Critical sinking of hydrophobic micron particles

Chemical Engineering Science 207 (2019) 17–29

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Critical sinking of hydrophobic micron particles Bingqiang Ji, Qiang Song ⇑, Ao Wang, Qiang Yao Key Laboratory of Thermal Science and Power Engineering of Ministry of Education, Department of Energy and Power Engineering, Tsinghua University, 100084 Beijing, China

h i g h l i g h t s

g r a p h i c a l a b s t r a c t


Critical sinking of micron

The accuracy of the proposed criterion (C0) is significantly higher than that of previously reported criteria (C1–C3) comparing with the experimental results, because it considers the fluid force, TPCL movement, and other factors reasonably.

hydrophobic particle is studied by detailed simulation.
Effect of Re, contact angle, and density ratio on critical sinking is revealed.
Both fluid force and surface tension play important roles in critical sinking.
Empirical formulas for accumulated work done by each force are correlated.
A criterion for critical sinking is proposed based on energy conversion analysis.

a r t i c l e

i n f o

Article history: Received 8 March 2019 Received in revised form 31 May 2019 Accepted 6 June 2019 Available online 7 June 2019 Keywords: Micron particles Impact Liquid surface Sinking Criterion

a b s t r a c t Particle impacting on a liquid surface is frequently observed in natural and industrial processes. This study numerically investigates the critical sinking behaviour after the impact of hydrophobic micron particles on liquid surfaces. The effects of contact angle h, density ratio D, and the Reynolds number Re on the motions of the particle and TPCL are analyzed. The dimensionless penetration depth of the particle increases with increasing h and D, whereas the effect of Re is negligible. The relationship between the TPCL position on the particles and the normalised particle displacement is only determined by h and almost unaffected by Re and D. For the critical sinking of hydrophobic micron particles, both fluid force and surface tension play important roles in the conversion of particle kinetic energy. A dimensionless energy conservation equation is constructed, based on which a simple criterion of critical sinking is proposed. The expression of this criterion is obtained based on the semi-empirical expressions describing the relationships of each dimensionless accumulated work with Re, h, and D obtained from simulation results. Compared with existing criteria, the proposed criterion is more accurate because it considers the fluid force, TPCL movement, and other factors reasonably. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction

⇑ Corresponding author. E-mail address: [email protected] (Q. Song). https://doi.org/10.1016/j.ces.2019.06.009 0009-2509/Ó 2019 Elsevier Ltd. All rights reserved.

Particle impacting on a liquid surface is frequently observed in natural and industrial processes. Particles with different sizes

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B. Ji et al. / Chemical Engineering Science 207 (2019) 17–29

Nomenclature TPCL a0 Bo Cf D dp F F0 Fg m Re T t ts U u W W0 W00 We X

three-phase contact line dimensionless acceleration Bond number coefficient of fluid force density ratio particle diameter, m force, N dimensionless force gravity, N mass, m3 Reynolds number dimensionless time time, s time duration of critical sinking process, s dimensionless velocity particle velocity, m s1 work, J dimensionless work relative work Weber number dimensionless particle displacement

exhibit various impact behaviours that involve different mechanisms. Large particles sink into the liquid after impact when they are denser than the liquid (Truscott et al., 2013), or float finally when they are lighter than the liquid (Bormashenko et al., 2013). By contrast, hydrophobic micron particles, which are very common in wet deposition and wet scrubbing, may float on a liquid surface after impact due to the significant surface tension effect even when the density of the particle is significantly greater than of the liquid (Vella, 2015; Liu et al., 2010; Bormashenko, 2015). Superhydrophobic particles may bounce off the liquid surface (Lee and Kim, 2008; Wang et al., 2015). Elucidating the characteristics and physical mechanism of particles’ impact could widen our understanding of natural phenomena, such as insect movement on water (Bush and Hu, 2005; Vella and Metcalfe, 2007) and dust removal during wet deposition (Bae et al., 2010; Maria and Russell, 2005), and promote the development of technologies, such as dust removal by wet scrubbing (Jaworek et al., 2006; Park et al., 2005), mineral flotation (Nguyen and Schulze, 1993; Mitra et al., 2015), and production of particulate-reinforced metal matrix composites (Kaptay, 1996; Kaptay, 2001). Research on the behaviour of particles after impacting on a liquid surface mainly focused on the millimetre-sized or larger spheres because of the limit of research methods. Lee and Kim (2008) observed that after impacting on a liquid surface, millimetre-sized hydrophobic particles exhibit three impact modes, namely, oscillation, rebound, and submergence with increase in impact velocity. However, only submergence and oscillation were observed in general research on the particle’s impact (Liu et al., 2010; Wang et al., 2017; Doquang and Amberg, 2009; Liu et al., 2012). Force analysis (Lee and Kim, 2008; Lee and Kim, 2011; Aristoff and Bush, 2009) shows that the impact behaviour of millimetre-sized particles is determined by the overall action of surface tension, gravity, buoyancy, and hydrodynamic force. The behaviours of particle, fluid, gas–liquid interface, and threephase contact line (TPCL) are determined by the coupled dynamics of gas, liquid, and solid (Ding et al., 2015; Aristoff et al., 2010). Lee and Kim (2011) investigated the impact of millimetre-sized particles on a liquid surface when Re<<1 and calculated the hydrodynamic force exerted on the particles by solving Stokes equation

X0 xp

normalised particle displacement particle displacement, m

Greek symbols h contact angle, degree ll viscosity coefficient of the liquid, Pa s ql liquid density, kg m3 qs particle density, kg m3 r surface tension coefficient of the liquid, N m1 / angular position of the TPCL (TPCL angle, for short), rad Subscript 0 cr f m p s t

