International Journal of Heat and Mass Transfer 219 (2024) 124875
Available online 3 November 2023
0017-9310/© 2023 Elsevier Ltd. All rights reserved.
Bouncing modes and heat transfer of impacting droplets on textured superhydrophobic surfaces
Shusheng Zhang
a, Li-Zhi Zhang
a,b,*aKey Laboratory of Enhanced Heat Transfer and Energy Conservation of Education Ministry, School of Chemistry and Chemical Engineering, South China University of Technology, Guangzhou 510640, China
bState Key Laboratory of Subtropical Building and Urban Science, South China University of Technology, Guangzhou 510640, China
A R T I C L E I N F O Keywords:
Superhydrophobic surface Droplet impact Heat transfer
Lattice Boltzmann method Anti-icing
A B S T R A C T
Superhydrophobic modifications could effectively minimize heat exchange at the interfaces of impacting droplets and solid surfaces. Previous studies have lacked numerical explorations regarding the effect of bouncing modes on the heat transfer characteristics during droplets impacting on textured superhydrophobic surfaces with micro- pillars. To address this issue, a multiple distribution function phase-field lattice Boltzmann model is developed to numerically study dynamic behaviors and heat transfer during droplet impact. Comparisons between the sim- ulations and previous experimental results validate the computation model. Subsequently, the dynamic behav- iors of impacting droplets and the effects on the heat transfer were studied using the proposed model. The effects of the textured surface structural parameters on the dynamics and heat transfer are discussed in detail. The numerical results indicate four possible bouncing modes of the impacting droplets: Cassie bouncing, partially penetrated bouncing, pancake bouncing and Wenzel bouncing. These modes depend on the surface energy stored in the penetrating droplet in the microstructures cavities of the surface. Moreover, the synergistic effects of contact time and contact area affect the heat transfer performance. Further, the developed theoretical model to predict the total transferred heat is based on the identified droplet dynamics. Finally, the effects of roughness parameters on total transferred heat are studied, and the design principles of textured superhydrophobic surfaces for heat transfer suppression are given for two application scenarios. The results demonstrate that the control of microstructures would be crucial for the dynamics and heat transfer of impacting droplets on textured super- hydrophobic surfaces.
1. Introduction
Droplets that impact a solid surface are widespread in both natural and industrial processes. The dynamics and heat transfer of impacting droplets on solid surfaces have attracted significant interest due to their extensive potential applications [1–3], such as spray cooling [4–6], metal quenching [7], and inkjet printing [8]. In general, controlling the wettability of solid surfaces is considered an effective method to manipulate droplet dynamics and heat transfer during impact. A specific application is the preparation of hydrophilic surfaces to increase the wetting area of droplets and improve the heat exchange efficiency during spray cooling [9,10]. However, rapid heat transfer between droplets and a solid surface is not always desirable. Undesirable con- sequences caused by rapid heat exchange include icing processes on
aircraft [11], hypothermia of birds in sleety climates [12] and ice for- mation on outdoor industrial equipment [13,14]. As opposed to the rapid heat exchange caused by hydrophilic modifications, the heat ex- change between droplets and solid surfaces can be reduced by applying superhydrophobic modifications that rapidly shed droplets from the surface [15]. Therefore, the superhydrophobic surfaces have attracted extensive attention in anti-icing and thermal management.
The wettability of superhydrophobic surfaces is affected by the intrinsic contact angle and micro-nano structures [16]. Due to the simplicity and controllability of constructing rough structures, a key factor when preparing various functional superhydrophobic surfaces is to construct various rough structures. Extensive designs for various structured superhydrophobic surfaces focus on regulating droplet dy- namics [17–22], where the significant evaluation criterion is the contact time between the droplet and the surface. Droplet contact time
* Corresponding author at: Key Laboratory of Enhanced Heat Transfer and Energy Conservation of Education Ministry, School of Chemistry and Chemical En- gineering, South China University of Technology, Guangzhou 510640, China.
E-mail address: Lzzhang@scut.edu.cn (L.-Z. Zhang).
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer
journal homepage: www.elsevier.com/locate/ijhmt
https://doi.org/10.1016/j.ijheatmasstransfer.2023.124875
Received 26 April 2023; Received in revised form 19 September 2023; Accepted 23 October 2023
reduction is observed on these designed superhydrophobic surfaces because of the asymmetric rebounding dynamics caused by rough structures. Generally, researchers believe that the transferred heat de- creases with shorter droplet contact time. Thus, the design principle of the superhydrophobic surfaces is to minimize contact time in numerous studies [17,19,23]. However, recent studies [24,25] have noted that in addition to the contact time, the contact area is another factor affecting heat transfer for droplets that impact surfaces. Girard et al. [24] and Lin et al. [25] defined an interaction parameter that couples the contact time and contact area into a single parameter to evaluate the transferred heat as droplets impact superhydrophobic surfaces (smaller interaction parameters lead to less heat transfer). Most previous studies focused on droplet dynamics, while the heat transfer between an impacting droplet and a surface was indirectly evaluated by calculating specific parame- ters of the droplet dynamics.
Some preliminary studies have reported the heat transfer process of
impacting droplets on superhydrophobic surfaces. Shiri et al. [15]
measured the heat exchange between a bouncing droplet and a super- hydrophobic substrate and proposed an associated model. A super- hydrophobic surface was shown to cool faster via small droplets. Guo et al. [26] experimentally investigated the dynamics and heat transfer of bouncing droplets on superhydrophobic silicon wafers. A model for the transient heat flux at the interface was proposed based on the similarity solution. Moon et al. [27] studied the dynamic behaviors during the spreading and receding phases of impacting droplets on heated textured surfaces. The modified equations for the cooling effectiveness of textured surfaces with three wetting states were suggested. In these studies, high-speed cameras captured the dynamic behaviors, and thermocouples and high-speed infrared thermography measured the temperature variations of the droplets and solid substrates. The heat transfer performance was calculated using the temperature distribution.
