behavior of a droplet impacting on a horizontal plate

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5th International Symposium on Advanced Science and Technology in Experimental Mechanics, 4-7 November, 2010, Kyoto, Japan

Behavior of a Droplet Impacting on a Horizontal Plate

Yusuke SAKAI1, Yoshiaki UEDA1, Toshio ISHII2 and Manabu IGUCHI1

1 Graduate School of Engineering, Hokkaido University, Sapporo 060-8628, Japan 2 JFE Steel Corporation, Kanagawa, Japan

(Received 7 December 2010; received in revised form 27 April 2011; accepted 5 June 2011)

AbstractThe aim of this study is to investigate an unsteady behavior of a droplet impacting on a horizontal plate with poor wettability. Visualization techniques employed in this study include Computational Fluid Dynamics (CFD) and experimental one using a high-speed camera. The contact angle of the plate is varied by coating a water repellent material on its surface. This paper presents snapshots of bouncing motion of a droplet, the deformation rate at the impact on the plate, the coefficient of restitution, and flow in the droplet.

Key wordsDroplet, Wettability, Deformation Rate, Coefficient of Restitution, Surface Tension, Experimental Visualization, CFD

1. Introduction Information on a single droplet impacting on a solid surface is of practical importance in many engineering fields. Thus, many researchers have investigated the dynamic behavior of a droplet impacting on a solid surface [1 3]. The results of these studies are applicable to fabrication of semiconductor, cleaning of ink-jet printers, development of steelmaking processes and so on [4 6]. In the steelmaking processes, for example, the refractory is usually chosen to be poorly wetted by molten steel to prevent metallurgical reactions with the steel [7]. The wettability of the refractory therefore plays an important role for the refining of molten steel.

A recent review on the wettability and surface tension physics is found in the textbook of de Gennes, Brochard-Wyart & Quéré [8]. Kato et al. investigated the critical inclination angle of a plate for sliding down of a liquid drop [9 11]. Sonoyama and Iguchi calculated the advancing and receding contact angles of a bubble or a droplet using Finite Element Method (FEM) [12]. Kagawa and Iguchi investigated a detachment shape of a single silicone oil or water droplet from the exit of a wetted or a poorly wetted single-hole nozzle [13]. Akao et al., Wachters et al. and Qiao et al. investigated the behaviors of a droplet impacting on a heated substrate surface [14 16]. Many researchers have investigated the maximum spread of the droplet impacting on a rigid surface [17 19]. Gunjal et al. researched the influence of the diameter of a droplet, impact velocity, surface tension, and contact angle on the dynamic behavior of the droplet by experiment and numerical simulation using the VOF method [20]. Mao etal. compared the experimentally measured maximum

spread of a droplet impacting on a flat surface of c = 30 ~ 120 deg. with the numerically calculated one [21], where

c is the equilibrium contact angle. Previous researchers focused on the dynamic behavior

of a droplet impacting on a rigid surface. Investigations on the bounce-off height after the impact and coefficient of restitution are very limited. For example, one of research groups in France dealt with dynamic behavior of a droplet on a super-hydrophobic inclined plate of a contact angle of approximately 170 deg. [22], the detailed mechanism is still open as far as the authors know.

This study investigates the dynamic behavior of a water droplet impacting and bouncing on a hydrophobic horizontal plate and then compares the experimental results obtained in this study with the existing equation proposed by Mao et al. Furthermore, a particular attention is paid to the influence of the number of impacts on the bounce-off height and the coefficient of restitution. Empirical equations are proposed for the coefficient of restitution as a function of the Weber number.

2. Experimental Procedure The experimental apparatus and physical setting is shown in Fig. 1. The flat plate made of transparent acrylic resin was 200 mm long and 200 mm wide. The original contact angle, c, of de-ionized water on the acrylic plate in the air was 70 deg. This experiment used a water repellent to vary the contact angle c to 144 deg. A single water droplet was generated at the exit of a single-hole nozzle. The inner nozzle diameter, dni, was 0.10, 1.0, and 1.7 mm and the corresponding outer nozzle diameter was 0.26, 2.0, and 3.1 mm. The volume of the droplet, VD, was calculated by measuring the weight of 10 droplets. The volume of the

Fig. 1 Experimental apparatus and physical setting

droplet h0

high-speed camera

tank

cock

Journal of JSEM, Vol.11, Special Issue (2011) SS25-SS30Copyright Ⓒ 2011 JSEM

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droplet was found to be 4.3, 33.2, and 45.4 mm3. The droplet was released from a predetermined height, h0. The Weber number, We, was changed from 0.0011 to 36.1. The Weber number is defined as follows:

We = L dD vD2 /

where L is the liquid density, dD is the equivalent volume diameter, vD is the velocity of the droplet at the impact, and

is the surface tension. The Weber number means the ratio of the inertial force to the force due to the surface tension.

