Dynamic Leidenfrost temperature behaviors on uniformly distributed micropillars

Experimental Thermal and Fluid Science 111 (2020) 109954

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Dynamic Leidenfrost temperature behaviors on uniformly distributed micropillars

T

Dong Eok Kim School of Energy System Engineering, Chung-Ang University, Seoul, Republic of Korea

A R T I C LE I N FO

A B S T R A C T

Keywords: Dynamic Leidenfrost temperature Microstructured surface Dynamic pressure Capillary pressure Vapor hydraulic resistance

In this study, we conducted single droplet impinging experiments to measure dynamic Leidenfrost temperatures (TLF ) on surfaces with the uniformly distributed circular micropillars (diameter: 10 μm, height: 20 μm), where the pillar pitches varied from 15 to 1000 μm. Interestingly, the experimental results for two cases of Weber number showed that the TLF have the maximum values at a certain pillar pitch (i.e., 120 μm). In other words, TLF increases monotonically as the pitch increases to a certain value but decreases above that pitch. To explain quantitatively the experimental results, a model for the dynamic Leidenfrost phenomenon is presented with considerations of the wetting pressure exerted by droplet kinetics and capillary force, and the nonwetting pressure due to vapor flow between the droplet and the solid surface. In the model, TLF can be determined by the competing effects of the wetting and nonwetting pressures. According to the model, the wetting pressure decreases with an increase in the pillar pitch; however, it can reduce more largely the hydraulic resistance for the vapor flow, decreasing the nonwetting pressure and thereby increasing the TLF . For resolving the peak behavior of TLF , the concepts of the droplet contact radius and related time scale are introduced. Additionally, we conjecture that the droplet contact radius at the moment that the dynamic Leidenfrost phenomenon occurs increases with the pillar pitch, and it contributes to lengthen the vapor flow path and directly affects the nonwetting pressure. The present model procedure is technically supported by simple numerical simulations and theoretical bases, and it can explain appropriately the experimental TLF behavior.

1. Introduction In engineering applications that demand surface cooling for high heat flux and high temperature, such as thermal power plants, highly integrated electronic chip cooling devices, and compact heat exchangers, liquid-vapor phase-change heat transfer mechanisms involving nucleate boiling and spray cooling have been broadly adopted as reliable solutions. Among these mechanisms, the direct injection of small-sized droplets in subcooled state (i.e., spray cooling) is the most effective and prompt technique for the heat removal from a largely superheated surface [1,2]. In particular, spray cooling technique can simultaneously provide various heat transfer mechanisms on a hot surface, such as nucleate boiling, thermal conduction to liquid, and strong evaporation at the liquid-vapor interface, with a large area of heat transfer [3,4]. However, beyond a certain superheating level of the surface, the liquid droplets cannot make contact with the heated surface because of the formation of a thin vapor layer between the liquid and surface, which is generated by strong evaporation of the liquid at the droplet bottom. This is called the dynamic Leidenfrost phenomenon [5–7]. Above the dynamic Leidenfrost temperature (TLF ), the film

boiling mode governed by thermal conduction through the vapor layer is triggered, and the heat transfer from the surface to the droplet thus drastically deteriorates. The quantitative prediction of the TLF is therefore an important issue in the design and analysis of cooling systems using droplet impingement [8,9]. The dynamic Leidenfrost phenomenon can be distinguished from the static Leidenfrost phenomenon [10] by a nonzero droplet velocity (or nonzero kinetic energy). The static Leidenfrost temperature has been analytically calculated using a simple equation that balances the weight of droplet (gravity force) and the pressure force due to the thin film vapor flow generated by evaporation at the liquid-vapor interface [11–13]. In those studies, the order of magnitude of the vapor film thickness ranged from 10 to 100 μm. However, for the dynamic Leidenfrost phenomenon on a flat surface, the force (pressure) components affecting the droplet impact behavior become more complicated owing to the nonzero droplet kinetic energy. One pressure is the dynamic pressure of the droplet, written as PD = ρl vd2/2 , where ρl and vd are respectively the liquid density and droplet impact velocity. Another pressure may be exerted by transient behavior of the droplet. This pressure (PH ) originates from the abrupt stopping of fluid particles

E-mail address: [email protected]. https://doi.org/10.1016/j.expthermflusci.2019.109954 Received 8 February 2019; Received in revised form 15 September 2019; Accepted 15 October 2019 Available online 18 October 2019 0894-1777/ © 2019 Elsevier Inc. All rights reserved.