However, the experimental data still suffer from high uncertainty in Nomenclature
Roman symbols
A contact area, m2
cp constant-pressure specific heat capacity, J/kg⋅K cs lattice sound velocity, m/s
Cs sound velocity in droplet, m/s dp maximum penetration depth, m
d* dimensionless maximum penetration depth D0 initial diameter, m
Ds spreading diameter, m
D* dimensionless spreading diameter eα discrete velocity, m/s
fα velocity distribution function fs solid fraction
F the force vector, N
Fα the hydrodynamic forcing term Fb body force
Fμ viscous force Fp pressure force Fs surface tension force
gα temperature distribution function h heat transfer coefficient, W/(m2⋅K) h1 thickness of flat substrate, m h2 height of micro-pillars, m hα phase-field distribution function h* dimensionless height of micro-pillars I dimensionless total contact area parameter k water hammer pressure constant
m droplet mass, kg
M mobility
M orthogonal transformation matrix
̂nw the normal unit vector of the solid wall p pressure, Pa
p* normalized pressure PC capillary pressure, Pa PD dynamic pressure, Pa Pr Prandtl number
PWH water hammer pressure, Pa q heat flux, W/m2
Q global heat flow, W Qtotal total transferred heat, J r0 droplet radius, m rs roughness factor Re Reynolds number
S distance of micro-pillars, m
S ̂ diagonal relaxation matrix sv relaxation factor
t time,s
t* dimensionless time tc contact time, s
te capillary emptying time, s ti inertia-capillarity time, s T temperature, K
u velocity vector, m/s U0 initial velocity, m/s v kinematic viscosity, m2/s Vi impact velocity, m2/s W width of micro-pillars, m wα weight coefficient We Weber number Greek symbols
β coefficient related to surface tension and interface thickness
χ thermal diffusivity, m2/s δt time step, s
ϕ order parameter
γ the ratio of the thermal effusivity between the droplet and the solid surface
κ coefficient related to surface tension and interface thickness
λ thermal conductivity, W/(m⋅K) μ dynamic viscosity, Pa⋅s μϕ chemical potential θi intrinsic contact angle, ◦ ρ density, kg/m3 σ surface tension, N/m τ hydrodynamic relaxation time τϕ phase-field relaxation time τT temperature relaxation time Ωα collision operator
ξ interfacial thickness Subscript
d droplet
H heavy fluid L light fluid Superscript
eq equilibrium
transient processes for droplets impacting superhydrophobic surfaces [26,27]. The velocity and heat flux distributions during impact are difficult to observe experimentally, which hinders the revelation of the underlying mechanisms of droplet dynamics and heat exchange.
Therefore, it is still a challenge to investigate the droplet dynamics and heat transfer by experimental techniques.
Fortunately, the past 30 years have seen increasingly rapid advances in computer simulations, and simulations are now considered reason- able alternatives to experimental research. In the context of impacting droplet on superhydrophobic surface, Samkhaniani et al. [28] studied heat transfer between a droplet and superhydrophobic surface at mod- erate temperature based on OpenFOAM, which included the Marangoni effect. Jaiswal and Khandekar [29] investigated heat transfer phenom- ena during drop-on-drop impacts on superhydrophobic surface. They summarized the dynamic behaviors of two droplets and the corre- sponding heat transfer performance. However, albeit of the worthy studies was implemented, the droplet impact dynamics and heat transfer on superhydrophobic surface are still far from completely understood.
The previous numerical studies paid the most attention to the heat transfer phenomena of droplet impacting on smooth superhydrophobic surface. However, the microstructures of textured superhydrophobic surfaces significantly affect the dynamic behaviors of impacting drop- lets, which leads to the unique bouncing modes. Therefore, the associ- ated heat transfer as droplets impact textured superhydrophobic surfaces differs from smooth superhydrophobic surface, which should be further investigated. To fill this research gap, this work aims to imple- ment a comprehensive numerical study on the dynamics and heat transfer of droplets impacting on textured superhydrophobic surfaces with a focus on tracking the actual heat transfer process.
In recent years, there has been an increasing interest in the lattice Boltzmann method (LBM). This is an effective alternative to conven- tional computational fluid dynamics models because of its flexibility when studying multiphase flows with complex interfacial dynamics [30]. Furthermore, the LBM is an effective approach to handle of com- plex boundaries. Therefore, the LBM is suitable for the simulations of droplets that impact superhydrophobic surfaces with rough structures [31–36]. In this work, a multiple distribution function phase-field LBM is developed to study the dynamics and heat transfer as droplets impact textured superhydrophobic surfaces.
In summary, the present study aims to further investigate the dy- namics and heat transfer of droplet impacting on the textured super- hydrophobic surface, with particular interest in the various dynamic behaviors and their effects on the actual heat transfer process. The remainder of this manuscript is organized as follows: Section 2 is a description for research object. Section 3 is an overview of the numerical method. The results and discussion are presented in Section 4. Finally, the conclusions are given in Section 5.
2. Problem description
Due to the simplicity in controlling roughness parameters, micro- pillar structures were commonly employed to manipulate the impact- ing droplet dynamics on functional superhydrophobic surfaces. There- fore, the textured superhydrophobic surfaces with micro-pillars were investigated in this work. Fig. 1 shows the computational domain for low-temperature droplets impacting a high-temperature textured superhydrophobic surface. A single droplet with an initial diameter D0 and initial temperature Td,0 = 20 ◦C was initialized above a super- hydrophobic substrate with an initial temperature Ts,0 =60 ◦C. The bottom of the initial droplet was tangent to the substrate. Subsequently, the droplet impacted the superhydrophobic surface with a pre- determined velocity U0. In general, the Weber number (We) is employed to characterize the droplet impact process on superhydrophobic sur- faces, which is calculated as:
We=ρdU02D0
σ , (1)
where ρd and σ denote the density and surface tension of the droplet, respectively. In this work, we focus solely on the rebound of impacting droplets as a whole and disregard the break-up of impacting droplets with high We. Therefore, the We is limited to the range of 5–40 to avoid droplet break-up [37,38]. The size of the computational domain is Nx × Ny ×Nz =240 ×240 ×200 in the lattice unit (LU). The micro-pillars are at the bottom of the computational domain. A schematic diagram and geometrical parameter are presented in Fig. 1(b), where h1 and h2
denote the thickness of the flat substrate and micro-pillars height, respectively. The width of micro-pillars and distance between neigh- boring micro-pillars are defined as W and S, respectively. The solid fraction fs is defined to describe the sparsity of micro-pillars, which is calculated by:
fs=W2
S2 (2)
A dense arrangement of micro-pillars leads to a larger fs. A smooth solid surface can be observed when fs =1. In present simulation, the parameters are set as D0 =100 LU, h1 =6 LU, and h2 =30 LU. The intrinsic contact angle θi is set as 150◦In this work, the gap and width of micro-pillars are within an order of magnitude smaller than the impacting droplet size to trigger various bouncing modes. For the sake of modeling simplicity, S/D0 is controlled within the range of 0.1–0.16.
