Droplet Growth Model for Dropwise Condensation on Concave Hydrophobic Surfaces.

In this paper, mathematical models for predicting the droplet growth and droplet distribution of dropwise condensation on hydrophobic concave surface are developed and a theoretical analysis of the results of the model simulation is made. Under the assumptions of the Cassie-Baxter wetting mode and the consideration of noncondensable gases, the droplet growth model is not only established by heat transfer through a single droplet but also considered the thermal conductive resistance of the surface promoting layer. In addition, a droplet distribution model has been built based on the population balance theory. According to the calculation, the main thermal resistance in the droplet growth process is the conductive resistance inside the droplet. With the increase of contact angle, the above-mentioned thermal resistance increases; thus, the higher the hydrophobicity is, the slower the droplet growth and the less the droplet density are. Besides, the lower the temperature of the condensing surface is, the faster the droplets grow and the less the droplet density is. The models provide a mathematical tool for predicting the droplet radius at the initial stage of dewing on the concave surface and contribute to the design of functional surfaces in the field of water harvesting.

Introduction

Water nucleation and growth of water droplets are an important phenomenon both in nature such as dewing on surfaces of plants and in many fields of technologies such as on textured surfaces for water harvesting[1−6] or surface self-cleaning[7−9] purposes. In the field of water harvesting materials treated with surface modification, the wettability and structure of the surface have a remarkable influence on the efficiency of droplet formation and droplet collection. Hydrophobic or superhydrophobic surface with a water contact angle (CA) higher than 90 and 150°, respectively,[10−12] has attracted much attention and been investigated by researchers due to its low sliding angle of droplets and high surface renewal rate[13,14] for their special objective, such as collecting liquid water rapidly. On the other hand, it is verified that the tiny pits or grooves on surfaces enhance the water nucleation[15] and droplet motion. Dai[16] has found in his research that a directional slippery rough surface is capable of fast droplet removal. Many studies have also found that biomimetic micromorphology structure on the surface of materials exhibits excellent water harvesting rate.[17,18] By roughening the material surfaces or surface micro/nanofabrication technology, cavities or channels of different wettabilities can be constructed, which are suitable for droplet growth[19] and transportation.[20−22] Therefore, hydrophobic concave surfaces may lead to facilitation of dewing[19] and droplet removing, which displays considerable potential for investigation of water collection or anticorrosion.

To design surfaces with efficient water condensation rate and surface renewal rate, mathematical models that simulate the droplet growth process are desirable for predicting the droplet distribution on hydrophobic concave surfaces in the first place. Bintein measured and summarized the condensation rate on surfaces with grooves and reported that submillimetric grooves accelerate dew shedding.[1] It is worth noting that Kim[14] and Lu[23] quantified the growth of a single droplet on hydrophobic flat surfaces and grooves, respectively. To the best of our knowledge, there were a few studies on establishing a droplet growth model on hydrophobic concave surfaces. In view of this, the objective of the current study is to develop a mathematical model for droplet growth calculation on hydrophobic concave surfaces while the thermodynamical influence of surface roughness and noncondensable gases is taken into consideration as parameters. Furthermore, the drop size distribution on concave surfaces with various hydrophobicities is to be found and the effect of the surface wettability on water condensation is to be explored by simulation in terms of contact angle and population density at certain drop size.

Droplet Growth Model

Water condensation is a process through which vapor molecules accumulate and form initial condensation nuclei. The nuclei grow mainly by adsorbing ambient vapor molecules until two adjacent nuclei contact and merge into a larger nucleus. The process above is usually omitted and the calculation is directly started, from where each nucleus has its initial droplet radius. During droplet growth, each drop contributes to the condensation by transferring heat through itself. In this section, with consideration of noncondensable gases, a growth model of the radius of a single droplet by water condensation on a concave hydrophobic surface will be established based on thermodynamics theory.