initial critical sinking fluid force maximum particle surface total

and predicting the profile of the gas–liquid interface with a quasi-static assumption. The numerical results of the particle movement and gas–liquid interface evolution agree with the experimental results. However, significant deviation occurs in the prediction of particle movement after the collision of millimetresized particles on a liquid surface when Re>>1 while calculating hydrodynamic force by assuming a potential flow and predicting the profile of the gas–liquid interface with the previous quasistatic assumption (Lee and Kim, 2008). This deviation may be due to the neglect of the viscous effect during the calculation of hydrodynamic force when Re is between 102 and 103 and that the quasi-static assumption of the gas–liquid interface does not apply to the condition of We > 1. Kintea et al. (2016) numerically captured the particle movement and gas–liquid interface of Lee’s experiment by directly solving the Navier–Stokes (N–S) equation and determining the gas–liquid interface with the volume-offluid (VOF) method. They found that the gas–liquid interface dynamically evolves when the impact We is large, indicating that the prediction of quasi-static assumption causes a large deviation. Although numerical simulation can accurately predict the impact behaviour of millimetre-sized particles, developing a simple method to distinguish the impact mode of the particle is necessary for industrial applications. Verezub et al. (2005) observed the impact behaviour of >3 mm hydrophilic particles on a liquid surface. They ignored the distortion of the liquid surface and considered fluid force as the drag force, which was calculated using Newton’s resistance formula for a sphere in single-phase flow, and proposed a criterion for particle’s critical sinking by energy balance to effectively predict experimental results. Using the same methods, Liu et al. (2010) proposed another criterion considering the gas–liquid interface shape with quadratic curve fitting based on experimental observation and obtained good agreement with the experimental data of the critical sinking velocities of 1 mm particles. The relative importance of forces changes with decreasing particle size. In a numerical research on the impact of micron particles, Wang et al. (2015) considered the effects of surface tension, gravity, drag force, added mass force, and buoyancy on the particle. They calculated the TPCL position using Young–Laplace (Y–L)

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B. Ji et al. / Chemical Engineering Science 207 (2019) 17–29

equation based on the quasi-static assumption and showed three impact modes similar to those of millimetre-sized particles. Wang et al. (2015) found through reduced force model that surface tension exceeds other forces by at least one to two orders of magnitude and becomes the dominant force during impact when the particle size is <10 lm. On the basis of energy balance with the assumption that surface tension uniquely dominates, a set of simple criteria of critical velocities was proposed to distinguish the three impact modes. However, only two impact modes are experimentally observed for the impact of submillimetre-sized PMMA particles (Wang et al., 2017), and the submergence/oscillation critical velocity is significantly higher than the results predicted by the simple criterion proposed previously (Wang et al., 2015). This prediction deviation can be attributed to the neglect of other forces and the simplification of the gas–liquid interface shape and TPCL motion (Wang et al., 2015). The energy conservation equation describes the transformation between particle kinetic energy and works contributed by forces acting on the particles along the way, which involves not only the reasonable expression of forces but also the correct calculation of forces, displacement, and TPCL motion. Ji et al. (2017) developed a simulation method to investigate the experimentally observed impact of submillimetre-sized particles by Wang et al. (2017). The simulation method describes the gas– liquid interface by using the VOF method combined with the dynamic meshing technique, and directly solves the N–S equation, accurately capturing the particle motion and gas–liquid interface evolution. Further investigation on the impact dynamics shows that the hydrodynamic force dominates the early period of impact, and its accumulated work is at the same order as that of the surface tension and thus should not be neglected. The gas–liquid interface dynamically evolves and the actual TPCL motion is different from that predicted by Y–L equation. The hydrodynamic force initially increases and then decreases as particle displacement increases; moreover, it is largely different from the calculation based on potential flow and Newton’s resistance formula for a sphere in single-phase flow. The same conclusion was obtained by the simulation of Abraham et al. (2014) and the experiment of Mouchacca et al. (1996). Therefore, a highly accurate physical model for the systematic investigation of particle and fluid dynamics during impact should be adopted to establish a criterion with a high accuracy and a wide application scope for distinguishing different impact modes of micron particles. In this study, the simulation method established in our previous study (Ji et al., 2017) is adopted to investigate the impact of hydrophobic micron particles under different parameters and to reveal the influence of different parameters on the motions of the particles and TPCL, the main forces acting on the particles, and energy conservation. On the basis of the simulation results, a criterion for the critical sinking of the particles is proposed, and its accuracy is evaluated from experimental results.

u0

2.1. Theory of particle’s critical sinking Here, a particle is considered as a rigid sphere and moves vertically towards the liquid surface without rotation. As shown in Fig. 1(a), when a solid particle with radius rp and density qs impacts a liquid surface with density ql, surface tension r, and viscosity ll at an initial velocity of u0, the gas–liquid interface becomes deformed after time t, as illustrated in Fig. 1(b). xp and up are the displacement and velocity of the particle, respectively. / is the angle between the TPCL and the particle centerline (TPCL angle for short). h is the contact angle between the particle surface and TPCL. Due to the hydrophobicity of the particle, the liquid surface self-intersects before the three phase contact line (TPCL) reaches the top of the particle, making the particle immerge into the liquid and a little bubble sealed on its top, corresponding to a penetration time of ts and a penetration depth of xm, as depicted in Fig. 1(c). The equation of particle motion can be written as

mp €xp ¼ F f þ F s þ F g ;

ρs

F f ¼ C f

xp

(a)

8

ql d2p u2p ;

ð2Þ

where Cf is the coefficient of fluid force, which is different from the resistance coefficient of Newton’s resistance formula. Although Cf varies with the increase in the contact area of the particle and liquid, as well as the decay of particle velocity during impact (Abraham et al., 2014), fluid force is expressed as Eq. (2) for convenience with Cf, which represents the combined action of drag force and added mass force during impact. The surface tension acting on the particle can be expressed as

F s ¼ pdp rsin/sinðh þ /Þ:

ð3Þ

Substituting the dimensionless displacement, time, and forces

X¼

x up

ρl μl Liquid

p

xp tu0 ; T¼ dp dp

ð4Þ

Gas

Interface σ

0

ð1Þ

where mp is the particle mass. Ff, Fs, and Fg are the fluid force, surface tension, and gravity acting on the particle, respectively. Gravity is at least two orders of magnitude less than the two other forces for the impact of micron particles and can thus be completely neglected (Ji et al., 2017). For the calculation of fluid force, only the act of the liquid phase to the particles is considered because the act of the gaseous phase to the particle can be neglected due to the large density and viscosity ratio of the liquid phase to the gaseous phase. The fluid force exerted on the particle consists of buoyancy and hydrodynamic force, and the buoyancy can be completely neglected. Thus, the fluid force can be represented by hydrodynamic force as follows:

Gas

rp rp

2. Theory and methodology

y

(b)

x=0

xm

Pinch off

θ Liquid

t

Gas

ts

up = 0

Liquid

(c)

Fig. 1. Schematic of particle impacting on a liquid surface at (a) initial time, (b) time t, and (c) time ts when the particle submerges.