The fs is controlled by adjusting the gap and width of micro-pillars, which is specified in the following sections. The thermophysical prop- erties applied in the simulations are listed in Table 1.
3. Method
The thermal regime during droplet impacting on a heated wall de- pends on the solid surface temperature [39]. Four distinct thermal re- gimes are presented with increasing wall heat: film evaporation, nucleate boiling, transition boiling, and film boiling. Considering the present research background of ice formation on outdoor industrial equipment, the temperature of the solid substrate remains below the droplet saturation temperature. Thus, the thermal regime of the impacting droplet is considered to be film evaporation. However, the short contact time between the impacting droplet and the Fig. 1.Computational domain for the low-temperature droplet impacting a high-temperature textured superhydrophobic surface: (a) Entire computational region. (b) Geometrical parameters of the textured superhydrophobic surface.
Table 1
Summary of the thermophysical properties used in the simulations.
Properties Droplet Air Solid surface
ρ (kg⋅m−3) 998 1.29 2710
μ (Pa⋅s) 9.01×10−4 1.48×10−5 –
cp (J⋅kg−1⋅K−1) 4200 1006 902
k (W⋅m−1⋅K−1) 0.6 0.026 236
superhydrophobic surface indicates that the evaporative heat transfer is negligible. Therefore, phase change is ignored in the numerical model.
Herein, a multiple distribution function phase-field LBM is developed, to model the real vapor/liquid density ratio. Three lattice Boltzmann equations are employed to describe the evolution of the velocity field, phase field, and temperature field, respectively. The specific imple- mentation is given in the following sections.
3.1. Governing equations
The continuity and momentum equations of incompressible multi- phase flows, interface tracking equation (Allen–Cahn equation), and energy equation are given by [40]:
∂ρ
∂t+ ∇⋅ρu=0, (3)
ρ (∂u
∂t+u⋅∇u )
= − ∇p+ ∇(
μ[∇u+ (∇u)T])
+Fs+Fb, (4)
∂ϕ
∂t+ ∇⋅ϕu= ∇⋅M [(
∇ϕ− ∇ϕ
|∇ϕ|
[1− 4(ϕ− ϕ0)2] ξ
)]
, (5)
∂T
∂t+ ∇⋅(uT) = ∇⋅(χ∇T), (6)
where ρ, p, ϕ, and T denote the fluid density, pressure, order parameter, and temperature, respectively. The μ, M, and χ denote the dynamic viscosity, mobility, and thermal diffusivity, respectively. The location of the interface ϕ0 is defined as ϕ0 =(ϕL +ϕH)/2, and the subscripts L and H denote values for light fluids and heavy fluids, respectively. The order parameters ϕL =0 and ϕH =1 are used in the numerical simulation.
Furthermore, u, Fb, and ξ are the velocity vector, body force, and interfacial thickness, respectively. The surface tension force Fs is defined as:
Fs=μϕ∇ϕ, (7)
where the chemical potential μϕ is given by:
μϕ=4β(ϕ− ϕL)(ϕ− ϕH)(ϕ− ϕ0) − κ∇2ϕ, (8) where β and κ are coefficients related to the surface tension σ and interfacial thickness ξ by β =12σ /ξ and κ =3σξ / 2.
3.2. Lattice Boltzmann equations
The lattice Boltzmann equation for hydrodynamics is given by:
fα(x+eαδt,t+δt) =fα(x,t) +Ωα(x,t) +Fα(x,t), (9) where fα denotes the velocity distribution function, and Fα denotes the hydrodynamic forcing term, which can be described as:
Fα(x,t) =δtwα
eα⋅F
ρc2s, (10)
where cs and δt denote the lattice sound velocity and time step, respectively. In present work, the D3Q19 lattice model is employed, and the particle velocity eα and weight coefficients wα are:
eα=
⎧⎨
⎩
(0,0,0) α=0
(±1,0,0),(0,±1,0),(0,0,±1) α=1− 6 (±1,±1,0),(±1,0,±1),(0,±1,±1) α=7− 18
, (11)
wα=
⎧
⎪⎪
⎪⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎩ 1
3 α=0
1
18 α=1− 6 1
36 α=7− 18
. (12)
In Eq. (10), the force vector F can be divided into the surface force Fs, body force Fb, pressure force Fp, and viscous force Fμ, which is calculated by:
F=Fs+Fb+Fp+Fμ, (13)
with
Fp= − p∗c2s∇ρ, (14)
Fμ=v[
∇u+ (∇u)T]
⋅∇ρ, (15)
where the normalized pressure is defined as p∗ =p/ρc2s. The v denotes the kinematic viscosity. The Ωα in Eq. (9) denotes the collision operator.