Experimental Thermal and Fluid Science 111 (2020) 109954

D.E. Kim

splashing, were investigated in the systematic manner. Generally, the existence of the micropillar arrays on a heated surface affects the nonwetting pressure by modifying the heat conduction and the vapor flow structures within the thin vapor layer; e.g. under the approximation of uniform vapor film thickness, an increase in the micropillar spacing reduces the conductive heat transfer area of the heating surface, and, moreover, a sparser post array reduces the flow resistance of the vapor [8,9,14,22]. These effects contribute to the increase in the Leidenfrost temperature lowering the pressure buildup of the vapor flow. Some studies [8,9] argued that the micropillar arrangements affect the transient pressure PH . The wettability effect on the Leidenfrost phenomenon has been considered by a few studies [14,23]. These studies postulated that at the early stage of the droplet impact on the heated surface, the droplet base makes slight contact with the top regions of the micropillars, and the hemispherically shaped (concave) menisci form through rapid evaporation at the liquid-vapor interface. At the triple contact line of the liquid, vapor, and micropillar side surface, a downward surface tension force (capillary pressure, PC ) is generated by the wetting nature, and it can be added to the wetting pressure (Pw = PD + PH + PC ). This work quantitatively explains TLF behaviors on surfaces with uniformly distributed micropillars, where the pillar spacings (about 5–1000 μm) are precisely controlled with the same diameter and height. According to theoretical considerations and numerical analysis results obtained with the help of commercial computational fluid dynamics software [26], the effects of the pillar spacing on the wetting and nonwetting pressures were quantified. In particular, in our experiments, as the pillar spacing increased, TLF did not increase monotonically as reported in previous literatures. The experimental results show that there is an optimal pillar spacing maximizing the TLF , and a further increase in the spacing reduces the TLF . These interesting experimental finding can be explained by considering the competing effects of the pillar spacing on wetting and nonwetting pressures.

within the droplet over a short time scale (< 1 < ms). These two pressure components drive the droplet to be in contact with the surface, and are thus referred to as the wetting pressure (Pw = PD + PH ). Meanwhile, as the droplet base approaches the thermal boundary layer of the heated surface, strong evaporation from the liquid-vapor interface occurs by conductive heat transfer. The vapor flows experience a viscous resistance within the thin film vapor layer, resulting in the buildup of the vapor pressure. The pressure (PV ) can push the droplet upward, and is thus referred to as nonwetting pressure (Pnonw = PV ). Several studies [8,9] quantitatively evaluated the TLF by considering the balance between the wetting and nonwetting pressures (PD + PH ≈ PV ). Several experimental studies of the Leidenfrost phenomenon on various micro-textured surfaces have recently been reported [8,9,14–22]. Among those studies, the most recent ones [8,9,14,22], in which the micropillars on the heating surface were precisely controlled with well-defined geometries and dimensions with the help of a microfabrication technique, found that the Leidenfrost temperature increases with the spacing between posts having the shape of a circular or rectangular pillar. However, different reasons based on disparate physical mechanisms were given to explain the results. In Jerng and Kim’s study [8], they argued that the wetting pressure, particularly in the transient pressure PH , on a micro-textured surface might be reduced compared with that of the smooth surface due to acoustic wave absorption into the additional vapor region formed with the micropillars. The existence of the structures strongly affects the nonwetting pressure due to modifying the hydraulic resistance for the vapor flow and the thermal conduction heat transport from the surface to droplet. According to them [8], the dynamic Leidenfrost on a surface might be determined by the interactions between the two mechanisms. Li et al. [9] also considered the balance of the wetting and nonwetting pressures to predict the dynamic Leidenfrsot temperature on textured surfaces. In their study, however, the PH was modeled with the capillary pressure generated by the wetting nature between the droplet and surface structures. The capillary pressure effect had been treated as an important parameter to explain the triggering mechanism of dynamic Leidenfrost temperature on textured surfaces in the works of Adera et al. [14] and Kwon et al. [23]. Tran et al.’s study [22], interestingly, reported significant reduction of the dynamic Leidenfrost temperatures on textured surfaces compared with that on the smooth surface. For reasoning the results, they suggested that the enhancement of heat transfer area due to the surface textures might increase the vapor mass flow rate generated by conductive heat transfer, i.e. the increase of nonwetting pressure. Recently, Liang and Mudawar [24,25] had intensively reviewed the droplet impact behaviors on solid and liquid surfaces. In those studies, the relevant phenomena postulated by the droplet impact, such as dynamic Leidenfrost point, spreading, and

2. Sample preparations and experiments 2.1. Sample fabrications and characterizations The micropillar structures were fabricated on polished silicon wafer surfaces (thickness: 525 μ m ± 25 μm ) by using DRIE (Deep Reactive Ion Etching) process. Seven types of micropillar sample with different pitches (Lp ) between pillars (i.e., different spacings (Sp)) were prepared, and the diameters (Dp) and heights (Hp) of the pillars were precisely controlled to be uniform; i.e., Dp = 10 μm and Hp = 20 μm (see Fig. 1). Additionally, a smooth surface without pillars was prepared as a reference sample. All samples were given a final coating of a thin SiO2

Fig. 1. Scanning electron microscope (SEM) images of a test sample (P30 sample). 2

Experimental Thermal and Fluid Science 111 (2020) 109954

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length, Lc = (σl/ ρl g )1/2 where σl , ρl , and g are the liquid surface tension, liquid density and gravitational acceleration, was calculated as 2.73 mm, and the droplet shape was assumed to be approximately spherical. To vary the droplet impact velocity (vd ), (where the Weber number We = ρl D0 vd2/ σl , with D0 being the droplet diameter immediately before impact), two different heights of the needle tip were chosen. For the two cases, the average droplet velocities were 0.18 m/s and 0.44 m/s, and the corresponding Weber numbers were 1.35 and 8.07 according to the properties of saturated water liquid at 1 atm. The droplet impact behaviors on the sample surfaces were recorded using a high-speed CCD camera (M320, Phantom) set to a spatial resolution 1,280,800× pixels and frame rate of 3200 frames per second. The TLF was determined to be the temperature at which the disturbance of the droplet shape due to interplay with the surface, such as boiling and nucleation, completely disappears and the droplet smoothly spreads on the surface and then bounce off gently in visualization images (Fig. 4(a) and (b)), i.e. at Ts = TLF , the contact between the liquid and solid exist no longer by strongly evaporated vapor from the droplet, and it denotes the inception of film boiling regime.