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B. Ji et al. / Chemical Engineering Science 207 (2019) 17–29

F 0f ¼

Ff

pql d2p u20 =8

; F 0s ¼

Fs

ð5Þ

pdp r

into Eq. (1) yields

dU 3 0 6 0 ¼ F þ F; dT 4D f WeD s

ð6Þ

where U = up/u0 is the particle dimensionless velocity and D = qs/ql is the particle density ratio. We = qlu20dp/r is the impact Weber number, which reflects the importance of surface tension with respect to the initial inertia of the liquid. a0 s = 6F0 s/WeD and a0 f = 3F0 f/4D are the dimensionless acceleration components of surface tension and fluid force, which represent the contribution of the two forces to the changes in particle dimensionless velocity, respectively. By integrating Eq. (6) on X from the initial time t0 (X = 0, U = 1) to time t (X, U) during the sinking process, we can obtain the energy conservation equation as follow.

1 2 3 6 ðU  1Þ ¼  W0  W0 ; 2 4D f WeD s

ð7Þ

where W0 f and W0 s are the dimensionless accumulated works of fluid force and surface tension from time 0 to time t, respectively:

W 0f ¼ 

Z

X 0

F 0f dX; W0s ¼ 

Z 0

X

F0s dX:

ð8Þ

W00 f = 3W00 f/4D and W00 s = 6W0 s/WeD are the relative works of surface tension and fluid force, which represent the contribution of the two forces to the changes in particle dimensionless kinetic energy, respectively. For the critical sinking process, up (or U) is almost equal to 0 at the moment when the liquid surface pinches off and the particle submerges into the liquid. Considering the energy conservation during the whole critical sinking process, assuming U = 0 and substituting it into Eq. (7) yields

Wecr ¼

24W 0st : 2D  3W 0ft

ð9Þ

Eq. (9) is the criterion for particle’s critical sinking, where Wecr is the critical Weber number, corresponding to the We calculated with the critical impact velocity ucr, above which the particle will completely sink into the fluid. W0 ft and W0 st represent the dimensionless accumulated works of fluid force and surface tension from t0 to ts of the critical sinking process, hereafter referred as fluid force work and surface tension work, respectively. Combining Eqs. (2), (3), (5), and (8) yields

W 0ft ¼ 

Z 0

Xm

F 0f dX ¼

W ft

pq

3 2 l dp u0 =8

; W0st ¼ 

Z 0

Xm

F0s dX ¼

Wst

pd2p r

; ð10Þ

where Xm = xm/dp is the dimensionless penetration depth of the particle. Wft and Wst are the accumulated works of fluid force and surface tension during critical sinking, respectively. Clearly, the crux of the calculation of Wecr is to determine Wft and Wst. In this study, the critical sinking processes at different impact conditions are simulated with the numerical method (Ji et al., 2017) we previously established, which can accurately capture the impact behaviours of the particles and fluids, and it is introduced in Section 2.2. Therefore, Wft and Wst during the critical sinking processes (when We = Wecr) at different impact conditions can be calculated, to obtain the expression of Wecr based on Eq. (9).

2.2. Numerical method The fluid is assumed to be incompressible, and the flow around the particle is regarded as axisymmetric flow. The flow near the sphere is considered laminar because Re  1000 (Richter and Nikrityuk, 2012; Johnson and Patel, 1999). The fluid flow is computed with an incompressible N–S equation, whereas the motion of the gas–liquid interface is treated by the VOF method (Hirt and Nichols, 1981), and the surface tension force acting on the fluids around the gas–liquid interface is counted with the continuum surface force model (Brackbill et al., 1992). The governing equations of fluid flow were presented in our previous work (Ji et al., 2017). After obtaining information on flow field, we can compute the fluid force acting on the particle by integrating the pressure and viscous stresses over the wetted particle surface area. The surface tension acting on the particle can be calculated using Eq. (3), after which the motion equation of the particle can be solved. As a dynamic meshing technique, the layering scheme (ANSYS Inc., 2012) is adopted to reconstruct the mesh in case of solid particle movement to achieve the coupling of particle and fluid motions. Given the axisymmetric flow assumption, a 2D symmetric computational domain is employed, as shown in Fig. 2(a). An independent analysis is conducted to determine the reasonable domain, mesh sizes and time step. The computational domain width is decided to be 10dp, whereas the total cell number is 96,400 with corresponding minimum and maximum mesh sizes of 0.01dp and 0.13dp, as shown in Fig. 2(b). The dimensionless time step DT is between 1  105 and 1  103 to ensure a global Courant number of <0.2. The gaseous and liquid phases are still at the initial moment when the particle moves from the gaseous phase to the gas–liquid interface at a speed of u0 in a position of 1.5dp above the gas–liquid interface. The domain boundary and particle surface are considered as walls, and a no-slip wall boundary condition is adopted. The contact angle of the domain wall is 90°. Though the contact angle between the particle and TPCL is dynamically changing during the sinking process, the experimental observation on the sinking of small hydrophobic spheres found that the difference between the dynamic contact angle and static contact angle is very small during the most time of the sinking process (Wang et al., 2017; Kim et al., 2017). The simulation research by Ding et al. (2015) and Kintea et al. (2016) showed that a constant contact angle can also capture the dynamics of the particle and gas–liquid interface. Therefore, we adopt a constant static contact angle h for the particle wall during critical sinking (detailed analysis is shown in the Supplementary material). Since the TPCL advances during the sinking process, h equals to the advancing contact angle in this study, and the contact angle hysteresis won’t affect the critical sinking behaviour of the particle. A second-order upwind scheme is used to discretise the momentum equation, whereas the volume fraction parameter and pressure are discretised using Geo-Reconstruct (Youngs, 1982) and body force weighted scheme, respectively. Pressure–velocity coupling is achieved with the PISO scheme. A residual of 106 is set for the convergence of continuity, momentum, and volume fraction equations. The fluid force exerted on the particle is also monitored and it is found to converge faster than the convergence of residuals. A commercial finite volume solver (ANSYS FLUENT V14.5) (ANSYS Inc., 2012) is used to solve the fluid flow equation. The particle motion is calculated by incorporating a user-defined function. Dimensionless analysis (Aristoff and Bush, 2009) shows that the impact behaviour of particles can be determined by the five dimensionless parameters of D; h, Re, Bo, and We, where Re = qlupdp/l and Bo = qld2pg/r are the impact Reynolds number and Bond number, which reflect the importance of inertia with respect to the viscos-