In this work, the multiple-relaxation-time (MRT) model is employed for numerical stability. The MRT collision operator is given by:
Ωα= − M−1̂SM( fα− feqα)
, (16)
where M and ̂S are the orthogonal transformation matrix and diagonal relaxation matrix, respectively. The diagonal relaxation matrix is defined as:
Ŝ= (1,1,1,1,sv,sv,sv,sv,sv,1, ...,1), (17) with
sv= 1
τ+0.5, (18)
where τ denotes the hydrodynamic relaxation time as:
τ= v
c2sδt. (19)
The modified equilibrium distribution feqα in Eq. (16) is calculated by:
feqα =fαeq− 1
2Fα, (20)
with
fαeq=p∗wα+ (Γα− wα), (21)
Γα=wα
[ 1+eα⋅u
c2s +(eα⋅u)2 2c4s − u⋅u
2c2s ]
. (22)
Then, the normalized pressure and velocity are updated at each time step, which is described as:
p∗=∑
α
fα, (23)
u=∑
α
fαeα+F
2ρδt. (24)
For interface tracking, the phase-field distribution function hα is obtained from another lattice Boltzmann equation:
hα(x+eαδt,t+δt) =hα(x,t) − hα(x,t) − heqα(x,t)
τϕ+0.5 +Fαϕ(x,t), (25) where the equilibrium phase-field distribution function heqα is calculated by:
heqα =heqα − 1
2Fαϕ, (26)
with
heqα =ϕΓα. (27)
where τϕ denotes the phase-field relaxation time as:
τϕ= M
c2sδt. (28)
The phase-field forcing term Fϕα(x,t)is calculated by:
Fϕα(x,t) =δt
[1− 4(ϕ− ϕ0)2] ξ wαeα⋅∇ϕ
|∇ϕ|. (29)
Herein, the order parameter is updated by:
ϕ=∑
α
hα. (30)
A Neumann boundary condition scheme is used to impose the intrinsic contact angle θi on the solid surface and obtain the order parameter on the fluid–solid interface (ϕw) as [41]:
̂nw⋅∇ϕw= −
̅̅̅̅̅
2β κ
√
ϕw(1− ϕw)cosθi, (31)
where ̂nw is the normal unit vector of the solid wall. The temperature field is updated using the lattice Boltzmann equation for the temperature distribution function as:
gα(x+eαδt,t+δt) =gα(x,t) − gα(x,t) − geqα(x,t)
τT+0.5 , (32)
where gα denotes the temperature distribution function. The tempera- ture equilibrium distribution function geqα is written as:
geqα(x,t) =wαT [
1+eα⋅u c2s +(eα⋅u)2
2c4s − u⋅u 2c2s ]
. (33)
The temperature relaxation time τT is calculated by:
τT= k
ρcpc2sδt, (34)
where cp and k denote the constant-pressure specific heat capacity and thermal conductivity, respectively. Then, the temperature is updated by:
T=
∑
α
gα. (35)
The boundaries of the computational domain (Fig. 1) for the three distribution functions are divided into four categories: side (x =0, x = Nx, y =0, y =Ny), atmosphere (z =Nz), bottom (z =0), and interface (the fluid-substrate interface). The boundary conditions are summarized in Table 2. Herein, constant pressure and zero gradient boundaries are implemented in the non-equilibrium extrapolation scheme, and the half- step bounce back scheme is employed at the bottom and interface [42].
The Neumann boundary (Eq. (31)) is used to control the intrinsic contact angle at the interface for the phase-field distribution function [41],
while the conjugate thermal boundary is implemented at the interface to ensure the continuity of the heat transfer [43]. It should be noted that the present numerical model can be applied to simulate surfaces with any rough structures. The key step is the accurate calculation of the normal unit vector of the solid structures.
Based on the aforementioned discussion, a flowchart of the multiple distribution function phase-field lattice Boltzmann procedure is given in Fig. 2.
3.3. Model validation
Comparing the numerical results with existing experimental results verifies the reliability of the proposed model. Two validation cases were implemented to validate the droplet dynamics and heat transfer simu- lations. First, the dynamic behaviors of a droplet impacting a textured superhydrophobic surface are validated against the experiments of Moevius et al. [44]: the modeling parameters are the same as in experiment. Fig. 3 shows that the model reproduces the two types of droplet bouncing states obtained by Moevius et al. [44] in experiments, including conventional bouncing and pancake bouncing. Similar to previous computational fluid dynamics study [32], the nanostructures in the experiment were neglected in the validation due to their large dif- ferences in scale with the micro-pillars and droplet. Snapshots of the droplet shape obtained via simulations at different instants are nearly identical to those captured in the experiments, indicating that the pre- sent model is believed to track the dynamics of bouncing droplets on the superhydrophobic surfaces.
Subsequently, the heat transfer process of a cold droplet impacting a hot hydrophobic surface was simulated to further validate the model reliability for simulating droplet-solid heat exchange. The simulation parameters were consistent with the experimental conditions described in Ref. [26]. Herein, the cooling effectiveness ε is a characteristic parameter that describes the droplet-solid heat exchange process, which is calculated by [7]:
ε=
∫tc
0
∫A
0q(t)dAdt
mcpΔT , (36)
where tc and A denote the contact time and contact area, respectively.
The q is the heat flux at the droplet–solid interface. The m, cp, and ΔT are the droplet mass, specific heat, and temperature difference between the impacting droplet and solid surface, respectively. Fig. 4 compares the cooling effectiveness of the present simulations and the previous
Table 2
Summary of the boundary conditions used in the model.
Distribution
function Side Atmosphere Bottom Interface
Velocity Periodic Constant
pressure Bounce
back Bounce back
Phase-field Periodic Zero gradient Zero
gradient Neumann boundary Temperature Periodic Zero gradient Zero
gradient Conjugate
boundary Fig. 2.Flowchart of the simulation procedures for a droplet impacting a textured superhydrophobic surface.
experiment [26]. In this validation, the processes of cold droplet impacts on hot solid surfaces were modeled. Herein, three different surfaces in experimental study [26] were numerically reconstructed, including the smooth surface with θ =120◦, the textured surface with θ =140◦and the textured surface with θ =155◦ As shown in Fig. 4, the cooling effec- tivenesses obtained by the two methods are very close, and the average relative error of the simulations is 5.77 %. Therefore, the proposed model is considered accurate when simulating the heat transfer process.
Thus, the validation cases indicate high accuracy in the multiple dis- tribution function phase-field lattice Boltzmann model for simulating the dynamics and heat transfer in bouncing droplets on super- hydrophobic surfaces.