Consider a droplet with radius r on a hydrophobic concave surface with radius R and contact angle θ, as shown in Figure . The contact angle θ is divided into two parts, θu and θd, where the subscripts u and d indicate the upper and bottom parts of the droplet, respectively. Hl represents the height from the bottom of the concave surface to the midpoint of the boundary line that separates θu from θd. As the droplet radius is increased by Δr, θu is assumed to be constant and the tangent of φ can be calculated by Rp/Hl. The vapor temperature and surface temperature are indicated as Tv and Ts, respectively. Water condensation is an exothermic process by which vapor forms liquid water on surfaces and releases heat. In this process, the heat flows from the environment through the gas–liquid interface and then passes through the droplet and eventually reaches the condensing surface. All temperature drops will be derived as follows by considering the heat transfer rate q and the thermal resistance for the region involved. The thermal resistances through which the heat flow passes are: interfacial thermal resistance Rin, thermal resistance due to the droplet curvature Rc, thermal conductive resistance of the droplet Rd, and thermal conductive resistance of the promoting layer Rco (as shown in Figure ).

Figure 1

Geometric structure of the droplet growth model.

Figure 2

Thermal resistances of heat transfer through a single droplet.

The vapor–liquid interfacial resistance Rin is responsible for the temperature drop ΔTi between the vapor and liquid phases. It is represented as[14]where hi is the interfacial heat transfer coefficient. Tanasawa[24] showed that for dropwise condensation with consideration of noncondensable gases, hi equals 15.7 MW/(m2 K) for 1 atm.

The droplet acts as a resistance to heat conduction between vapor and solid surfaces. According to Singh’s and Kim’s works,[14,25] the temperature drop caused by thermal conduction through the upper part of the droplet, as shown in Figure , can be written aswhere kc is the thermal conductivity of water. In this study, the heat conduction through the body of the condensing surface is assumed to be a transient process, that is, the temperature at any point in the body of the condensing surface is equivalent to Ts throughout the condensation process. Based on this assumption, the solid conduction is neglected. By simplifying the shape of the bottom part of the droplet to a cylinder instead of a spherical sector, the height of the cylinder is assumed to be Hl/2, as shown in Figure , for which case, the temperature drop for the bottom part of the droplet can be written asand the total temperature drop through the droplet is given by

The curvature of the vapor–liquid interface leads to an equilibrium saturation temperature, which is lower than the saturation temperature at a flat interface. The temperature drop due to the droplet curvature[26] is found to bewhere ΔTc corresponds to the difference in the equilibrium temperature of vapor in saturated state at a convex interface, σ is the surface tension of water, Hfg is the latent heat of vaporization, and ρ is the water density. The minimum viable radius of a droplet can be written as[25]Eq can be reduced to

All surfaces are rough to some extent, and therefore, only a portion of surfaces comes into contact with the droplet, as shown in Figure . The heat flow from the droplet to the surface is therefore retarded by this incomplete contact, and the thermal conductive resistance of the rough promoting layer is given by combining the thermal resistance of solid phase of the condensing surface and the thermal resistance of the air trapped in grooves of the surface microstructure.

Figure 3

Geometry when a single droplet wets the condensing surface.

The way in which the droplet wets the concave surface is simplified to the Cassie–Baxter mode, that is, the cavities are not wetted by the droplet, and the temperature drop due to the thermal conductive resistance is given by[27]where ke is the effective thermal conductivity, fs is the fraction of surface area occupied by solid phase, kw and ka are thermal conductivities of the solid phase and air, respectively, δe represents the average height of the micromorphology on the surface, and Sw indicates the contact area between the bottom part of the droplet and the concave rough surface. To reduce the calculation steps, Sw is idealized as the area of a circle with the diameter of 2R, as shown in Figure , and thus is given as

In addition to thermal effect, microstructure on surface also enhances the vapor nucleation at the beginning of condensation under special conditions, that is, the surface characteristics (e.g., the arithmetic mean deviation of the surface roughness) and the ratio between droplet size and the size of the cavity should be in certain ranges.[15,28] At present, investigations on the mathematical expression between surface roughness and the droplet radius are insufficient, as the result of which, the geometrical influence of the surface roughness is not within the scope of this study.

