product of heat transfer coefficient and the fin arc length, to ensure an efficient surface tension drainage.

An extensive review by Bergles [7] and Shah et al. [8] on the passive methods to enhance film condensation heat transfer rate with the employment of surface extensions has shown that the internally grooved or knurled tubes, spirally fluted tubes, and conventional microfin tubes are successful in exhibiting effective filmwise condensation of steam and other fluids. The approach of using structured grooves on the condensation surface is attractive because it leads to high localized surface heat transfer coefficients near the crest where the liquid film is thin. Panchal and Bell [9] showed, with theoretical analysis for condensation on a vertical fluted (Gregorig) surface with cosine-shaped grooves, that the liquid film ascends over the crest of the groove owing to Gregorig effect and the drainage of the film in the trough region is completely gravity-controlled.

The aforementioned studies on structured surfaces for filmwise condensation focused on enhancement of heat transfer rate and relied on gravity for film drainage. The objective of the current work is to determine whether asymmetry introduced into the structures could, in addition to the higher heat transfer coefficient over a plain surface, be used to cause preferential drainage of the film on a horizontal surface. If such a surface could be devised, a net self-generated directional motion of liquid during condensation on horizontal surfaces or under microgravity conditions could be attained.

The role of surface asymmetry in heat transfer has been studied in the context of single phase and boiling heat transfer. In nonevaporating thin films, Stroock et al. [10] demonstrated that a macroscopically horizontal silicone oil liquid film on asymmetric heated ratchets exhibited a net directional motion, in addition to the cellular convection typically observed in Marangoni–Bernard flows. They observed that the directional motion reverses depending on the thickness of the liquid film and the vertical temperature difference imposed across the film. The net motion increased rapidly for Ma larger than the critical Ma. The velocity of the net motion was up to 2 mm/min, while the convection cell velocities were typically two to three orders of magnitude higher. In the film boiling regime, Linke et al. [11] demonstrated self-propelled droplet motion on asymmetrically profiled saw-tooth surface (or a ratchetlike topology). They attributed the droplet motion to the viscous force exerted by the flow of vapor between the droplet and the asymmetric ratchet surface. A similar study was performed by Lagubeau et al. [12] depicting that dry ice-disks, acting as Leidenfrost solids, self-propel on hot structured surfaces.

Kaspenberg et al. [13] observed passively generated liquid motion on asymmetrically textured surfaces during nucleate boiling of water. The asymmetry was devised using referential nucleation sites on one face of the repeating textures, which caused bubble growth and departure from these cavities. They developed a semi-empirical model, with the use of a simple force-balance model and experimental data, to demonstrate the effect of the nonvertical bubble growth on the net lateral liquid velocity. Thiagarajan et al. [14] conducted identical pool boiling experiments under reduced gravity (∼ 0.01 g) using such a surface. They observed bubbles sliding laterally along the surface at velocities of ∼27.4 mm/s.

The focus of the aforementioned studies has been exclusively on use of asymmetric structures in thin heated liquid films and boiling. The goal of the present work is to study the effects of asymmetry in surface texturing on the filmwise condensation process. Figure 1 shows a schematic of the test surface and introduces the geometrical parameters of relevance. The test surface consists of millimeter-sized ratchets on which condensation occurs. To assist the film drainage on such a surface, periodic drainage pathways are located in the spanwise direction. In the previous studies leading to this work [15,16], it has been shown that a condensate film developing from the trough on a periodic asymmetric ratchet rises to the crest of the steeper slope prior to that of the shallow crest. The hypothesis tested herein is whether the condensate film would further develop in a manner so as to provide a net directional motion of the film.

Specific quantities of interest are the directional mass condensation rate and total mass condensation rate for asymmetric ratchet surfaces with variable drainage pathway densities, ranging from 133 to 400. The density of the drainage pathways is defined as the number of drainage pathways per meter width of the condensation surface, as shown in Fig. 1. Hence, the 2, 4, and 6 drainage pathways would constitute a density, $WDP,$ of 133, 267, and 400, respectively. Experiments are performed over a range of subcooling varying from 1 to 9 °C. To compare results against a control surface, experiments are also performed on a 45 deg–45 deg symmetric ratchet with a of 400. To explain the global experimental results, visualization experiments of the liquid film profiles at the drainage pathways are performed at various subcooling levels for the symmetric and asymmetric ratchet surfaces. Furthermore, for small $WDP,$ a 2D analytical model is used to explore the parametric effects of ratchet angle and pitch on preferential film drainage.

## Experimental Methods

###### Test Substrate Preparation.

The test sections consisted of a brass test surface and a polycarbonate (PC) housing as shown in Fig. 2(a). The brass test surfaces were epoxied to a polycarbonate (PC) substrate that included two vial holders and enclosed by PC side walls. The bottom side of the PC substrate had a rectangular 15 mm wide × 50 mm long slot opening to permit thermal contact of the brass test surface on the cooled copper block (see Sec. 2.2).

The multiple-asymmetric grooved test surface, illustrated in Fig. 2(b), consisted of a series of ratchets placed right-side up, with gravity acting down on the grooves. Prior numerical simulations [15,16] on the condensate rise on such periodic ratchets indicated that the asymmetry in the rise of condensate to the crest of a ratchet increased with the contrast in the angles on the ratchet walls. Hence, an asymmetric ratchet with 75 deg–15 deg angles was selected for the present experiments. Furthermore, in order to permit ease of fabrication and visualization, the pitch of the asymmetric ratchet was selected to be 3 mm with a total length of 60 mm measured from crest to crest. The width of the test surface was 15 mm. To keep the end boundary conditions identical, an equal slope of 15 deg was prescribed at each end to eliminate the difference in film curvature induced by the asymmetry. A symmetric 45 deg–45 deg ratchet test surface (see Fig. 2(c)) with thirty-three 2 mm pitch ratchets was also fabricated to serve as a baseline control. The reduced pitch of the symmetric ratchet was a geometrical compromise to ensure that the ratchet heights between the 75 deg–15 deg and 45 deg–45 deg at 0.75 mm versus 1 mm, respectively, were comparable with each other.

The two PC walls that bound the test surface formed two drainage pathways for the fluid to the either of the vials at the ends. These side walls were attached to the bras ratchet using laser-cut double-sided tape that matched the profile of the ratchets. Additional PC walls could be affixed to the test surface; if one wall were inserted in the middle of the test surface, it would result in four drainage pathways, and if two walls were inserted on the test surface, it would constitute six drainable pathways. Drainage pathway densities of 133, 267, and 400 corresponded to a width between the drainage pathways of 15 mm, 7.5 mm, and 5 mm, respectively. Figure 2 illustrates a test surface with of 400 with six drainage pathways. The PC walls of the inner drainage pathways were machined at the bottom face to match the ratchet cross section and attached to the test surface using double-sided Kapton® tape. Precision blocks were used to ensure that the internal walls were spaced parallel to the side walls.

