Time-averaged droplet size distribution in steady-state dropwise condensation

International Journal of Heat and Mass Transfer 88 (2015) 338–345

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Time-averaged droplet size distribution in steady-state dropwise condensation Maofei Mei a,⇑, Feng Hu a, Chong Han a, Yanhai Cheng b a b

School of Physical Science and Mathematics, Xuzhou Institute of Technology, Xuzhou 221000, Jiangsu, China School of Mechatronics Engineering, China University of Mining and Technology, Xuzhou 221116, Jiangsu, China

a r t i c l e

i n f o

Article history: Received 17 November 2014 Received in revised form 22 April 2015 Accepted 24 April 2015 Available online 14 May 2015 Keywords: Dropwise condensation Droplet size distribution Fractal Simulation

a b s t r a c t This work focused on the time-averaged droplet size distribution in steady-state dropwise condensation. Through theoretical analysis, we derived an expression to describe the droplet size distribution, which is a function of the surface fraction, fractal dimension, and the maximum radius of droplets. Compared with the well-known semi-empirical model by Le Fevre and Rose (1966), the present model has no empirical constant and all parameters in the model have clearly physical meaning. Computer simulations were further conducted for different initial configuration of primary droplets, growth rate of an individual droplet and time step. It was found that the simulation results corresponded well with the available experimental data. The predictions from the semi-empirical model by Le Fevre and Rose and from the present model were also compared with the simulation results and the measured data. A good agreement between them implies that both the two models are valid to be used to describe the time-averaged droplet size distribution under various conditions used in experiments and simulations. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Aggregation phenomena are ubiquitous in many fields including material sciences and chemical engineering [1–5]. A remarkable case is dropwise condensation, i.e., water vapor condenses into droplets on substrates [6–11]. It has long been an engineering concern due to higher heat transfer coefficients in dropwise condensation than that in filmwise condensation [12]. Fundamental understandings of the droplet growth process in dropwise condensation include nucleation, growth, coalescence with renucleation, and removal with renucleation [13,14]. After the first removal, the evolution of the droplet pattern reaches a statistically steady state and some quantities describing the features of dropwise condensation will fluctuate with growth time. Considerable efforts have been directed toward better understanding the time-averaged droplet size distribution in the steady-state dropwise condensation [15–23]. In the droplet growth process, absorbing water molecules and coalescing neighboring droplets are two mechanisms to promote the droplet growing. The former is to fit for primary/smallest droplets formed at nucleation sites, the latter for droplets whose radius exceeds the average distance of neighboring primary ⇑ Corresponding author. E-mail address: [email protected] (M. Mei). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.04.087 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.

droplets. The average distance, denoted as Rad , depends on the initial nucleation density N s on a L  L substrate. Assuming a square arrangement of the primary droplets, Rad is half the mean spacing between mass centers of two neighboring droplets and can be pffiffiffiffiffiffiffiffi expressed as Rad ¼ L= 4N s . More densely packed nucleation sites lead to a smaller average distance. The unit of Rad is the same with that of characteristic size L of the considered surface. For instance, Rad ¼ 0:0005 cm when the initial nucleation density is 1  106 =cm2 . According to the nucleation theory, the primary droplet over a condensing surface has the magnitude of nanometer [24–26]. Therefore, the growth of droplets with size less than Rad mainly depends on condensation of vapor on its surface. Due to the resolution limit of optical microscope, it is difficult to accurately count the number of droplets under micrometer through taking images of droplet distribution. Most experimental investigations on the droplet size distribution direct at droplets larger than micrometer [16,22,23]. To keep in line with other investigators, in this context, the term ‘‘small droplets’’ represent those droplets growing mainly by direct condensation, while ‘‘large droplets’’ grow predominantly by coalescence with their neighbors. Up to now, there are several expressions to describe the time-averaged size distribution of the large droplets in steady-state dropwise condensation. Among them, the semi-empirical model expressed by Le Fevre and Rose [15] and the fractal model by Mei et al. [20] were of

