Study on the wettability and condensation heat transfer of sine-shaped micro-grooved surfaces

Experimental Thermal and Fluid Science 90 (2018) 28–36

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Study on the wettability and condensation heat transfer of sine-shaped micro-grooved surfaces

MARK

⁎

Baojin Qia,b, , Jiasen Zhoua, Jinjia Weia,b, Xiang Lia a b

School of Chemical Engineering and Technology, Xi’an Jiaotong University, Xi’an 710049, China State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Dropwise condensation Micro-groove Hydrophobicity Heat transfer enhancement Dynamic analyses

In this study, sine-shaped micro-grooved surfaces with depth of 12–24 μm and width of 30–60 μm were precisely and smoothly fabricated using dry etching technique on aluminium surfaces. After hydrophobic modification, the wettability and the heat transfer characteristics of dropwise condensation on the micro-grooved surfaces were investigated experimentally, and the coalescence and sweeping processes of droplets on micro-grooved surfaces were dynamically analyzed. As the results show, the wetting behavior and heat transfer characteristics on the micro-grooved surfaces presented anisotropic characteristics, the static contact angle in perpendicular direction θ⊥ was significantly larger than that in parallel direction θ∥, and same trends can also be observed for contact angle hysteresis. In heat transfer experiments, the plates were set vertically and the grooves were arranged in two positions, vertical and horizontal. For the vertically grooved surface, the sweeping effects of falling droplets were enhanced by the vertical grooves and the heat transfer during dropwise condensation was increased to 30–50%. Better heat transfer performance can be achieved when the ratio of height to pitch, A/P, increased. Different from vertical grooved surfaces, the experimental results obtained from horizontal grooved surfaces were similar to the results of smooth surface. Both net force and sliding velocity increased as droplets grew, and larger geometrical size was favorable to droplets falling for same ratio of A/P. The velocities of sliding down on horizontal grooved surface were only 60–70% of that on smooth surface, while the velocities of sliding down on vertical grooved surfaces can reach 1.2 times or higher than that on smooth surface.

1. Introduction Condensation involves change of phase from the vapor state to the liquid. It is associated with heat and mass transfer, during which vapor migrates towards the liquid–vapor interface and is converted into liquid. Apart from natural phenomena, condensation is an essential part of energy conversion, water harvesting, and thermal management systems. When vapor comes in touch with a surface below the saturation temperature, dropwise condensation is preferred when surface is not wetted by the liquid. The heat transfer coefficient of dropwise condensation is an order of magnitude larger than for filmwise mode that occurs when the surface is wetted. This makes dropwise condensation a very attractive mechanism for industrial applications [1]. Dropwise condensation begins with drop formation at preferred nucleation sites at the atomic scale. The small droplets start to grow up through direct condensation of steam and then coalesce with adjacent droplets. When droplets become large enough, they are generally removed from the surface by the action of gravity or drag forces resulting from the motion of surrounding gas. As the drops roll or fall from the surface they merge ⁎

with droplets in their path, effectively sweeping the surface clean of droplets. Droplets then begin to grow anew on the freshly exposed solid surface. Through the analysis of the heat transfer process of the dropwise condensation, it can be found that the heat transfer performances of dropwise condensation are significantly affected by the departure diameter and the sweeping cycle of the droplets. Tanasawa and coworkers’ experiments [2] showed that the heat transfer coefficient, which was proportional to the scale of droplets, decreased with the increase of drop sizes. Their conclusions were supported by the research of Rose [3], Ma et al. [4,5] and Lee et al. [6]. Moreover, Yamali et al. [7] and Izumi et al. [8] investigated the effect of sweeping cycle on dropwise condensation heat transfer with theoretical analyses and experiments. They found that the sweeping frequency of droplets was a very important factor affecting the heat transfer process of the dropwise condensation. Lee et al. [6,9] fabricated pin structures and unique micro/nano-scale porous surfaces to promote dropwise condensation, their experimental results showed that higher heat transfer rate can be obtained from the modified surfaces. From visual observations, they found that micro/nano-scale porous surfaces can effectively initiate

Corresponding author at: School of Chemical Engineering and Technology, Xi’an Jiaotong University, Xi’an 710049, China. E-mail address: [email protected] (B. Qi).

http://dx.doi.org/10.1016/j.expthermflusci.2017.09.002 Received 16 April 2017; Received in revised form 24 August 2017; Accepted 2 September 2017 Available online 07 September 2017 0894-1777/ © 2017 Elsevier Inc. All rights reserved.