The condensate was collected at the ends of the test surface using glass vials that were placed within the PC vial holders shown in Fig. 2. O-rings seated between the rim of the vials and the PC vial holders provided the necessary sealing between the vials and the holder. The liquid in each vial was measured using an Acculab precision mass balance at the end of the experiment.

###### Experimental Chamber and Testing Facility.

The test section described earlier was placed in a test chamber, illustrated in Fig. 3 via a three-dimensional model. In order to reduce the influx of heat from the ambient environment, the test chamber (38 × 51 × 204 mm3) and the entire flow loop was fabricated using low thermal conductivity transparent PC. A 25 mm-diameter calcium fluoride window is placed on a side wall for visualization of the condensate film.

A separate boiler section with working volume capacity of 90 cm3 was linked to the condensation chamber using two 12 mm insulated lines and was used to introduce vapor from the left and right sides of the chamber. Fluid was drawn through a tube that is located in the fluidic line connecting the boiler to the chamber as shown in Fig. 3. The copper block (15 × 50 × 75 mm3) was cooled using a NESLAB ThermoFlex™ 5000 chiller and helped maintain the brass substrate of the test section at a constant surface temperature by means of a 5 mm-diameter cooling channel. Three K-type thermocouples were laterally placed 3.2 mm from the top surface of the copper block. Chamber pressure was measured using a PX409-015A pressure transducer. Steady-state visualization images of the condensate film were recorded using a Photron FASTCAM Mini AX-100 camera with a long working distance microscope objective lens backlit by a custom-built high-intensity 26 × 26 light emitting diode (LED) array. A vacuum pump was used to reduce the pressure and to remove any noncondensable gases from the chamber.

###### Experimental Procedure.

Film condensation experiments were performed utilizing a highly wetting fluid, PF5060, under subatmospheric conditions: over a pressure range of 0.19–0.24 bar, corresponding to saturation temperatures between 13 and 24 °C. At the start of test, the test section was affixed to the copper block using a thermally conductive Dow Corning® TC-5622 grease (k = 4.3 W/m K) [17], after which the test chamber was completely sealed. The substrate was precisely leveled and referenced at the surface of the copper block using a Digi-Pas® DWL-3000XY dual-axis high-precision digital leveler with an accuracy of 0.01 deg (at angles 0–10 deg) to negate the effects of gravity on condensate drainage.

Vacuum was pulled from the chamber in two sequential stages. The first stage involved decreasing the pressure inside the chamber so that the fluid fills the line from the reservoir to the boiler. This step ensured that all the noncondensable gases were evacuated from the fill line. In the second stage, the chamber pressure was further reduced to ∼0.01 bar and then held constant for nearly 300 s. The chiller was turned on to bring the surface temperature of the copper block to a predetermined temperature. Once the temperature reached a steady state (with fluctuations less than 0.1 °C), recording of temperature and pressure data was initiated using labview.

Each mass-collection test commenced with the filling of reservoir with 30 ml of PF-5060, at a rate of 1 ml/s. Condensation began immediately upon introduction of fluid into the system. The heater was turned on to generate vapor, which entered the chamber through the two ports at the bottom on either side of the test section. For steady-state conditions, the chamber pressure had to be maintained constant by varying the heat input to the boiler corresponding to the rate of condensation. A proportional–integral–derivative (PID) control of the heater power was attempted to maintain the chamber pressure constant; however, due to system inertia and pressure oscillations, precise control could not be achieved. Hence, manual adjustment of the heater power had to be performed to ensure that the chamber pressure was maintained approximately constant (with a variation of ±0.02 bar). The corresponding variation in saturation temperature was between 0.8 °C and 3.5 °C. This variation was in the form of a systematic slow change in pressure and not fluctuations about a mean. Hence, data are reported with a band that describes the variation of subcooling corresponding to the variation in saturation temperature during the test condition.

The experiment ended after a predetermined lapse of time for each subcooling, set to ensure low uncertainty in the collected mass rate. Power to the chiller was then turned off and the test section was then removed to retrieve the vials from both ends for mass-measurement.

## Data Reduction and Uncertainty Analysis

Subcooling was defined as the difference between the saturation temperature and the surface temperature. The saturation temperature for PF-5060 was extracted from temporal data of the chamber pressure saved during the experiment using the expressionDisplay Formula

(1)$Tsat=15629.729−log10p∞$

where the saturation temperature and experimental chamber pressure are specified in Kelvin and Pascal, respectively [18]. The surface temperature was measured as the average temperature of the three thermocouples. Estimates of the surface temperature at the base of the ratchets were performed by using a one-dimensional heat transfer analysis. The heat rate was determined using the sensible and latent heat rate from mass collection. The conductive resistances in the copper block and brass test section were determined based on the distances and thermal conductivities. The interfacial contact resistance was evaluated based on the manufacturer-provided thermal conductivity of the grease and multiple measurements of the thickness of the paste using a consistent repeatable application process. The resulting ratchet base surface temperature was less than 0.15 °C different from the measured temperatures. Given the added uncertainty involved in estimation of the base surface temperature, and the small difference between the measured and estimated temperatures, it was deemed satisfactory to use the measured thermocouple temperature as the representative surface temperature for subcooling estimation.

The total mass condensation rate (in $g/m2s$) was estimated based on the sum of the mass collected in the vials and the measured time of collectionDisplay Formula

(2)

The surface area used in Eq. (2) is based on the total surface area of the cooled brass surface for condensation. The net mass flow rate of the condensate over the structured surface of the ratchet is determined based on the difference between the mass of the fluids in the vialsDisplay Formula

(3)

One would expect that this time-averaged mass flux rate would be close to zero for a symmetric surface and that it would vary as a function of subcooling for asymmetric surfaces with different drainage pathway networks. To evaluate the heat transfer characteristics of the ratcheted surface, an average surface heat transfer coefficient was calculated using the total mass flux rate from Eq. (2), the temporally averaged subcooling at the surface, and the fluid propertiesDisplay Formula

(4)$havg=m˙total″[hlv+Cp(Tsat−Tw)Tsat−Tw]$

Both latent as well as sensible cooling terms were evaluated for determination of heat flux. Since the film flow rates are low, it was assumed that the condensate is sensibly cooled to the temperature of the surface prior to draining into the vials.

Table 1 shows the uncertainty estimates for all the variables in this experiment. Uncertainties in the measured quantities included calibration error for the thermocouple data and manufacturer specified errors for the pressure transducer. Uncertainties in the determined quantities such as total and net mass flux rate and heat transfer coefficient were determined using a propagation of errors [19].