The droplet growth process is driven by the temperature drop ΔT between ambient and surface temperatures. The total temperature difference between the vapor and the surface is the sum of the four temperature drops found in eqs , 4,7, and 8, and is given aswhere

The total heat transfer rate through a droplet of radius r equals the change of energy required for the formation of an infinitely small amount of droplet volume in an infinitely small amount of time,where the droplet volume V and the derivative of the droplet volume with respect to time are, respectively, given in eqs and 15

Substituting eqs and 14 into eq , and the droplet growth rate as a function of droplet radius r is given aswith

Droplet Size Distribution Model

The purpose of establishing the droplet growth model is to calculate the amount of droplets with different drop sizes when condensation on hydrophobic concave surface reaches equilibrium. The condensation efficiency of a material surface can be evaluated by the distribution of the droplets number. Condensation occurs at proper nucleation sites. The initial droplet forms mainly by direct deposition from the vapor phase onto the surface and grows by absorbing ambient water molecules. As the droplet becomes larger and the distance between neighboring droplets becomes closer, coalescence starts to be the dominating mechanism for droplet growth until droplets roll down. The falling droplets sweep away other droplets in its path so that the condensing surface is cleaned and new initial droplets can be formed. During water condensation, there are droplets of different sizes on the condensing surface. For large droplets, the drop size distribution M(r) was established by Wen[3]

The maximum droplet radius rmax, which indicates the size of the departure drop, can be calculated based on the force balance between gravity that acts on the droplet and liquid–solid interfacial capillary force. The latter can be approximately calculated aswhere c is the constant that relates to the steepness of the surface and the drop shape.[29] θr and θa represent the receding and advancing contact angles, respectively. By assuming that the condensing surface is perpendicular to the horizontal direction, the gravitational force on the body of the droplet is approximately calculated as follows

By balancing between eqs and 24, the radius of the departure droplet can be derived as

In this study, an experimental value of rmax has been adopted to improve the calculation efficiency by reducing the calculation steps of advancing and receding contact angles on different structures. The concept of the experiment and the experimental setup is given in Figure . A spherical shell made of stainless steel with a diameter of 5 cm was used to provide the condensing surface. The inner surface of the spherical shell was deposited with a hydrophobic material (solution of polysiloxane and propylamine polymer) and fixed on a frame, which is attached to an electronic condenser. The contact angle of the condensing surface was 108°. The ambient temperature and the temperature of the condenser were 30 and 0 °C, respectively. The condensation experiment lasted 4 h, and the radius of every departure droplet was observed by a camera. The statistical value of the size of droplets that rolled off from the condensing surface was approximately 2 mm by taking the average of the sizes of departured droplets in the whole period of the test.

Figure 4

Experimental determination of the statistical value of rmax.

For small droplets that grow mainly by absorbing surrounding vapor molecules, the population balance theory can be applied to determine the drop size distribution.[30] The idea of the theory is that when the condensation reaches steady state, the derivative of the number of droplets with a certain radius is kept in conservation, that is, the number of droplets enters a certain drop size interval equal to the number of droplets leaving that size interval. By denoting the population density of droplets, i.e., the number of droplets with radius r per unit area as m(r), the number of droplets that enter an arbitrary size range (r1, r2) in an infinitely small amount of time dt can be written as Am1(r)J1(r) dt, while the number of droplets leaving that range can be written as Am2(r)J2(r) dt, where A is the area of an arbitrary section of the condensing surface. Considering the sweep off effect by departure droplets, the number of drops be swept off can be written as Sm12(r2 – r1) dt, where S is defined as the sweeping rate at which the surface is renewed by falling droplets and m12 is the average population density in the range from r1 to r2. According to the population balance theory, the number of droplets entering the size range by droplet growth must be equal to the sum of the number of droplets leaving by growth and the number of droplets be swept off

when Δr approaches zero, m2 approaches infinitely close to m1 and thus m12 becomes an exact value; thus, eq can be simplified aswhereIntegrating eq with respect to r gives