Table 1 Experimental samples and geometrical dimensions. Samples

Pillar diameter Dp (μm)

Pillar height Hp (μm)

Pillar pitch Lp (μm)

Pillar spacing Sp (μm)

Porosityε

Smooth P15 P30 P60 P120 P300 P500 P1000

– 10 10 10 10 10 10 10

– 20 20 20 20 20 20 20

– 15 30 60 120 300 500 1000

– 5 20 50 110 290 490 990

1.0 0.6509 0.9127 0.9782 0.9945 0.9991 0.9997 0.9999

layer having thickness of 5000 Å. The pillar pitches Lp on the samples broadly ranged from 15 μm to 1 mm (Sp = 5~990 μm ), and the porosity of the samples ε , defined as ε = 1 − πDp2 /4Lp2 within a square unit cell surrounded by the pillars (Fig. 1), was changed from about 0.65 to 0.9999 (see Table 1). To estimate the wetting character of the SiO2 surface, we measured the contact angle of a sessile water droplet on the smooth sample surface cleaned by rinsing with acetone, ethanol and de-ionized water in that order. Both static and dynamic (receding and advancing) contact angles were measured, with the static contact angle (θs ) of the smooth SiO2 surface being 70.6°, and the receding (θr ) and the advancing (θa ) contact angles respectively being 56.9° and 81.6° (Fig. 2).

3. Results and discussion To observe the interaction between the impact droplet and solid surface for various surface temperatures, experiments were conducted with Ts ranging from 155.7 °C to TLF . Analysis of high-speed visualization images for the droplet impact behavior indicated that the dropletsolid interaction can be classified into three modes depending on the surface temperature; i.e. complete-contact mode, partial-contact mode, and Leidenfrost droplet mode. In the complete-contact mode, a falling droplet initially wets the surface resulting in complete contact between the droplet bottom and the surface, and the droplet spreads on the surface finally boils with complicated liquid interfacial motion (Fig. 5(a)). As the surface temperature increases, the interfacial motion due to boiling becomes faster and more rigorous. In this mode, the droplet does not bounce off from the surface. The complete-contact mode can thus be regarded as a regime for which Pw > Pnonw . The impact droplet has different behavior in the partial-contact mode than in the complete-contact mode. The impact droplet in the partial-contact mode is interrupted to wet the surface through strong nucleation and evaporation immediately after its initial contact (Fig. 5(b)). And, the interface between the droplet and vapor irregularly fluctuates and with the impact, the liquid at the bottom of the droplet makes intermittent contact with the surface. After maximum spreading, the droplet is lifted above the surface, and vapor bubbles of various size formed by nucleation at droplet-solid interface are observed inside the droplet. In this mode, the relationship between the wetting and nonwetting pressures is still considered to be Pw > Pnonw with the existence of bubble nucleation as clear evidence of liquid-to-solid surface contact. In the Leidenfrost droplet mode, the impact droplet undergoes smooth deformation to the maximum spreading condition, and it bounces off from the surface without any contact behavior, such as vapor–liquid interface fluctuation or bubble nucleation (Fig. 5(c)). It is considered that in the Leidenfrost droplet mode, the downward wetting pressure (or force) exerted by the droplet is not enough to overcome the upward nonwetting pressure generated by strong evaporative vapor flow from the bottom surface of the liquid; i.e., Pw ⩽ Pnonw . According to the recent studies by Liang et al. [27,28], the droplet impact behaviors could be more sophisticatedly classified into four modes depending on the surface temperature and droplet Weber number; i.e. nucleate boiling, explosive detachment, explosive rebound, and reflection rebound modes. Phenomenologically, the complete contact and Leidenfrost droplet modes in present study might correspond with the nucleate boiling and reflection rebound modes, respectively. And, the explosive detachment and rebound modes in Liang et al. [28] might be equivalent to the partial-contact mode. However, they clearly indicated that the contact between droplet and surface might occur at the initial stage even in the

2.2. Experimental method A precision hot plate (PC-420D, Corning) with a reading accuracy of ± 2% was used to heat the samples to the Leidenfrost temperature. Considering the temperature difference between the position of the hot plate sensor and the sample top surface due to thermal conduction, we calibrated the temperature given by the hot plate sensor using the real temperature of the top surface of a smooth sample measured by a surface temperature probe with a reading accuracy of ± 1.2% (80PK27, FLUKE). It was found that, though the hot plate indication temperature had good linearity with the real surface temperature (Ts ), the former was higher than the latter by up to 17% in the Ts temperature range of about 40–410 °C. The surface temperature in each experiment case was thus calculated from the linear fitting relation (see Fig. 3(a)). The de-ionized water droplet was generated from a 25 gauge microneedle (OD: 0.514 mm, ID: 0.260 mm) connected to a precision syringe pump (NE-1000, New Era Pump Systems, Inc.). The droplet with an average diameter of 2.56 mm impinged under gravity onto the sample surface (Fig. 3(b)). For an impinged droplet, the capillary

Fig. 2. Static and dynamic contact angles of a water droplet on the smooth sample surface. 3

Experimental Thermal and Fluid Science 111 (2020) 109954

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Fig. 3. Experimental setup: (a) Sample surface temperature calibration curve; (b) Schematic diagram of the experiment.

