A spherical Leidenfrost droplet with translation and rotation

International Journal of Thermal Sciences 129 (2018) 254–265

Contents lists available at ScienceDirect

International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts

A spherical Leidenfrost droplet with translation and rotation a

b

T

a,∗

Zhi-Hao Wu , Wan-Hsin Chang , Chen-li Sun a b

Department of Mechanical Engineering, National Taiwan University, 1 Sec. 4 Roosevelt Road, Taipei, 10617, Taiwan Department of Mechanical Engineering, National Taiwan University of Science and Technology, 43 Sec. 4 Keelung Road, Taipei, 10607, Taiwan

A R T I C LE I N FO

A B S T R A C T

Keywords: Leidenfrost wheel Motion of droplet Film boiling Phase-change heat transfer Spray quenching

This study presents a theoretical model for a spherical Leidenfrost droplet with rotational and translational motion. Scaling analysis is carried out to validate the lubrication approximation, from which the velocity, the pressure field, and the temperature profile in the vapor film underneath the mobile Leidenfrost droplet are solved. The velocity field in the rotating Leidenfrost droplet is measured by microscale particle image velocimetry, and used as the boundary condition. The pressure and the rotational speed is then linked by the balance of shear force at the liquid-vapor interface. For a given wall temperature and an initial radius of the Leidenfrost droplet, our model is able to determine the minimal thickness of vapor film, the rotational speed, the translational speed, the heat transfer rate, and the variation of droplet radius with time by numerical iteration. During the evaporation process, the Leidenfrost droplet spins and moves faster and faster, and droplet shrinkage also accelerates. In addition, there exists a critical radius below which the levitation of droplet reduces rapidly. As the wall temperature increases, this critical radius decreases and the speed-up of the droplet shrinkage becomes more apparent. The results also show that heat transfer of a Leidenfrost droplet is dominated by thermal conduction through the vapor film, but contribution of thermal radiation from the hot surface to the entire droplet grows as the wall temperature heightens. Due to the lower thermal conductivity of oil vapor, a mineral oil droplet leads to a heat transfer rate smaller than that of a water droplet. Comparing to the results of a stationary droplet, incorporating the droplet motion into the model is able to predict the characteristics of the Leidenfrost droplet more accurately.

1. Introduction When droplets are deposited over a hot surface exceeding a specific temperature, liquid evaporates so fast that a vapor layer is produced to levitate the droplets. This is known as the Leidenfrost phenomenon and usually associated with the onset of film boiling [1–4]. As a Leidenfrost droplet shrinks during evaporation, the bottom surface of the droplet becomes highly unstable and the breakup of symmetry makes it drift. Recent investigations have focused on using a textured surface to manipulate the direction that a Leidenfrost droplet followed [5–8]. Linke et al. [5] milled the heated surface to a ratchet profile and found that a Leidenfrost droplet could perform self-propelled motion. Dupeux et al. [6] repeated the experiment with a sublimating platelet of dry ice and confirmed the resulting self-propelling force was originated from the asymmetric vapor flow between the hot surface and the levitating object. Despite the poor heat transfer, Leidenfrost effect occurs in numerous high-temperature applications. For instance, spray quenching for heat treatment of metal parts usually involves immersing hot alloy in a mist

∗

of atomized spray [9]. While the metal part is heated up to 600 °C–900 °C, the spray is composed of a large number of fine droplets in the range of 100 μm. Another example is the impinging fuel spray in direct injection engines [10]. Under heavy-load condition, improper cooling can cause incomplete combustion in fuel rich regions that is responsible for soot emissions. In these aforementioned applications, the droplets are so small that they retain a spherical shape [11]. Smaller droplets carry less kinetic energy and strike the hot surface at lower velocity [12]. As a result, the momentum is insufficient to penetrate the vapor layer and film boiling dominates. Despite its importance, earlier models that analyzed the Leidenfrost effect usually focused on large droplets that geometrical assumption such as a flat disk [13] or a hemispherical shape [14] was conveniently reasonable. Baumeister and Hamill [13] neglected the inertia term in the governing equations and obtained the creeping-flow solution of a stationary Leidenfrost droplet with flat-disk geometry sitting on top of a uniform vapor layer. By balancing heat conduction across the vapor film with convection of evaporating vapor, a relationship between the heat transfer coefficient and the wall superheat was established. Later, Baumeister and

Corresponding author. E-mail address: [email protected] (C.-l. Sun).

https://doi.org/10.1016/j.ijthermalsci.2018.02.033 Received 9 September 2017; Received in revised form 26 February 2018; Accepted 26 February 2018 1290-0729/ © 2018 Elsevier Masson SAS. All rights reserved.

International Journal of Thermal Sciences 129 (2018) 254–265

Z.-H. Wu et al.

Nomenclature

ϕ

a D F F FT h

θ μ ρ σ τ ω

h0 hlv k P Q˙ q r T t u U x y z

radius of the spherical Leidenfrost droplet drag view factor total interchange factor thrust the distance between heated surface and lower surface of droplet minimal thickness of the vapor film latent heat of vaporization thermal conductivity pressure heat transfer rate heat flux radial coordinate temperature time velocity translational velocity of droplet lateral coordinate longitudinal coordinate vertical coordinate

superscript *

dimensionless

subscript cond l low r rad sat θ up v w z

Greek α ε

polar angle measured from the moving direction of the droplet azimuth in the cylindrical coordinate system dynamic viscosity density Stefan-Boltzmann constant, σ = 5.67 × 10−8 W m−2 K−4 torque rotational speed of the Leidenfrost droplet

conduction liquid lower surface of droplet r-component radiation saturation θ-component upper surface of droplet vapor hot surface z-component

thermal diffusivity emissivity

assumed a similar configuration of the vapor layer to investigate the validity of the lubrication approximation by dynamic van der Walls theory. Comparing the results of the Leidenfrost case and the case of an injected flow that levitated the liquid drop, Taylor and Qian [23] found the underside of the droplet was insensitive to the injected flow. Karami et al. [24] employed the lattice Boltzmann method for solving the evaporation and deformation of the drop in Leidenfrost regime, and extended the convective Cahn-Hilliard equation to capture the liquidgas interface. A radial outflow of gas was found underneath a stationary droplet and two symmetric counter-rotating vortices were generated inside the droplet by evaporation. This toroidal vortex trajectories was also observed experimentally by Snezhko et al. [25]. Despite these aforementioned efforts, a Leidenfrost droplet inevitably rotates and undergoes spontaneous translation parallel to the wall during the film evaporation process (see Appendix A). For tiny droplet with a radius smaller than 1.25 mm, Bouillant et al. [26] reported that the inner flow pattern transformed from a toroidal axisymmetric structure to rolling. In addition, this inner rolling propelled the droplet which they called a Leidenfrost wheel, and an acceleration as high as 80 mm s−2 was found. Our preliminary experiment (see Appendix A) also shows that the magnitude of the rotating velocity is on the order of 100 mm s−1, elucidating that the internal circulation is not negligible as Sen and Law [17] suggested. The novelty of this study lies on addressing the influences of this shear-induced, single circulation in a spherical Leidenfrost droplet and its associated translational motion, which are never dealt with in previous models. For a given wall temperature, the gap distance, the rotational and translational speeds are solved under the quasi-steady condition to determine the heat transfer rate and the rate of droplet shrinkage. The outcome can help us to predict the heat transfer characteristics of spray quenching or fuel spraying processes more accurately, and have better understanding of the effects of internal convection on Leidenfrost dynamics of a tiny droplet.