M. Mei et al. / International Journal of Heat and Mass Transfer 88 (2015) 338–345

better theoretical foundation. Through observing the photographs of dropwise condensation, Le Fevre and Rose [15] found that the geometrical pattern of photographs looks much the same irrespective of the magnification used. They related this feature to the packing problem of circles on a plane [27], and suggested an expression of the surface fraction covered by the large droplets. Based on the expression, Rose and Glicksman [19] later derived a power-law function to describe the droplet size distribution. The power-law relation has been proved to be in fair accord with the results from computer simulations [28] and experiments [16,17]. Although the power-law relation has been widely used to predict the theoretical heat transfer of dropwise condensation combing the heat flux of a single droplet, the exponent in the expression has no clear physical meanings and is estimated to be 1/3 according to experimental data [19]. In fact, the similarity of the geometrical pattern of droplet-distribution photographs under different magnification found by Le Fevre and Rose is exactly the concept of self-similarity, a feature of fractal objects later proposed by Mandelbrot [29,30]. Fractal objects exist extensively in nature and science, such as islands on earth, sandstones in oil reservoir, packed beds in chemical engineering and so on. These objects are called fractal porous media in a general sense because the cumulative size distribution of pores or particles inside them has been found to abide by the power-law behavior [31]. Based on the phenomenon, Yu et al. has developed a fractal geometry theory and technique to investigate the fluid flow and transports in porous media [31]. Because the size distribution of droplets over a condensing surface is analogous to that of pores in porous media, Mei et al. recently applied the fractal geometry theory to model dropwise condensation, and also proposed a fractal expression to describe the droplet size distribution [20]. The prediction from the fractal model is in good agreement with the experimental data. However, most recently, Watanabe et al. showed that their measured data was consistent with the semi-empirical model but disaccorded with the fractal model [22]. After careful checking of the comparison, we found the discrepancy was caused by comprehension of the concept of the fractal model. Therefore, the fractal model will be further illustrated in the present work for using more clearly. Although explaining the discrepancy is the original purpose of the present work, we later realize that it is more significant to resolve the question of whether both the semi-empirical and the fractal models are universal, irrespective of ambient conditions of dropwise condensation. Because computer simulation, compared with the experiments, can be carried out with less expense and controlled much better to explore a much wider range of conditions, it is usually preferred over experiments for the purpose of testing theoretical ideas. In this paper, we focus on the analytical and numerical study of the droplet size distribution. Specifically, we first analyzed and compared the semi-empirical model from Le Fevre and Rose and the fractal model from Mei et al. in detail. Then, the predictions from the two models were compared with the experimental data measured by different investigators. After that, we developed a computer simulation of dropwise condensation in which several key factors influencing the droplet growth were taken into account. Finally, the simulation results of the droplet size distribution were compared with recently experimental data, the semi-empirical and the fractal models, respectively.

2. Two models for the size distribution of the large droplets To describe the time-averaged droplet size distribution in the steady-state dropwise condensation, Le Fevre and Rose proposed a well-known expression that follows the scaling relation [15,19]

N¼

e ðR Þe Rð3eÞ p max

339

ð1Þ

where, N represents the number of large droplets per unit radius and surface area, e is an empirical parameter, generally, which is predicted to be 1/3 based on the experimental data. In Eq. (1), Rmax is an average radius of maximum droplets emerged at different instant. The lower cutoff of the scaling function is the average distance Rad between the primary droplets [32]. Eq. (1) is used frequently to calculate the heat transfer rate of the steady-state dropwise condensation by incorporating the heat flux of an individual droplet [32–34]. Eq. (1) shows that the droplet size distribution has a power-law decay in the droplet radius, i.e., N  Rð3eÞ . Turning Eq. (1) into Log–Log linear relation gives

LogN ¼ ð3  eÞLogR þ C

ð2Þ

where ð3  eÞ and C ¼ Log pe  eLogRmax are the slope and intercept, respectively. By using the empirical result e ¼ 1=3, then the slope in Eq. (2) is 8/3 which reflects the scaling behavior of the droplet size distribution in steady-state dropwise condensation. Recently, a theoretical model on the droplet size distribution was established by Mei et al. based on the fractal geometry theory [20]. The fractal characteristic of the steady-state dropwise condensation means that the photographs of the droplet distribution at different instant or scales appear self-similarity. In particular, there are two aspects to exhibit the self-similarity. One is that a fractal zone is a representative distribution and determines the droplet size distribution over the entire surface. Analogous to the cumulative size distribution of pores in fractal porous media, the cumulative size distribution of droplets on the condensing surface also follows the power law [31]