## Results and Discussion

###### Total Mass Flux Rate.

The total mass flux rate, calculated using Eq. (2), is depicted in Figs. 4(a)4(d) as a function of subcooling for four test surfaces. In Fig. 4(a), as a baseline for contrast, the total mass flux rate for a symmetric ratchet test surface with six drainage pathways ($WDP$ = 400). It should be noted that the drainage occurs on both sides of the ratchet and the total mass flux rate accounts for the sum of mass collected in both vials placed at either ends of the test section. Figures 4(b)4(d) depict trends of the total mass flux rate for the three different drainage pathway densities of $WDP$ of 133, 167, and 400, respectively. The data are plotted in different figures with identical scales for ease of comparison. Note that, as discussed in a prior section, the bands on the subcooling in the data do not represent uncertainty in the data (which would be given as standard deviation of the mean of the data), rather they represent the range of variation of subcooling in the experiments due to slight and gradual variations in system pressure during the time of mass collection. Also, plotted with the data in Figs. 4(a)4(c) are best fit trend lines as well as the lower and upper bound linear trend-lines for the data. It has to be emphasized that the mass flux rate is based on the actual brass condensate test surface area for each surface; and hence, the variations in surface area between the test surfaces is taken into account.

For the symmetric ratchet test surface (Fig. 4(a)), the total mass flux rate is seen to vary between 2 and 5 g/m2 s, with a roughly increasing trend with subcooling from 2 to 8 °C, with the exception of one data point at a subcooling of 5.5 °C. For identical drainage pathway density, the slope of the mass flux rate with subcooling for the asymmetric test surface (Fig. 4(d)) is 1.8 times that of the symmetric ratchet test surface, suggesting that surface asymmetry significantly changes the condensate film dynamics and heat transfer rate.

As seen in Figs. 4(b)4(d), the total mass flux rate of condensate on asymmetric test surfaces also follows an approximately linear increase with subcooling between 0 and 10 °C. This increasing trend indicates that the drainage pathways are capable of removing the condensed fluid from the surface effectively for all drainage densities. In comparing the total mass flux rate across different drainage pathway densities, in Figs. 4(b)4(d) at a fixed subcooling, one notices higher mass flux rate (and consequently a higher slope) for the smallest drainage density of $WDP$ = 133 and progressively lower mass flux rates for the larger drainage densities. This trend suggests that the liquid film resistance progressively increases with an increase in $WDP$ from 133 to 400. It is also clear from the slopes of the mass flux rate with subcooling that the percent reduction in total mass flux between $WDP$ = 267 and $WDP$ = 133 (at 4.5%) is smaller than that between $WDP$ = 400 and $WDP$ = 267 (at 13.2%).

Preliminary studies confirmed that in the absence of the drainage pathways (i.e., $WDP$ = 0), the condensate film would rise to the steep slope but remain pinned and not cascade to the next ratchet. It is possible that an optimum density exists below $WDP$ = 133; however, as seen from Fig. 2, this density corresponding to two drainage pathways bounding the test section was the lowest possible in current experiments. While it is clear that an increase in liquid resistance with drainage density is the cause for the trend observed, the exact cause for the increase in liquid resistance with $WDP$ is as yet unclear. It is conjectured that the curvature in the depth (spanwise) direction causes a net increase in film thickness in the depth for higher density of drainage pathways, thereby resulting in a reduction in total mass flux rate.

###### Heat Transfer Coefficient.

The linear trend lines of total mass flux rates and the sensible cooling rate of the condensate were used to estimate the area-average heat transfer coefficient (see Eq. (4)); the resulting condensation heat transfer coefficient is plotted in Fig. 5. For the symmetric ratchet, a decreasing trend in heat transfer coefficient is observed with subcooling, similar to that expected for a flat surface, due to an increase in film thickness. In contrast, for all three asymmetric test surfaces, a slightly increasing trend in heat transfer coefficient with subcooling in the range of the experiments is discerned. As an example, the heat transfer coefficient of the asymmetric $WDP$ = 133 test surface increased by 11% over the range of subcooling from 1 to 11 °C. Since the total mass flux rate exhibits a linear trend with subcooling, the increasing trend is a result of an increase in the sensible cooling rate with subcooling. Comparing across the three drainage pathway spacing, the heat transfer coefficient was higher for the lowest drainage density of $WDP$ = 133 and decreased monotonically for the other two densities. For a fixed spacing of $WDP$ = 133, the average heat transfer coefficient, $havg,$ of the asymmetric test surface was 29% larger than of symmetric test surface at low subcooling of 2 °C and increased with subcooling.

If a purely conductive resistance were assumed to exist in the liquid film for the asymmetric test surfaces, the heat transfer coefficient trend with drainage spacing would suggest that, at any subcooling, a $WDP$ = 133 has smaller average film thickness ($δl=k/havg$) over the entire test surface as compared with $WDP$ of 267 and 400. For any given $WDP$, the increasing trend of $havg$ with subcooling would further suggest that the film thickness decreases with subcooling. However, visualization experiments, shown later in this section, indicate that the condensate film thickness increases with subcooling. Hence, the validity of a pure conductive assumption within the liquid layer for asymmetric ratchets is inadequate.

However, for any given subcooling, $havg$ is higher for a sparse drainage pathway spacing. Although this observation cannot be explained from the visualization experiments shown later in this section, it is believed that as decreases, the average film thickness in the depth decreases. This reduces the conductive resistance due to the condensate film and promotes a higher heat transfer coefficient on the surface.

###### Net Mass Flux Rate.

The net mass flux rate, calculated using Eq. (3), is depicted in Figs. 6(a)6(d) as a function of subcooling for four test surfaces. In Fig. 6(a), as a baseline for contrast, the net mass flux rate for a symmetric ratchet test surface with six drainage pathways ($WDP$ = 400). The dotted and solid lines in Figs. 6(b)6(d) correspond to trend lines of the upper and lower limits of variation of subcooling, respectively, in the experiments.

The net mass flux rate for symmetric surface (see Fig. 6(a)) is near zero irrespective of subcooling and does not show any consistent preferential side of net motion, as indicated by the positive and negative values. Such a trend would be expected for the baseline symmetric surface where the fluid would drain equally into the vials at the pathways. In contrast to the symmetric test section results, the net mass flux rate for all asymmetric test surfaces (Figs. 6(b)6(d)) show an increasing trend with subcooling for the lower subcooling range ($ΔTsub>4 °C$). In the prior work [15] on the initial rise of the condensate on periodic ratchets, it was seen that the film rose to the crest of the steeper ratchet wall prior to the shallow wall. The results in Figs. 6(b)6(d) further demonstrate that a steady-state film on asymmetric ratchets leads to a cascading effect that provides a net mass flow in the direction of the steeper ratchet slope.