Schoessow [15] found that a relative velocity existed between the drop and the heated wall in the boiler of the liquid metal space power Rankine system, and took that into account in their analytical model. By combining stagnation and Couette flows to represent the velocity profile in the vapor film between the Leidenfrost droplet and the moving surface, the results showed that vaporization time decreased with the increase in the velocity of the plate with respect to the droplet. For droplets with a volume smaller than 0.1 cm3, Gottfried et al. [16] presumed a spherical droplet model to predict the evaporation rate. Sen and Law [17] also developed the solution in the creeping-flow limit for film evaporation of spherical droplets but neglected the shearinduced circulation in the droplet. Their results showed the levitating force grew with a decreasing gap. Considering the quasi-steady Leidenfrost evaporation of spherical droplets, Nguyen and Avedisian [18] obtained the analytical solution in the limit of potential flow which revealed that the droplet progressively moved away from the plate. Similar trends were predicted by the model of Zhang and Gogos [19], which further elucidated that pressure force contributed more to support the spherical droplet than the viscous force at smaller separation distance. Bleiker and Specht [20] assumed a disk-like shape for Leidenfrost droplets with volumes on the order of 1 cm3 and regarded droplets with volumes smaller than 0.5 mm3 as spherical. For most droplets, they found the evaporation time was bound by the limiting cases of spherical and disk-like droplets. Sobac et al. [21] presented a simple fitting-parameter-free theory to model the stable shapes of Leidenfrost drop. Supported by interferometric measurement [22], Sobac et al. [21] divided the vapor layer into a concave cavity, an annular neck, and an outer region. Vapor flow in the cavity and the neck regions were solved by lubrication theory with evaporation, matched to the outer drop region that maintained the equilibrium shape. Sobac et al. [21] found that the concave depression in the drop interface was not apparent and the drop became quasi-spherical when the drop radius was smaller than half the capillary length. Taylor and Qian [23]

255

International Journal of Thermal Sciences 129 (2018) 254–265

Z.-H. Wu et al.

2. Formulation

liquid, and the closest point of the droplet to the hot surface is at a distance h0. Hence, the thickness of the vapor film h is a function of r: h (r) = h0 + a – (a2 – r2)1/2. Due to the surface shear induced by the vapor flow, the droplet tumbles and moves along the hot surface. In reality, the circulation inside the Leidenfrost drop does not rotate in any specific direction, and the course of the droplet can change accordingly (see supplementary video S7). Based on experimental observation (see

2.1. A spherical Leidenfrost droplet with rotation and translation As shown in Fig. 1, a spherical droplet with a radius of a evaporating over a hot surface at temperature Tw is considered. The droplet and the hot surface is separated by a vapor film generated by evaporation of the

Fig. 1. Spherical model of a Leidenfrost droplet with rotation and translation: (a) 3D diagram, (b) side view and top view, and (c) vector projections of velocities.

256

International Journal of Thermal Sciences 129 (2018) 254–265

Z.-H. Wu et al.

To solve Eq. (10), we need to link the rotational velocity ω to P by applying the balance of shear force at the liquid-vapor interface:

Appendix A), we assume the droplet spins counterclockwise about an axis (the dot-dashed line in Fig. 1 (a) and (b)) parallel to the y-axis at a rotational velocity of ω and moves at a translational speed of U to the right. Herein, relative-motion analysis is employed so the hot surface is considered to move at U to the left in the rest frame of the rotating droplet as illustrated in Fig. 1 (b). An auxiliary polar angle ϕ measured from the x-axis is defined so the projected speed of the droplet's rotation on the horizontal plane and the translational speed can be split to the rand θ-components as exhibited in Fig. 1 (c). The droplet is at its saturation temperature and spatial variation of temperature in the liquid droplet is neglected due to its diminutive size and the high wall temperature [19]. Supplementary video related to this article can be found at http:// dx.doi.org/10.1016/j.ijthermalsci.2018.02.033. According to previous studies [11,22,27], the orders of magnitude of h0 and a are 10 μm and 1 mm, respectively. The scaling analysis of the momentum equations is provided in Appendix B to validate the use of lubrication theory in describing the vapor flow in the gap. Assuming that the vapor flow is viscous and incompressible, the continuity and the Navier-Stokes equations can be simplified to:

∂uz ∂ur u 1 ∂uθ =0 + r + + ∂r r r ∂θ ∂z

0=−

0=−

0=

μl ω = μ v

1 h 0 dP ω = ⎜⎛ ⎝ 2 μ v dr

(4)

where μv is the viscosity of vapor, P is the pressure, ur and uθ are the velocity components in the radial and azimuthal directions, respectively. The no-slip condition implies that vapor moves with the wall at z = 0 and has zero velocity relative to the bottom surface of the rotating droplet at z = h. Because the Leidenfrost droplet rotates as a solid body (see Appendix A), the tangential velocity of vapor at the droplet surface is aω for a rotational speed of ω. Therefore, the boundary conditions can be expressed as

ur = −U cos θ and uθ = U sin θ

(5)

At z = h,

ur = ω a2 − r 2 cos θ and uθ = −ω a2 − r 2 sin θ

(6)

uθ =

∫0

r = −4πρ l hlv a2

⎜

+ rσ (Flow + Fup)(Tw 4 − Tsat 4 ) ⎤ ⎥d ⎦

da dt

(14)

⎟

(15)

−k v (Tw − Tsat ) ⎡ (a + h 0)ln(1 + a/ h 0) − a ⎤ da = dt ρ l hlv a2 ⎣ ⎦ (Flow + Fup) σ (Tw 4 − Tsat 4 ) − 2ρ l hlv