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B. Ji et al. / Chemical Engineering Science 207 (2019) 17–29

Wall

Axis Gas Particle wall 15dp

y

9.5dp

O x

Gas-liquid interface

8dp

Liquid Wall 10dp

(a)

(b)

Fig. 2. Sketch of (a) computational domain and (b) mesh.

x/dp

ity, and the importance of gravity to the surface tension of the liquid, respectively. Gravity can be neglected completely in comparison with other forces for the impact of micron particles, that is, Bo<<1, and exhibits minimal effect on the impact behaviour. Therefore, the main dimensionless parameters that control the impact behaviour of micron particles are We, D, h, and Re. During critical sinking, these four dimensionless parameters must be satisfied in Eq. (9), that is, Wecr is a function of D, h, and Re. Therefore, the relationship of Wecr to D, h, and Re must be established to propose the

0

Slamming stage

1

T = 0.01, 0.07, 0.15, 0.25, 0.36

0

critical sinking criterion. According to the parameter range of the common impact processes of hydrophobic micron particles on the liquid surface, we conduct numerical simulations when D = 1, 2, 5, 10; h = 90°, 120°, 150°, 180°; and Re = 100, 200, 500, 1000, producing a total of 64 cases. In our simulation, the physical parameters of the gaseous phase are specified to the physical parameters of air at normal temperature (293.15 k) and pressure (0.1 MPa). To ensure that the act of the gaseous phase can be ignored compared with the liquid phase, we set the density ratio of the liquid to the gas at ql/qg = 1000 and ensure that the viscosity of the liquid to the gas ll/lg at > 20. To make D, h, and Re meet the above values, dp, ll, r, and u0 are adjusted in different cases. The simulation also ascertains that Bo < 0.01 to ensure that gravity can be completely ignored. The impact processes at different Weber numbers are simulated, and the critical sinking process is obtained as the particle velocity equals to 0 at the moment the meniscus pinches off (as shown in Fig. 3) at a specific impact Weber number, i.e., Wecr, which is determined by dichotomy. 3. Results and discussion

1

Cavity developing stage

The works of forces acting on the particle are crucial for the consideration of energy conservation. The work done by each force is not only related to its action displacement but also to the calculation of the force. Therefore, we discuss the motions of particle and TPCL during the critical sinking process in Section 3.1 and the forces acting on the particle and the energy transformation in Section 3.2. Basing from this discussion, we develop a criterion for particle’s critical sinking in Section 3.3.

T = 0.50, 1.00, 1.50, 2.00, 3.00

2 0 1 Cavity collapsing stage

2 0

3.1. Motions of particles and TPCL

T = 4.00, 4.52, 5.03, 5.54, 5.63

1

2

3

4

5 y/dp

Fig. 3. Time series of different stages during the critical sinking process of micron particles (D = 5, h = 150°, and Re = 200). The dashed line represents the wetted part of the particle, and the solid line represents the gas–liquid interface. Time increases in the direction of the arrow.

Fig. 3 shows the typical features during the critical sinking process of a micron particle, taking the case of D = 5, h = 150°, and Re = 200 as an example. As found in our previous work (Ji et al., 2017), the critical sinking process of a micron particle can be divided into three stages, namely, slamming stage, where the particle velocity declines the fastest and the TPCL rapidly slips at the particle surface; cavity developing stage, where the decline in par-

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B. Ji et al. / Chemical Engineering Science 207 (2019) 17–29

ticle velocity slows down and the TPCL slips slowly at the particle surface; and cavity collapsing stage, where the particle velocity is extremely slow and the TPCL moves rapidly, resulting in a meniscus pinch-off and particle submergence finally. The particle velocity (kinetic energy) almost equals to zero when the meniscus pinches off (T = 5.63 in Fig. 3) at the end of cavity collapsing stage for critical sinking process. Fig. 4(a), (b), and (c) show the particle displacement and TPCL angle over time under different Re, h, and D, respectively, and the three stages during the critical sinking process are distinguished from the vertical dash dot line. The simulation and experimental results of the particle and TPCL motions during critical sinking agree well, which verifies the accuracy of the simulation method, as shown in Fig. 4(b) and 4(c). For all cases, the particle displacement rapidly increases at first and then gradually slows down to plateau, whereas the TPCL angle quickly increases at first from 0, gradually slows down, and finally rapidly increases to near p, which is consistent with our previous research (Ji et al., 2017). Particle displacement is mainly completed in the slamming and cavity developing stages, whereas the displacement in the cavity collapsing stage is negligible. Re exhibits no obvious effect on the displacement at each stage, as shown in Fig. 4(a). Fig. 4(b) illustrates that large variations occur at the displacement curve of different h at the latter period of the cavity developing stage and that the penetration depth of the particle increases with increasing h. Fig. 4(c) reveals large variations in the particle displacement curve of different D at the early period of the cavity developing stage, and the penetration depth of the particle

0.0

increases with increasing D. Fig. 5(a) gives the penetration depths of the particle in all 64 cases, with values between 1.75 and 2.6. The penetration depth of the particle increases with increasing h and D but is slightly influenced by Re. We fit the penetration depths of all critical sinking processes and construct the following expression:

X m ¼ ð1:991 þ 0:148 cos 2hÞD0:081 :

Eq. (11) describes the penetration depths of all cases very well, and the correlation index of R2 = 0.948 is obtained, as illustrated in Fig. 5(b). Eq. (11) shows that X m / D0:081 , which increases Xm by 20% when D is increased by an order of magnitude, whereas Xm increases by 16% when h is increased from 90° to 180°. As shown in Fig. 4(a), Re exerts no obvious effect on the evolution of the TPCL angle and thus on the penetration time. Fig. 4(b) indicates that h does not affect the evolution of the TPCL angle at the early period of the slamming stage when / increases from 0 to near 1. At the latter period of the slamming stage and cavity developing stage, the greater the h, the slower the speed of TPCL, thereby prolonging the penetration time. As illustrated in Fig. 4 (c), D exerts no obvious effect on the evolution of the TPCL angle at the slamming stage and the early period of the cavity developing stage. At the latter period of the cavity developing stage, the greater the D, the faster the speed of TPCL and the shorter the penetration time. Fig. 4(a) and (c) show that under different Re and D, the particle almost completed its total displacement of the whole sinking