Differences are observed in the trends of the net mass flux rate for different drainage spacing at higher subcooling. For the $WDP$ = 133 surface (Fig. 6(b)), the net mass flux rate reaches a plateau at a subcooling of ∼6 to 9 °C. However, as seen in Fig. 4(a), the total mass flux rate showed increasing trend at all subcooling levels tested, indicating that while the net motion reaches a limiting condition, the drainage is sufficient to maintain steady-state conditions even at the highest subcooling tested of 9 °C. For the $WDP$ = 267 surface (Fig. 6(c)), the trend is similar to $WDP$ = 133 with a plateau at subcooling of 6–8 °C. However, the net mass flux rate at the plateau is larger for the $WDP$ = 267 spacing, at 1 g/m2 s compared to the plateau value of 0.8 g/m2 s for the $WDP$ = 133 spacing. In contrast to the trends of the $WDP$ = 133 and 267, there is no plateau in net mass flux rate for the $WDP$ = 400 spacing, with a linear trend being exhibited over the range of subcooling from 1 °C to 8 °C. The average deviation over the entire subcooling range depicted by the band (dashed and solid lines on either side of the data points) is around 6.5% for $WDP$ = 133, 7.1% for $WDP$ = 267, and 19.3% for $WDP$ = 400.

From the data in Figs. 4 and 6, the ratio of the net to total mass flux rate was calculated to provide an indication of the secondary flow variation with drainage pathway density and subcooling. With an increase in surface subcooling from 1 °C to 8.5 °C, this ratio of net to total mass flux rate decreases monotonically for $WDP=133$ from 22% to 4% and $WDP=267$ from 28% to 9%. However, the ratio remains constant at ∼ 18% for , suggesting that ratchet with six drainage pathways is equally effective at preferential drainage at all subcooling levels.

###### Visualization of the Condensate Film.

In order to explain the differences in trends of total and net mass flux rates between the symmetric and asymmetric ratchet test sections, visualization of the liquid film was performed in a region (into the depth of the ratchet) surrounding the drainage pathway. The film near the drainage pathway is complex and three-dimensional in nature; two edges of this film were captured in the recorded images. The thicker film edge corresponds to the one near the drainage wall and is termed the drainage film edge. The thinner film edge, imaged through the depth of the ratchet, corresponds to the film on the brass wall, i.e., the condensate film edge.

The development of the film at the drainage wall for a symmetric ratchet with $WDP$ = 400 is shown in Figs. 7(a)7(c) for three different subcooling conditions. With the increase in subcooling, condensate is seen to pool up inside the ratchet, and the radius of curvature at the trough of the drainage edge increases. At each of the six subcooling conditions, the location of the trough of the drainage edge of the film coincides with the midpoint of the ratchet (shown in dashed–dotted line). This does not, however, reduce the ability of a symmetric ratchet to drain the liquid film, which occurs equally in both directions. However, there would be no net mass flux rate in one direction or the other, as corroborated by Fig. 6(a) where the net mass flux rate is nearly zero for all the test cases performed.

Figures 8(a)8(f) presents results of film visualization for the asymmetric ratchet for $WDP$ = 133 (Figs. 8(a)8(c)) and $WDP$ = 400 (Figs. 8(d)8(f)) at different subcooling levels, respectively. The net mass flux rate observed in asymmetric ratchets is caused by the asymmetry in the drainage film edge. At high subcooling levels, the curvature of the drainage film becomes almost zero (very large radius of curvature) as observed in Fig. 8(b). From Fig. 8(c), it is seen that with an increase in subcooling, the location of the trough of the drainage film edge, obtained using Canny's edge detection algorithm [20], moves to the right (toward the midpoint of the ratchet pitch) and away from the ratchet trough. With the movement of the drainage film edge trough toward the midline of the pitch, a more symmetric film configuration at the drainage wall is attained. However, in the depth (i.e., in the direction of the camera view), the curvature on the film edge remains asymmetric even at higher subcooling, suggesting that a complex three-dimensional curvature change with subcooling. The result of this curvature change is a plateau in net mass flux at the higher subcooling conditions reported in Fig. 6(b).

For the higher drainage pathway density $WDP$ = 400 (Figs. 8(d)8(f)), it is seen that an increase in the subcooling does not change the location of the trough of the drainage film edge at the same rate as that of the drainage film edges for $WDP$ = 133. The location of the trough of the drainage film is still closer to the trough of the ratchet than the midpoint of the ratchet, and this is also evident from the drainage edges plotted in Fig. 8(f). This ensures that the asymmetry in the film required for the net drainage of the liquid film is maintained in that plane. This phenomenon is also observed at higher subcooling levels, where the liquid film does not pool up inside the ratchet, corroborating the net mass flux rate results obtained in Fig. 6(d), where the net mass flux rate does not plateau with increase in subcooling.

###### Analytical Modeling of the Condensate Film.

In order to further analyze the effect of ratchet geometry on the film profile and net preferential motion generated, a mathematical two-dimensional model to study the characteristics of a steady-state adiabatic liquid film on a periodic ratchet microstructure was developed. Because the model is two dimensional, its validity would be limited to conditions of sparse drainage pathways. In the experiments, the pitch and the ratchet angles were kept constant and the effect of drainage pathways was characterized. The intent of this model is to vary parameters like pitch, ratchet angles, and Bond number, and study their effects on the liquid film profile and the net preferential motion.

The model is based on the idea that surface tension (pressure gradient due to change in the film curvature), viscous forces, and gravity are the only three dominant forces in the ratchet geometry. Steady-state momentum and mass-conservation fluid equations are solved in a hyperbolic coordinate system, similar to the approach presented by Stocker and Hosoi [21]. The hyperbolic coordinate system is given in terms of the Cartesian coordinates asDisplay Formula

(5)$ϕ=x2−y2, η=2xy$

Figure 9 shows the lines of constant $ϕ$ and constant $η$ to demonstrate the coordinate system. The $x$ -axis and $y$ -axis represent the shallow and steep walls of the ratchet, respectively. This coordinate system is useful in representing flow inside the ratchet because the $ϕ$ -coordinate is parallel to both walls of the ratchet (along the length of the ratchet), while the $η$ -coordinate is perpendicular to both walls of the ratchet. This would mean that the liquid film thickness (sample shown in Fig. 9) would only be a function of the $ϕ$ -coordinate, unlike the Cartesian system where the film thickness is a function of both x and y.