By solving eq , the population density of droplets with radius r is given as

The boundary between the size of small droplets and large droplets[14] is defined as rewhere Ns is the number of nucleation sites on unit area of condensing surface.[31] The population density m(r) and M(r) are equivalent at re, i.e., m(r) = M(r) at r = re; furthermore, the derivatives of In(m(r)) and In(M(r)) with respect to r are equivalent at re. The above two boundary conditions are, respectively, shown in eqs and 33

With eq , (Jm)min can be solved, and the expression of m(r) is shown as followswheref can be calculated using eq aswhere

Using eqs and 35, the drop size distribution of water condensation in equilibrium on hydrophobic concave surface can be solved numerically.

Results and Discussion

Using eqs , 22, and 35 from the previous section, the variation of droplet size with respect to time and the distribution of droplet number on a hydrophobic concave surface can be calculated.

The variation of droplet radius with respect to time is shown in Figure a. The calculation was executed under the condition where the vapor and surface temperature are 21 and −8 °C, respectively, the contact angle of the surface is 108°, and the initial droplet radius is 1.1 times rmin. The comparison between results using the prior growth model[27] and results using the current droplet growth model is illustrated in the figure. It is demonstrated that the current droplet growth model has a higher accuracy compared to the prior model, as the thermal conductive resistance of the lower part of the droplet has been considered. The curve also shows that the droplet size on the surface increases with time, and the droplet size obtained by simulation increases faster than that of the measurement. The reason for this is that in the theoretical model, the initial value of the droplet radius was set to 1.1 times larger than that of the minimum viable radius for the purpose of initiation of the numerical iteration;[27] therefore, the simulated value of the droplet radius was larger than that in experiment at time 0 s. Second, the results of radius increase in the model were calculated by iterative summation at every time step, while under real conditions, phase transition from vapor to liquid is required before the increase of the droplet radius, resulting in delay of nucleus growth. As shown in Figure a, the deviation between the results of the experiment and those of the current growth model is still large, due to the above-mentioned initial value of droplet growth and influence of the phase transition, and also the assumptions made by modeling. In the current model, the liquid–solid interfacial thermal resistance has been neglected and the shape of the lower part of the droplet has been considered as a cylinder to avoid the nonintegrable in eqs and 30 when modeling the droplet distribution; the thermal contact resistance between droplet and the condensing surface has also been neglected for lacking of the mathematical expression that relates the heat flux and the temperature drop.

Figure 5

Droplet growth with different models and the proportion of each thermal resistance. (a) Comparison of models and (b) the proportion of each thermal resistance.

By configuring the initial condition as Tv = 30 °C, Ts = 0 °C, and contact angle = 108°, Figure b illustrates the proportions of thermal resistance to total thermal resistance with respect to time. The result shows that the ratio of each thermal resistance to the total thermal resistance can be arranged in a decreasing order as follows: thermal conductive resistance due to the droplet, thermal conductive resistance of the promoting layer, interfacial thermal resistance, and thermal resistance due to curvature. The sum of the proportion of the conductive resistance of the droplet and the promoting layer accounts for more than 90% of the total thermal resistance.

The results of the droplet radius on surfaces with various hydrophobicities are shown in Figure a with Tv = 30 °C and Ts = 0 °C. The corresponding interfacial thermal resistance and thermal resistance of heat conduction inside the droplet with respect to time are plotted in Figure b,c. As demonstrated in Figure a, the higher the hydrophobicity on the concave surface is, the slower the droplet radius increases. The reason is that, during water condensation, as shown in Figure b,c, the larger the contact angle is, i.e., the higher the hydrophobicity of the surface is, the larger the interfacial thermal resistance and internal thermal conductive resistance of the droplet are.