4. Results and discussion
4.1. Bouncing modes on textured superhydrophobic surfaces
The dynamic behaviors when a droplet impacts a textured super- hydrophobic surface depend on the Weber number of the droplet and the surface structure parameters. According to our extensive simulations,
four possible bouncing modes as droplet impacting on textured super- hydrophobic surface are identified. The wetting regime of the surface is an important classification criterion for droplet bouncing modes. There are two significant wetting regimes for droplets on textured super- hydrophobic surfaces: Cassie regime [45] and Wenzel regime [46]. In the Cassie regime, the droplet stays on the top of the rough structure, and air is trapped in the structure cavities. In the Wenzel regime, the droplet infiltrates the cavities of the rough structure and fully contacts the solid surface. The time evolution of the droplet shape under different bouncing modes is shown in Fig. 5. In this manuscript, the time scale is normalized by the inertia-capillarity time (t*=t/(ρdD03/8σ)0.5) [47]. The four bouncing modes are summarized as:
(1) Cassie bouncing (CB): the droplet spreads and retracts on the top of the microstructures without penetrating into the cavities, as shown in Fig. 5(a).
(2) Partially penetrated bouncing (PPB): The droplet penetrates into the cavities of the microstructures during spreading and then retracts and rebounds from the superhydrophobic surface, as shown in Fig. 5(b).
(3) Pancake bouncing (PB): The droplet penetrates into the cavities of the microstructures during spreading and then rebounds from the surface before completely retracting. A pancake-like droplet shape is observed, as shown in Fig. 5(c).
(4) Wenzel bouncing (WB): The droplet completely fills the micro- structures cavities and contacts the bottom of the surface. The droplet maintains contact with the bottom before rebounding, as shown in Fig. 5(d).
It should be noted that due to the superhydrophobic nature of the surfaces considered here, droplet will be bounced off from the surfaces finally in present simulation conditions. So only bouncing modes will be considered.
A dimensionless spreading diameter D* and dimensionless maximum penetrating depth d* are used to characterize the bouncing process of droplets as they impact textured superhydrophobic surfaces to better understand the fundamental dynamic characteristics of the bouncing modes. These are defined as:
D∗=Ds
D0
. (37)
d∗=dp
D0
. (38)
where Ds and dp denote the spreading diameter and maximum pene- tration depth, respectively. A schematic of the characteristic parameters is presented in Fig. 6.
Fig. 7 illustrates the time evolution of the dimensionless spreading diameters and dimensionless maximum penetrating depths for different droplet bouncing types. The dimensionless spreading diameters for the CB, PPB, and WB on textured superhydrophobic surfaces present char- acteristic patterns of initially increasing and then gradually decreasing.
For PB, the dimensionless spreading diameter of the droplet during spreading exhibits a pattern similar to the other bouncing modes.
However, after reaching the maximum spreading diameter, the droplet rebounds directly with a pancake shape, rapidly decreasing the spreading diameter. The penetrating depth is a key parameter for determining the bouncing mode. When dp =0, the droplet is in the CB mode and is always on the top of the microstructures. When dp =h2 the droplet is in the WB mode and completely fills the cavities of the mi- crostructures and contacts the bottom of the surface. When 0 <dp <h2
the droplet may be in either of the PPB and PB bouncing modes. In this situation, the bouncing mode depends primarily on whether the surface energy stored in the penetrating droplet is sufficient to overcome the adhesion force between the droplet and surface with sufficient Fig. 3. Comparison of snapshots from experimental impacting droplet dy-
namics (the first and third lines) [44] and present simulations (the second and the last lines): (a) We =9.4. (b) We =17.4.
Fig. 4. Comparison of the proposed simulations with previous cooling effec- tiveness results.
conversion into kinetic energy. For the PB mode, the maximum pene- trating depth is larger than PPB; thus, there is more kinetic energy during rebound to detach the droplet from the surface. In PPB, the surface energy stored in the penetrating droplet is insufficient to fully detach the droplet from the surface due to adhesion forces, leading to re- impacting and re-penetrating the droplet. The released surface energy of the penetrating droplet causes its center to rebound slightly. Subse- quently, the droplet impacts the textured superhydrophobic surface and re-penetrates the microstructure cavities before rebounding from the surface via conventional retracting.
As a droplet impacts a textured superhydrophobic surface, the Fig. 5. Four bouncing modes of a droplet impacting a textured superhydrophobic surface: (a) Cassie bouncing, We =10 and fs =0.8. (b) Partially penetrated bouncing, We =10 and fs =0.5. (c) Pancake bouncing, We =20 and fs =0.5. (d) Wenzel bouncing, We =25 and fs =0.25.
Fig. 6.Schematic of the characteristic parameters for bouncing droplet: (a) Spreading diameter. (b) Penetration depth.
penetrating state is determined by the magnitude relationship between the wetting pressure Pwet and non-wetting pressure Pnwet [20]. The wetting pressure Pwet is composed of the water hammer pressure PWH
and the dynamic pressure PD. The water hammer pressure is attributed to liquid compression behind the shock wave envelope [20,48,49], which is defined as:
PWH=kρdCsVi, (39)
where k is the water hammer pressure constant that is determined by the impact conditions, which is empirically determined as 0.003 [50]. The Cs is the sound velocity in the droplet, and Vi is the impact velocity. The dynamic pressure during spreading stage is defined as:
PD=ρdVi2
2 , (40)
The non-wetting pressure is the capillary pressure that resists
penetration, which is defined as [50]:
PC=− 4σcosθiW
S2− W2 . (41)
The wetting state can be divided into three types: the fakir state, the partially impaled state, and the Wenzel state, which correspond to the bouncing modes in present simulations [20,50]. Therein, the CB mode is PC >PWH >PD, the PBB and PB modes are PWH >PC >PD, and the WB mode is PWH >PD >PC.