###### Mathematical Formulation.

The flow in a ratchet, as sketched in Fig. 9 (where x and y represent the walls of the ratchet), is considered and the steady, laminar, incompressible momentum, and continuity equations in vector-form are, respectively,Display Formula

(6a)$0=−∇p+μ∇2u−g cos θstî−g sin θstĵ$
Display Formula
(6b)$∇⋅u=0$

Considering a two-dimensional system, i.e., assuming no variation in the third dimension, the two stress boundary conditions (normal and tangential) at the edge of the condensate film areDisplay Formula

(7)$n̂⋅Π⋅n̂=σκ, t̂⋅Π⋅n̂=0$

The curvature of the free surface is given by $κ=−∇⋅n̂$. The Newtonian stress tensor is given by $Π=μ[∇u+(∇u)T]−pI$. The governing equations (6a) and (6b) can be rewritten in hyperbolic coordinates (see the Appendix for details) asDisplay Formula

(8a)$∂u∂ϕ+∂v∂η−12r2(uϕ+vη)=0$
Display Formula
(8b)$∂p∂ϕ=μs[∂2u∂ϕ2+∂2u∂η2−1r2(η∂v∂ϕ−ϕ∂v∂η+u4)]−ρg(∂x∂ϕcos θst+∂y∂ϕsin θst)$
Display Formula
(8c)$∂p∂η=μs[∂2v∂ϕ2+∂2v∂η2−1r2(−η∂u∂ϕ+ϕ∂u∂η+v4)]−ρg(∂x∂ηcos θst+∂y∂ηsin θst)$

The number of terms in the governing equations in hyperbolic coordinates increases because the unit vectors $ϕ̂$ and $η̂$ are position dependent. The curvature is computed from the unit normal vectorDisplay Formula

(9)$κ=h″s(1+h′2)32+12sr2(1+h′2)12(−η+ϕh′)$

where $η=h(ϕ)$ is the position of the free surface in the hyperbolic coordinates and a prime denotes a derivative with respect to $ϕ$. It is important to note that the liquid film thickness in the hyperbolic coordinates has units of square of length. The conservation of mass (or volume, since constant density is assumed) needs to be imposed, i.e., the volumetric flow rate, $Q$, remains constantDisplay Formula

(10)$∂Q∂ϕ=0, Q=∫0h(ϕ)u(ϕ,η)sdη$

Lubrication theory is applied to the full governing equations (8) and (10) to obtain a single ordinary differential equation defining the variation of film thickness $h(ϕ)$ inside the ratchet. Rescaling the governing equations by making the variables nondimensionalDisplay Formula

(11)$ϕ=L2ϕ̃, u=Uũ, p=Pp̃, η=LHη̃, v=Vṽ$

Here, $L$ is the length scale of the ratchet, i.e., its pitch, $H$ is the characteristic liquid film thickness, $U$ and $V$ are the characteristic velocities along and across the liquid film, respectively, and $P$ is the pressure scale. Since only nondimensional variables shall be used henceforth, all tildes are dropped. In the lubrication limit, assuming that the parameter $ϵ=H/L$ is small, all variables can be expanded in the powers of $ϵ$ using Taylor series. Applying this limit onto Eqs. (8a)8(c), assuming $ϵη/ϕ$ < 1 and neglecting the higher order terms in $ϵ$, the governing equations would reduce toDisplay Formula

(12a)$V=ϵU$
Display Formula
(12b)$∂p∂ϕ=2Ca|ϕ|12∂2u∂η2−ϵ2Bo(∂x∂ϕcos θst+∂y∂ϕsin θst)$
Display Formula
(12c)$∂p∂η=−ϵ2Bo(∂x∂ηcos θst+∂y∂ηsin θst)$

Here, $Ca=μU/σ$ is the capillary number and $Bo=ρgL2/σ$ is the Bond number, and they represent the relative importance of gravity, surface tension, and viscous forces. The stress boundary conditions at the free surface $η=h$ reduce toDisplay Formula

(13a)$−p=ϵ3κ$
Display Formula
(13b)$∂u∂η=0$

Upon integrating equation (12c) and applying the boundary condition (13a), we getDisplay Formula

(14)$p=−ϵ3κ−ϵ2Bo((xη−xh)cos θst+(yη−yh)sin θst)$

where subscript denotes the value of $η$ at which $x$ and $y$ are estimated. Substituting this expression into Eq. (12b), and integrating twice with respect to $ϕ$, will result in the velocity along the liquid filmDisplay Formula

(15)$u=ϵ2h2ψ(ϕ)2Ca|ϕ|12[12(ηh)2−ηh], ψ(ϕ)=−ϵκ′+Bo(∂xh∂ϕcos θst+∂yh∂ϕsin θst)$

It should be noted that the velocity is parabolic in the new coordinate $η$.

Incorporating this expression for the velocity into the volumetric flow rate expression in Eq. (10) results inDisplay Formula

(16)

Equating expressions for $ψ(ϕ)$ from Eqs. (15) and (16), rearranging, and using curvature gradient expression (A18), we obtain a differential equation for $h(ϕ)$ asDisplay Formula

(17)$h″′=β32α[6ϵ2αβ5h′(h″)2+3(h−ϕh′)(ϕ+ϵ2hh′)2α7β+12Ca|ϕ|Qϵ3h3+Boϵ(∂xh∂ϕcos θst+∂yh∂ϕsin θst)]$

The expressions for $β$ and are specified in Eqs. (A19a) and (A19b), respectively. matlab's BVP5C function is used to solve this differential equation by splitting it into three first-order ordinary differential equations. The function implements a four-stage Lobatto-IIIa collocation formula that is fifth-order accurate uniformly in the $ϕ$ domain using an implicit Runge–Kutta method. Since equation is of third-order and $Q$ is treated as an unknown parameter, four boundary conditions are required to solve the set of three first-order ordinary differential equations (ODEs): Dirichlet condition on both crests of the ratchet (crest of steep and shallow walls), i.e., ; and zero flux at both crests of the ratchet, i.e., $h″=0$. Here, $D$ is the nondimensional thickness in the hyperbolic coordinates that is to be prescribed at the crest of the ratchet.

###### Comparison of Analytical Model Solution With $WDP$ = 133 Surface Experiments.

Since the model is two dimensional and there is no variation in the liquid film in the third dimension, it is important to choose the lowest drainage pathway density to compare the results against the model. This is because the lowest drainage pathway density would have the largest width between the drainage pathways, and it is likely that the behavior of the condensate film would compare with the results from the model, as opposed to higher drainage pathway densities. Also, it is important to note that the comparison is performed only against the condensate edge (see Fig. 8(b)) of the film, because the drainage edge of the film is the film edge at the drainage pathway surface, and the model does not account for the no-slip condition at the drainage walls (in the third-dimension). Figures 10(a)10(d) shows the comparison of the condensate film edges obtained from the experiment at various subcooling levels against the liquid film profiles for a 75 deg–15 deg ratchet at different boundary conditions. The intent behind this comparison was to determine, if the model and experimental profiles matched when the boundary condition at the crest of the steep wall was matched. Since this model does not include mass or heat transfer, the effects of subcooling are captured by varying the boundary condition at the crest to match the experiments. The condensate film edges are obtained from the images shown in Figs. 8(a) and 8(b) using edge detection techniques in matlab, and the coordinates of the condensate film edge at the crest are used to estimate D (boundary condition at the crest) that needs to be imposed in the model. The pitch of the ratchet used in the experiment was 3 mm and it is used as the length scale to compute the Bond number. Only a third of the shallow wall is shown in these plots because the condensate edge of the film cannot be completely resolved on the shallow wall (see Figs. 8(a) and 8(b)).