Nt ðP RÞ ¼



R

df

Rmax

ð3Þ

where, df is the fractal dimension, and 0 < df < 2 for droplets on 2-dimension condensing surface. Eq. (3) indicates the cumulative number of droplets with sizes equal to or greater than R on a fractal zone. Note that, for a fractal porous media, the pore structure is stable and the pore size distribution is independent of the time. At this point, Eq. (3) is fit for the instantaneous droplet size distributions. But on a condensing surface, the maximum radius, the fractal dimension, as well as the cumulative number of droplets in Eq. (3) frequently vary with the time due to the continuous growth. Fortunately, in the steady-state dropwise condensation, both the maximum radius and the cumulative number of droplets fluctuate around certain average values with the growth time [21]. This means that Eq. (3) can be used to express the time-averaged droplet size distribution. Thereby, the maximum radius and the fractal dimension in Eq. (3) are statistical averages during the steady-state dropwise condensation. The other is that we cannot distinguish geometry patterns of the droplet distribution photographed under various resolutions. This feature can be realized by differentiating Eq. (3) with respect to R. This leads to the number of droplets within the infinitesimal interval R to R + dR d

f dN ¼ df Rmax Rðdf þ1Þ dR

ð4Þ

where dN > 0, the negative sign implies a decrease of the droplet number with an increasing droplet size. Rearranging Eq. (4) gives the number of droplets per unit radius as d

f N ¼ df Rmax Rðdf þ1Þ

ð5Þ

Eq. (5) also follows a power-law behavior. Eq. (5) implies that the droplet size distributions under different resolutions obey the same scaling relation which exactly reflects the self-similarity of

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Fig. 1. Comparison of the predicted results from Eqs. (1) and (8) with the measured data from Tanasawa and Ochiai [16]. The inclinations of the substrate are (a) 90° and (b) 45°, respectively.

The surface fraction of droplets predicted by Eq. (7) has been proved to be in good agreement with the measured data [22]. Based on Eqs. (6) and (7), the area of the fractal zone is thus calculated to be A¼

pdf ð1/Þ 2 ð2df Þ/

Rmax . Consequently, through dividing Eq. (5) by the area

of the fractal zone, Eq. (5) is converted to the droplet size distribution per area of surface as follows

N¼

Fig. 2. Comparison of the predicted results from Eqs. (1) and (8) with the measured data from Wanatabe et al. [22].

the droplet size distribution. Note that Eq. (5) denotes the droplet size distribution over a fractal zone, but Eq. (1) on a surface with unit area. The area of a fractal zone, generally, is not equal to unit. For comparing Eq. (5) with Eq. (1), Eq. (5) should connect the droplet size distribution to the unit area. From Eq. (4), we can obtain the total area covered by all large droplets by integrating Eq. (4) from the average distance Rad to the largest radius Rmax

Ad ¼

Z

Rmax

dNpR2 ¼

Rad

pdf ð1  /Þ ð2  df Þ

R2max

ð6Þ

where / is the surface fraction of droplets in a fractal zone and is expressed as [31]

 /¼

Rad Rmax

2df

ð2  df Þ/ ð2df Þ ðdf þ1Þ R R pð1  /Þ max

Eq. (8) is a function of the surface fraction /, fractal dimension df , and the maximum radius Rmax . The fractal dimension can be calculated by Eq. (7) based on the measured surface fraction, the minimum and maximum radii. Both the fractal dimension df and the surface fraction / increase with the temperature difference DT based on the existing expressions [20] where df and / are related to DT. The droplet number per unit radius and surface area also increases with the temperature difference according to Eq. (8). This result is in accord with the physical situation because higher temperature difference provides more powerful driving force for water vapor to condense on the substrate with producing more droplets. In form, Eq. (8) is quite similar with Eq. (1). Eq. (8) also shows a power-law decay of the droplet size distribution with respect to the droplet radius, i.e., N  Rðdf þ1Þ . However, compared with Eq. (1), Eq. (8) has no empirical constant and all parameters in Eq. (8) have clearly physical meanings. Turning Eq. (8) into Log–Log linear relation gives

LogN ¼ ðdf þ 1ÞLogR þ C 1 ð2d Þ/

ð7Þ

Fig. 3. Comparison of the predicted results from Eqs. (1) and (8) with the measured data from Ma et al. [23].