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Nomenclature Alv Asl Cpl Cf Fr Fs G h hfg M Mg n q rb rd

Ts Tw U V Wa α Δa ΔT η θ θadv θavg θrcd λ ξ ρ σ

liquid–vapor interface [m2] solid-liquid interface [m2] specific volume [kJ/kg·K−1] coefficient of friction [–] retention force [N] viscous force [N] gravity [N] heat transfer coefficient [W/m2·K−1] latent heat of vaporization [J/kg] mass flow rate [kg/s] gravity moment [N·M] number of micro-grooves [–] heat flux [W/m2] base radius [mm] departure radius [mm]

surface temperature [°C] wall temperature [°C] sliding velocity [mm/s] volume of the deformed drop [mm3] adhesive work [N/m] initial point of the crest [mm] wetting width on crest [mm] subcooling temperature [°C] surface wetting ratio [–] contact angle [degree] advancing contact angle [degree] average contact angle [degree] receding contact angle [degree] thermal conductivity [W/m2·K−1] azimuthal angle [rad] density [kg/m3] liquid–vapor interfacial tension [N/m]

results indicated that the contact angle increased with the increasing of groove depth and V-angle. The increase of the groove pitch was conducive to droplets sliding, and wetting behavior was also anisotropic obviously. Zamuruyev et al. [15] designed microscale trapezoid grooves on hydrophobic surface to enhance its dropwise condensation heat transfer. Their experimental results indicated that micro-grooved geometry can enhance droplets transfer from the Wenzel to the Cassie state and directional transport over long distances, and these trapezoidal grooves created a capillary pressure gradient which enabled droplet transfer, right after nucleation, from the “pinned” state, inside the groove, to the upper surface of the crest. A theoretical model was developed by Lu and co-workers [16] to predict the heat transfer efficiency of dropwise condensation for surface with tiny sizes of triangular grooves (2, 5 and 10 μm), and the calculations agreed well with their experiment results. Both experiments and simulations were employed by Zhong et al. [17] to study the wettability of homogeneous and heterogeneous surfaces with rectangular microgrooves 20–40 μm in spacing and 20–180 μm in depth. They found that groove geometry had a profound impact on the drainage behavior of condensed droplets, and the sliding of droplets on micro-grooved surface presented obvious anisotropy. Ma et al. [18] studied the anisotropic wettability of droplets during the sliding process on rectangular microgrooves experimentally. Their results showed that the contact angle in the parallel direction was

dropwise condensation by generating smaller condensates and limiting the growth of ‘large’ condensate drops and by improving surface renewal rate [9]. They also indicated that a thinner nano- or sub microscale pins surfaces was required to increase condensation heat fluxes [6]. The adoption of microscale grooves on surface could reduce the resistance and adhesion work of droplets sliding, and therefore was recognized as an effective method for enhancing the heat transfer of dropwise condensation. Watanabe et al. [10] firstly proposed that the departure diameter and sweeping cycle of droplets can be significantly decreased by utilizing micro-grooves with special geometry on superhydrophobic surface. Sommers and Jacobi [11,12] described photolithographic techniques to obtain micropatterns on aluminum surfaces with parallel grooves, 30 μm wide and tens of microns in depth. The experimental results showed that critical droplet size was nearly 50% smaller on micro-grooved surfaces than on the same surface without micro-grooves. Droplets movement in superhydrophobic grooves were performed by Xu and co-workers [13] experimentally. They found that droplets on V-shaped grooves translated from the immersed state to the suspended state as the cross sectional angle of the groove decreased and the suspended droplet departed from the groove bottom as the droplet volume increased. Li et al. [14] also investigated the sliding process of droplets on micro-V-grooves with different pitches. Their experimental

Fig. 1. Schematic diagram of the experimental apparatus. 1-steam generator; 2, 4, 6, 10-trim valve; 3-superheater; 5-diaphragm pump; 7rotor flow meter; 8-three direct links; 9, 15-tank; 11-condensation cavity; 12-LED light source; 13-stem trap valve; 14-meter tank; 16-data acquisition instrument; 17-high speed camera.