3. Numerical Procedure In all numerical simulations, FLUENTTM ver.6.2.16 was employed on 2.66 GHz Intel (R) Core (TM) 2 Quad CPU Q9400 processor with 3.25 GB RAM. The construction of the three-dimensional computational grids was made by GAMBIT.

The computational grid was made of cubic 312,130 elements ( 2.5dD < x < 2.5dD and 0 < y < 6.5 dD and 0 < z <3.5dD, dD: diameter of an initial droplet), as shown in Fig. 2. To save CPU time, the droplet and flow field were assumed to be symmetric with respect to the symmetry face shown in Fig. 2. The contact angle of de-ionized water on the plate was 144 deg. The spherical water droplet was set at the certain height from the horizontal plate, which was identical with the above-mentioned experimental condition.

In the FLUENT code, we employed a segregated implicit solver and first-order upwind interpolation scheme. A small time-step of t = 1.0 10-4 (s) was adapted to achieve a convergence in every iteration. A free surface behavior (i.e. gas-liquid interface behavior) was tracked by the VOF model. The contribution of a volume force from a wall adhesion was added to the momentum equation as the source term, i.e., Fvol = ij ( i grad i) / [(1/2) ( i + j)] with the curvature i = div n and the surface normal n =grad i. Here, i is the density of the liquid i, j is the density of the gas j, is the volume-averaged density, i is the volume fraction of the liquid i, and ij is the surface

tension. The wettability at the meniscus was taken into account using the geometrical condition

n = nw cos c + tw sin c (2)

on the contact line where nw and tw are the normal and tangential vectors on a wall. We employed the following Laplace-Young equation.

p = ij (1/R1+ 1/R2) (3)

where p is the pressure difference between the two fluids and, 1/R1 and 1/R2 are the surface curvatures as measured by two radii in orthogonal directions (see FLUENT 6.2 User’s Guide [23] for more details). Laplace Young equation is available only in the case of the static condition. However, if the time-step t is chosen to be much shorter than the whole deformation period of the droplet, a quasi-static condition would be realized during the time-step tbecause the droplet would deform a little. This is the reason why Eq. (3) was adopted in this study.

In our previous study, we affirmed this numerical procedure was in agreement with experimental results of dynamic behavior of a bouncing droplet on an inclined plate [24].

4. Results and Discussion 4.1 Comparison of the measured diameter of a falling

droplet from the nozzle exit and calculated value from an empirical equation

Table 1 shows the relationship between the nozzle diameter and the diameter of a single droplet falling in the atmosphere. The following empirical equation proposed by Ohtake et al. [25] could predict the diameter, dD, of a single droplet falling in the atmosphere under the critical flow rate.

dD = 1.62 dn [ / ( gdn2 ) ]0.35 (4)

where ( = ij) is the surface tension, is the density difference between liquid and gas, and g is the acceleration due to gravity. When the wettability of the nozzle is good, the outer diameter, dno, should be substituted into dn in Eq.

Fig. 2 Computational grids and initial setting (VD = 33.2 mm3, h0 = 15.4 mm)

x

y

z

droplet

symmetric face

hydrophobic plate

grids points 70 91 49

3.5 dD5.0dD

6.5dD

dno = 0.26 mm

experiment empirical equation

dno = 2.0 mm

dD = 2.17 (mm)

dD = 3.99 (mm)

dD = 4.59 (mm)

dD = 2.01 (mm)

dD = 3.99 (mm)

dD = 4.43 (mm)

outer nozzle diameter

dno = 3.1 mm

Table 1 Comparison of the measured diameter of falling droplet from the exit nozzle and empirical equation (4)

Y. SAKAI, Y. UEDA, T. ISHII and M. IGUCHI

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(4), while when the wettability of the nozzle is poor, the inner diameter, dni, should be into dn. In this study, the wettability of the nozzle was good, so we adopted dno as dn.As shown in Table 1, all experimental results for dD were in good agreement with Eq. (4).