0.0

3

3

0.5

1.5

2

Re=100 Re=200 Re=500 Re=1000

X

1.0

φ

X

X

φ

φ (rad)

X

1.0

1.5

1

2.0

2

1

2.0

0

1

2

3

T

4

5

2.5

0

0

2

4

(a) 3

3

2

2

φ

X 1.5

D=1 D=2 D=5 D=10

Re=100 Re=200 Re=500 Re=1000

1

1

2

Τ

(c)

4

6

0

0 0.0

D=1 D=2 D=5 D=10

θ=90° θ=120° θ=150° θ=180°

2.0

0

0 10

8

φ (rad)

X

φ (rad)

0.5 1.0

T

6

(b)

0.0

2.5

θ=90° θ=120° θ=150° θ=180°

φ (rad)

0.5

2.5

ð11Þ

0.2

0.4

0.6

0.8

1.0

X'

(d)

Fig. 4. Particle displacement and TPCL angle over time during critical sinking with different dimensionless impact parameters of (a) D = 5 and h = 120°, (b) D = 1 and Re = 500, (c) h = 120° and Re = 500.  and + in (b) and (c) represent the evolutions of particle displacement and TPCL angle with time obtained by experiment (A. Wang, Research, 2016) when D = 1.18, h = 115°, and Re = 684. (d) Evolution of the TPCL angle with X0 , where X0 =X/Xm is the normalised displacement of particle.

23

2.6

2.6

2.2

2.2 Xm/D0.081

Xm

B. Ji et al. / Chemical Engineering Science 207 (2019) 17–29

1.8

1.0

Re=100 Re=200 Re=500 Re=1000

θ=90° θ=120° θ=150° θ=180°

1.4

0

2

4

6

8

1.8

D=1 D=2 D=5 D=10 Eq. 11

1.4

10

D

(a)

1.0 90

120

Re 1000 500 200 100

150 θ (degree)

180

(b)

Fig. 5. (a) Penetration depths of the particle at all cases. (b) Comparison of the prediction of Eq. (11) with simulation data.

process at the slamming and cavity developing stages, whereas the evolution of the TPCL angle exhibits no difference. If we normalise the particle displacement X with its penetration depth Xm and plot the evolution curve of / with X0 , as depicted in Fig. 4(d), the curves of /ðX 0 Þ will only be influenced by h and coincide with each other totally at different Re and D when h is of the same value. The effect of h on the relationship between / and X0 can be explained by the force balance of the meniscus around TPCL. Forces exerted on the meniscus around TPCL consist of surface tension, fluid pressure and viscous force, among which surface tension is the dominant force since the length scale of the meniscus is extremely small, as shows in Section 3.2. Therefore, the movement of TPCL is determined by the balance between the surface tension and the inertia of the meniscus around TPCL (Ding et al., 2015). The larger the contact angle h, the greater the surface tension exerted on the meniscus by the particle at the same X0 (as Fig. 8(a) shows), and the greater the deceleration of the movement of TPCL around particle surface, resulting in a smaller / at the same X0 in the middle and late periods of impact. The corresponding relationship between the TPCL angle and the normalised particle displacement at the same h indicates the similarity of the TPCL movement during critical sinking, which is greatly important to the derivation of critical sinking criterion in Section 3.3. 3.2. Dominant forces and energy transformation For the impact of the micron particle, gravity can be neglected relative to the other forces, and the work done by gravity can also be neglected relative to those by the other forces. Therefore, only fluid force and surface tension and their work are analysed. Comparison of the dimensionless values of these two forces and their work in Eqs. (5) and (8) cannot reflect their actual relative magnitude because of the different characteristic forces and work they divide. Thus, the dimensionless acceleration components a0 f and a0 s are chosen to represent and compare the contribution of the two forces to the changes in particle movement. The relative works W00 f and W00 s are chosen to represent and compare the contribution of the two forces to energy transformation. The particle displacement in the cavity collapsing stage is negligible, as previously mentioned. Thus, the changes in particle kinetic energy and work done by each force in this stage can also be neglected. Hence, we focus on the forces acting on the particle and their work in this section. Fig. 6(a), (c), and (e) show the dominant forces acting on the particle over time during critical sinking under different Re, h, and D, respectively. Fluid force mainly acts during the early period,

whereas surface tension mainly acts on the middle and late periods of the impact. a0 f sharply decreases at first and then rapidly increases, presenting a negative value at the early period of the impact, increasing slowly to positive, and then decreasing slowly until the submergence of the particle. a0 s is positive at the initial period after particle collision with the liquid surface and then quickly reduces to negative. It decreases at first, increases slowly for a long time, and then increases quickly at the end of the sinking process. Fig. 6(a) illustrates that Re mainly affects the forces acting on the particle at the cavity developing stage. The increase in Re increases a0 s and decreases a0 f, and the two effects cancel each other out, causing Re to exhibit a negligible effect on total particle acceleration. This phenomenon explains why the particle displacement curve coincides with each other at different Re in Fig. 4(a). Fig. 6(c) reveals that the increase in h significantly reduces the positive a0 s and slightly increases the negative a0 f, thereby increasing the absolute value of negative total acceleration at the slamming stage. At the cavity developing stage, the increase in h increases the negative a0 s and slightly decreases the positive a0 f, thereby decreasing the absolute value of negative total acceleration. These results can be attributed to the fact that an increase in h makes the particle displacement increase slowly at the early period and then increase quickly at the middle and late periods of the impact, thereby increasing the particle penetration depth. Fig. 6(a) shows that the change in D does not affect the value and evolution of a0 s but greatly reduces the negative a0 f, resulting in a large absolute value of the negative total acceleration at the early period. However, an increase in D increases the positive a0 f, resulting in a small absolute value of the negative total acceleration at the middle and late periods. These results are because an increase in D enlarges the growth of particle displacement and finally increases the particle penetration depth. In summary, an increase in Re weakens the acceleration component of surface tension and enhances that of fluid force. The two effects cancel each other out, resulting in the slight influence of Re on the particle movement during critical sinking. An increase in h mainly weakens the acceleration component of surface tension at the middle and late periods to affect particle movement, resulting in a large penetration depth, whereas an increase in D mainly weakens that of fluid force at the early period to affect particle movement, causing a larger penetration depth. Fluid force and surface tension work with particle sinking. The effects of the three dimensionless parameters on the work of the two forces are determined by their combined effect on the evolution of the two forces and particle displacement. Fig. 6(b), (d), and (f) show the relative works of the two forces over time during