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top of the condensing plate to eliminate the sweep effect of steam flow to the condensation droplets. The condensate on the test surface was collected by a funnel and then flowed into a measuring tube. A window was mounted at the chamber for visual observation of the condensation process. The chamber was insulated by rock wool with thickness of 50 mm to diminish the condensation that occurred at other place. By doing this, the influence of the unwanted condensation was very limited and can be neglected in the experiments. The excessive steam and the condensate condensed on the other surfaces of the condensing chamber flowed into an auxiliary condenser and were collected by another measuring tube. Water was used to cool the test surface. It was kept at a constant temperature in a cooling-water tank and was sprayed on the backside of the test condensing plate by means of a pump and a nozzle. Heat flux through the heat transfer plate was regulated continuously by adjusting the pressure and flux of cooling water. The inlet and outlet water temperature were also measured by two platinum resistance thermometers (Pt 100). The uncertainty of temperature measurement was estimated as ± 0.1 °C.

Fig. 2. Sample structure and temperature measurement point layout.

larger than the perpendicular direction which indicated that the anisotropy was significant on the groove-liked textures. The difference of contact angle value between these two directions ranged from 15.7° to 47.4°. Kannan et al. [19] compared static contact angle on U type grooves with different direction. They pointed out that the contact angle perpendicular to the groove direction was larger than that parallel to the groove direction, and both of them were far larger than the contact angle on smooth surface. Lara and co-workers [20] studied the effects of semicircular groove size on dropwise condensation heat transfer. Their experimental results showed that a greater heat flux was obtained on the smaller sizes of semicircular groove. The theoretical analyses and experimental studies above were mainly focused on discontinuous micro-grooves (e.g. rectangular groove, trapezoid groove, V and U type grooves, and semicircle groove). However, the numerical analyses performed by Park [21] and Promraksa et al. [22] showed that the continuous micro-grooves such as waved grooves and cosine grooves were in favour of the coalescence and sweeping of droplets and could greatly improve the efficiency of condensation heat transfer, but they did not give experimental verification. Based on this reason, the experimental studies on the wettability of sinusoidal micro-grooved surfaces with different geometrical parameters were carried out firstly. Then, the influences of the microgrooves on departure diameter, sliding process and sweeping cycle of the droplets were analysed dynamically. Finally, the influences of micro structure of grooves on dropwise condensation heat transfer were investigated through the comparison of experiments.

2.2. Sample preparation Details of the test plate are shown in Fig. 2. A 3 mm-thick square condensing plate, with a test area of 50 mm × 50 mm was made of Aluminium alloy (AA6061) (Al 97.9 wt%, Mg 1.0 wt%, Si 0.60 wt%, Cu 0.28 wt%, Cr 0.20 wt%) . Two holes of 1 mm in diameter were drilled at the condensing plate. Their depths were 15 mm and 25 mm, respectively. Two sheathed copper-constantan thermocouples (type T, Omega Engineering Inc.) were inserted into the two holes to measure the wall temperature. The uncertainty of temperature measurement was estimated as ± 0.5 °C. Experimental data were collected using Agilent 34970A data acquisition unit, and all the data including the pressure, thermocouple readings, surface subcooled temperature, heat flux, condensation heat transfer coefficients can be calculated by date reduction software, and the real-time data profiles can be displayed on computer monitor. Finely polished aluminium plate surfaces were used to prepare the test surfaces, five sinusoidal micro-grooved surfaces with different structure parameters (height and pitch of grooves) were fabricated using the dry etching technique (Etching in Cl2 gas). The geometrical sizes and of micro-grooves are listed in Table 1 and their SEM images are shown in Fig. 3. Super hydrophobic film was deposited on the micro-grooved surfaces by molecule self-assembly in the following steps: (1) The samples were etched with 5.2 wt% hydrochloric acid (HCl) for 5 min at 42 °C, and the etched samples were ultrasonically cleaned with deionized water to remove any residual dust particles from their micro-grooves; (2) The etched clean samples were immersed in 30% H2O2 solution for oxide film fabrication at 25 °C, stirred solution constantly to ensure the uniformity of concentration and temperature, and then cleaned the samples with deionized water after 10 h oxygenizing; (3) The samples were immersed into 1 vol% silane/ethanol solution (PFDS) for 5 min at room temperature, and then taken out and heated at 120 °C in an oven for 10 min. After these procedures, a selfassembled monolayer of silane can form over sample surfaces. The aim of steps (1) and (2) was to increase the adsorption strength of the selfassembly films and to improve their life time.