(9)

where P0 is the reference pressure at r = 0 and p is a function of r only. By substituting Eq. (7), Eq. (8) and Eq. (9), Eq. (1) can be integrated over z to obtain an ordinary differential equation for p(r):

p 1 h3 ⎛ dp + − ⎞ + rω ⎤ ⎥ 12μ v r ⎝ dr r⎠ ⎦

z=0

where εl is the emissivity of liquid droplet, εl = 0.98 and 0.95 for water and mineral oil, respectively. Fup and Flow represent the fractions of radiation leaving the upper-half and lower-half surfaces of the droplet to strike the plate. Herein, Fup = 0.318 and Flow = 0.682 [16]. Substituting Eq. (13) for the temperature profile, Eq. (14) can be rewritten as

(7)

(8)

2 ⎡− h dh dp − ⎛ ω a2 − r 2 − U ⎞ dh − ω ⎢ 4μ dr dr 2 ⎠ dr 2 ⎝2 v ⎣

⎡−rk ∂T v ⎢ ∂z ⎣

1 1 1 = ⎛ − 1⎞ + Fup (low) Fup (low) ⎝ εl ⎠

From Eq. (4), we know that P does not vary with z. Following the convention, the dependence of the pressure field on r and θ can be separated intuitively as:

12μ v d 2p = dr 2 h3

a

where Q˙ cond and Q˙ rad are the heat transfer rates by conduction and radiation, kv is the thermal conductivity of vapor, σ is the Stefan–Boltzmann constant, ρl is the density of liquid, and hlv is the latent heat of vaporization. Fup and Flow are the total interchange factors associated with the geometric view factors Fup and Flow [16]:

1 1 dP 2 z (z − hz ) − (ω a2 − r 2 sin θ + U sin θ) + U sin θ 2μ v r dθ h

P = P0 + p (r )cos θ

(13)

Q˙ = Q˙ cond + Q˙ rad = 2π

In order to obtain the radial and azimuthal components of the velocity in the vapor film, we integrate Eq. (2) and Eq. (3) with respect to z and apply the boundary conditions Eq. (5) and Eq. (6):

z 1 dP 2 (z − hz ) + (ω a2 − r 2 cos θ + U cos θ) − U cos θ ur = h 2μ v dr

(12)

In reality, temperature distribution can be more complicated and there exists an “evaporative region” immediately below the droplet that nonlinearity of temperature variation is present [23]. However, Taylor and Qian [23] demonstrated that local evaporation rate scaled as 1/h and justified the lubrication approximation. In this study, both thermal conduction across the vapor film and thermal radiation from the hot surface are considered to contribute to the total heat transfer due to the high temperature of the hot surface. If we neglect the radiative absorption of the vapor film, the total heat transfer rate Q˙ is consumed by evaporation of liquid at the droplet surface:

(3)

At z = 0,

r=0

T − Tsat z =1− Tw − Tsat h

(2)

∂P ∂z

−1

U ⎞ ⎛ μl a⎞ − ⎟ ⎟⎜ h0 ⎠ ⎝ μ v h0 ⎠

+

By replacing ω in Eq. (10) with Eq. (12) and assuming that dp/ dr = 0 at r = a, p(r) can be solved numerically for a given h0. Consequently, P, ur and uθ can be determined from Eq. (9), Eq. (7), and Eq. (8), respectively. To estimate the heat transfer rate, energy equation is considered. The detailed scaling analysis is provided in Appendix B and we find that heat conduction term perpendicular to the hot surface apparently dominates the energy equation. Therefore, temperature varies linearly with z in the vapor film and its profile is given by

∂ 2u

1 ∂P + μ v 2θ r ∂θ ∂z

(11)

r=0

where μl is the viscosity of liquid. After substituting Eq. (7) for ur in Eq. (11), the rotational velocity of the droplet can be written as

(1)

∂P ∂ 2u + μ v 2r ∂r ∂z

∂ur ∂z

(16)

To solve Eq. (16), we need to determine h0 first. This is done by applying the mass conservation for the vapor layer between r = 0 and r = a on the vertical plane of θ = 0:

rh

∫0

a2 − r 2

h0 + a

ur

r=a

dz −

∫0

h0

ur

r=0

dz = −

ρ l πa da ρv 2 dt

(17)

We can substitute ur, uθ, and Eq. (16) for da/dt in Eq. (17), and solve for h0 iteratively. Once h0 is known, we are able to determine Q˙ and da/

(10) 257

International Journal of Thermal Sciences 129 (2018) 254–265

Z.-H. Wu et al.

the measurements quite well. The reduction in droplet volume slows down during the film boiling process, and the differentiation between the experimental data and our prediction is smaller than 10%. For a Leidenfrost droplet of water undergoing rotation and translation, Fig. 3 depicts the variations of minimal thickness of vapor film h0 and droplet shrinkage |da/dt| with the droplet radius a. Similar to the results from previous measurements [11,22], we find that there exists a critical radius above which the change in thickness of the vapor film is very mild at a given wall temperature. Once the droplet is smaller than this critical size, decrease in h0 becomes more evident during the evaporation process. In addition, a hotter surface leads to a smaller critical radius and a thicker vapor cushion. Because droplet evaporates faster at higher wall temperature (greater droplet shrinkage as shown in Fig. 3 (b)), venting the excess vapor requires a wider gap. Intriguingly, higher rate of shrinkage is found at smaller a, revealing that droplet size reduces more rapidly at the later stage of evaporation. Actually, this trend does not contradict the smaller variation of droplet volume which is dominated by the droplet size at smaller a, and is consistent with the finding of previous study [11]. In particular, the acceleration of shrinkage becomes more evident as the wall temperature heightens. According to Eq. (7), a larger pressure gradient is needed to discharge the vapor from a narrower gap. Fig. 4 illustrates the pressure variation in the lateral direction underneath a rolling droplet. As shown in Fig. 4, we find that pressure is negative for x/a > 0, but turns positive for x/a < 0. At the edge of the droplet, pressure at x/a = −1 is slightly higher than that at x/a = 1, which drives a vapor flow from left to right. The flowing direction agrees with the shear force exerted by the counter-clockwise rotating droplet shown in Fig. 1. As x/a → 0, the magnitude of pressure first grows rapidly, reaches a peak value, then diminishes. Vapor underneath the left hemisphere of the droplet accelerates due to the narrowing gap and experiences a growing back pressure. However, the deceleration underneath the right hemisphere of the droplet provokes a pressure recovery effect. At x/a = 0, a negative slope is obliged to produce a net flow of vapor in the +x direction. With the decrease in a, the maximal pressure augments and the magnitude of this pressure gradient increases as well. Fig. 4 (b) and (c) exhibits the effect of wall temperature on the pressure profile in the early and late stage of evaporation. At a = 900 μm, increasing the wall temperature leads to a higher maximal pressure which occurs further away from x/a = 0. Because the Leidenfrost droplet vaporizes more quickly above a hotter surface, high

dt by providing an initial radius of the Leidenfrost droplet and the wall temperature Tw. 2.2. A stationary spherical Leidenfrost droplet To examine the influence of the droplet motion, we also consider a stationary spherical Leidenfrost droplet with a radius of a separated from the hot surface by a layer of vapor for comparison. Without the rotation and translation of the Leidenfrost droplet, the flow field is symmetric about the z-axis and the boundary conditions become