the dimensional arrangement of the micropillars. Postulating a hypothetical condition in which the vapor-liquid-solid triple contact line is stuck to the pillar top-side surface near the TLF point, the liquid-vapor interface is deformed as a concave shape through strong evaporation at the bottom of the droplet (Fig. 7). A receding contact angle θr thus forms at the triple contact line and the surface tension force in turn pulls down the liquid interface. The capillary pressure acting on the liquid–vapor interface in a unit cell (see the inset of Fig. 7) of the micropillar structured surface can be expressed as

reflection rebound mode, i.e. in the view from [28], the Leidenfrost mode could be regarded as belonging into the partial-contact mode. In this study, however, for the sake of clarity and simplicity, the Leidenfrost mode is defined to a mode where there may be no direct contact between the droplet and surface or it may be negligible even if there is momentary contact because of its very short time scale. The dynamic Leidenfrost temperature which points out the initiation of the Leidenfrost droplet mode varies with the value of Lp . The TLF increased monotonically as the pillar spacing increased from 15 to 120 μm. However, a further increase in the pillar spacing reduced the TLF (Fig. 6(a)). In other words, there is an optimized pillar spacing for which TLF is a maximum. To our knowledge, these results have not been reported previously. The increase in the pillar spacing increased the temperature of the complete-contact mode to the partial-contact mode transition (TC − P ), and the region of the complete-contact mode thereby broadened with an increase in the pillar spacing. After the maximum TLF on the P120 sample, the region of the partial-contact mode became narrower through a decrease in TLF and increase in TC − P (Fig. 6(b)). Qualitatively, the unique behavior of TLF on the micropillar samples can be explained through the relationship between Pw and Pnonw . Among the components of Pw , PD does not change for the same droplet impact velocity vd . However, the PH and PC can be varied with

PC = πDp σl cos θr / εAUC

(1)

where AUC is the cross-sectional area of the unit cell; AUC = Lp2 . Eq. (1) shows that the capillary pressure decreases as the pillar spacing increases. In this study, PH is defined as the pressure induced by rapid deceleration of liquid inside a droplet during short time scale at early impact stage. In previous studies [8,9,29,30], general expression for the PH considers the effect of liquid compressibility with the term of speed of sound (Cs, l ), i.e., PH ~ρl vd Cs, l . However, the onset of compressibility effect requires very short time scale τ ~L/ Cs, l , where L is a relevant length scale. With L = D0 , τ is calculated to about 1.7 × 10−6 s for the droplet used in this study. This time scale may be much smaller than that for 4

Experimental Thermal and Fluid Science 111 (2020) 109954

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Fig. 4. Droplet behavior on the heated surface: (a) Ts < TLF ; (b) Ts ⩾ TLF .

i.e. L/ τ ≈ Cs, l . According to the scaling given by [31,32], we define PH as PH = αH ρl vd2 , where αH is an adjustable constant varying with the surface structure. However, to measure or quantify analytically the value of αH on a microstructured surfaces is challenging. Dash et al. [33] proposed a simple empirical correlation for the PH on the microstructured surfaces as PH = (2.54 × 10−7PC + 7.53 × 10−4) ρl vd Cs, l . Based on the PH values obtained from this correlation, αH can be linearly fitted with the porosity of the textured surfaces ε (inset of Fig. 8). It indicates that αH (i.e., PH ) decreases as the pillar spacing increases. As seen in Fig. 8, at smaller pillar spacing Sp = 5 and 20μm (corresponding to P15 and P30 samples), PC has the highest values among the wetting pressure components; however, PC becomes lower than PH from Sp = 50 μm (corresponding to the P60 sample) and decreases to about ~ 1 Pa at Sp = 990 μm while PH converges to a constant value of about ~ 487 Pa. The total wetting pressure Pw = PC + PH + PD largely decreases to Sp = 110 μm , and the slope of the decrement reduces rapidly. The nonwetting pressure Pnonw depends entirely on fluid dynamics of the vapor flow generated from the liquid–vapor interface. The pillar structures obviously act as hydraulic resistance to the flow and, intuitively, Pnonw increase largely as the pillar spacing decreases. To quantify the pressure acting on the microstructured surface, porous media analysis has been in previous studies [14,34–36]. The essential part of the analysis is to find an analytical or empirical relationship between the permeability based on Darcy’s law (Eq. (2)) and the geometry of the microstructures:

the deceleration of liquid drop, and little effect of the liquid compressibility is therefore expected. According to Roisman et al. [31] and Eggers et al. [32], the PH scales to PH = 1.7ρl vd2 e−3.1t ~ρl vd2 e−τ and PH ~ρl vd2 τ / t = ρl vd2 , where t is the time after impact, and t = τ is applied. From their numerical results [31,32], the order of magnitude of PH is similar with the dynamic pressure PD ~ρl vd2 . Assuming one-dimensional inertia-dominated (negligible viscous friction) liquid flow inside a droplet immediately after impact, we can write the NavierStokes equation as ρl ∂u/ ∂t + ρl u∂u/ ∂x ≈ −∂p / ∂x , where ρl , u , x , and p are the liquid density, fluid velocity, space coordinate, and pressure, respectively. The gravity effect is neglected. Equating the second term (convective term) on the left-side of the equation with the pressure gradient term on the right gives a pressure scaling, p~ρl vd2/2 . This pressure scale is consistent with the definition of dynamic pressure (PD ). Meanwhile, through postulating the balance between the first term (unsteady term) on the left-side and the pressure gradient term, we can obtain another pressure scaling as p~ρl vd L/ τ . Physically, the time scale τ can be evaluated as a time interval from droplet impact to its abrupt stop by rapid deceleration. From our high-speed visualization, the time scale was roughly estimated into 10-5 ~ 10-4 s. And, the length scale L might be scaled to the droplet diameter. Although the time scale τ is larger than 10-6 s calculated above, the pressure scale shows that the droplet can generate significantly larger pressure impact than the dynamic pressure. If the droplet velocity is very high so that the time scale for the deceleration is quite small, the magnitude of resulting pressure might approach to well-known water hammer pressure, PH = ρl vd Cs, l ,