2. Experimental apparatus and sample preparation 2.1. Experimental apparatus The experimental apparatus, shown schematically in Fig. 1, consisted of four parts: the steam generating system, cooling system, steam condensing system, and data acquisition/control system. Steam at about 100 °C was supplied from a steam generator, and then passed through a superheater to remove mist and to ensure the steam have reached saturation or 1–2 °C super-heated. The dry steam was led into the condensing chamber and condensed on the surface of the test condensing plate. The steam velocity in the duct was obtained by dividing total flow rate of steam through the duct by the cross-sectional area and was maintained at approximately 5 ± 0.5 m/s. The steam temperature was measured by a platinum resistance thermometer (Pt 100), with an accuracy of ± 0.1 °C. The steam pressure was monitored by a manometer, and maintained at atmospheric pressure by adjusting the valve installed at the outlet of the condensing chamber. The condensing chamber can freely adjust angle (from horizontal to vertical) by rotating device to satisfy experiment requirement. A baffle was set at

Table 1 Geometrical sizes of micro-grooves on sample surfaces. Samples #

1 2# 3# 4# 5# Smooth

30

Height, A (μm)

Pitch, P (μm)

Ratio of A/P

12 18 24 24 24 0

30 30 30 40 60 ∞

0.4 0.6 0.8 0.6 0.4 0

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Fig. 3. SEM image of micro-grooved structure and surface.

trough but kept on the crest of grooves partly. Compared with the smooth surface, it can be found that the contact angle perpendicular to groove direction, θ⊥ = 130.6°, was significantly greater than the contact angle on smooth surface, θ = 105.5°. On the contrary, the contact angle along the sinusoidal groove, θ∥ = 101.2°, was slightly smaller than that on smooth surface. Therefore, the wetting behavior of droplet on micro-grooved surface presented obvious anisotropy. This was mainly due to the difference of the energy barrier for droplet to overcome during the wetting process in different directions. When wetting along groove direction, the droplet only needed to overcome surface tension, which was very similar to the wetting process on smooth surface. However, if a droplet wetted the grooved surface perpendicular to groove direction, besides overcoming surface tension, this droplet also needed to overcome the potential energy barrier caused from the crest when sliding over the grooves. So it was more difficult for droplet to wet the surface in this direction. Owing to the above reasons, the contact angle hysteresis (Δθ = θadv − θrcd) along groove direction (Δθ⊥ = 18.5°) was also smaller than that perpendicular to groove direction (Δθ∥ = 7.9°). The values of contact angle and wetting depth of droplets on various micro-grooved surfaces are listed in Table 2. Under the combined action of surface tension and gravity, the wetting depth of droplets on microgrooved surfaces remained almost unchanged with the increase of groove height, but a pronounced increase of wetting depth was observed with larger groove pitch. Comparing the wettability on different micro-grooved surfaces, we found that the droplets on various microgrooved surfaces were all presented Cassie wetting mode. However, all contact angles both perpendicular and parallel to groove direction (θ⊥, θ⊥, rcd, θ⊥, adv, θ∥, θ∥, rcd and θ∥, adv) increased slightly with the increase of groove height but deceased as increasing the values of groove pitch. That is, the hydrophobicity of micro-grooved surface would be improved as groove height increasing, but degraded with the increase of groove pitch. The contact angle hysteresis was also influenced by the structures of micro grooves, and the similar trends as contact angle can be obtained. In addition, we also found from the test that the influence of ratio A/P on θ⊥ was more obvious than that on θ∥.

2.3. Experimental data reduction According to heat transfer theory, the heat transfer coefficient h is defined as [23].

h=

q q = ΔT (Ts−Tw )

(1)

In order to calculate the heat transfer coefficient with Eq. (1), it is necessary to obtain the heat flux q and surface temperature Tw. In the current research, the heat produced by condensation on the condensing surface was transferred to the cooling water through the heat transfer plate. The heat transfer rate calculated from the amount of the condensate per unit time can be expressed as.

Q1 = M·hfg

(2)

and the heat transfer rate calculated from the flow rate and temperature increment of the cooling water was obtained from the following equation [23].