ð8Þ

ð9Þ

f where C 1 ¼ Log pð1/Þ  ð2  df ÞLogRmax is intercept. In Eq. (9), when / ¼ 1=2 and df ¼ 5=3, it can be seen that Eqs. (2) and (9) are exactly the same. Eqs. (1) and (8) are compared with the measured data in different experimental conditions. In the experiments, due to the geometrical self-similarity of droplet pattern, instead of considering the entire condensing surface, a representative large enough zone can be chosen to measure the droplet size distribution. In measurements, a square zone is preferred whose side length is, generally, about ten times larger than the maximum radius [16,22,23]. Figs. 1–3 show the comparison of Eqs. (1) and (8) with the measured data. The three figures keep the coordinates consistent with that used by authors in their papers. Measured data such as the surface fraction /, the maximum radius Rmax and the average distance Rad are listed in Table 1 under different experimental conditions. In Table 1, DT is the temperature difference between the substrate and vapor or vapor–air mixture, P vapor or vapor–air mixture pressure, X non-condensable gas mole concentration. The fractal dimension in Eq. (8) is worked out based on Eq. (7)

M. Mei et al. / International Journal of Heat and Mass Transfer 88 (2015) 338–345 Table 1 Experimental conditions.

a

Authors

DT (K)

P (kPa)

X (%)

Rad (mm)

Rmax (mm)

U

Tanasawa et al.

1.0 2.0 1.3

101 101 101

– – –

0.005 0.005 0.005

0.96 0.96 0.96

0.75 0.75 0.75

Watanabe et al.

5.0– 35.1

101

0– 25

0.05

0.7

0.56– 0.7

Ma et al.

1.0 1.0

0.015 101

0 0

0.01 0.01

1.32 1.1

0.7 0.8a

The value is guessed based on the data from Ma et al.

combing the experimental data in Table 1. The empirical constant in Eq. (1) equals to 1/3 as suggested in the experiment [19]. Note that the average distance Rad in Table 1 is the lower cutoff of the measured data due to the limit of the resolution of optical microscope. The experimental data in Fig. 1 were measured by Tanasawa and Ochiai [16] under two different substrate inclinations (90° and 45°). Droplets larger than 5 lm were counted over the photographs of droplet pattern. The maximum radius of the measured droplet was 0.96 and 1.45 mm on the 90° and 45° surface, respectively, as shown in Table 1. The surface fraction of the counted droplets was approximately 0.75 on both surfaces. As seen in Fig. 1, the predicted results from both the present model and the semi-empirical model were found to be in excellent agreement with the measured data. Fig. 2 shows the comparison of the predicted results from Eqs. (1) and (8) with the measured data by Wanatabe et al. [22]. In the experiments conducted by Wanatabe et al., droplets less than 10 lm were not counted. The maximum radius used in Eqs. (1) and (8) was 0.7 mm obtained by averaging all the largest droplets in the eight instantaneous droplet patterns considered by Wanatabe et al. The size distributions of the large droplets, particularly in the size range of 0.05 mm to 0.7 mm, were irrespective of the non-condensable gas mole concentration and the temperature difference. In the range of 0.05 mm < R < 0.7 mm, it can be found that droplet size distributions predicted from Eqs. (1) and (8) are very consistent with that from experiments under various conditions. This result is different from that obtained by Wanatabe et al. Such a discrepancy is possibily due to the fact that the fractal model used by Wanatabe et al. was the droplet size distribution on a fractal zone not on a surface of unit area. But with the droplet radius increasing from 0.07 mm to the maximum, the gap between the prediction and the experimental results enlarges, and the measured data at the tail in Fig. 2 falls behind the prediction. Two reasons may trigger the departure of the fractal model from the experimental data at larger radius. One is contact angle hysteresis (CAH). Compared with the smaller droplets, the larger droplets have more remarkable pinning effects due to the CAH. This