At z = 0,

ur = 0

(18)

At z = h,

ur = 0

(19)

At r = 0,

ur = 0

∂P / ∂r = 0

∂T / ∂r = 0

(20)

We modify the similarity variables used in the disk model of Baumeister and Hamill [28] for the spherical geometry so that the velocity and the pressure distributions can be written as

ur = r ∂f (r , z )/ ∂z

(21)

uz = −2f (r , z ) − r ∂f (r , z )/ ∂r

(22)

P=

1 2 β ρv (a2 − r 2) + Psurr 2

(23)

where f is the transformation variable, β is constant, and Psurr is the ambient pressure. By substituting Eq. (21) and Eq. (23) into Eq. (2), we integrate the momentum equation in the r-direction twice over z and apply the boundary conditions (Eq. (18) and Eq. (19)) to obtain ur.

ur =

ρv β 2 r (hz − z 2) 2μ v

(24)

To determine β, force balance in z-direction is considered: the levitating force produced by pressure is equal to the weight of the Leidenfrost droplet so that

πa2Psurr +

4 πρ ga3 = 3 l

∫0

a

2πrP (r ) dr

(25)

From Eq. (25), we find that β = 16gρl/3aρv and the pressure distribution and radial velocity are given by 2

P=

8gρ l 2 (a − r 2) + Psurr 3a

(26)

ur =

8gρ l r [(h 0 + a − 3aμ v

(27)

a2 − r 2 ) z − z 2]

For a stationary Leidenfrost droplet, Eq. (13) to Eq. (16) are also valid to calculate Q˙ cond , Q˙ rad , and da/dt. However, the mass conservation to determine h0 is slightly different. Vapor originated from evaporation over the bottom half of the droplet is balanced by vapor vented out from the gap:

ρ da =− v dt aρ l

∫0

h0 + a

ur

r = a dz

(28)

By substituting Eq. (27) in Eq. (28) and making the right-hand sides of Eq. (28) and Eq. (16) equal, h0 is determined for a given a and heat transfer rate can be evaluated at each time step. Subsequently, da/dt is estimated to update the droplet radius for the following time step. 3. Results and discussion 3.1. An evaporating droplet Fig. 2. Comparison of the experimental results [29] to our model for a water droplet with rotation and translation. The black lines and symbols denote a wall temperature of 280 °C, and the red ones refer to a wall temperature of 316 °C. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

During the evaporation process, the droplet continues to shrink with time. To validate our model, Fig. 2 compares the results obtained by the equations shown in Section 2.1 and the experiments [29]. We find that the evolution of the droplet volume predicted by our model conforms to 258

International Journal of Thermal Sciences 129 (2018) 254–265

Z.-H. Wu et al.

Fig. 3. Variations of (a) minimal thickness of vapor film, and (b) |da/dt| with the radius for a Leidenfrost droplet of water.

pressure gradient persists over a broader part of the widening gap so that the surplus vapor can be ejected properly. When the radius of droplet goes down to 300 μm, however, the influence of wall temperature in the pressure profile is not monotonic. At small a, the heattransfer area of the droplet reduces drastically and the effect of wall temperature on the evaporation rate becomes weak. Although h0 increases with Tw, the maximal pressures at Tw = 616 °C and Tw = 416 °C are comparable and increasing the wall temperature to 816 °C even results in a decrease in the peak pressure. At x/a = 0, the pressure variation is also more gentle at higher Tw, elucidating that the increase in vapor flow can be accommodated by the wider gap alone. From Fig. 4, it is noted that the vapor pressure in the gap is on the order of 1 Pa, which is much smaller than the Laplace pressure (approximately 10 Pa–100 Pa). This justifies the assumption of a spherical droplet and the neglect of interfacial dynamics in our study [30]. 3.2. Translational and rotational velocities Fig. 5 shows the variations of translational and rotational velocities of a Leidenfrost droplet of water with its radius at Tw = 816 °C. During the evaporation process, a decreases with time and the mobile droplet accelerates. As the droplet shrinks, the droplet not only moves but also spins faster and faster. This complies to the acceleration of a Leidenfrost wheel previously observed [26]. Herein, the asymptotic approach of lubrication theory is taken to determine the drag D and the resistant torque τ exerted on the rotating and moving droplet near a wall [31]:

Fig. 4. Comparison of pressure variations in the lateral direction for a Leidenfrost droplet of water: (a) Tw = 816 °C (a) a = 900 μm, and (c) a = 300 μm.

259

International Journal of Thermal Sciences 129 (2018) 254–265

Z.-H. Wu et al.

Fig. 6. Comparison of radial velocity distributions in the vapor film underneath water droplets, r = 0, Tw = 816 °C.

We find that ur is negative at z/h0 = 0 due to the movement of the hot surface relative to the droplet, whereas ur becomes positive at the liquid-vapor interface (z/h0 = 1) because of the rotation of the droplet. As the droplet size reduces, the thickness of the vapor film decreases and the magnitudes of ur at the two boundaries (z/h0 = 0 and 1) both heighten, reflecting on a faster spinning and propelling droplet. Hence, a higher pressure gradient is needed at r = 0, and the acceleration of the droplet also increases. In Fig. 6, the steeper slope of ur at smaller a also elucidates a stronger shear force, which leads to a faster rolling droplet.

3.3. Heat transfer rate Fig. 7 illustrates the heat transfer characteristics of Leidenfrost droplets running and spinning above a hot surface with different wall temperature. We find that heat transfer rate decreases as the droplet becomes smaller, and the reduction is more severe at higher wall temperature. Because tinier droplet leads to smaller surface area, less thermal energy can be removed from the hot surface. On the other hand, increasing the wall temperature results in larger heat transfer rate due to the higher temperature difference between the droplet and the hot surface. The influence of the wall temperature is particularly obvious for larger droplets. When Tw increases from 416 °C to 816 °C, the heat transfer rate amplifies nearly three-fold, increasing from 0.65 W to 1.83 W at a = 900 μm. But Q˙ only experiences a double growth for a droplet with a radius of 200 μm. Although increasing the droplet size is an effective way to accelerate cooling, the heat transfer deficit of finer droplets can be improved by using a denser spray during quenching. In Fig. 7 (b), the fractions of heat transfer contributed by different mechanisms are shown. We find that thermal conduction delivers more than 78% of total heat transfer, while thermal radiation is in charge of transporting up to 22% of the thermal energy. For water vapor, the contribution of radiation can be even less due to its very strong absorption in the infrared range (1 μm–10 μm) [33] where the emissive power from the hot surface peaks. Despite the growing importance of thermal radiation at higher wall temperature, heat conduction across the vapor layer still controls the film boiling process. While the droplet vaporizes, the quantities of heat transported by conduction and radiation both decreases. The decrease in Q˙ rad is solely caused by the reduction of heat-transfer area, but Q˙ cond also depends on the thickness of vapor film, which diminishes with decreasing a. The narrower gap leads to a larger temperature gradient in the vapor layer, and more heat can be conducted per unit area. As a result, the fraction of thermal conduction grows slightly with decreasing a.