4.2 Behavior of a droplet impacting on a horizontal plate

Details of the behavior of a droplet impacting on the hydrophobic plate can be observed by a high-speed camera. The photographs displayed in Fig. 3 were taken at 500 frames per second. The volume of the droplet was 33.2 mm3, and it was set at a height of 9.7 mm from the horizontal plate. The Weber number was 7.3. At first, the droplet impacted on the hydrophobic plate (Fig. 3(a)), and then deformed like a pan-cake (Fig. 3(c)). After the impact, the droplet bounced off the hydrophobic plate several times (Fig. 3(i)).

On the other hand, a droplet was observed to pinch off when the Weber number was 16.7, as shown in Fig. 4(m). A similar phenomenon was reported by Akao et al [14]. They investigated the behavior of a droplet impacting on a hot plate. They concluded that disintegration of the droplet was caused by the instability of the liquid column.

The deformation rate, ed, of the droplet was defined as follows:

ed = dmax / dD (5)

where dmax denotes the maximum width of the droplet during the impact motion and dD is the droplet diameter. Figure 5 shows a comparison of ed with the number of impacts, N. When the Weber number was large (We = 31.5), ed decreased with an increase in the number of impacts. The values of ed , however, were kept constant when the Weber number was small (We = 1.16). The reason is that the droplet hardly deforms like the feature in Fig. 6.

Mao et al. proposed the following empirical equation for the deformation rate, ed. The range of the impact

velocity is from 1000 to 6000 mm/s.

.032)1

12We(]

ReWe2.0)cos1(

41[ 3

33.0

83.0

ddc ee (6)

Figure 7 shows the comparison between the measured values of ed and the calculated ones. The deformation rate can be predicted by Eq. (6) for ed approximately greater than 1.6. Figure 8 shows the relationship between the Weber number and the deformation rate ratio, edEXP / edCAL,where the subscripts EXP and CAL denote the measured and calculated values, respectively. This figure demonstrates that Eq. (6) is valid for the Weber number greater than about 10, although Eq. (6) was originally derived for c 120 deg.

(a) (b) (c) (d) (e)

(f) (g) (h) (i) (j)

(l) (n) (o)(m) (k)

Fig. 4 Behavior of a droplet pinching off after the impact on the hydrophobic plate. The volume of the droplet is 33.2 mm3, the Weber number is 16.7, and the time interval of each photograph is 4 ms

0 1 2 3 41

1.5

2

2.5

N

e d

(a) (b) (c) (d) (e)

(f) (g) (h) (i) (j)

(k) (l) (m) (n) (o)

Fig. 3 Photographs of a droplet impacting on the hydrophobic plate. The volume of the droplet is 33.2 mm3, the Weber number is 7.3, and the time interval of each photograph is 2ms

Fig. 5 Deformation rate of a droplet impacting on the hydrophobic plate. The Weber number is 1.16 (circle), 12.6 (triangle), and 31.5 (square), respectively

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The coefficient of restitution (COR), e, of a droplet was defined as

e = (hn+1 / hn)1/2 (7)

where hn is the n-th peak height from the center of the droplet to the plate. Figure 9 shows the relationship between the number of impacts, N, and COR, e. The volume of the

droplet was 33.1 mm3. As shown in Fig. 9, e1 became greater with a decrease in h0. Here, the subscript 1 denotes the first impact. The reason is that the droplet deforms more significantly at the impact for higher falling velocity (see Fig. 4 (a) (g)) and dissipates energy due to the intense surface deformation. The value of e3 was not dependent on the number of impacts. This is because energy loss at the third impact is very small, as the droplet hardly deforms (Fig. 6).

Figure 10 shows the relationship between the Weber number, We, and COR, e, obtained in the experiments. The solid circle and solid triangle are the results obtained by Richard et al. [18]. A water droplet, whose diameter was 0.8mm or 1.0mm, impacted on a super-hydrophobic surface of c = 170 deg. The plane was slightly tilted at an angle of about 1 deg. They mentioned that when the Weber number was smaller than around 0.14, the error increased. The reason is that it becomes hard to determine the heights from the plate from which the drops are detaching. The present data are in good agreement with those of Richard et al. except the Weber number smaller than 0.14. When the Weber number was smaller than unity, the force due to the surface tension was dominant to the behavior of the droplet. In this region (We < 1.26), the droplet hardly loses

(a) (b) (c) (d) (e)

(f) (g) (h) (i) (j)

(k) (l) (m) (n) (o)

Fig. 6 Photographs of a droplet impacting on the hydrophobic plate taken by high-speed camera. The volume of the droplet is 4.3 mm3, the Weber number is 0.63, and the time interval is 1ms

1 1.2 1.4 1.6 1.8 21

1.2

1.4

1.6

1.8

2

Measured ed

Cal

cula

ted e d

Fig. 7 Comparison between the measured deformation rate, ed, and the calculated one proposed by Mao et al