Figure 6

Droplet size and thermal resistance on surfaces with various hydrophobicities: (a) droplet radius on surfaces with various wettabilities; (b) interfacial thermal resistance of various wettabilities; and (c) thermal conductive resistance of various wettabilities.

At time t = 0 s, the interfacial thermal resistance and the conductive resistance inside the droplet of the surface which has a contact angle of 145° are the largest, resulting in a decline of the droplet growth. As the condensation proceeds, the droplet radius on the surface with 145° contact angle is always smaller than that on surfaces of 120 and 108° contact angles, which leads to a smaller area for heat transfer. A small heat transfer area makes interfacial and conductive resistances higher than those of the 120 and 108° surfaces at a later stage. High thermal resistance in turn restrains droplet growth. Therefore, it is concluded that the larger the contact angle is, the higher the vapor–liquid interfacial resistance and internal thermal conductive resistance are, which reduce the increment of the droplet radius at the beginning of the condensation.

Figure a–c demonstrates the influence of hydrophobicity on the droplet density and the influence of subcooling on droplet density and droplet size under the condition where contact angle equals 108°. The correlation between the amount of droplets at each droplet size and the hydrophobicity of the surface is shown in Figure a. The result indicates that the larger the contact angle is, the less the droplet amount can be found. This can be explained by Figure a, which shows that high hydrophobicity leads to a low droplet growth rate, thus resulting in a low droplet density. Figure b,c shows the tendencies of droplet radius and droplet density by different subcoolings with respect to time and droplet size, respectively. As shown in Figure b, the lower the surface temperature is, the higher the droplet growth rate is. This is because a lower surface temperature represents a higher driving force of phase transition, which makes vapor condensation easier and thus the droplet growth becomes faster. Sequentially, as shown in Figure c, the larger the subcooling is, the less the droplet density is. The explanation is that a high subcooling provides more driving force supply to the droplet growth. While the vapor pressure is limited, more vapor is consumed by droplets with a high growth rate, resulting in a lack of vapor for condensation and growth of new drops, thus less droplets can be spotted.

Figure 7

Droplet radius and droplet density by various hydrophobicities and subcoolings. (a) Relationship between droplet density and hydrophobicity; (b) relationship between subcooling and droplet size; and (c) relationship between subcooling and droplet density.

Conclusions

In this study, a droplet growth model was developed for water condensation on hydrophobic concave surface based on thermodynamics theory. The wetting behavior between liquid and the surface was assumed to be the Cassie–Baxter mode, and the effect of noncondensable gases was considered. In comparison to the prior model, the current model takes into account the thermal conductive resistance in the lower part of droplets, which makes the current model more compatible for simulation of droplet growth on concave hydrophobic surfaces. Furthermore, based on the droplet growth model, the population density model for condensation in steady state was established. The models provide effective mathematical tools for the prediction of condensation on hydrophobic concave surfaces, which will reduce the consumption in surface design.

According to the simulation of the models, the following results were obtained:

By droplet growth process in the early stage of condensation, the main thermal resistance includes the heat conductive resistance inside the droplet. With the increment of the surface contact angle, i.e., the increment of the surface hydrophobicity, the thermal conductive resistance of the droplet and the interfacial thermal resistance increase, resulting in a decrease of droplet growth rate and droplet density on the concave surface. Therefore, the higher the surface hydrophobicity is, the slower the droplet growth and the less the amount of droplets are.

The lower the surface temperature is, the higher the droplet growth rate is, which leads to the decrease of water concentration above the droplet surface. This retards the nucleation and condensation of new initial nuclei, so the amount of droplets decreases by lower surface temperature. In conclusion, by reducing the surface hydrophobicity and restricting the surface temperature, an enhancement to the droplet amount and droplet growth on hydrophobic concave surfaces will be achieved.