The profiles from the model match closely with that of the experiment, especially close to the steep wall. A magnified view of the profiles in Fig. 10(a) shows that the thickness at the steep wall is on the order of tens of microns, based on images in the experiment. The curvatures are very similar, and it shows that as the subcooling rises, the film curvature changes and the model accounts for that change by computing the pressure force at the free surface of the film. The relative percentage error between the film profiles from the model and the experiment is estimated at all subcooling levels along the entire length of the film. For a subcooling of 9.29 °C, the error, when averaged over the length of the film, is around 3.7%.

Some differences in the condensate film edge are also observed, particularly close to the shallow wall. It is believed that the deviations at the trough are due to the three-dimensional effects in the experiments and use of the condensate edge of the film that is in the relative vicinity of the drainage wall. Second, the model is adiabatic and does not account for interfacial mass transfer. Therefore, the differences appear because the model does not account for the liquid film variation due to high interfacial and wall heat fluxes. Given the overall fair prediction of the film profiles, the model is used to study the effect of parametric variations of the ratchet profile.

###### Model Predictions: Variation of Liquid Film Thickness.

The effect of variation of ratchet angle on the film profile for three different ratchet angle configurations: symmetric (45 deg–45 deg), 60 deg–30 deg and 75 deg–15 deg is presented in both hyperbolic (Fig. 11(a)) and Cartesian (Fig. 11(b)) coordinate system. Before discussing the results from the model, it is important to note that negative side of $ϕ$ denotes the steep wall, while the positive side of $ϕ$ denotes the shallow wall (from Eq. (5)).

From Figs. 11(a) and 11(b), it can be observed that the thickness increases as the film proceeds toward the trough of the ratchet ($x=y,$ or $ϕ=0$) and then decreases as it goes back to the crest of the shallow wall (toward the positive extent of $ϕ$). The curve for the symmetric ratchet (solid line) shows that the film profile is symmetric in $ϕ$ and the maximum in $h$ occurs at $ϕ=0$. But, the curves for the asymmetric ratchets (60 deg–30 deg and 75 deg–15 deg) show that the length of negative side of $ϕ$ reduces and the film profile is asymmetric about $ϕ=0$, with maximum in $h$ approximately at $ϕ=0.2$ for 60 deg–30 deg and at $ϕ=0.3$ for 75 deg–15 deg. This means that as the ratchet steep wall angle increases the location of the maximum liquid film thickness moves closer to the crest of the shallow wall.

As the same boundary condition is specified at the crests of the ratchet, the average film thickness computed over $ϕ$ decreases as the steep angle increases from 45 deg (symmetric) to 75 deg. These observations are in agreement with the film images in the experiment, where the average thickness of the condensate film edge for a symmetric ratchet was greater than that for an asymmetric ratchet. From Fig. 11(b), it is also evident that the curvature is more pronounced for the asymmetric ratchets than the symmetric ratchet. This means the radius of curvature is lower for an asymmetric ratchets, and assuming the same vapor pressure, results in a larger capillary pressure difference (driving force) for the asymmetric ratchets.

###### Model Predictions: Velocity Profiles and Bond Number Effect.

The relative importance of surface tension and gravity can be quantified by studying the effect of Bond number of the liquid film profile. Figures 12(a) and 12(b) show the film profiles at different corresponding Bond numbers for a symmetric and an asymmetric (75 deg–15 deg) ratchet, respectively. The direction of gravity is always the vertical line joining the trough of the ratchet (as shown by small ratchets in the right corner of Fig. 12) and its respective components along the ratchet walls is depicted in Fig. 12. Because the ratchet is symmetric in Fig. 12(a), film profiles at all Bond numbers are symmetric about line $y=x$. But, as the Bond number increases, the liquid starts pooling up at the trough of the ratchet because equal components of gravity are forced at the ratchet walls.

However, in the case of the asymmetric ratchet in Fig. 12(b), as the Bond number is increased, the liquid starts to pool up more on the shallow side of the ratchet than the steep side of the ratchet. This is because the gravity component is larger on the shallow wall, and this allows more fluid to drain on the shallow wall. The slope of the liquid film thickness also increases along the shallow wall as the Bond number is increased, but the slope along the steep wall does not change substantially. This changes the radius of curvature at the shallow wall, and it is expected that at higher Bond numbers (i.e., low surface tension forces), the liquid film would flatten out and the net motion would decrease. The observations made regarding the film profiles in this section for both symmetric and asymmetric ratchet shows the ability of the model to capture the physical phenomena that occurs in a variety of length scales (from microscale to meter-scale structures) for both low and high surface tension fluids and with variations in gravity.

Figure 13 shows the variation of the interfacial velocity along the entire length of the liquid film for a symmetric and an asymmetric ratchet. The values in the plot represent a nondimensional velocity $u/U$. For a symmetric ratchet (45 deg–45 deg), the values are identically zero at all locations. For an asymmetric ratchet (75 deg–15 deg), it can be seen that the values of the interfacial velocity are negative. This means that the flow occurs in the negative $ϕ$ direction, i.e., from the shallow wall to the steep wall. This is corroborated by the net mass flux results observed in the experiments and is in agreement with the hypothesis of net preferential motion in the liquid film. The flow decelerates as it approaches the trough of the ratchet ($ϕ=0$) from the crest of the shallow wall ($ϕ≅0.9)$. There is a discontinuity at $ϕ=0$ because expression for the velocity along the film is inversely proportional to $|ϕ|1/2$ (see Eq. (15)). The flow then accelerates from the trough to the crest of the steep wall ($ϕ≅0.1$) to an amplitude that is lower than that at the shallow wall. This is because the liquid film thickness at the crest of the steep wall is higher than the thickness at the crest of the shallow wall (refer Fig. 11(b)).