0 10 20 30 400

0.4

0.8

1.2

1.6

We

e dEX

P / e d

CAL

Fig. 8 Relationship between the Weber number and the deformation rate ratio

Fig. 9 The coefficient of restitutions of a droplet of 33.1mm3 at the impact on the hydrophobic plate

0 1 2 3 40

0.2

0.4

0.6

0.8

1

N

e

h0 = 4.8mm h0 = 8.0mm h0 = 9.8mm h0 = 13.0mmh0 = 15.6mm

0.001 0.01 0.1 1 100.1

0.2

0.30.40.50.60.70.80.91

We

e e1 e2 e3

Richard et al. (dD = 0.8mm) Richard et al. (dD = 1.0mm)

Fig. 10 Relationship between the Weber number and the coefficient of restitution. Circles indicate the value of e1, triangles the value of e2, and squares the value of e3. The values of e1 for We 1.26 is fitted as e = We 0.22

+20%

20%


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its energy due to the viscosity of the droplet; the droplet slightly deforms, as shown in Fig. 6. As seen in Fig. 10, when the Weber number was smaller than about unity, e was slightly smaller than unity. When the Weber number was larger than about unity, the experimental results of the first impact decreased sharply. The following empirical equations therefore were derived.

e = 0.95 (We < 1.26). (8)

e = We 0.22 (We 1.26). (9)

The scattering of the data around Eq. (9) was 20%. The dynamic viscosity of water ranged from 0.957 10 3 kg /(m s) to 1.084 10 3 kg / (m s).

4.3 Internal flow of a droplet impacting on a hydrophobic plate

In this subsection, we focus on the internal flow of a droplet impacting on a hydrophobic plate. Figures 11(a) ~ (e) show the dynamic behavior of small particles pre-mixed in the droplet. The volume of the droplet was 33.2mm3, and the Weber number was 12.1. In the experiment, tracer particles of a density s of 1040 kg/m3 were mixed in the droplet. The mass percentage of the particles was 0.1 %. We confirmed that a small amount of the particles does not affect the dynamic behavior of the droplet. The behavior of the particles in the droplet can not be observed clearly in the experiment. This is because the free surface of the droplet is distorted due to the impact. In the numerical simulation, the particles were successively put in the droplet to observe the internal flow of the droplet impacting on the hydrophobic plate. The physical properties of the particles are identical with those used in the above-mentioned experiment.

Figures 11 (f) ~ (j) show the results obtained by the numerical simulation. The profile of the droplet at the

symmetric face is drawn to understand an instantaneous situation of the impact motion of the droplet. At first, the droplet impacts on the hydrophobic plate (Fig.11 (f)), and then it deforms like a disk, as shown in Fig. 11 (h) of the computational results. After deforming, the droplet is ring-shaped pan-cake due to gravity, as seen in Fig. 11 (i). The droplet returns to nearly spherical shape due to the surface tension (Fig. 11 (j)) and then it bounces off the hydrophobic plate.

5. Conclusions This study has investigated the dynamic behavior of a water droplet impacting on and bouncing off a horizontal hydrophobic plate. The visualization techniques employed were the high-speed camera and the computation with the aid of the FLUENT numerical software. Main findings are summarized as follows: (1) The empirical equation proposed by Mao et al. was

applicable to predicting the deformation rate, ed, for the Weber number greater than about 10.

(2) When the Weber number was large, the deformation rate, ed, decreased with an increase in the number of impacts, N. The reason is that the droplet deforms and then dissipates energy due to the intense surface deformation. The values of ed, however, was kept constant when the Weber number was smaller than about unity. This is because the droplet hardly deforms.

(3) The relationship between the Weber number, We, and the coefficient of restitution, e, was described by the following equations. The dynamic viscosity of water ranged from 0.957 10 3 kg/(m s) to 1.084 10 3 kg / (m s).

e = 0.95 (We < 1.26). (8)

e = We 0.22 (We 1.26). (9)

(a) (b) (c) (d) (e)

(f) (g) (h) (i) (j)

Fig. 11 Comparison of the particle trajectory in the droplet between (top) experimental results and (bottom) CFD ones. The volume of the droplet is 33.2 mm3 and the Weber number is 12.1. In the CFD, the photographs show the profile of the droplet at the symmetric face. The line in the droplet consists of the particles ( s = 1040 kg/m3). The particles did not affect the behavior of the droplet

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behavior of a droplet impacting on a horizontal plate

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