Fig. 5. Classification of droplet-solid interaction modes: (a) Complete-contact mode; (b) Partial-contact mode; (c) Leidenfrost droplet mode. 5

Experimental Thermal and Fluid Science 111 (2020) 109954

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Fig. 6. Experimental results for various micropillar surfaces: (a) Leidenfrost temperature distributions with variation of the pillar pitch at two We cases; (b) Regime map of droplet-solid interaction modes for the test surfaces at We = 8.07.

−

μ dPV = v U¯v, r dr K

(2)

where r , μ v , K , and U¯v, r are the radial coordinate, dynamic viscosity of vapor, permeability of the porous media, and volumetric averaged radial velocity of vapor flow. Although the pressure gradient under the droplet can be obtained using Eq. (2), the flow length of the vapor should be known in calculating the quantitative pressure distribution. It can be defined to the characteristic length scale R c , called the contact radius (Fig. 7), which refers to the radius of the circular plane formed by the contact of the droplet and surface near the TLF point. Quantified prediction of the TLF on the micropillar structured surfaces therefore requires suitable physical models for each pressure component constituting the wetting and nonwetting pressures. Here, we present the model procedure and analysis results. As a governing relation for prediction of the TLF on the micropillar surface, we use the balance of the wetting and nonwetting pressures, Pw = Pnonw . Among the components of Pw , PC is defined by Eq. (1), and PH is expressed as Eq. (3). Also, the droplet dynamic pressure PD is expressed as PD = ρl vd2/2 .

PH = αH ρl vd2

Fig. 8. Wetting pressure behaviors with a variation in pillar spacing.

pressure PV ), we should consider the fluid (vapor) momentum transport inside the micropillar structures. According to the principle of energy conservation, the area-averaged vapor velocity downward from the liquid interface (z = Hp ) at the bottom of the droplet (see Fig. 7) can be

(3)

where αH ≈ 20.29 − 17.72ε . To obtain a suitable expression for Pnonw (i.e., built-up vapor

Fig. 7. Schematic diagram of the relationship between droplet, vapor flow, and solid surface near the TLF point.

6

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plates. A new approach for calculating the permeability on the micropillar structures is proposed here to avoid the limitation of Eq. (8). The method is based on not the Brinkman equation but the original Darcy’s law Eq. (2). From Eq. (2), the total effective permeability for the vapor flow in a unit cell KUC can be written as

written as

Uv¯, z =

qUC ¯

QUC = ρv hlv AUC ρv hlv

(4)

where QUC , ρv , hlv , and q¯ UC are the heat transfer rate from the solid surface to the liquid interface in a unit cell, vapor density, latent heat for liquid-to-vapor phase change, and average heat transfer rate per unit area. By continuity consideration for the vapor flow (Eq. (5)), the volumetric averaged radial vapor velocity at radial position r can be derived by substituting Eq. (4)–(6). In Eq. (5), the z-directional velocity component is neglected by the thin film vapor flow assumption based on Hp = 20 μm .

∫0

Hp

u v, r (z )(2πr ) dz=πr 2Uv¯, z

KUC =

∫0

Hp

u v, r (z ) dz =

(5)

¯ qUC 2Hp ρv hlv

r

(6)

Next, a proper momentum equation is established and solved for calculation of the radial pressure distribution inside the vapor flow region. Assuming that a region of vapor flow through several micropillars can be simulated to a porous medium, Darcy’s law (Eq. (2)) might be applied. However, because Darcy’s law was originally based on the infinite arrangement of dense small particles and pores, the simple application of the law to the circumstances of this analysis where the pillar structures are confined by top (i.e., liquid interface) and bottom (i.e., base surface of the pillar) walls is not appropriate. In other words, viscous shear losses due to velocity gradients induced by both pillar side walls and the top and bottom confinement (z-direction) should be simultaneously considered. For this purpose, the Brinkman equation (Eq. (7)) [37] has been applied to the confined micropillar arrays in previous studies [14,38]:

−

μ d 2u v, r μ dPV (r ) = v u v, r − v ε dz 2 dr K

KUC ≈

∫0

Hp

γp (1 − γp) Kpillar KTBW γp Kpillar + (1 − γp) KTBW

=

γp (1 − γp) Kpillar Hp2/12 γp Kpillar + (1 − γp) Hp2/12

(10)

where γp = Dp / Lp is the diameter-to-pitch ratio of the pillar. According to this equation, as γp approaches to zero (i.e., ε → 1), KUC approaches KTBW . In Fig. 10(b), the results of Eq. (10) are compared with KUC values obtained by numerical analysis. The numerical results are best fitted with the model of Van der Westhuizen and Du Plessis [46], Kpillar = πDp2 ε (1 − 1 − ε )2 /96(1 − ε )1.5 , which is used for the present model procedure. Next, from Eqs. (2) and (6), we obtain an explicit formulation for the vapor pressure gradient in the r-direction as