Q2 = G·Cpl·(Tout −Tin)

(3)

The difference between Q1 and Q2 was less than 5%, and therefore it is reasonable to calculate the heat flux using the following equation.

q=

Q1 + Q2 2A

(4)

The temperature of the condensing surface Tw was calculated from Eq. (4) by using the heat flux q obtained in Eq. (3) [23].

Tw = Ti +

q·Δl λ

(5)

where Δl is the distance between the condensing surface and the measuring points of the sheathed thermocouples inserted into the heat transfer plate. 3. Experimental results and discussion 3.1. Study on surface hydrophobicity of sample

3.2. Droplets condensation morphology Fig. 4 shows the wetting morphology of droplet on the surface of sample take sample 4# and smooth surface as examples) and droplet volume is 1.2 × 10−2 mL. Cassie-Baxter regime can be clearly observed on surface of sample 4# from Fig. 4(a), and the droplet did not fully wet

Images of typical experimental phenomena under subcooled temperature ΔT = 9.2 K ± 1.0 K are shown in Fig. 5. During the experiments, the condensing surface was fixed and test with micro grooves

Fig. 4. Contact angle of droplets on the sample 4# and smooth surfaces.

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Table 2 Values of contact angle and wetting depth on various surfaces. Samples

#

1 2# 3# 4# 5#

Contact angle (°) θ⊥

θ⊥,

122.7 127.3 133.4 130.6 126.3

128.8 133.5 140.1 137.3 132.6

adv

Wetting depth (μm) θ⊥,

rcd

111.6 115.8 121.4 118.8 114.9

θ∥

θ∥,

99.4 100.5 102.3 101.2 101.7

102.3 103.5 105.3 104.1 104.7

adv

θ∥,

rcd

94.4 95.5 97.2 96.2 96.6

1.06 1.08 1.08 1.22 1.23

vertically (droplets sliding parallel to grooves, defined as vertical grooved surface) and horizontally (droplets sliding perpendicular to grooves, defined as horizontal grooved surface) respectively. As shown in the figure, stable dropwise condensation occurred on micro-grooved surface and smooth surface, but the coalescence, sweeping and sliding velocity of droplets were significantly affected by the structure parameters and the direction of micro grooves. For vertical grooved surface, the lateral movement and merge of droplets were obviously suppressed by the energy barrier so the growth rate of droplets were slightly lower than the on smooth surface, as shown in Fig. 5(a). However, the vertical micro grooves can drive droplet depart more frequently in a smaller size than those on smooth surface, which were highly advantageous to heat transfer of dropwise condensation. Because of the combined action of adhesive force and gravity, the sliding droplet changed from sphere to ellipsoid. Moreover, as the groove height increasing, the sliding velocity increased resulting in the decrease of the departure radius and sweeping cycle. Conversely, the departure radius and sweeping cycle increased with the increase of groove pitch but sliding velocity decreased. On the surface with horizontal grooves, the drops grew larger in a “tadpole” shape and contact angle hysteresis became very clear, as shown in Fig. 5(b). During sliding process, droplets flowed downwards at an angle oblique owing to the dual effect of horizontal grooves, promoting droplets lateral coalescence and hindering the departure process. The velocity of sliding down on horizontal grooved surface was much slower than that on vertical grooved surface due to the highly resistant of micro grooves against the droplet falling. In addition, the departure radius of droplets was significantly larger than that on smooth surface, but the increase of groove height and pitch had little influence on them. In the experimental, the sweeping cycles of droplets on various test surfaces were decreased with the increase of the subcooling temperature in form of power function, as shown in Fig. 6. Through the comparisons of the experimental phenomena, we found that the sweeping cycle on the surface with horizontal micro grooves was longer than that on the surface with vertical micro grooves. Moreover, it can be found that the sweeping cycles were reduced as the ratio of height to pitch of micro grooves, A/P, increasing at the same subcooled temperature.

Fig. 6. Sweeping cycle of droplets on the various test surfaces.