24

B. Ji et al. / Chemical Engineering Science 207 (2019) 17–29

0.1 0.0 0.0 -0.2 -0.1

Re=100 Re=200 Re=500 Re=1000

a'

W''

-0.4

-0.2

-0.6 -0.3 a's -0.4

a'f 0

1

2

3

Re=100 Re=200 Re=500 Re=1000 4

W''s W''f

-0.8 -1.0

5

0

1

2

T

3

(a)

5

(b)

0.50

0.25 W''s

0.25

0.00

0.00

-0.25

W''f

θ=90° θ=120° θ=150° θ=180°

W''

a'

4

T

-0.25

-0.50

θ=90° θ=120° θ=150° θ=180°

a's -0.50 -0.75

a'f 0

2

4

6

T

8

-0.75

10

-1.00

0

2

4

(c)

6

T

8

10

(d)

0.2 0.00

W''s W''f

0.0 W''

a'

-0.25 -0.2

-0.50

-0.6

a'f

D=1 D=2 D=5 D=10

4

6

a's

-0.4

0

2 T

(e)

D=1 D=2 D=5 D=10

-0.75

-1.00

0

2

4

6

T

(f)

Fig. 6. Dominant forces and their relative work during critical sinking with different dimensionless parameters of (a), (b) D = 5 and h = 120°; (c), (d) D = 1 and Re = 500; (e), (f) h = 120° and Re = 500.

critical sinking under different Re, h, and D, respectively. The work of fluid force is mainly performed at the early period because fluid force mainly acts during this period, whereas the work of surface tension is mainly exerted at the middle and late periods for the same reason. W00 f is negative throughout the impact. It steadily decreases at the slamming stage and reaches its maximum negative value and then slightly increases for a long period until the particle submerges. W00 s slightly increases and reaches its maximum positive value at the slamming stage, decreases to a positive value, and continues decreasing to near its maximum negative value at the cavity developing stage. Fig. 6(b) shows that as Re

increases, W00 s increases and W00 f decreases because of the effect of Re on these two forces. As illustrated in Fig. 6(d), an increase in h decreases the positive W00 s and increases the negative W00 f at the early period of the impact. Although an increase in h weakens the negative a0 s at the middle and late periods, it also increases the particle displacement at the middle and late periods, thereby increasing the work of surface tension at the middle and late periods, increasing the absolute value of final negative W00 s, and decreasing the corresponding absolute value of final negative W0 0 f. Fig. 6(b) reveals that an increase in D increases W00 s and decreases W00 f during critical sinking. The effect of D on W00 f mainly occurs at

B. Ji et al. / Chemical Engineering Science 207 (2019) 17–29

the early period because an increase in D mainly weakens the dimensionless component of fluid force during the early period. The effect of D on W00 s mainly arises during the middle and late periods because an increase in D increases the particle displacement during the middle and late periods, which is the time when surface tension dominates. Fig. 7 shows the proportion of surface tension-accumulated work to total work during critical sinking at different conditions, which reflects the proportion of particle initial kinetic energy consumed by the work of surface tension. The proportion of surface tension-accumulated work increases as Re, D, and h increase. The accumulated work of surface tension exceeds 50% of the total work in most cases. The effect of h on the proportion of surface tensionaccumulated work is more significant compared with the two other dimensionless parameters. For the critical sinking of superhydrophobic particles (h  150°), the accumulated work of surface tension accounts for over 73% of the total work. Notably, for common hydrophobic particles (h  120°), the proportion of the accumulated work of surface tension is <80% even if D reaches 10, and distributes between 33% and 77% for the particle with a common density of D  2. This finding indicates that for the impact of hydrophobic micron particles in natural and industrial processes, the contribution of fluid force to the energy transformation of particle kinetic energy cannot be ignored. Thus, the work of fluid force must be considered when distinguishing whether the particle will submerge based on energy balance. 3.3. Criterion for critical sinking Eq. (9) is derived from the dimensionless energy conservation equation, describing the relationship of particle kinetic energy to the works done by the two forces during critical sinking. The surface tension-accumulated work and fluid force-accumulated work of the critical sinking process must be determined to distinguish different impact modes with Eq. (9). Given Eq. (10),

W 0st ¼ X m

Z 0

1

F 0s dX 0 ;

ð12Þ

where F 0s ¼ sin/sinðh þ /Þ. As illustrated in Fig. 4(d), the relationship between / and X0 is only related to h, indicating that the relationship between F0 s and X0 is also only related to h. As shown in Fig. 8(a), the effect of h on the evolution of F0 s with X0 is highly significant. When h increases from 90° to 180°, the maximum value of F0 s increases from 0.5 to 1, and the integral area of F0 s on X0 also significantly increases. The effect of Re and D on the evolution of F0 s with X0 can be neglected in comparison with the effect of h,

1.0

Wst/(Wst+Wft)

0.8

25

revealing that the integral on the right of Eq. (11), that is, W0 st/Xm, is only a function of h. As exhibited in Fig. 8(b), the values of W0 st/ Xm coincide well with each other at the same h and increase as h is increased. According to Eqs. (9), (3), and (5), W0 st/Xm must be a trigonometric function of h. We fit these data with MATLAB R2016a and propose the following equation:

W 0st =X m ¼ ð0:232  0:608 cos hÞ;