(7)

Adera et al. [14] presented a solution (Eq. (8)) for U¯v, r from Eq. (7) by applying the no-slip boundary condition at the pillar top and bottom walls (i.e., u v, r (0) = 0 and u v, r (Hp) = 0 ) and then integrating the solution of u v, r from 0 to Hp :

1 U¯v, r = Hp

(9)

ΔPV AC

where Q v , ΔPV , and AC are the vapor volumetric flow rate, vapor pressure loss across the unit cell, and cross-sectional area for the vapor flow. As mentioned above, KUC should be determined by considering the velocity gradient effects in both y- and z- directions inside the unit cell. We newly define the permeabilityKpillar , which is contributed only by the y-direction velocity gradient, and the permeability KTBW , which is contributed by the z-direction velocity gradient induced by top and bottom walls (see Fig. 10(a)). Using Eq. (9), KTBW can be analytically calculated as KTBW = Hp2/12 , and the various prediction models for Kpillar might be selected from the existing literature (Table 2) [34,40–46]. KUC With the permeabilities, can be given as 1/ Lp KUC ≈ 1/ Lpillar Kpillar + 1/ LTBW KTBW , where Lpillar and LTBW are the length scales respectively related to Kpillar and KTBW . Using simple approximation Lpillar ≈ Dp and LTBW ≈ Sp , the final relation for KUC is obtained as

where u v, r (z ) is the local radial vapor velocity at z position.

1 U¯v, r = Hp

μ v Q v Lp

μ v qUC ¯ dPV = r dr 2KUC Hp hlv ρv

−

(11)

By integrating Eq. (11) as seen in Eq. (12), the gauge pressure of the vapor at position r can be calculated as Eq. (13).

tanh[(Hp ε /K )/2] ⎞ K dP ⎛ u v, r (z ) dz = ⎜⎛ ⎟⎞ ⎛ V ⎞ ⎜−1 + ⎟ (Hp ε / K )/2 ⎝ μ v ⎠ ⎝ dr ⎠ ⎝ ⎠

∫P

Pv (r )

atm

∫R

dPV = −

r

¯ μ v qUC

c

2KUC Hp hlv ρv

(8) To verify the applicability of the solution, Eq. (8), to the porosity ε range in this study, we conducted numerical simulations by using ANSYS FLUENT 16.1 [26]. The calculation domain for the numerical analysis contained the vapor flow region and the surrounding walls consisting of connected single unit cells repeated in the flow direction, where the total length of flow region was adequately determined considering the flow region structure (see Fig. 9). For inlet and outlet surfaces, the boundary conditions were respectively set to the mass flow inlet and pressure outlet (0 Pa) and the curved pillar side surfaces and the top and bottom surfaces to no-slip wall conditions. A symmetry condition was applied to the side walls of the domain. The radial pressure gradient dPV / dr was obtained from the numerical solution and substituted into Eq. (8). Eq. (8) was numerically solved using Wolfram Mathematica 10.0 [39] to find the solution of permeability K. As a result, the Eq. (8) is not valid for cases of higher ε . For the cases of P15, P30, P60, and P120, K values were respectively calculated as 1.31 × 10−12 , 2.55 × 10−11, 1.48 × 10−10 , and 7.89 × 10−10 m2. However, for the cases of P300, P500, P1000 in which the porosity is close to unity, Eq. (8) had no solutions. In fact, for those cases, the permeability might be similar to that for laminar flow between two infinite parallel

PV (r ) − Patm =

μ v qUC ¯ 4KUC Hp hlv ρv

rdr

(12)

(Rc2 − r 2)

(13)

where Patm is the atmospheric pressure. Thus, the nonwetting pressure Pnonw = PV can be obtained by averaging the expression in Eq. (13) from r = 0 to r = R c :

Pnonw = PV =

1 Rc

∫0

Rc

[PV (r ) − Patm] dr =

μ v qUC ¯ Rc2 6KUC Hp hlv ρv

(14)

q¯ UC can be evaluated by considering the thermal conduction resistance between the solid surface and liquid-vapor interface. As shown in Fig. 11(a), the thermal resistance consists of two components arranged in parallel; one is the resistance from the pillar bottom and side surfaces acting on the interface through the vapor region (Rth, v ) while the other is the resistance from pillar top surfaces acting on the interface through the liquid region (Rth, l ). Then, q¯ UC can be written as

¯ = qUC

QUC 1 ⎛ ΔTs ΔTs ⎞ = + ⎜ ⎟ AUC AUC ⎝ Rth, v Rth, l ⎠

where ΔTs is the wall superheat. 7

(15)

Experimental Thermal and Fluid Science 111 (2020) 109954

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Fig. 9. Numerical analysis domain and boundary conditions for vapor flow simulation inside micropillar structures.