Therefore, it was concluded the heat and mass transfer process of dropwise condensation of micro-grooved surface was significantly affected by the geometrical structure and setting direction of the micro grooves. 4. Dynamic analyses of droplets falling off process Inclining the substrate causes an imbalance in the forces and results in drop deformation, to achieve necessary static balance. The critical radius of a deformed pendant droplet on a vertical substrate is shown in Fig. 7. Accordingly, the retention force arising from contact angle hysteresis, the difference in the advancing angle and receding angle, is equal to the component of weight parallel to the substrate. Thermodynamically, when the adhesion work of retention force is equal to or less than the total gravity moment the droplet began to slide-off on the substrate. The contact angle hysteresis, namely, the variation in the advancing to receding contact angle, is taken to vary linearly along the contact line with respect to azimuthal angle. The base of the droplet is taken to be circular as discussed earlier. The variation of contact angle, with respect to azimuthal angle along the drop contact line is formulated as [24]

cosθrcd−cosθadv ⎞ cosθ = cosθadv + ⎛ ξ π ⎝ ⎠

(6)

Thermodynamically, droplets will slide down only when the external forces moment overcome the adhesion work. For the convenience of calculation, suppose that placing a drop of liquid on a vertical plate leads to the formation of a segment of a sphere macroscopically, characterized by the average contact angle, θavg, which is equal to the

Fig. 5. Condensation morphology on test surfaces.

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Fig. 7. Force analysis of droplet on vertical surface.

condensing surface. For smooth surface η = 1, and for micro-grooved surface, it can be calculated from a + Δa

η=

∫a

1 + 0.5A2sin2 ωx dωx (9)

2π

where α is the initial point of the crest, Δa is the wetting width on crest. Obviously, the value of η on micro-grooved surface is less than 1. The adhesive work of the droplet on the condensing surface can be written as

Wα =

∫0

π

σ (1 + cosθ)cosξdξ

(10)

So the total work required to overcome for departing drops can be given by

W = Wα Asl =

∫0

π

πηr 2σ (1 + cosθ)(1−cos2 θavg )cosξdξ

(11)

Substituting Eqs. (6) and (7) into Eq. (11) and integrating, one obtains

Fig. 8. Comparisons between the experimental data and calculations of droplets departure radii.

W = −2πηr 2σ (1−cos2 θavg ) ⎛ ⎝

cosθrcd−cosθadv ⎞ π ⎠

(12)

Total gravity moment of the drop can be obtained as

Mg =

π

∫0 ∫0

θ

[(ρl −ρv ) gπr 4sin3 α (cosα−cosθ)]cosξdαdξ

(13)

When the adhesion work equals to the total gravity moment, the departure radius was obtained as 1/2

ησ ⎤ rd = K ⎡ ⎢ (ρ −ρ ) g ⎥ ⎣ l v ⎦

(14)

where 1/2

12(1 + cosθavg )sin2 θavg ⎤ K=⎡ ⎢ (3−8cosθavg + 6cos2 θavg −cos4 θavg ) ⎥ ⎣ ⎦

Before sliding, the droplet is in equilibrium so the retention force arising from contact angle hysteresis need to be balanced to gravity. After reaching the departure size, the gravity force acting on the droplet exceeds the adhesive force between the liquid and condensing substrate, and the droplet begins to move. The retention force arising from contact angle hysteresis is

Fig. 9. Forces acting on a droplet versus the base radii.

average of advancing angle and the receding angle, as shown in Fig. 7.

θavg = (θrcd + θadv )/2

(15)

(7)

⎧− G π ⎨ 2 ∫0 rb σηcosθcosξdξ ⎩

Resting

The area of the micro-grooved surface corresponding to the interface on vertical plate is

Fr =

Asl = ηπrb2 (1−cos2 θavg )

Once droplets start sliding, the other retention force, which associated with the relative velocity, U, between the fluid and the substrate, starts acting between the wall and fluid and can be obtained as follows [25]

(8)

where η is the surface wetting ratio, which is relate to structure of 33

Sliding

(16)

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Fig. 11. Relationship between heat flux and subcooled temperature for various test surfaces.

G = (ρl −ρv ) Vg

(19)

here, rb is the base radius of drop and is related to the drop radius as rb = r sinθavg. Assuming the droplet slides with a initial velocity, U0, before sweeping, the gravity, retention force of the deformed drop at the three phase contact line owing to surface tension, and wall friction are in balance. Hence (20)

Fr + Fs−G = 0

Substituting Eqs. (16), (17) and (19) into Eq. (20) and integrating, the sliding velocity is obtains

U0 =

2[(ρl −ρv ) Vg−4/ π·σrb (cosθrcd−cosθadv )] Cf ρl Asl

(21)

During droplets sliding, both the gravity and adhesive force are increasing as droplets merging, and the net force on droplet can be written as (22)