Table 2 Significant simulation factors. Type

L

R0

Dt

l

Rmax

Ns

1 2 3 4 5 6 7 8 9 10 11

6000 6000 6000 6000 6000 6000 6000 6000 6000 6000 7600

1.0 1.0–2.0 1.0–5.0 1.0–5.0 1.0–5.0 1.0 1.0 1.0 1.0–5.0 1.0–5.0 1.0–5.0

500 500 500 500 500 500 100 1 500 500 500

1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 2/3 1/3 1/3

600 600 600 600 600 600 600 600 600 375 1000

50,000 50,000 50,000 40,000 30,000 20,000 50,000 50,000 50,000 50,000 80,000

341

phenomenon keeps the larger droplets from expanding in size. So the evolution of the size distribution for the larger droplets lags behind that of the smaller droplets. The other is the starvation of nucleation of smaller droplets that occurs around the larger droplets which dramatically decreases their growth rate. This also leads to the evolution hysteresis of the size distribution of the larger droplets. Compared with Fig. 2, the power-law behavior of the measured data is better in Fig. 3 where the measured data by Ma et al. [23] are compared with the predicted from Eqs. (1) and (8). The experimental results display excellent linear Log–Log plots. The slope slightly depends on the vapor pressure as can be seen from Fig. 3. The predicted from the semi-empirical model is in good accord with the data measured in the condition of high vapor pressure (P = 101 kPa), but both the present model and the semi-empirical model slightly overestimate the droplet size distribution in the experiment under lower pressure (P = 1.5 kPa). Based on the compared results in Figs. 1–3, Eqs. (1) and (8) present good predictions to those data measured under various experimental conditions (including the temperature difference, vapor pressure, non-condensable gas). The compared results imply that both the semi-empirical model from Le Fevre and Rose and the present model are valid under various experimental conditions. In addition, the measured data effectively confirm the reliability of computer simulation of the droplet growth process in dropwise condensation. This will be further discussed in next section.

3. Simulation of droplet growth process 3.1. General procedure The first step of the procedure is to establish an initial configuration of primary droplets. The configurations were generated by placing droplets sequentially at random sites. The sites were taken as the centers of the primary droplets. Sites that would result in overlaps were rejected under the limit of the distance between two neighboring sites. Once the initial configuration is completed, growth time is then incremented and all droplets grow according to a power-law model. When two droplets come into contact, they coalesce and are replaced by a new droplet whose radius is computed on the assumption that the droplets are hemispheres. The search for coalescences is carried out by calculating the distance between the mass centers of two droplets and comparing it to the sum of their radii. Once the latter were larger, the coalescence will take place. Based on the experimental observation, the new droplet is found to be centered at the center of mass of the coalescing pair [35]. We take the assumption that the new droplet from coalescence relaxes instantly to a hemispherical cap. After the coalescence, a check is made for contact with other droplets that occur as a result of the coalescence. In the simulation, the check of the coalescence is most time-consuming. When the check is done, the growth time is again incremented and single-droplet growth continues until the next contact and coalescence. Among the coalescence events, new primary droplets have renucleation chances at the sites exposed owing to coalescence. When the largest droplet reaches a sufficiently large size at which the gravity force exceeds the surface resistance, the droplet slides down the surface sweeping other droplets contacted with it. New primary droplets appear in the swept area and the cycle begins anew. Then a statistical steady state of droplet growth is reached. The simulations end after finishing a given number of time steps that is sufficient enough to obtain a time-averaged statistical result of the droplet size distribution. Main simulation factors are listed in Table 2, where L is the side length of the square simulation zone, R0 the radius of the primary droplets, Dt the time step, and l the growth

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Fig. 4. (a) Droplet size distribution affected by different primary droplet distribution, (b) R0 ¼ 1:0, (c) R0 ¼ 1:0—2:0, (d) R0 ¼ 1:0—5:0.

Fig. 5. (a) and (b) Droplet size distribution affected by number of initial droplets, (c) N s ¼ 20; 000, (d) N s ¼ 30; 000, (e) N s ¼ 40; 000, (f) N s ¼ 50; 000.

exponent of an isolated droplet. As a computer experiment, simulation is influenced by many factors. Making clear the role of various factors in simulation is indispensable for comparing the results from the simulations with that from theoretical models and experimental measurements. Therefore, we first investigate the effects of four key factors on the droplet size distribution, including polydispersity of initial droplet distribution, number of initial number, time step, and growth rate of an isolated droplet. 3.2. Key factors in simulation 3.2.1. Polydispersity of the primary droplet distribution The first concern is the size distribution of the primary droplets. In previous simulations, the primary droplet distribution was