seen in Eq. (16),

Because it is necessary to solve the three-dimensional heat diffusion equation in calculating the thermal resistances, we conducted numerical analysis using ANSYS FLUENT 16.1 [26]. For Rth, v , the vapor region inside the unit cell was considered as the calculation domain (Fig. 11(b)). The boundary conditions of the pillar bottom and side walls, and the liquid-vapor interface were set to a constant wall temperature condition while the side walls set to a symmetry condition. The liquid-vapor interface temperature was fixed to the saturation temperature at 1 atm. (100 °C) while the temperature of the solid walls was appropriately controlled in an approximate range of 200–500 °C. For Rth, l , a liquid region on the unit cell was established for the calculation domain, and the thickness of the liquid region was set uniformly to 1 mm. The geometry and applied boundary conditions are shown in Fig. 11(c). The temperatures of the pillar top walls were changed in the range of 200–500 °C while the temperature at the liquid top surface was set to the saturation temperature, 100 °C. In particular, convection heat transfer was set as a boundary condition of the liquid-vapor interface, and the interfacial evaporative heat transfer coefficient (hi, ev = 230 kW/m2K ) based on the kinetic theory of gases was used for the heat transfer coefficient [12,14]. In the analyses, the conduction resistances were obtained by dividing the wall superheat by the total heat transfer rate to the liquid-vapor interface. To find a suitable relation for Rth, v , it can be expressed as Rth, v = keff εAUC / Hp according to the definition of thermal conduction resistance, where keff is the effective thermal conductivity of the vapor region including additional heat transfer from the pillar side walls. And, keff / k v might have a linear relationship with the ratio of the enlarged heat transfer area due to the pillar side walls to the pillar bottom area as

keff kv

= aF + 1

(16)

where F = πDp Hp/(Lp2 − πDp2 /4) The numerical results are best fitted by Eq. (16) when a = 2.775 (see Fig. 12(a)). Next, the numerical results for Rth, l are fitted to a polynomial function of the porosity as in Fig. 12(b) and Eq. (17).

1 = −5.6789 × 10−6ε 2 + 2.404 × 10−5ε Rth, l

(17)

As a result, we obtain a relation for q¯ UC as

εk ⎛ 2.775πDp Hp ⎡ 10−6 × ε (−5.6789ε + 24.04) ⎞⎤ + v⎜ 2 + 1⎟ ⎥ ΔTs qUC ¯ =⎢ Hp ⎝ Lp − πDp2 /4 Lp2 ⎠⎦ ⎣ ΔTs = ′ R′ th, UC (18) where Rth′ ′, UC is the total thermal conduction resistance per unit area in the vapor region in the unit cell. Thus, from Eqs. (1), (3), (14), (18), and the definition of PD , we can establish an equation for the balance between Pw and the Pnonw ,

μ v Rc2 (TLF − Tsat ) 6KUC Hp hlv ρv R′ ′th, UC

=

πDp σl cos θr εLp2

+ (αH +

1 ) ρ vd2 2 l

(19)

where Tsat is the saturation temperature. Although the TLF can be explicitly solved using Eq. (19), the contact 8

Experimental Thermal and Fluid Science 111 (2020) 109954

D.E. Kim

Fig. 10. Comparison of Darcy permeability between the numerical results and the approximate relation of Eq. (10).

liquid at the droplet bottom and the solid surface with the microstructures (pillars) form the contact radius R c ; otherwise, the liquid can penetrate the space between the pillars and wet the bottom surface of the pillars through relatively higher wetting pressure (i.e., the partialcontact mode). However, little of the literature attempting TLF modeling has focused on the length scale. Although Kwon et al. [23] mentioned the importance of R c , which they called the contact patch length, they used an approximated constant value (~60 μm) derived from

radius R c is unknown. R c would strongly affect to the nonwetting vapor pressure Pnonw (i.e., the left-hand side of Eq. (19)) because it corresponds to the total length of the vapor flow passage between the vapor–liquid interface and solid surface. Phenomenologically, R c can be referred to a critical length scale at which the dynamic Leidenfrost condition emerges on a microstructured surface. In other words, the dynamic Leidenfrost condition will occur if the balance between the wetting and nonwetting pressures (i.e., the equality of Eq. (19)) can be established when the 9

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Table 2 Darcy permeability models from the literature. Author Happel [40] Kuwabara [41] Sangani and Acrivos [42] Drummond and Tahir [43] Tomadakis and Sotirchos [44] Kaviany [45] Van der Westhuizen and Du Plessis [46]

Model expression

Kpillar = Kpillar = Kpillar =

Kpillar = Kpillar = Kpillar =

Kpillar =

Gunda et al. [34]

Kpillar =

ρl hlv Sp3 Rth" , UC ΔTs Lp2

− ε) +

(1 − ε )2 − 1 ⎤ (1 − ε )2 + 1

⎦

Dp2 [−ln(1 − ε ) − 1.5 + 2(1 − ε )] 32(1 − ε ) Dp2 [−ln(1 − ε ) − 1.476 + 2(1 − ε ) − 1.774(1 − ε )2 + 32(1 − ε ) 2 Dp 2(1 − ε ) − 0.796(1 − ε )2 ⎤ ⎡−ln(1 − ε ) − 1.473 + 32(1 − ε ) ⎣ 1 + 0.489(1 − ε ) − 1.605(1 − ε )2 ⎦

4.076(1 − ε )3]

Dp2 ε (ε − 0.33)2.707 ⎤ ⎡ 8(ln ε )2 ⎣ 0.670.707 [1.707ε − 0.33]2 ⎦ 0.0606πDp2 ε5.1 4(1 − ε )

πDp2 ε (1 − 1 − ε )2 96(1 − ε )1.5 π 0.16Dp2 ⎡ ⎢ 4(1 − ε ) − ⎣

π +3− 4(1 − ε ) ε 0.5

4(1 − ε ) ⎤ ⎥ π ⎦

hand side term of Eq. (19). However, for the final prediction of TLF , the effects of the vapor permeability KUC and thermal conduction resistance Rth′ ′, UC in the denominator on the left-hand side should be considered, where one can know that those values would obviously decrease as Dp increases. Therefore, the geometry of the surface textures may give significant effects to TLF on the surface in a coupled manner. The quantitative validation of the present model for various surface geometries is recommended as a future work.