Fig. 5. Variations of (a) translational velocity, and (b) rotational velocity of the Leidenfrost droplet of water with its radius at Tw = 816 °C.

D = 6πμ v a ⎧ ⎡ 15 ln ⎨ ⎩⎣ 8

2

− aω ⎡− 15 ln ⎣

( )+ h0 a

( )− h0 a

64 h 0 375 a

86 h 0 375 a

ln

ln

( ) − 0.952⎤⎦ U h0 a

( ) − 0.257⎤⎦ ⎫⎬⎭ h0 a

2 h 66 h 0 ⎛ h 0 ⎞ τ = −8πμ v a3ω ⎡ ln ⎛ 0 ⎞ + ln − 0.371⎤ ⎥ ⎢ 5 a 125 a ⎝a⎠ ⎝ ⎠ ⎦ ⎣

(29)

(30)

For Tw = 816 °C and a = 900 μm, D and τ are on the order of 10−7 N, and 10−4 N μm, respectively. On the other hand, the thrust FT is estimated from the momentum flux of the produced vapor leaving the 2 gap: FT ∼ πaQ˙ (h 0 + a)/ hlv2 ρv , where Q˙ is the heat transfer rate that is covered in next section. We find that FT is about 10−6 N, one order of magnitude larger than the drag. Similarly, the associated driving torque is on the order of 10−3 N μm, also much larger than the resistant torque. As the droplet radius reduces from 900 μm down to 200 μm, the translational velocity experiences a five-fold growth herein, increasing from 86.1 mm s−1 to 347.2 mm s−1. The rotational speed also increases one order of magnitude, augmenting from 581.2 rad s−1 at a = 900 μm to 8601.6 rad s−1 at a = 200 μm during the evaporating process. As explained in previous section, the spinning of a Leidenfrost droplet is induced by the shear stress of the vapor flow. In Fig. 8, we compare the distributions of radial velocities along the centerline of the vapor film (r = 0) for droplets with different sizes. 260

International Journal of Thermal Sciences 129 (2018) 254–265

Z.-H. Wu et al.

Fig. 8. Comparisons of (a) heat transfer rate, and (b) droplet shrinkage |da/dt| for a Leidenfrost droplet of water and mineral oil.

represented by the slopes in Fig. 7 (c). According to Nukiyama's boiling curve [32], film boiling heat flux exhibits a power law dependence on the wall superheat, and the exponent is augmented by considering the combined effects of radiation and convection [34]. Therefore, convective heat transfer and radiative absorption of vapor maybe approximated with conduction, which is described by the equivalent conduction coefficient. As shown in Fig. 7 (c), the exponent increases with the increase in droplet radius, verifying a stronger influence of radiation for larger droplets. 3.4. Influence of working fluid Fig. 8 (a) exhibits the comparison of a water droplet and a mineral oil droplet, both are commonly used in spray quenching. Since the Leidenfrost temperature of mineral oil is much higher than water, heat transfer with an oil spray is considered to be more effective due to the suppression of vapor film [12]. However, we find that using an oil droplet leads to a heat transfer rate 50–60% lower than a water droplet once a vapor blanket is formed. For a given droplet radius, the vapor cushion underneath an oil droplet is much thinner than that of a water droplet. While h0 remains on the order of 10 μm for a water droplet of a = 200 μm, an oil droplet results in h0 down to 100 nm. Despite the very short path for conduction, heat transfer is rather poor for oil droplets due to its low vapor conductivity. Consequently, the slow cooling rate makes the oil spray ideal quenchant to prevent surface cracking.

Fig. 7. (a) Variations of heat transfer rates and (b) fractions of heat transfer rates with the radius of a Leidenfrost droplet of water with rotation and translation, (c) predictions of our model depicted as the Nukiyama curve [32] for Leidenfrost droplets of water with various sizes.

Fig. 7 (c) illustrates the boiling curves, a plot of heat flux q" versus wall superheat Tw-Tsat, for Leidenfrost droplets of various sizes. The heat flux is determined by dividing the heat transfer rate over the projection area of the droplet. In liquid-vapor phase-change application, a heat transfer coefficient defined by the ratio of input heat flux and wall superheat is often used in correlations [34] and can be 261

International Journal of Thermal Sciences 129 (2018) 254–265

Z.-H. Wu et al.

900 μm to 200 μm (Fig. 9 (b)). In contrast, droplet motion leads to a much thicker vapor layer and h0 experiences a nearly 30% reduction when the radius decreases below 400 μm. Comparing to the results reported in previous studies [11,22,27], we find that incorporating the droplet motion leads to a more reasonable prediction of h0. The difference in the thickness of vapor film is caused by the distinction of vapor pressure distributions between a mobile droplet and a stationary droplet. Unlike those shown in Fig. 4, the pressure profile is axisymmetric about r = 0 for a stationary droplet. Evaporated vapor vents out radially so that net drag force exerted on the droplet is zero. Nevertheless, pressure distribution exhibits a peak and a valley in proximity to the center of a rotating droplet because a non-zero pressure gradient at r = 0 is required to squeeze vapor through the narrowest location. As a result, the pressure-induced force pushes the droplet towards the right (equivalent to a relative motion of the hot surface to the left). In addition, the value of the pressure gradient at r = 0 is inversely proportional to h0. When the translation and rotation of the droplet are considered, the non-zero pressure gradient at r = 0 may make satisfying the boundary condition (dp/dr = 0) at r = a difficult if its magnitude is too large. To reach numerical convergence, a larger h0 is necessary for a mobile droplet. Comparing to the values found in previous studies [11,22,27], the assumption of a stationary droplet greatly underestimates the thickness of the vapor film. Since thermal conduction is controlled by the thickness of the vapor film, incorporating the movement of a Leidenfrost droplet to the model is particularly critical to the accurate prediction of its heat transfer rate.