## Conclusions

A combined experimental and analytical study of filmwise condensation on asymmetrically structured millimeter-sized ratchets was performed. The results of total and net mass condensation rate were compared against that of a symmetric ratchet control surface. Periodic drainage pathways of densities $WDP$ = 133, 267, and 400 were characterized over a range of subcooling varying from 1 °C to 10 °C. Visualization of the liquid film was performed to explain the results of global data. A 2D analytical model, applicable to low drainage pathway densities, was used to perform parametric studies of ratchet angle and Bond number variations on the film profile and net condensate flow. Key findings from the section are as follows:

1. (1)Effective drainage of horizontal films was demonstrated by a novel surface design consisting of asymmetric or symmetric millimeter-sized ratchets and periodic drainage pathways located in the spanwise direction.
2. (2)For test surfaces with identical $WDP$ = 400, the total mass flux rate for the asymmetric ratchets was larger than that of the symmetric case. The estimated condensation heat transfer coefficient was 1.8 times larger for the asymmetric surface compared to the symmetric surface.
3. (3)Among the asymmetric ratchets, the surface with the least dense drainage pathway of $WDP$ = 133 exhibited the highest total mass flux rate at any given subcooling. The reason for this trend is not tangible from the visualization of the liquid film, but it is believed that as $WDP$ decreases, the average liquid film thickness in the depth decreases, thereby decreasing the conductive thermal resistance in the liquid film. This might be the reason why higher heat transfer coefficients (or higher total mass flux rates) are observed for sparse $WDP$. Future studies shall be devoted to developing a three-dimensional model that would be able to capture the variation of the liquid film in the depth and optimize the surface heat transfer based on $WDP$.
4. (4)While the fluid drained from both ends of the test surface, the net mass flux rate for the asymmetric test surfaces indicated a net preferential direction of the steeper ratchet wall. No net mass flux rate was observed for the symmetric test surface.
5. (5)The net mass flux rate for the asymmetric ratchets with drainage pathway spacing of $WDP$ = 133 and 267 exhibited a plateau at higher subcooling values in the range of 7–9 °C. However, no such plateau was observed for the densest spacing of $WDP$ = 400. Visualization of the drainage film for the $WDP$ = 133 asymmetric surface indicates that the asymmetry in the film edge near the wall decreases with an increase in subcooling, potentially causing a plateau observed in the net mass flux rate at higher subcooling for these spacing.
6. (6)The mathematical model provides insight into the flow inside the ratchet, showing a net directional flow from shallow to steep wall for an asymmetric ratchet and no net flow for a symmetric ratchet. The liquid film profiles are also in good agreement with the condensate edges of the film for the $WDP$ = 133, exhibiting an error of 3.7%. The flow inside the ratchet is parabolic in , and the shear stress is higher on the shallow wall than the steep wall. The model shows that the preferential motion diminishes at higher Bo because the gravity forces dominate and render the film curvature flat.
7. (7)Results from the study indicate that different drainage pathway density needs to be chosen depending on whether total condensation rate or directional film drainage is important in the application. Within the range of parameters studied, for increased heat transfer coefficient and total mass flux rate, a less dense network of drainage pathways with asymmetry is preferred; if preferential drainage is important, a denser spacing is recommended.

## Acknowledgements

Funding for this work was provided by NASA through the Early State Innovation (ESI) Grant No. NNX15AE58G.

## Nomenclature

• $As$ =

total surface area of the ratcheted test article (m2)

• Bo =

Bond number

• $Cp$ =

specific heat (J/kg K)

• Ca =

Capillary number

• D =

boundary condition for the film thickness (m2)

• g =

acceleration due to gravity (m/s2)

• $h$ =

liquid film thickness in hyperbolic coordinates (m2)

• H =

characteristic liquid thickness (m)

• $havg$ =

area-averaged surface heat transfer coefficient (W/m2 K)

• $hlv$ =

latent heat of condensation (J/kg)

• I =

identity matrix

• $î$ =

unit vector along x-axis

• $ĵ$ =

unit vector along y-axis

• L =

characteristic length scale for the model (m)

• $m˙″$ =

mass flux rate (g/s m2)

• Ma =

Marangoni number

• $n̂$ =

normal vector to the free surface

• $p$ =

pressure (Pa)

• P =

characteristic pressure scale for the model (Pa)

• Q =

volumetric flow rate (in hyperbolic coordinates) (m3/s)

• r =

distance of a point from the origin in the hyperbolic coordinates

• s =

scale factor

• $t$ =

time (s)

• T =

temperature (K)

• $t̂$ =

tangential vector the free surface

• u =

$ϕ−$ component of the velocity (m/s)

• $u$ =

velocity vector in hyperbolic coordinates (m/s)

• U =

characteristic velocity scale (along the ratchet walls) (m/s)

• v =

$η−$ component of the velocity (m/s)

• V =

characteristic velocity scale (across the ratchet walls) (m/s)

• W =

density per meter depth (of drainage pathways)

• x =

abscissa in Cartesian coordinate system

• y =

ordinate in Cartesian coordinate system

Greek Symbols
• $α$ =

variable in ODE for h

• $β$ =

variable in ODE for h

• $δ$ =

film thickness in Cartesian coordinates (m)

• $ϵ$ =

ratio of characteristic length scale to characteristic film thickness

• $η$ =

ordinate for the hyperbolic coordinates (m2)

• $θ$ =

angle of the ratchet (deg or rad)

• $κ$ =

film curvature (1/m)

• $λ$ =

pitch of the ratchet (m)

• $μ$ =

dynamic viscosity of the fluid (Pa·s)

• $Π$ =

Newtonian stress tensor

• $ρ$ =

fluid density (kg/m3)

• $σ$ =

surface tension coefficient (N/m)

• $ϕ$ =

abscissa for the hyperbolic coordinates (m2)

• $ψ$ =

function of $ϕ$ in the expression for u

Subscripts
• c =

condensation

• DP =

drainage pathway

• f =

film

• lv =

liquid–vapor

• net =

net (mass flux rate)

• s =

surface

• sat =

saturation

• sh =

shallow

• st =

steep

• sub =

subcooling

• total =

total (mass flux rate)

• v =

vapor

• w =

wall

• − =

lower limit (of net mass flux rate)

• + =

upper limit (of net mass flux rate)

• $∞$ =

system

## Appendices

###### Appendix

This Appendix provides details involving the derivation of the governing equations in the hyperbolic coordinates. The change of coordinates was performed on the basis of two equations, $ϕ=x2−y2$ and $η=2xy$, and the following expressions were required to derive the flow equations and boundary conditions in the hyperbolic coordinates:

Distance of a hyperbolic coordinate point from the originDisplay Formula

(A1)$r=(ϕ2+η2)12$

Cartesian coordinates as a function of hyperbolic coordinatesDisplay Formula

(A2a)$x=(r+ϕ2)1/2$
Display Formula
(A2b)$y=(r−ϕ2)1/2$

Scale factorDisplay Formula

(A3a)$sϕ=[(∂x∂ϕ)2+(∂y∂ϕ)2]12$
Display Formula
(A3b)$sη=[(∂x∂η)2+(∂y∂η)2]12$
Display Formula
(A4)$s=sϕ=sη=12r$