experimental results. However, R c might depend on the geometry of the microstructures and the thermal state of the solid surface (temperature). R c values in our experiments are calculated using Eq. (19) and the experimental TLF data, and Fig. 13(a) clearly shows that R c increases with the pillar spacing Sp . To find a suitable relation for R c , it is assumed that R c is governed by a power law, R c ~τcn where τc is a time scale from droplet impact to Leidenfrost phenomenon initiation, and n is a positive constant smaller than 1. The power law relation between the droplet spreading diameter and time had been verified by several studies, and from theoretical considerations, the value of n is predicted to be 1/7 [47–49]. Therefore, in defining R c , the determination of an appropriate time scale would be essential. Here, we hypothesize that the time scale τc can be regarded as the time period from the droplet impact to the moment that the triple contact line attached to the pillar side surfaces recedes under the effect of the evaporative reaction force at the liquid-vapor interface. At the time of the droplet impact, an advancing contact angle forms at the triple contact line, and the interface then becomes more concave as the liquid evaporates from the interface, and the contact angle eventually approaches the receding contact angle. Therefore, τc can be defined as the time at which the liquid with a volume (ΔVl ) equivalent to the concave region evaporates completely (see Fig. 13(b)); i.e. τc ~ρl ΔVl hlv / Qs , where Qs is the heat transfer rate from the surface to the liquid. Through considerations of the micropillar geometry and the heat transfer from the structured surface, a simple relation for τc can be derived as

τc

Dp2 ⎡−ln(1 32(1 − ε ) ⎣

4. Conclusions According to our model, the wetting pressure induced by the droplet transient behavior (deceleration), droplet inertia (dynamic pressure) and capillary effect decreases with an increase in the pillar spacing. However, the increase in pillar spacing reduces exponentially the flow resistance per unit length of the vapor while the permeability considerably increases, reducing the nonwetting vapor pressure. The competing effect of the wetting and nonwetting pressures can enhance the dynamic Leidenfrost temperature (TLF ) by increasing the pillar spacing. However, the experimental TLF data showed a maximum value at a certain pillar spacing (Lp = 120 μm ), and the TLF were lower for larger pillar spacings. This behavior of the TLF can be explained by considering the critical length scale R c (i.e., the contact radius). As the pillar spacing increases and the time required to evaporate the liquid volume for the formation of the concave meniscus between pillars becomes longer, R c increases following a power law of the time scale. The increase in R c augments the total flow resistance of the vapor beneath the droplet. According to the Eq. (14), the vapor pressure increases in proportion to ~Rc2 . It is conjectured that the augmentation of R c reduces the slope of the nonwetting vapor pressure reduction, and this effect accompanied with the reduction of the wetting pressure by the increase in the pillar spacing can reasonably explain the experimental TLF behavior. Additionally, lower TLF values for lower Weber number cases can be simply predicted on the basis of the right-hand side terms in Eq. (19), where the last term (the sum of PD and PH ) varies in proportion to vd2 . Furthermore, additional study might be required to establish the physical relationship between the contact radius and time scale τc and thus obtain a more quantitative and generalized solution with which to predict the TLF on microstructured surfaces.

. (20)

The calculated R c values are plotted with the time scale relation of Eq. (20), showing that the curve is well-fitted by a power law R c ~τcn with the exponent n ≈ 1/5 (Fig. 14). However, although the model of Eq. (20) can be used to predict our experimental TLF data well, an experimental validation with a more elaborate experimental setup or a theoretical procedure based on more insightful physics might be needed for a general prediction model of the TLF . Additionally, the proposed model might need to be validated with experimental dataset with various textured surfaces particularly with different pillar heights and diameters. For the goal of design optimization of the droplet-cooled surfaces, the geometrical parameters can play an important role to determine the TLF on the designed surface. According to the present model (Eq. (19)), the variations of Hp and Dp might significantly affect the wetting and nonwetting pressures for a dynamic droplet. For instance, with the same Hp and Lp values, the increase of Dp can strengthen PC and PH on the right-hand side of Eq. (19), and it may work to increase the Leidenfrost temperature in the numerator on the left-

Declaration of Competing Interest None declared. 10

Experimental Thermal and Fluid Science 111 (2020) 109954

D.E. Kim

Fig. 11. Descriptions of the thermal conduction resistance calculation between the solid surface and liquid-vapor interface: (a) Schematic heat flow path and thermal resistance diagram; Numerical calculation domain and boundary conditions for heat conduction through vapor (b) and liquid (c) regions.

11

Experimental Thermal and Fluid Science 111 (2020) 109954

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Fig. 12. Numerical calculation results for the effective thermal conductivity of the vapor region (a) and the thermal resistance of the liquid region (b).

Fig. 13. (a) Calculated contact radius versus the pillar spacing and (b) Schematic of the characteristic time scale for the contact radius from droplet impact to the concave liquid-vapor interface formation with the receding contact angle.

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Acknowledgements This research was supported by the Chung-Ang University Research Grants in 2018 and the individual basic science and engineering research program through the NRF funded by the Ministry of Education, South Korea (Grant No. 2018R1D1A1A09082838). 12

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