4. Conclusion In this study, we propose a theoretical model to study the characteristics of a spherical Leidenfrost droplet rotating and moving horizontally over a hot surface. Droplet evaporation is considered a quasisteady process and lubrication theory is applied to analyze the flow and temperature fields of the vapor layer in the absence of external convective heat transfer and radiative absorption of vapor. Several remarks are summarized as follows: Fig. 9. Effects of droplet motion on the variations of (a) heat transfer rate and (b) minimal thickness of vapor film.

1. Our model agrees well to the experimental results. The deviation of evolution of droplet volume is less than 10%. In addition, the existence of the critical droplet radius, below which thickness of vapor film decreases drastically, is predicted successfully. 2. During the evaporation process, the Leidenfrost droplet both spins and moves faster and faster. Although the decrease in droplet volume flattens with time, the variation of droplet radius accelerates. 3. Despite the high wall temperature, more than 78% of total thermal energy is transported by thermal conduction, while thermal radiation is responsible up to 22% of heat transfer rate. 4. In spray quenching, the differentiation of using different working fluid is mainly ascribed to the distinction in thermophysical properties. Despite the much thinner vapor film caused by an oil droplet, the heat transfer rate is 50–60% lower than a water droplet because of its lower thermal conductivity of vapor. On the other hand, the higher shrinkage rate of an oil droplet is resulted from its smaller latent heat of vaporization. 5. Incorporating the droplet motion into the model makes a better estimation of the thickness of vapor film. Due to its non-axisymmetric pressure profile, a thicker vapor cushion is required for a mobile droplet. This leads to a heat transfer rate lower than that obtained under the assumption of a stationary droplet.

Fig. 8 (b) compares the droplet shrinkage for a water droplet and a mineral oil droplet. We find that an oil droplet evaporates and shrinks much faster than a water droplet above a hot surface with a given temperature. Herein, the latent heat plays a more important role in the shrinkage rate than the total heat transfer rate. Since mineral oil has a much lower latent heat of vaporization, an oil droplet vanishes more rapidly than a water droplet. Although the heat transfer rate associated with an oil droplet is also lower, it only helps to reduce the distinction in shrinkage rates of different fluid at large droplet radius. As a decreases, the rates of heat transfer become closer (Fig. 8 (a)), and the gap between shrinkage rates widens (Fig. 8 (b)) due to the large deviation in latent heats of mineral oil and water. 3.5. Effect of droplet motion Fig. 9 depicts the effects of droplet motion on heat transfer rate and shrinkage rate for a Leidenfrost droplet of water. From Fig. 9 (a), we find that less thermal energy can be transported when a mobile Leidenfrost droplet is considered. This is because droplet motion leads to a much thicker vapor film (Fig. 9 (b)), which deteriorates thermal conduction. According to Eq. (14), only thermal properties of working fluid, droplet size, and wall temperature may alter the heat transfer rate by radiation. Therefore, thermal radiation is independent of the droplet motion. For a stationary droplet, the minimal thickness of vapor film h0 remains less than 10 μm and changes very little as a decreases from

Acknowledgements This study is supported by the Ministry of Science and Technology of Taiwan under grant number MOST 105-2628-E-002-011-MY3. 262

International Journal of Thermal Sciences 129 (2018) 254–265

Z.-H. Wu et al.

Appendix A. μPIV measurement We employ microscale particle image velocimetry (μPIV) [35] to measure the velocity field in a rotating Leidenfrost droplet levitating above a hot surface. A silicon wafer, micromachined by the DRIE (Deep Reactive-Ion Etching) process, is used as the heated substrate so that translation of the Leidenfrost droplet is prevented during the evaporation process [29]. As illustrated in Fig. A1, the microstructure is composed of twenty 50 μm height radial wedges with their spikes pointed toward the center point and aligned along a circle of 1059 μm in diameter. The expanding angle between the edges of two adjacent wedges is 1.6°. Prior to experiments, the substrate is cleaned by immersing in Piranha solution and a buffer solution of hydrofluoric acid and ammonium fluoride to remove organic contamination and native oxide. To supply heat flux to the silicon substrate, a heater element made of copper is used to hold the two electric cartridge heaters. The surface temperature of the heated substrate is controlled by adjusting input heat transfer rate via a power supply connecting to the cartridge heaters. Six K-type thermocouples are embedded in the upper portion (with a cross section of 1 cm × 1 cm) of the copper element. The measurements of the four vertically-aligned thermocouples are utilized to extrapolate the wall temperature by applying the Fourier's Law with the linear least-square regression. A syringe (Diamond MS Syringes, SGE Analytical Science) with a cone-shaped needle of 0.15 mm inner diameter (N25/500-C/T-5/0.63C, SGE Analytical Science) is used to release a liquid drop onto the heated substrate from a height of 1 cm. Measured by a semi-micro analytical balance (MS205DU, Mettler Toledo), the average weight of a water drop is 10.07 mg with a 1.6% deviation. For flow visualization purpose, the working fluid (water) is seeded by microparticles. In order to record the image sequence of a Leidenfrost droplet, a highspeed camera (NX7-S1, IDT) equipped with a variable magnification lens (0.75X-4.5X) is mounted to a tripod. Three xenon lamps are utilized to provide auxiliary illumination from different angles and a black screen is placed behind the droplet to avoid reflection. After the release of the drop, the drop-wall interaction first leads to drop bouncing and shape oscillation is induced by the strong shear stress at the liquid-vapor interface [36]. As the drop size reduces, number of oscillation spikes decreases and the drop becomes rounder as viewed from the top [37]. In addition, the drop shape is rather stable from the side view and two counter-rotating vortices are seen inside the Leidenfrost droplet, suggesting a toroidal flow structure. Nevertheless, the sizes of these two vortex are not identical and the asymmetry deteriorates, resulting in the larger vortex dominating over the smaller one (see supplementary video S1). Eventually, circulation only occurs in one direction. The same experiment is conducted on a smooth silicon wafer with the droplet pinned to the needle tip to ensure similar flow pattern is exhibited (see supplementary video S2). Supplementary video related to this article can be found at http://dx.doi.org/10.1016/j.ijthermalsci.2018.02.033. Supplementary videos S3-S6 display a Leidenfrost drop of water levitating above the microstructured surface heated to a wall temperature varying from 195 °C to 315 °C. As the wall temperature heightens, the rotation speed of the droplet increases. Although the droplet radius is smaller than the capillary length (∼2.5 mm for water) [38], the drop shape is oblate (a snapshot at Tw = 315 °C is given in Fig. A2 (a)). This may be attributed to the proximity of the wedge tips of the microstructures to the vapor-liquid interface. Near the apparent three-phase contact line, disjoining pressure may act to deform the drop shape. Although a Leidenfrost drop levitating above a smooth surface has a more spherical appearance (supplementary video S2), it moves rigorously if not being held by a needle. The spontaneous translation with rotation of the droplet is clearly seen in supplementary video S7. Moreover, the course of the droplet is changeable and no specific direction of rotation is found. To avoid complexity of experiment, the microstructured surface is employed to facilitate the measurement of the rotating flow field inside a Leidenfrost droplet. The counter-clockwise rotation of the inner flow is chosen to link the droplet translation in the +x direction as observed in supplementary video S7. Supplementary video related to this article can be found at http://dx.doi.org/10.1016/j.ijthermalsci.2018.02.033. Figure A2 (b) and A2 (c) depict the results of the μPIV diagnosis. Despite the optical distortion and image blur caused by the curvature of the droplet, tangential velocity increases linearly with the radial distance and the droplet rotates like a solid body. Appendix B. Scaling Analysis We perform the scaling analysis on the momentum equations and the energy equation by using the following nondimensional variables: T* = (T – Tsat)/(Tw – Tsat), ur* = ur/U, uθ* = uθ/U, r* = r/a, z* = z/h. The z-component of the nondimensional velocity is obtained by substituting other nondimensional variables into the continuity equation: uz* = auz/hU. Zhang and Gogos [19] noted that the influence of the pressure was much greater than that of the viscous force if the gap was small. Therefore, the pressure term is not negligible and the nondimensional pressure should be P* = h2P/μvUa. With these scaling parameters, the governing equations can be written as follows: ρ v Ua μv