FN = G−Fr −Fs So the instantaneous velocity, U, at time t is given

U = U0 +

dFN t ρdV

(23)

The work to overcome energy barrier during droplet sliding can be written

W=n

1 Cf Asl ρl U 2 2

(17)

The drop volume (V), area of liquid–vapor interface Alv, and area of solid-liquid interface Asl of deformed drop on vertical surface are calculated using the spherical cap approximation. Accordingly, the volume of the deformed drop is [26]

V=

πrb3 (2−3cosθavg + cos3 θavg ) 3sin3 θavg

P

Aπ 2 2 π ⎞ cos x (1 + cosθ) ησ cosξdξdx 1+⎛ P ⎝ P ⎠

(24)

It can be found from Eq. (11) that the departure radius of droplet rmax was the function of surface wetting ratio and the average contact angle, and both of the them were impacted by the structure parameters of micro grooves. As mentioned in Section 3.1, the wetting depths of droplet on various micro-grooved surfaces were always remained constant. So the solid-liquid interface between droplet and condensing surface shrunk as increasing the height of grooves, and the value of η also decreased accordingly. On the contrary, if the groove pitch increased, both the solid-liquid interface and surface wetting ratio would also increase. Fig. 8 illustrates the comparisons between the experimental data and theoretical results of droplets departure radii on various test surfaces. The experimental data for horizontal surfaces were in good agreement with the calculations, with the largest difference less than 20%. From the Eq. (14), it can be found that the departure radius, rd, of droplet was related to the surface wetting ratio, η, and the average contact angle, θavg, both of these two parameters were affected by the

Fig. 10. Relation curves of sweeping velocities varies with time.

Fs =

π

∫0 ∫0 2

(18)

so the force due to gravity is 34

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sweeping cycle, but this height increment had negative effect on heat transfer of horizontal grooved surface due to the rise of energy barrier. These experimental results demonstrated that the heat transfer process on the micro-grooved surface presented obviously anisotropic, which were also confirmed by the condensation morphology on microgrooved surfaces.

micro-grooved structures. We assumed that the wetting depth of the droplets on various test plates were the same and the static contact angle was equal to the average contact angle. In this case, the solidliquid interface on micro-grooved surface decreased with increasing the groove height, whereas the solid-liquid interface increased with the increase of groove pitch. Therefore, the total work (Eq. (10) and (24)) to overcome for droplet falling were decreased with the increase of the values of A/P, which led to the volumes and departure radius of droplets declined gradually with the values of A/P increasing, as shown in Fig. 8. In addition, it can be found from the experimental data that the larger geometrical size was favorable to droplets falling for same ratio of two distance components owing to its lower adhesive work and energy barrier. Further more, the departure radii on vertical grooved surfaces were always smaller than those on smooth surface, but almost completely opposite conclusions came to horizontal grooved surfaces. The main reasons for these phenomena were: (1) For vertical grooved surfaces, the surface wetting ratio, η, was less than smooth surface, that was to say, the solid-liquid interface on micro-grooved surface was smaller than that on smooth surface, so the adhesive work between the droplet and condensing surface was accordingly weakened; (2) The contact angle on vertical micro-grooved surface was larger than that on smooth surface, so the center of droplet gravity was moved far from the condensing surface, and the gravity moment became larger. Under the coeffect of the above two factors, the departure radii on vertical grooved surface became smaller than that on smooth surface. Contrarily, although the solid-liquid interface also reduced on horizontal grooved surface, there were lots of energy barrier to overcome during the sliding process of droplets, see Eq. (24). This eventually led to a larger departure radii on horizontal grooved surface compared with smooth surface. Fig. 9 shows the forces acting on a droplet for various base radii. It was shown that all the forces increased with increasing of basic radii, but obvious difference existed in their increasing trends. At the initial stage (r < rd), the retention force Fr was balanced to the droplet gravity so net force FN was zero. When started moving, the values of Fr rose linearly as the basic radius increased whereas the other retention force Fr was proportional to the square of the basic radius. For gravity G, it grew cubically as a function of basic radii, but in opposite direction to the retention forces. Therefore, the net force FN increased as droplet grow and drove the droplet sliding down. Relation curve of sweeping velocities vary with time are shown in Fig. 10. As shown in the figures, the theoretical results were in good agreement with the experimental data with the largest difference less than 13%. Although the all the velocities increased over time, the acceleration was gradually reduced. By comparison, it can be found that the velocity of sliding down on horizontal grooved surface was only 60–70% of that on smooth surface, while the velocity of sliding down on vertical grooved surface can reach 1.2 times or higher than that on smooth surface. Besides the experimental phenomena, the theoretical analyses also indicated that sliding velocities were increased with the increase of the values of A/P, and larger geometrical size was favorable to droplets sliding for same ratio of two distance components.