Gradients of the Cartesian coordinates with respect to the hyperbolic coordinatesDisplay Formula

(A5a)$∂x∂η=−∂y∂ϕ=y2r$
Display Formula
(A5b)$∂x∂ϕ=∂y∂η=x2r$

Gradients of the scale factorDisplay Formula

(A6a)$∂s∂ϕ=−ϕ4r5/2$
Display Formula
(A6b)$∂s∂η=−η4r5/2$

Unit vectors in hyperbolic coordinatesDisplay Formula

(A7a)$ϕ̂=(xrx̂−yrŷ)$
Display Formula
(A7b)$η̂=(yrx̂+xrŷ)$

Unit vectors in Cartesian coordinatesDisplay Formula

(A8a)$î=(1s∂x∂ϕϕ̂+1s∂x∂ηη̂)$
Display Formula
(A8b)$ĵ=(1s∂y∂ϕϕ̂+1s∂y∂ηη̂)$

Single derivatives of the coordinate unit vectorsDisplay Formula

(A9a)$∂ϕ̂∂ϕ=η2r2η̂$
Display Formula
(A9b)$∂η̂∂ϕ=−η2r2ϕ̂$
Display Formula
(A9c)$∂ϕ̂∂η=−ϕ2r2η̂$
Display Formula
(A9d)$∂η̂∂η=ϕ2r2ϕ̂$

Double derivatives of the coordinate unit vectorsDisplay Formula

(A10a)$∂2ϕ̂∂ϕ2=−η24r4ϕ̂−ηϕr4η̂$
Display Formula
(A10b)$∂2ϕ̂∂η2=−ϕ24r4ϕ̂+ηϕr4η̂$
Display Formula
(A11a)$∂2η̂∂ϕ2=ηϕr4ϕ̂−η24r4η̂$
Display Formula
(A11b)$∂2η̂∂η2=−ϕηr4ϕ̂−ϕ24r4η̂$

(A12)$∇p=1s[∂p∂ϕϕ̂+∂p∂ηη̂]$

Divergence of velocity vector in hyperbolic coordinatesDisplay Formula

(A13)$∇⋅u=1s[∂u∂ϕ+∂v∂η−12r2(uϕ+vη)], u=uϕ̂+vη̂$

$ϕ−$component ofDisplay Formula

(A14a)$1s2[∂2u∂ϕ2+∂2u∂η2−1r2(η∂v∂ϕ−ϕ∂v∂η+u4)]$

$η−$component ofDisplay Formula

(A14b)$1s2[∂2v∂ϕ2+∂2v∂η2−1r2(−η∂u∂ϕ+ϕ∂u∂η+v4)]$

Gravity termDisplay Formula

(A15)$−gs[(∂x∂ϕcos θst+∂y∂ϕsin θst)ϕ̂+(∂x∂ηcos θst+∂y∂ηsin θst)η̂]$

Normal vector to the interfaceDisplay Formula

(A16)$n̂=−h′ϕ̂+η̂(1+h′2)1/2$

Newtonian stress tensorDisplay Formula

(A17)$Π=[−p+2μs(∂u∂ϕ−ηv2r2)μs(∂v∂ϕ+∂u∂η+ηu+ϕv2r2)μs(∂v∂ϕ+∂u∂η+ηu+ϕv2r2)−p+2μs(∂v∂η−ϕu2r2)]$

(A18)$κ′=2αβ3h″′−6ϵ2αβ5h′(h″)2−3(h−ϕh′)(ϕ+ϵ2hh′)2α7β$
Display Formula
(A19a)$β=(1+ϵ2h′2)12$
Display Formula
(A19b)$α=(ϕ2+ϵ2h2)14$

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## References

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## Figures

Fig. 1

Liquid flow over an asymmetric millimeter-sized textured condensing surface, assisted by a network of drainage pathways

Fig. 2

(a) Isometric views of the test section with a network of six drainage pathways, (b) close up of an asymmetric ratchet, and (c) close up of a symmetric ratchet

Fig. 3

An exploded-view of the experimental chamber

Fig. 4

Total mass flux rate as a function of wall subcooling with respect to saturation temperature. Asymmetric 75 deg–15 deg ratchets with (a) symmetric ratchet test surface with six-drainage pathways WDP = 400, (b) two-drainage pathways, WDP = 133, (c) four-drainage pathways, WDP = 267, and (d) six-drainage pathways, WDP = 400.

Fig. 5

Trend lines of condensate film average heat transfer coefficient with surface subcooling for the asymmetric ratchet surfaces with WDP of 133, 267, and 400, and symmetric ratchet with WDP of 400

Fig. 6

Net mass flux rate as a function of wall subcooling with respect to saturation temperature. Asymmetric 75 deg–15 deg ratchets with (a) symmetric ratchet test surface with six-drainage pathways, WDP = 400, (b) two-drainage pathways, WDP = 133, (c) four-drainage pathways, WDP = 267, and (d) six-drainage pathways, WDP = 400. The net mass flux rate is in the direction of the steeper slope of the ratchets.

Fig. 7

Condensate film profile for a symmetric ratchet (with six-drainage pathways) at varying subcooling. Dashed–dotted line represents the center of the ratchet and bold lines represent the walls of the ratchet. (a) 2.81 °C, (b) 5.69 °C, and (c) 8.96 °C.

Fig. 8

Condensate film profile for an asymmetric ratchet with two-drainage pathways/ WDP = 133 (left) and six-drainage pathways/ WDP = 400 (right) at varying subcooling. Dashed–dotted line represents the center of the ratchet, bold lines represent the walls of the ratchet, and the arrow indicates the position of minima of the drainage edge of the film. (a) 1.58 °C, (b) 8.38 °C, (d) 2.42 °C, (e) 9.57 °C.

Fig. 9

Lines of constant- ϕ (solid) and constant-η (dashed) and a sample liquid film profile (bold-dashed) on a Cartesian coordinate system

Fig. 10

Comparison of condensate film edge profiles from experiment (WDP = 133) and liquid film profile from the model at various subcooling levels

Fig. 11

Liquid film thickness variation in (a) hyperbolic coordinates and (b) Cartesian coordinates. ϵ=0.1, D=2.5.

Fig. 12

Liquid film profile as a function of Bond number for (a) symmetric and (b) asymmetric ratchet

Fig. 13

Interfacial velocity as a function of ϕ for symmetric and asymmetric ratchet

## Tables

Table 1 Average uncertainty estimates in experimental measurement

## Errata

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