(u

∗

∗ ∂ur r ∂r ∗

+

uθ∗ ∂ur∗ r ∗ ∂θ

−

uθ∗2 r∗

∂u ∗

+ uz∗ ∂zr∗

a 2 ∂P∗ ∂r ∗ h

) = −( ) +

ρ v Ua μv

(u

∗

∗ ∂uθ r ∂r ∗

+

uθ∗ ∂uθ∗ r ∗ ∂θ

+

ur uθ∗ r∗

∂u ∗

+ uz∗ ∂zθ∗

a 2 ∂P∗ h ∂θ∗

) = −( ) +

ρ v Ua μv

(u

∗

∗ ∂uz r ∂r ∗

+

uθ∗ ∂uz∗ r ∗ ∂θ

∂u ∗

+ uz∗ ∂z∗z

) = −( )

a 4 ∂P∗ h ∂z∗

+

1 ∂2ur∗ r ∗2 ∂θ 2

∗ 1 ∂2uz r ∗2 ∂θ2

2 ∗ 1 ∂ uθ r ∗2 ∂θ2

∂u ∗ 1 ∂ + ⎡ r ∗ ∂r ∗ r ∗ ∂rr∗ − ⎢ ⎣

( )

−

∗ 2 ∂uθ ⎤ r ∗2 ∂θ ⎥

+

⎦

a 2 h

∗

( )−

2 ∂ur∗ ⎤ r ∗2 ∂θ ⎥

⎦

+

∂2ur∗

( )( )

∂u 1 ∂ + ⎡ r ∗ ∂r ∗ r ∗ ∂r ∗θ ⎢ ⎣

+

ur∗ r ∗2

∂z∗2

uθ∗ r ∗2

2 ∗ a 2 ∂ uθ ⎞ h ∂z∗2

( ) ⎛⎝

(A1)

⎠

(A2)

∂u ∗ 1 ∂ + ⎡ r ∗ ∂r ∗ r ∗ ∂r ∗z ⎢ ⎣

( ) ⎤+ ) ⎥ ( ) ( ⎦ a 2 h

∂2uz∗ ∂z∗2

(A3)

263

International Journal of Thermal Sciences 129 (2018) 254–265

Z.-H. Wu et al.

u ∗ ∂T ∗ ∂T ∗ ∂T ∗ aU ⎛ ∗ ∂T ∗ 1 ∂ 1 ∂T ∗ ⎤ a 2 ∂ 2T ∗ + uz∗ ∗ ⎞ = ⎡ ∗ ∗ ⎛r ∗ ∗ ⎞ + ∗2 + ⎛ ⎞ ⎛ ∗2 ⎞ ur ∗ + θ∗ ⎥ ⎢ ∂z ⎠ ⎣ r ∂r ⎝ ∂r ⎠ α v ⎝ ∂r r ∂θ r ∂θ ⎦ ⎝ h ⎠ ⎝ ∂z ⎠ ⎜

⎟

⎜

⎟

(A4)

The multiplier appearing on the left-hand sides of Eqs. (A1) – (A3) is the Reynolds number that governs the flow regime of vapor in the gap. For a translational velocity on the order of 10−1 m s−1, the corresponding Reynolds number is about 10−1. Fig. A3 compares the magnitudes of each term in Eqs. (A1) – (A4). It is evident that (a/h)4 » (a/h)2 » ρvaU/μv or unity except for r/a → 1 so that the momentum equations can be simplified to those forms given in Eqs. (2)–(4).

Fig. A3. Comparisons of the magnitudes of each terms in the momentum and the energy equations. U ∼100 mm s−1, h0 ∼ 10 μm, a ∼900 μm.

From Fig. A3, lubrication assumption is validated for a spherical Leidenfrost droplet smaller than 1 mm. Nevertheless, there also exists a lower bound in the droplet size below which the classical Leidenfrost lubrication regime breaks down and the droplet spontaneously lifts off the hot surface [39]. For a wall temperature between 416 °C and 816 °C, this critical radius of drop varies from 40 μm to 90.7 μm. In this study, only droplet with a radius between 200 μm and 900 μm is considered. Considering the energy equation, the last term on the right-hand side of Eq. (A4) clearly has larger order of magnitude than other terms. Hence, heat transfer is dominated by thermal conduction across the vapor film. Appendix C. Supplementary data Supplementary data related to this article can be found at http://dx.doi.org/10.1016/j.ijthermalsci.2018.02.033.