Fig. 8 presents the effects of We and fs on the bouncing mode and contact time. As seen, the solid fraction of the substrate and We can significantly affect the bouncing mode of impacting droplet. When the droplet impacts a textured superhydrophobic surface with a large solid fraction, the microstructure cavities generate a large capillary pressure that prevents the WB mode. The penetrating depth is a key parameter for determining the bouncing mode. When dp =0, the droplet is in the CB mode and is always on the top of the microstructures. When dp =h2 the droplet is in the WB mode and completely fills the cavities of the mi- crostructures and contacts the bottom of the surface. When 0 <dp <h2
the droplet may be in either of the PPB and PB bouncing modes. In this situation, the bouncing mode depends primarily on whether the surface energy stored in the penetrating droplet is sufficient to overcome the adhesion force between the droplet and surface with sufficient conver- sion into kinetic energy. For the PB mode, the maximum penetrating depth is larger than PPB; thus, there is more kinetic energy during rebound to detach the droplet from the surface. In PPB, the surface energy stored in the penetrating droplet is insufficient to fully detach the droplet from the surface due to adhesion forces, leading to re-impacting and re-penetrating the droplet. The released surface energy of the penetrating droplet causes its center to rebound slightly. Subsequently, the droplet impacts the textured superhydrophobic surface and re- penetrates the microstructure cavities before rebounding from the sur- face via conventional retracting.
To further reveal the energy conversion characteristic of droplets at different bouncing modes, the time evolution of the dimensionless sur- face energy and the dimensionless normal kinetic energy is plotted in Fig. 9. The snapshots represent the droplet at the moment it has a maximum surface energy. Herein, both the surface energy and normal kinetic energy are normalized by the initial surface energy of the droplet. In the CB mode, the droplet does not penetrate the micro- structure cavities, resulting in a relatively small increase in the surface Fig. 7. Time evolution of the dimensionless spreading diameters and maximum
penetrating depths for different droplet bouncing modes. The black and red lines correspond to D* and d*, respectively.
Fig. 8. Effects of We and fs on the bouncing mode and contact time: (a) The bouncing modes corresponding to different We and fs. (b) The normalized contact time as a function of We and f.
energy. The droplet bouncing relies on the energy released by large- scale retraction, which leads to longer contact times between the droplet and the surface. A similar phenomenon was reported by Yin et al.
[32]. In the PPB mode, some of the liquid penetrates the microstructure cavities, resulting in a greater maximum surface energy than in the CB mode. The penetrated liquid and retracting droplet release surface en- ergy together to enable droplet rebound, resulting in a shorter contact time of the PPB mode than the CB mode. In the PB mode, additional
surface energy is stored in the liquid trapped in the microstructure cavities. The surface energy of the penetrated liquid is converted into kinetic energy, which is sufficient to overcome surface adhesion force and causes the droplet to detach from the surface before retracting.
Therefore, the shortest contact time is observed in this bouncing mode.
In the WB mode, although most of the surface energy is stored in the liquid trapped in the microstructure cavities, the absence of significant capillary forces means that droplet bouncing relies primarily on the Fig. 9.Time evolution of the dimensionless surface energy and dimensionless normal kinetic energy for the different bouncing modes. The black, red, blue, and green lines correspond to the CB, PPB, PB, and WB modes, respectively. The snapshots represent the droplet when it has a maximum surface energy.
Fig. 10. The vertical velocity distribution and the velocity vector distribution for the different bouncing modes at the center slice of impacting droplets: (a) Cassie bouncing, We =10 and fs =0.8. (b) Partially penetrated bouncing, We =10 and fs =0.5. (c) Pancake bouncing, We =20 and fs =0.5. (d) Wenzel bouncing, We =25 and fs =0.25.
energy released by large-scale retraction. Therefore, it cannot generate the PB mode, and a longer contact time is observed.
Fig. 10 illustrates the vertical velocity distribution and velocity vector distribution for the different bouncing modes at the center slice of impacting droplets. The vertical velocity and the velocity vector are normalized by the initial droplet velocity. The positive vertical velocity of the droplet during rebound is distributed primarily at the upper half of the droplet for the CB, PBB, and WB modes. Therein, the droplet undergoes large-scale retraction and generates kinetic energy that lifts it from the surface. In contrast, the positive vertical velocity that drives the droplet rebound is distributed primarily at the bottom of the droplet (penetrated liquid) for the PB mode. This portion of the kinetic energy facilitates rapid separation of the droplet from the surface.
4.2. Heat transfer of bouncing droplet on textured superhydrophobic surface
Subsequently, the heat transfer characteristics of bouncing droplet on textured superhydrophobic surface are investigated for the different bouncing modes. Fig. 11 shows the time evolution of the temperature distribution for the different bouncing modes. Herein, an independent contour legend for the surface temperature is applied to illustrate the results clearly. Initially, the substrate is at a constant and uniform high temperature. The substrate temperature gradually decreases due to heat transfer between the impacting droplet and the substrate. The smallest interface temperature is at the center of the droplet–solid interface.
During the spreading stage of the impacting droplet, the low- temperature region increases with the dimensionless spreading diam- eter. In the simulations, the substrate of the WB mode exhibits the greatest heat transfer effect. Subsequently, the temperature distribution of the substrate tends to become uniform due to the thermal diffusion effect.
The time evolutions of the volume-averaged droplet (Td) and sub- strate (Ts) temperatures for the different bouncing modes are presented in Fig. 12 to quantitatively analyze temperature changes in the substrate and droplet. Among the four bouncing modes, the temperature change is the smallest in the CB mode, with the average temperature change for both the droplet and substrate being within 1.5 K. In contrast, the bouncing droplet takes away the most heat in the WB mode, causing the substrate temperature to change by over 3.0 K. Herein, the surface that triggers PB with the shortest contact time is not the surface with the smallest temperature change. This indicates that the contact time is not the sole criterion for designing superhydrophobic surfaces to suppress heat transfer. Fig. 13 depicts the time evolution of the global heat flow and heat transfer coefficient for different bouncing types. Here, the heat flux q, global heat flow Q, and heat transfer coefficient h are defined as:
q= − λ ∂T
∂̂nw
, (42)
Q=
∫
qdA, (43)
h= Q
A(Tw− Ts), (44)
where λ denotes the local liquid thermal conductivity.