ð13Þ

Eq. (13) is plotted in Fig. 8(b) with a dotted line, which is in good agreement with the simulation results, and its correlation index is R2 = 0.996. We can deduce from Eqs. (11) and (13) that when h increases from 90° to 180°, the actual accumulated work of surface tension increases by three times. Given that X m / D0:081 , then W 0st / D0:081 , which means that W0 st only increases by 20% when D increases by an order of magnitude. Compared with the effect of h, that of D on W0 st is insignificant. Re exerts a negligible effect on W0 st and is thus excluded in Eq. (13). RX The fluid force work can be expressed as W 0ft ¼ 0 m C f U 2 dX based on Eqs. (10) and (5). Considering the complexity of the evolution of fluid force and particle displacement, we analyse the influence of Re, D, and h on W0 ft to obtain its expression form. The influence of Re is first analysed. Re exerts negligible effects on the dimensionless displacement and the corresponding particle velocity during the critical sinking process, as previously discussed. Thus, the influence of Re on W0 ft is mainly reflected in the effect on Cf, which varies with the contact area of particle and liquid and particle velocity during impact. The evolution of W0 ft with Re at different conditions is plotted in Fig. 9(a) in logarithmic coordinates, indicating that W0 ft significantly decreases with Re, and a power relationship exists between them, that is, W 0ft / Rea . This relationship coincides with the power relationship between the resistance coefficient and Re in classical fluid mechanics. D and h exert minimal influence on the index, indicating that a is constant. D not only affects the evolution of particle displacement X and velocity U but also the value of Cf, leading to a complex influence of D. Herein, W0 ft is still expressed as the power function of D for brevity, that is, W 0ft / Rea Db . Fig. 9(b) illustrates that W0 ft increases with increasing D, and h significantly influences b, which means that b must be a function of h, that is, W 0ft / Rea DbðhÞ . Similarly, h also demonstrates a complex influence on W0 ft, as reflected in the variation of W0 ft at different conditions presented in Fig. 9(c). A trigonometric term about h is introduced to reflect the effect of h, that is, W 0ft / Rea DbðhÞ f ðhÞ, where b(h) and f(h) are the trigonometric functions of h. The changes in surface energy are positive as the particle submerges because of the hydrophobicity of the particle. For a particle with a D approaching 0, its initial kinetic energy is near 0, indicating that it cannot overcome the positive changes of surface energy and sink into the liquid (Verezub et al., 2005), that is, when D ? 0, We ? 1, which requires W0 ft ? 0. Therefore, the fluid force

0.6

work is expressed as W 0ft ¼ aRea DbðhÞ f ðhÞ. The simulation results are fitted with MATLAB R2016a, and the following equation is obtained.

0.4

W 0ft ¼ 1:968Re0:35 D½0:3250:112 sin ð2hþ0:794Þ ð1  0:127 cos 2hÞ:

θ=90° θ=120° θ=150° θ=180°

Re=100 Re=200 Re=500 Re=1000

0.2 0.0 0

2

4

6

8

10

D Fig. 7. Proportion of surface tension-accumulated work to total work during critical sinking at different conditions.

ð14Þ

Eq. (14) is plotted in Fig. 9(d) with a dotted line, which is in good agreement with the simulation results, and its correlation index is R2 = 0.959. The power relationship between W0 ft and Re with a constant index illustrates that the fluid force also possesses a specific power relationship with Re, which is almost unaffected by h and D. Notably, many researchers assume the flow as a potential flow for an impact process with Re > 100. However, Eq. (14) shows that W 0ft / Re0:35 , which decreases W0 ft by 55.3% when Re increases by an order of magnitude. This finding indicates that the contribution

26

B. Ji et al. / Chemical Engineering Science 207 (2019) 17–29

1.0

1.0

Eq. 13

-sinφsin(θ+φ )

0.8

W'st/Xm

0.5

0.0

-0.5 0.0

0.2

D=1 D=2 D=5 D=10 0.4

Re=100 Re=200 Re=500 Re=1000 0.6 0.8 X'

θ=90° θ=120° θ=150° θ=180° 1.0

0.6 Re 0.4

1000 500 200 100

D=1 D=2 D=5 D=10

0.2 0.0 90

120

150 θ (degree)

(a)

180

(b)

Fig. 8. Evolutions of (a) sin/sinðh þ /Þ with X0 and (b) W0 st/Xm with h during critical sinking process at different conditions.

D=1 D=2 D=5 D=10

Re=100 Re=200 Re=500 Re=1000

1

W'ft

W'ft

1

θ=90° θ=120° θ=150° θ=180°

0.1

0.1 100

300 Re

500

700 900

1

2

3 D

(b)

(a) D=1 D=2 D=5 D=10

Re=100 Re=200 Re=500 Re=1000

W'ft

0.9 0.6

0.5 0.4

0.2

10 5 2 1

Eq. 14 0.1

0.35

0.3 0.0 90

D

θ=90° θ=120° θ=150° θ=180°

0.3

W'ft / f(θ, D)

1.5 1.2

4

θ=90° θ=120° θ=150° θ=180° 5 6 7 8 9 10

1

120

150 θ (degree)

180

100

300 Re

(c) Fig. 9. Evolution of W0 ft with (a) Re, (b) D, f ðh; DÞ ¼ 1:968D½0:3250:112sinð2hþ0:794Þ ð1  0:127cos2hÞ.

500

700 900

(d) and

(c)

h

during

critical

of viscosity to the fluid force work cannot be neglected because the real-time Re decreases with the deceleration of particle though the initial Re is large. The assumption of potential flow causes a large derivation in the calculation of fluid force work. Fig. 10(a)–(c) show the variation of Wecr (dot) with dimensionless impact parameters. In Fig. 10(a), Wecr decreases with increase in D. As discussed before, D affects both Wst and Wft. The larger D is, the larger the penetration depth is, which makes Wst increase.

sinking.

(d)

A

comparison

of

Eq.

(14)

with

numerical

results.

Here,

Increase in D also increases Wft with the same impact velocity. Thus, the sum of surface tension work and fluid force work increases with increase in D, which means increase in initial impact kinetic energy for critical sinking. On the other hand, the initial impact velocity u0 decreases with increase in D for particles with the same initial kinetic energy, i.e. u20 / D1 . The trade off between the effects of D on the total work and on the initial impact velocity results in that ucr and corresponding Wecr decrease with

27

B. Ji et al. / Chemical Engineering Science 207 (2019) 17–29

75

100 Re=100 Re=200 Re=500 Re=1000

10

Re=100 Re=200 Re=500 Re=1000

60

D=1 D=2 D=5 D=10

Wecr

Wecr

45 30

1

θ=90° θ=120° θ=150° θ=180°

0.92

15

1 0 90

0.1 1

2

3

4

5 6 7 8 9 10

120

D

150 θ (degree)

(a)

(b) 20

D=1 D=2 D=5 D=10

θ=90° θ=120° θ=150° θ=180°

Wecr

30 20

Wecr(Criteria)

50 40

Exp. 1 [18] Exp. 2 [37] Exp. 3 [37]

15

C0 C1[17] C2[17] C3[7] Eq. 15

10

5

10 0

180

0

0

200

400

600 Re

800

1000

0

5

(c)

10 15 Wecr(Experiment)

20

(d)