6. Conclusions In the present study, five sinusoidal micro-grooved surfaces with depth of 12–24 μm and width 30–60 μm were precisely and smoothly fabricated using dry etching technique on aluminium surfaces, and then super hydrophobic film was deposited on the micro-grooved surfaces by molecule self-assembly. The wettability and the heat transfer characteristics of dropwise condensation on the micro-grooved surfaces were investigated experimentally. Subsequently, the coalescence and sweeping process of droplets on micro-grooved surfaces were dynamically analyzed. The following conclusions can be obtained from the above studies. (1) The experimental results showed that both wetting behaviors and heat transfer characteristics on the micro-grooved surfaces displayed obvious anisotropy. The static contact angle perpendicular to groove direction θ⊥ was significantly larger than that along groove direction θ∥ due to the difference of the energy barrier for droplet to overcome during the wetting process in different directions. Same trends can also be observed for contact angle hysteresis. (2) For heat transfer on vertically grooved surface, the sweeping effects of falling drop was enhanced by the vertical groove and the heat transfer during dropwise condensation was increased to 30–50%, and a better heat transfer performance can be achieved when the ratio of A/P increases. Different from vertical grooved surface, the experimental results obtained from horizontal grooved surface was similar to the results of smooth surface. (3) Both net force and sliding velocity increased as droplets grew. Droplets departure radii were decreased with the increase of the ratio of height to pitch of micro grooves, and larger geometrical size was favorable to droplets falling for same ratio of two distance components. (4) The velocity of sliding down on horizontal grooved surface is only 60–70% of that on smooth surface, while the velocity of sliding down on vertical grooved surface can reach 1.2 times or higher than that on smooth surface. Acknowledgment The authors are grateful to the financial supports by the National Natural Science Foundation of China under the Grants of 51776168 and 51306141. References [1] B. Bhushan, Y.C. Jung, Natural and biomimetic artificial surfaces for superhydrophobicity, self-cleaning, low adhesion, and drag reduction, Prog. Mater. Sci. 56 (1) (2011) 1–108. [2] I. Tanasawa, J. Ochiai, Y. Utaka, S. Enya, Experimental study on dropwise condensation effect of departing drop size on heat-transfer coefficients, Trans. JSME 42 (1976) 2846–2852. [3] J.W. Rose, Dropwise condensation theory and experiment: a review, Proc. Inst. Mech. Eng., Part A: J. Power Energy 216 (2) (2002) 115–128. [4] B. Peng, X. Ma, Z. Lan, W. Xu, R. Wen, Analysis of condensation heat transfer enhancement with dropwise-filmwise hybrid surface: droplet sizes effect, Int. J. Heat Mass Transf. 77 (2014) 785–794. [5] X. Ma, X.F. Chen, T. Bai, J.B. Chen, A new mechanism for condensation heat transfer enhancement: effect of the surface free energy difference of condensate and solid surface, J. Enhanc. Heat Transf. 11 (4) (2004). [6] S. Lee, H.K. Yoon, K.J. Kim, S. Kim, M. Kennedy, B.J. Zhang, A dropwise condensation model using a nano-scale, pin structured surface, Int. J. Heat Mass Transf. 60 (2013) 664–671. [7] C. Yamali, H. Merte Jr, A theory of dropwise condensation at large subcooling

5. Condensation of heat transfer Heat flux curves of test surfaces under various subcooled temperature are shown in Fig. 11, and the maximum experimental error is less than 10%. The experimental data showed that heat transfer increased as the values of A/P increasing, and the larger geometrical size was also favorable to heat transfer. Moreover, the heat flux for all test surfaces increased with increase of subcooled temperature. In the case of same subcooling, the heat fluxes through the horizontal grooved surfaces, which were very close to smooth surface, were about 30–50% lower than those through the vertical grooved surfaces. Heat transfer was enhanced when the groove height increased owing to the more frequent 35

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