As shown in Fig. 13, the global heat flow has a similar trend of initially increasing and then decreasing for all four bouncing modes.
However, dynamic differences in the droplets for the different bouncing modes cause variations in the heat transfer performance. For both the CB and WB modes, the droplets spend less time in the spreading stage than the retracting stage with large-scale droplet retraction, leading to a rapid increase followed by a gradual decrease. For the PB mode, the nearly equal durations of the growth and decay stages of the global heat flux is attributed to the rapid droplet rebound. For the PPB mode, the re- impacting and re-penetrating of droplets result in a slight increase in
the global heat flow as the heat flow decreases. Fig. 13(b) shows the maximum heat transfer coefficient at the early stage of the droplet impacting the substrate. The heat transfer coefficient decreases at different rates during contact for the various bouncing modes. A similar trend in the heat transfer coefficient was reported in previous simula- tions [51]. Furthermore, the maximum transient global heat flow and transient heat transfer coefficient are observed in the PB mode with the shortest contact time. Combining the analysis for the time evolution of the droplet and substrate temperatures indicates that the heat transfer performance as the droplet impacts the textured superhydrophobic surface is not a single-valued function of the contact time. This is consistent with the conclusions of Girard et al. [24] and Lin et al. [25].
The following section performs quantitative analyses on the relationship between the interaction parameter and the heat transfer performance.
In order to characterize the total amount of heat transfer during the process of droplet impacting on the textured superhydrophobic surface, the total transferred heat is defined as:
Qtotal=πD30ρdcp
(Td,b− Td,0
)
6 , (45)
where Td,b denotes the droplet temperature after bouncing. The distri- butions of Qtotal at different We and fs are plotted in Fig. 14(a). The re- sults of a bouncing droplet on a smooth superhydrophobic surface are also included. A greater heat transfer is observed when the bouncing droplet with a large We impacts a textured surface with a low fs. This phenomenon is explained by the synergistic effects of the contact time and contact area. A bouncing droplet with a large We that impacts a textured surface with a low fs generates a large PWH and low PC, which leads to the WB mode (PWH > PD >PC). Therefore, liquid fills the microstructure cavities with a greater contact area. Although the contact time of the WB mode is slightly shorter than those of the bouncing droplet on a smooth superhydrophobic surface and of the CB mode, this is insufficient to offset the significant increase in the contact area of the WB mode. Inspired by previous studies [24,25], the dimensionless total contact area parameter I couples the contact time and contact area as:
I=
∫tcontact
0 A(t)dt πD20ti
, (46)
where A(t) is the transient contact area at time t; and ti denotes the inertia-capillarity time, which is defined as ti =(ρdD03/8σ)0.5. Fig. 14(b) provides the relationship between I and the total transferred heat. The total transferred heat increases nearly linearly with the interaction pa- rameters for the superhydrophobic surfaces with various fs. Therefore, the dimensionless parameter is considered one of the characteristic pa- rameters for designing superhydrophobic surfaces that limits energy transfer. Furthermore, Fig. 14(b) shows that the slope of the lines for fs
=0.67 and 0.8 is noticeably larger than the other lines. These phe- nomena can be explained by the impacting droplet dynamics on these two surfaces. As shown in Fig. 8(a), the primary bouncing modes for the surface with fs =0.67 and 0.8 are PB mode and CB mode, respectively.
According to the above discussion, the contact time of the impacting droplet is short in the PB mode, while the contact area of the impacting droplet is small in the CB mode. Therefore, a more significant increase in the impacting velocity is required to increase the I for these two bouncing modes. When the impacting velocity increases, the convective heat transfer on the surface is enhanced. This results in a greater increase in transferred heat for the surfaces with fs =0.67 and 0.8 when the same I is increased.
4.3. Parametric study: micro-pillar height
The effect of microstructure height is crucial for droplets impacting a textured superhydrophobic surface [52]. The micro-pillar height significantly affects the dynamic behaviors and bouncing modes of the droplet, which impacts the heat transfer between the droplet and
Fig. 11. Time evolution of the temperature distribution for different bouncing modes: (a) Cassie bouncing, We =10 and fs =0.8. (b) Partially penetrated bouncing, We =10, fs =0.5. (c) Pancake bouncing, We =20, fs =0.5. (d) Wenzel bouncing, We =25, fs =0.25. The sequence diagram A is the temperature distribution for the front view of the impacting droplet. The sequence diagram B is the temperature distribution for the solid surface.The contour legends T and Ts describe the tem- perature distribution for the front views of the impacting droplet and solid surface, respectively.
superhydrophobic surface. Fig. 15 provides a typical case that illustrates the effect of micro-pillar height on a droplet impacting a textured superhydrophobic surface. When the micro-pillar height decreases from 30 LU to 18 LU, the bouncing mode of the impacting droplet changes from PB to WB. The contact area between the impacting droplet and surface increases significantly, which greatly affects the heat transfer of the droplet–solid interface.
Fig. 16 shows the effects of micro-pillar height on the bouncing mode distributions when a droplet impacts a textured superhydrophobic sur- face through three different micro-pillar heights (h2 =18 LU, h2 =30 LU, and h2 =42 LU). The proportions of various bouncing modes for these three textured superhydrophobic surfaces are presented in Fig. 16 (d). Overall, the number of droplets that exhibit the PB and PBB bouncing modes increases, while that in the WB mode decreases. The proportion of droplets in the CB mode remains unaltered. Increasing the micro-pillar height gives a longer deceleration process for the droplets before contacting the substrate, causing the droplet to have a velocity of zero before contact. Consequently, droplets that originally rebounded in
the WB mode now rebound in the PB or PBB modes. However, the penetrating state of the droplet in the CB mode is not affected by the increased micro-pillar height.
The distributions of the total transferred heat for textured super- hydrophobic surfaces with three different micro-pillar heights (h2 =18 LU, h2 =30 LU, h2 =42 LU) are depicted in Fig. 17 to investigate the effect of height on the heat transfer as droplets bounce. A high heat transfer (QH >0.0137 J) and a low heat transfer (QL <0.0074 J) region are defined and indicated by solid lines in Fig. 17 as boundaries. The area proportions of the region with high and low heat transfers in the total heat transfer distribution are denoted by SH and SL, respectively.