frequently monodispersity [21,36,37]. However, no proof hitherto demonstrated the initial droplets are identical in size at the beginning of condensation. So, in this simulation both monodispersity and polydispersity of the primary droplet distribution are taken into consideration to investigate the droplet size distribution. Monodispersity is shown in Fig. 4(b) where the radii of all primary droplets are set to be 1. The monodispersity results in the average radius of the primary droplets exactly equal to 1. Fig. 4(c) indicates that the radii of initial droplets are distributed uniformly between 1.0 and 2.0, and the solid line represents their average radius of 1.49. But when the radius is limited to the range from 1.0 to 5.0, more relatively smaller primary droplets were produced as shown at the right and lower part of Fig. 4(d), because less space is left for generating relatively larger primary droplet at the late stage. This

M. Mei et al. / International Journal of Heat and Mass Transfer 88 (2015) 338–345

Fig. 6. Droplet size distribution affected by time step.

results in their average lower than 3.0 denoted by the solid line as shown in Fig. 4(d). Fig. 4(a) shows the results from the simulations based on three types of the primary droplet distribution. In Fig. 4(a), the lower cutoff of the droplet radius is the average distance Rad ¼ 13:42 between the primary droplets. No obvious difference in the droplet size distribution is found to be related to the polydispersity of the primary droplet distribution. 3.2.2. Number of the primary droplets Generally, the number of the primary droplets is determined by the nucleation sites over the condensing surface, which is a function of temperature difference and surface preparation [38]. Compared with the experiments, computer simulation is more convenient to study the influence of initial configurations on the droplet size distribution. Increasingly primary droplets are deposited at random sites on a surface with area 6000  6000 as shown in Fig. 5(c)–(f). The radii of all droplets lie in the range from 1.0 to 5.0. The simulation results indicate that the scaling behaviors of the droplet size distribution are similar for different number of the primary droplets, but there are slight quantitative differences at the left and right ends as shown in Fig. 5(a). Surface with N s ¼ 50; 000 has more droplets in the size range between Rad to 60 than other surfaces due to more nucleation sites as shown in Fig. 5(b). The difference at the right end as shown in Figs. 1–3 is attributed to the nucleation density of the primary droplets around the larger droplets. 3.2.3. Time step In experimental observations, dropwise condensation consists of a series of the instantaneous images photographed using optical microscope. Time step is related to the frame rate of high-speed

343

camera. Higher frame rate denotes smaller time step. If much more growth details wanted to be revealed, smaller time step will be needed in simulation. But smaller time step will consume a mass of computer time to reach the steady-state dropwise condensation. In the present work, our concern focuses on the scaling behavior of the droplet size distribution. For saving simulation time, the standard to choose an appropriate time step is to keep the scaling behavior independent of the time step. The effect of the time step on the droplet size distribution was studied by simulating droplet growth process under the time step of 1, 100 and 500, respectively. It is observed from the simulation results in Fig. 6 that the droplet size distributions are almost consistent with each other in three types of simulation. The result demonstrates that it is important to substantially save computation time while keeping the same scaling behavior. 3.2.4. Growth exponent of an isolated droplet Growth rate of an isolated droplet is influenced by the flow rate of vapor, vapor pressure, non-condensable gas or temperature difference between a substrate and vapor. On the micro level, the isolated droplet grows by condensing of water molecules on droplet surface. Generally, the growth rate of an isolated droplet can be written as [14]

R ¼ R0 tl

ð10Þ

Here, the exponent l reflects condensing rate of water molecules on droplet surface. A larger exponent means a higher flow rate or condensable gas mole concentration. For a 3-dimensional droplet on a 2-dimensional surface, the exponent is about 1/3 based on the experimental data and theoretical analysis [13,39]. Fig. 7 shows the influence of the exponent on the merging time among three droplets whose coordinates and radii are inserted in Fig. 7(b). The left axis in Fig. 7(b) shows the merging time (solid points) of the three pairs, and the droplets 2 and 3 contact first of all with the exponent l ¼ 1=2. But the first coalescence occurs between droplets 1 and 2 when the exponent l ¼ 1=3. Because the growth exponent affects the local merging order, it is necessary to get insights into the role of the growth exponent played in the droplet growth process. We carried out two types of simulations with the growth exponents 1/3 and 2/3. The simulation results in Fig. 8 indicate that no obvious difference is found between the two distributions when l ¼ 1=2 and l ¼ 1=3, respectively. Although the exponent has no effect on the droplet size distribution, it enhances the growth rate of an isolated droplet and the coalescence frequency among droplets. This point is proved by the results that the first sliding events happened at growth time 1950 and 844,500 for the growth exponent 2/3 and 1/3, respectively.