Fig. 10. Comparison of Wecr calculated by Eq. (15) (line) with numerical results (dot) at different (a) Re, (b) D, and (c) h. (d) Comparison of Wecr calculated by Eq. (15) and existing criteria (Wang et al., 2015; Kaptay, 2001) shown in Table 1 with experimental results in literatures (Wang et al., 2017; A. Wang, Research, 2016). The horizontal coordinate represents the Wecr of experiment, whereas the longitudinal coordinate represents the Wecr calculated by criteria with the same impact parameters.

increase in D. In Fig. 10(b), Wecr increases with increase in h. The effect of h on Wst is more significant than its effect on Wft. The larger h is, the lager Wst is, resulting in a lager ucr and corresponding Wecr. In Fig. 10(c), Wecr increases with increase in Re. The decrease of Re results in the increase in fluid force, while has little effect on particle displacement and surface tension. Thus, Wft increases and Wst changes little, which make ucr and corresponding Wecr increase. The smaller Re is, the faster Wecr increases, indicating the significant viscous effect at small Re. Substituting Eqs. (11), (13), and (14) into Eq. (9), we can obtain the critical sinking criterion of the hydrophobic micron particle:

Wecr ¼

5:553ð1  2:614coshÞð1 þ 0:074 cos2hÞ D0:919  2:952Re0:35 D½0:2440:112 sin ð2hþ0:794Þ ð1  0:127cos2hÞ

:

ð15Þ For a specific impact process, ucr can be calculated with Eq. (15) when the physical properties of particles and fluids and are known. If u0  ucr, particle will sink into the fluid after impact, or the particle will not sink into the fluid. In this way, different impact modes of micron particles can be distinguished with Eq. (15). Fig. 10(a)–(c) show the evolution of Wecr (line) calculated using Eq. (15) with different dimensionless parameters. For the impact conditions considered by this study, the distribution of Wecr is between 0.4 and 50. Eq. (15) predicts the Wecr of our simulation cases very well, and its correlation index is R2 = 0.994. Fig. 10(d) shows a comparison of Wecr calculated using Eq. (15) with experimental results in literatures (Wang et al., 2017; Wang, 2016). For

the impact processes reported in literatures, the prediction error of Eq. (15) for the critical impact velocity is <7%, which demonstrates the reliability of our criterion. The three criteria for the critical sinking of micron particles reported in literatures are listed in Table 1. C1 (Kaptay, 2001) neglects the work of fluid force and assumes that the gas–liquid interface is not deformed during sinking. On the basis of C1, C2 (Kaptay, 2001) considers the work of fluid force calculated with the classical resistance coefficient for singlephase flow around a sphere. C3 (Wang et al., 2015) neglects the work of fluid force and calculates the TPCL position based on the quasi-static assumption of the gas–liquid interface. The criterion proposed in this study presented in Eq. (15) is denoted by C0. We take the experimental results in literatures (Wang et al., 2017; Wang, 2016) to compare the prediction precisions of these four criteria, as shown in Fig. 10(d). The prediction error of the three criteria in literatures shown in Table 1 for the critical impact velocity is between 12.4% and 47.5%, which is considerably greater than the prediction error of C0 (Eq. (15)). This result illustrates that the Table 1 Criteria for the critical sinking of micron particles reported in literatures (Wang et al., 2015; Kaptay, 2001). Criterion ID

Expression of criterion

Error for ucr

3ð1coshÞ2 D

37.2%–47.5%

C1

Wecr ¼

C2

3ð1coshÞ Wecr ¼ 2D3ð0:148þ1:13Re 1=2 þ9:71Re1 Þ

C3

2

Wecr ¼

cos4 ðh=2Þð1624cosðh=2ÞÞþ3sin4 ðh=2Þð0:339lnBoÞ2 D

25.4%–36.3% 12.4%–29.4%

28

B. Ji et al. / Chemical Engineering Science 207 (2019) 17–29

works by surface tension and fluid force should be considered and accurately calculated for the prediction of particle impact mode, and unreasonable assumptions lead to large errors. Compared with the existing criteria, Eq. (15) is proposed based on a direct, detailed, and accurate numerical simulation method by which the evolution of particle displacement, forces, and work is calculated precisely. Thus, our criterion reflects the main control mechanism of particle’s critical sinking and the influence of each impact parameter on the critical sinking process, applied to a wide scope. 4. Conclusion In this study, the critical sinking behaviour of a hydrophobic micron particle under different impact conditions is numerically investigated. The effects of Re, h, and D on the movements of particle and TPCL, forces acting on the particle, and energy transformation during the critical sinking process are analysed, and the following conclusions are obtained. The critical sinking process of a micron particle can be divided into slamming, cavity developing and cavity collapse stages. The motion characteristics of particle and TPCL during each stages are similar at different impact parameters. The penetration depth of the particle during critical sinking increases with the increase in h and D, but is almost unaffected by Re. The relationship between the TPCL angle and the normalised particle displacement is almost unaffected by Re and D, which means that the TPCL motion shows a universal behaviour at the same contact angle. Fluid force mainly works during the early period, whereas surface tension mainly works during the middle and late periods of critical sinking. An increase in Re weakens the role of surface tension and enhances the role of fluid force, and the two effects cancel each other out, making the particle motion during critical sinking unaffected by Re. An increase in h mainly weakens the role of surface tension and the particle’s deceleration at the middle and late periods. An increase in D mainly weakens the role of fluid force and weakens the particle’s deceleration during the early period. The proportion of surface tension-accumulated work to the total work increases as Re, h, and D increase. For the common impact process of hydrophobic micron particles, the accumulated works of fluid force and surface tension are at the same order and play an important role in the conversion of particle kinetic energy. A dimensionless energy conservation equation is constructed for critical sinking. The relationships between the dimensionless accumulated works of the two forces and the three dimensionless impact parameters are analysed, and the semi-empirical expressions of dimensionless accumulated works of surface tension and fluid force are provided, based on which the expression of critical sinking criterion is proposed. The criterion is in good agreement with the simulation results. The critical Weber number increases as Re and D decrease and h increases. Our criterion reasonably considers the fluid force, the TPCL movement, and other factors. Thus, the accuracy of this criterion is significantly higher than that of previously reported criteria, and the prediction error is <7% for the impact processes reported in literatures. Declaration of Competing Interest The authors declare that there is no conflict of interest. Acknowledgements This work was supported by funds from the National Key Research and Development Program of China (2017YFC0210704) and the National Natural Science Foundation of China (51576109).

Appendix A. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.ces.2019.06.009.

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