liquid droplets on a flat plate, Int J Heat Mass Tran 9 (1966) 1167–1188. [17] A.K. Sen, C.K. Law, On a slowly evaporating droplet near a hot plate, Int J Heat Mass Tran 27 (1984) 1418–1421. [18] T.K. Nguyen, C.T. Avedisian, Numerical solution for film evaporation of a spherical liquid droplet on an isothermal and adiabatic surface, Int J Heat Mass Tran 30 (1987) 1497–1509. [19] S. Zhang, G. Gogos, Film evaporation of a spherical droplet over a hot surface: fluid mechanics and heat/mass transfer analysis, J Fluid Mech 222 (1991) 543–563. [20] G. Bleiker, E. Specht, Film evaporation of drops of different shape above a horizontal plate, Int J Therm Sci 46 (2007) 835–841. [21] B. Sobac, A. Rednikov, S. Dorbolo, P. Colinet, Leidenfrost effect: accurate drop shape modeling and refined scaling laws, Phys Rev 90 (2014) 053011. [22] J.C. Burton, A.L. Sharpe, R.C.A. van der Veen, A. Franco, S.R. Nagel, Geometry of the vapor layer under a Leidenfrost drop, Phys Rev Lett 109 (2012) 074301. [23] M.T. Taylor, T. Qian, Numerical simulation of levitating liquid drops using the dynamic van der Waals theory, Comput Fluids 155 (2017) 76–88. [24] N. Karami, M.H. Rahimian, M. Farhadzadeh, Numerical simulation of droplet evaporation on a hot surface near Leidenfrost regime using multiphase lattice Boltzmann method, Appl Math Comput 312 (2017) 91–108. [25] A. Snezhko, E.B. Jacob, I.S. Aranson, Pulsating-gliding transition in the dynamics of levitating liquid nitrogen droplets, N J Phys 10 (2008) 043034. [26] A. Bouillant, T. Mouterde, P. Bourrianne, C. Clanet, D. Quéré, Leidenfrost wheels, 70th annual meeting of the APS division of fluid dynamics, Colorado, Denver, 2017, http://meetings.aps.org/link/BAPS.2017.DFD.E10.3. [27] H. Kim, B. Truong, J. Buongiorno, L.-W. Hu, On the effect of surface roughness height, wettability, and nanoporosity on Leidenfrost phenomena, Appl Phys Lett 98 (2011) 083121. [28] K.J. Baumeister, G.J. Schoessow, Creeping flow solution of Leidenfrost boiling with a moving surface, national aeronautics and space administration, NASA-TM-X52443, (1968). [29] W.H. Chang, The influence of patterned surface in Leidenfrost droplet motion and heat transfer, Master's Thesis, National Taiwan University of Science and Technology, Taipei, Taiwan, 2012. [30] Y. Pomeau, M. Le Merrer, F. Celestini, T. Frisch, The Leidenfrost effect: from quasispherical droplets to puddles, Compt Rendus Mec 340 (2012) 867–881.

References [1] J.G. Leidenfrost, On the fixation of water in diverse fire, Int J Heat Mass Tran 9 (1966) 1153–1166. [2] W.S. Bradfield, Liquid-solid contact in stable film boiling, Ind Eng Chem Fund 5 (1966) 200–204. [3] P.R. Chappidi, F.S. Gunnerson, K.O. Pasamehmetoglu, A simple forced convection film boiling model, Int Commun Heat Mass Tran 17 (1990) 259–270. [4] N. Nagai, S. Nishio, Leidenfrost temperature on an extremely smooth surface, Exp Therm Fluid Sci 12 (1996) 373–379. [5] H. Linke, B.J. Alemán, L.D. Melling, M.J. Taormina, M.J. Francis, C.C. DowHygelund, et al., Self-propelled Leidenfrost droplets, Phys Rev Lett 96 (2006) 154502. [6] G. Dupeux, M. Le Merrer, G. Lagubeau, C. Clanet, S. Hardt, D. Quéré, Viscous mechanism for Leidenfrost propulsion on a ratchet, Europhys Lett 96 (2011) 58001. [7] G. Lagubeau, M. Le Merrer, C. Clanet, D. Quéré, Leidenfrost on a ratchet, Nat Phys 7 (2011) 395–398. [8] Á.G. Marín, D.A. del Cerro, G.R.B.E. Römer, B. Pathiraj, A. Huis in 't Veld, D. Lohse, Capillary droplets on Leidenfrost micro-ratchets, Phys Fluids 24 (2012) 122001. [9] U. Alam, J. Krol, E. Specht, J. Schmidt, Enhancement and local regulation of metal quenching using atomized sprays, Quenching and Cooling 5 (2010) 91–103. [10] J.E. Eckhauses, R.D. Reitz, Modeling heat transfer to impinging fuel sprays in directinjection engines, Atomization Sprays 5 (1995) 213–242. [11] A.-L. Biance, C. Clanet, D. Quéré, Leidenfrost drops, Phys Fluids 15 (2003) 1632–1637. [12] G.E. Totten, C.E. Bates, N.A. Clinton, Handbook of quenchants and quenching Technology, ASM International, Materials Park, Ohio, 1992. [13] K.J. Baumeister, T.D. Hamill, Creeping flow solution of the Leidenfrost phenomenon, national aeronautics and space administration, NASA-TN-D-3133, (1965). [14] C.E. Brennen, Fundamentals of multiphase flow, Cambridge University Press, New York, 2005. [15] K.J. Baumeister, G.J. Schoeesow, Creeping flow solution of Leidenfrost boiling with a moving surface, NASA-TM-X-52443, (1968). [16] B.S. Gottfried, C.J. Lee, K.J. Bell, The Leidenfrost phenomenon: film boiling of

264

International Journal of Thermal Sciences 129 (2018) 254–265

Z.-H. Wu et al.

second ed., Taylor & Francis, New York, 2008. [35] W. Thielicke, E.J. Stamhuis, PIVlab - towards user-friendly, affordable and accurate digital particle image velocimetry in MATLAB, J Open Res Software 2 (2014) e30. [36] X. Ma, J.-J. Liétor-Santos, J.C. Burton, Star-shaped oscillations of Leidenfrost drops, Phys. Rev. Fluid 2 (2017) 031602 (R). [37] F. Celestini, T. Frisch, A. Cohen, C. Raufaste, L. Duchemin, Y. Pomeau, Two dimensional Leidenfrost droplets in a Hele-Shaw cell, Phys Fluids 26 (2014) 032103. [38] D. Quéré, Leidenfrost dynamics, Annu Rev Fluid Mech 45 (2013) 197–215. [39] F. Celestini, T. Frisch, Y. Pomeau, Take off of small Leidenfrost droplets, Phys Rev Lett 109 (2012) 034501.

[31] S.L. Dance, M.R. Maxey, Incorporation of lubrication effects into the force-coupling method for particulate two-phase flow, J Comput Phys 189 (2003) 212–238. [32] S. Nukiyama, The maximum and minimum values of the heat Q transmitted from metal to boiling water under atmospheric pressure, Int J Heat Mass Tran 9 (1966) 1419–1433. [33] L.S. Rothman, I.E. Girdibm, Y. Babikov, A. Barbe, D.C. Benner, P.F. Bernath, et al., The HITRAN2012 molecular spectroscopic database, J Quant Spectrosc Radiat Transf 130 (2013) 4–50. [34] V.P. Carey, Liquid-vapor phase-change phenomena: an introduction to the thermophyiscs of vaporization and condensation processes in heat transfer equipment,

265