The proportions of high and low heat transfer on the textured super- hydrophobic surfaces with various micro-pillar heights are summarized in Table 3. Increasing the micro-pillar height is shown to simultaneously increase SH and SL. When a droplet impacts a textured superhydrophobic surface with a lower We, increasing the micro-pillar height makes it more difficult for the droplet to contact the substrate, resulting in a decreased WB mode and an increased PB mode. Thus, the decreased contact area and shorter contact time lead to a lower heat transfer. Then, the increased SL is caused primarily by the lower heat transfer when droplets with smaller We impact surfaces with taller micro-pillars.
Moreover, an increased SH is explained by the dynamic characteristics of droplets with larger We impacting a textured superhydrophobic sur- face with a lower fs. When droplets with larger We impact a textured superhydrophobic surface with a lower fs, the dynamic pressure of the droplet is much larger than the capillary pressure of the textured surface.
Therefore, increasing the micro-pillar height does not prevent the WB mode, but instead causes the droplet to contact more solid surface and increase the contact area, resulting in additional heat transfer. The increased SH is caused primarily by the larger heat transfer when droplets with greater We impact surfaces with taller micro-pillars and lower fs.
4.4. Amount of transferred heat
Up to now, some studies have proposed correlations to predict the total transferred heat [7,15,26,27,53]. Most of these correlations are derived based on the heat transfer of impacting droplets on smooth surfaces [7,15,53], without considering the effect of textured structures.
Recently, Guo et al. [26] proposed a correlation for impacting droplets on superhydrophobic surfaces with micro-posts. However, they assumed that all droplets were in Cassie-Baxter state (i.e. CB mode in this manuscript) during the impacting process. Moreover, Moon et al. [27]
Fig. 12. Time evolution of the volume-averaged droplet and substrate tem- peratures for the different bouncing modes. The black and red lines correspond to Td and Ts, respectively.
Fig. 13. Time evolution of the global heat flow and heat transfer coefficient for the different bouncing modes. The black, red, blue, and green lines correspond to the CB, PPB, PB and WB modes, respectively.
modified the correlation from Strotos et al. [53] based on the three wetting states observed on textured superhydrophobic surfaces [20].
However, the ratio of the total-wetted area to the contact area was preset, which limits the application scenarios. Further, incomplete-retracting bouncing (such as the PB mode) is not discussed in Moon et al. [27]’s model. Therefore, further developments of the predictive models for the total heat transfer are necessary for the four bouncing modes identified in present study. Subsequently, a theoretical analysis model is proposed to predict the amount of heat transfer. The specific implementation is given as following:
According to previous analysis, the theoretical heat flux at the dro- plet–solid interface can be calculated as [26]:
q(t) =˙
̅̅̅5
√ eles
(Ts,0− Td,0
) ( ̅̅̅
√5 el+es
) ̅̅̅̅
πt
√ , (47)
where el and es denote the thermal effusivities of the droplet and solid surface, respectively. Then, the total transferred heat during the impacting droplet on the textured superhydrophobic surface is described as:
Qtotal=
∫tc
0
q(t)A(t)dt,˙ (48)
where tc is the contact time between the droplet and solid surface, and A (t) is the relationship between the contact area and time. The total transferred heat Qtotal can be normalized by the maximum possible heat transfer as:
Q∗= Qtotal
mcp,l
(Ts,0− Td,0
). (49)
As seen, the physical meaning is consistent with the cooling effec- tiveness defined in Eq. (36). The most crucial issue for predicting the total transfer heat is to determine the contact time (tc) and contact area (A(t)) as the droplet impacts the surface. The correlations between the four identified bouncing modes are discussed below
(1) For the CB mode, the contact area is calculated as A(t) =πD2(t)fs/ 4, where D is the spreading diameter of the droplet. The relationship between the spreading diameter and time can be approximated as [54]:
D(t) =2Dm
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
t/tc− (t/tc)2
√
, (50)
where Dm is the maximum spreading diameter. According to the energy balance, the relationship between the spreading diameter and initial droplet diameter is described as [28]:
Dm
D0
=
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
4 1− cosθCB
(0.038We+0.468)
√
, (51)
where θCB is the Cassie-Baxter contact angle of the droplet on the textured superhydrophobic surface, which is calculated as:
cosθCB=fscosθi− (1− fs). (52)
The contact time can be approximated as [55]:
tc= π
2 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
2(1− cosθCB)
√
̅̅̅̅̅̅̅̅̅̅
ρdD30 σ
√
. (53)
Then, substituting Eqs. (48) and (50)–(53) into Eq. (49) simplifies it as:
Q∗= 32 ̅̅̅
√5
fs(0.038We+0.468) 5(
2 ̅̅̅
√2)0.5
(1− cosθCB)1.25( ̅̅̅
√5 γ+1)
( We0.25 Re0.5Pr0.5
)
. (54)
where γ is the ratio of the thermal effusivity between the droplet and the solid surface (el/es). The Re and Pr denote the Reynolds number and Prandtl number of droplet, respectively.
(2) For the PPB mode, a schematic diagram of the droplet penetrating the micro-pillars is shown in Fig. 18 [56]. As seen, the penetrating droplet can be divided into cylindrical segment (Part I) and a partial spherical segment (Part II). Herein, the contact area of the droplet–solid interface is calculated by [56]:
Fig. 14. Heat transfer performance as a droplet impacts a textured superhydrophobic surface: (a) Total transferred heat as a function of We and fs. (b) Relationship between I and the total transferred heat (the straight line is obtained with a linear fit).
Fig. 15. Schematic diagram showing the effect of the micro-pillar height: (a) Dynamic behavior of a bouncing droplet on the textured surface with We =20, fs =0.5, and h2 =30 LU; (b) Dynamic behavior of a bouncing droplet on textured surface with We =20, fs =0.5, and h2 =18 LU.