Fig. 7. The merging order among three droplets influenced by the growth exponent.

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Fig. 8. Droplet size distribution affected by growth exponent.

Judging from the simulation results in Figs. 4–6 and 8, the droplet size distribution does not seem to vary greatly with the polydispersity and number of the initial droplets, time step, and growth exponent of an isolated droplet. Because the four key factors depend on experimental conditions, the present simulations indirectly illustrate that the scaling behavior of the droplet size distribution is not significantly affected by the various experimental conditions.

4. Results and discussion In the simulations, all parameters are dimensionless. To compare the simulation results with the experimental data, they will be dimensinonalized according to the measured data. In the measurement by Wanatabe et al., the maximum radius is 1.9 mm which is obtained by averaging the maximal droplets appeared in each observation [22]. In the simulation, the size ratio of the largest droplet to the primary droplet is 600:1. So the dimensionless maximum radius 600 represents 1.9 mm. This means that the radius of the primary droplet in simulation is 0.0032 mm (1.9 mm/600). Due to the limit of the resolution, the measured droplet is larger than 0.05 mm. In reality, the maximum radius of droplet can be influenced by the condensing materials and the substrate inclination. The maximum radius can be worked out from balancing the surface tension force to the droplet weight. We do not attempt to present how to calculate the maximum radius because the size ratio between the largest droplet and the smallest droplet is more crucial to the simulation of the droplet growth process as found below. Fig. 9 shows the comparison between the simulation results (solid circles, squares and triangles) and the experimental data (open circles). It can be found that the droplet size distribution within the range from 0.1 mm to 0.7 mm agree remarkably well with each other. However, obvious

Fig. 9. Comparison of droplet size distribution between the simulation results and the measured data.

Fig. 10. Droplet size distribution in the simulation compared with the predictions from the semi-empirical model and the present model.

differences between them appear at the two ends. The left difference is caused by the size ratio of the maximum to the primary droplet. The gap narrows with the ratio increasing, such as 1000:1 in Fig. 9, otherwise the gap enlarges, such as 375:1 in Fig. 9. Therefore, it is reasonable to believe that the gap will gradually fade away by simulating droplet growth process within a wide enough size range, such as magnitude of six. Unfortunately, simulation in such a broad range requires large amounts of computation time to reach the steady-state dropwise condensation. This is an essential difficulty encountered by all existing simulations. The gap at the right end is more noticeable as observed in Fig. 2. This is because the contact angle hysteresis and the starvation of the nucleation of the smaller droplet around the larger droplet are not considered in the simulation. The present simulation results are also compared with the present model and the semi-empirical model from Le Fevre and Rose as shown in Fig. 10. If the left discrepancy caused by the small size ratio was taken into account, the comparison in Fig. 10 indicates good agreement between them. In the previous section, it is proved that the droplet size distribution is independent of the polydispersity and number of the primary droplets, time step, and growth rate of an isolated droplet. Therefore, combining Figs. 1–3 and 10, it is concluded that the present model and the semi-empirical model hold under various experimental conditions. Both of them can be used to theoretically predict the heat transfer rate by combining the heat flux of an individual droplet. 5. Conclusions In this paper, we have studied the time-averaged droplet size distribution in the steady-state dropwise condensation. The considered droplets were distributed within the size range from the average distance to the maximum radius on the condensing surface. Two models (Eqs. (1) and (8)) were introduced to describe the droplet size distribution. The linear Log–Log relations transformed from Eqs. (1) and (8) were extremely similar with each other as shown in Eqs. (2) and (9). Good agreements between the predictions from the two models and the measured data from several experiments verified that the fractal geometry theory was a valid method to analyze the droplet size distribution in the steady-state dropwise condensation. Owing to their less expense and better controllability than experiments, computer simulations on dropwise condensation were carried out to explore the droplet size distribution under various simulation conditions. It was found that the droplet size distribution was almost independent of the choice of the polydispersity and number of the primary droplets, time step, and growth exponent. Good agreements were found in certain size range between the simulation results and the experimental data and the prediction from the two models. However,

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