Experimental and Numerical Study on Dynamic Characteristics of Droplet Impacting on a Hot Tailings Surface

Abstract

It is quite important to clearly understand the fluid dynamic process of water droplets impacting on a hot extracted titanium tailings surface for the recovery and utilization of tailings. In this research, the finite volume simulations of the droplet impingement were developed by applying the volume-of-fluid method and were validated against experimental results. Droplet-impact experiments were performed by using a high-speed camera. The effects of the Weber number, surface tension value, and contact angle on the spreading factor were quantitatively investigated, and the distributions of pressure, velocity contour, and temperature were analyzed in detail during the impact process. We found that the numerical results of the impact process and spreading factor conform to the experiments. Moreover, the surface tension, contact angle, and Weber number have important consequences for droplet dynamic characteristics. Finally, a new maximum spreading factor model that is governed by Weber numbers is proposed for the extracted titanium tailings surface based on the experimental and numerical results. These findings provide a pathway for controlling dynamic interactions of the droplets impacting on the tailings’ surfaces.

1. Introduction

The collision of the droplet on a hot surface is encountered in many engineering fields [1,2,3,4,5]. Examples include spray cooling of nuclear reactors, fuel injection and metal surfaces in particular. Vigorous heat and mass exchange occurs when the droplet impacts the surface. Moreover, thermodynamic and fluid dynamic behaviors during the impact are governed by the properties of the droplet and surface [6], such as the viscosity, density, and surface tension of liquid [7,8,9], surface temperature [10], surface roughness [11], and so on. The research on the impact behaviors of the droplets on hot surfaces can reveal the influence of surface and droplet properties on the droplet morphology and phase change. It also has important scientific significance and engineering value to further understand the heat transfer capacity between the droplets and the hot surface.

Numerous investigations indicate that the Weber number and surface temperature are the significant factors affecting fluid dynamics and heat transfer of the droplet [12,13]. The We is referred to as the Weber number, which is applied to assess the ratio of droplet inertial force and surface tension. It can be calculated as follows [14]: $We=\rho v_0^2{D}_0/\sigma$, where $D_0$, $\rho$, $v_0$, and $\sigma$ represent the initial droplet diameter, density, impact velocity, and surface tension value, respectively. The dynamic behaviors of the droplet on a heated solid surface have been researched by numerical simulation and experiments. Tran et al. [15] experimentally investigated a liquid-droplet impact on the heated surfaces above the boiling point. The impact behaviors were observed when the droplet contacts the surface at variable conditions of Weber number and surface temperatures. Quéré [16] discussed the impact behaviors of the droplet on a hot solid surface and quantitatively analyzed the influence of surface temperature and the Weber number on residence time and spreading diameter of the droplet. Rioboo et al. [17] studied the impact process of droplets impacting on the dry solid surface, using a high-speed camera, and quantitatively investigated the effect of experimental parameters on the spreading characteristics. The surface wettability and roughness have a significant influence on the droplet spreading. Ogata et al. [18] observed the impact phenomenon of a droplet on the steel-surface-coated SiO₂ nanoparticles through a high-speed camera. Cossali et al. [19] used photographic technologies to obtain the droplet morphology and secondary droplets’ formation. Naveen et al. [20] experimentally explored the effect of the droplet temperature on the impact morphology and spreading characteristics of the droplet impingement on the surface with different hydrophobicities.

With the rapid development of Computational Fluid Dynamics (CFD) calculation methods, numerical simulations have more applications in the field of gas–liquid two-phase flow and heat transfer. Diverse numerical models have been implemented to locate the gas–liquid interface, such as phase-field methods, volume of fluid (VOF) surface-tracking methods, level-set methods, etc. Ge et al. [21] numerically investigated the deformation and heat transfer between the droplet and the hot surface by using the level-set model with finite volume method. They studied the effect of the degree of droplet subcooling on heat transfer and liquid-film thickness. Strotos et al. [22] implemented a heat-transfer equation in VOF simulations. The Weber numbers, droplet sizes, surface temperatures, and material properties are considered to simulate the droplets’ impact process. Jeong et al. [23,24] investigated the dynamics behavior of a particulate droplet on a solid surface, using the level-set method. They concluded that the decrease in kinetic energy of the particulate droplet with energy dissipation and the viscosity increase occurred on account of the presence of particles. Malgarinos et al. [25] performed a simulation of liquid-droplet impingement normally on a solid surface by combing a wetting force model with the VOF method. They concluded that this model is a significant way to overcome the need for a predefined dynamic contact-angle law. Wang et al. [26] established the numerical model of the collision between the oil droplet and the surface based on the VOF method and discussed the effects of droplet diameters, impact velocity, and incident angle on the spreading length and the spreading width. Guo et al. [27] conducted a two-dimensional simulation by combing level set and VOF; they found that the droplet crown thickness increases with the film thickness. Jin et al. [28] combined the finite volume method with the volume of fluid (VOF) method to obtain the evolution of crown rim diameter, crown height, and thickness of the film at different Weber numbers. Abubakar et al. [29] conducted the experiments to measure the effect of droplet diameters on the impact properties and simulated the pressure distributions inside droplets during impact process.

Previous research studies have mainly been made on the droplets impacting conventional metal surfaces [30,31,32]. However, few studies have looked into the dynamic characteristics of the droplet impact on the extracted titanium tailings’ surface [33], since the collision of droplets on a tailing’s surface has a specific importance in cooling processes and resource utilization of the metallurgical industry. The primary objective of this research is to ascertain the hydrodynamic behaviors of water-droplet impact on a heated tailing’s surface by experiments and CFD simulation. In this paper, the dynamic behaviors of the droplet impingement on the hot tailings’ surface are observed by using the high-speed camera. The experimental setup is illustrated in Section 2. However, it is difficult to fully understand the complexity of dynamic characteristics in the impact process. Therefore, a numerical model was established by using the VOF method to simulate the droplet dynamics behaviors and spreading characteristics. Section 3.1 describes the momentum, the continuity, and the mass fraction conservation equations. The numerical method and boundary conditions used to solve these equations are reported in Section 3.2. A grid refinement is employed near the surface to precisely track the gas–liquid interfacial area. In order to validate the CFD simulation of the droplet-impact dynamics, comparisons between the numerical results and the available experimental measurement of droplets impacting on the tailings’ surface are presented in Section 4.1. The distribution of temperature, pressure, velocity vector, and flow field is discussed and analyzed in Section 4.2. Two kinds of dynamic behaviors with different Weber numbers are observed, including rebound and splash. The effects of droplet diameters, contact angles, and surface-tension coefficient on the spreading factors are investigated in Section 4.3. Finally, the predictions of the maximum spreading factors against the Weber number are discussed in Section 4.4.

2. Experimental Procedure

Figure 1 illustrates a diagram of the present experimental apparatus for observing interfacial behaviors and detecting reactor mass. The extracted titanium tailing used in this experiment was supplied by Pangang Group (Panzhihua, China). Extracted titanium tailings were prepared to form a cylindrical sample with a size of 3.0 cm × 3.0 cm and a thickness of 1.0 cm. Firstly, the cylindrical sample was placed on a reactor made of silica glass and then heated in a heating platform. The sample was lighted by a light source. The experiment begins when the sample temperature (T_s) has reached 573 K. Secondly, the syringe pump is used to feed the needle water with a volume rate of 1 mL/min. The height of a needle from the sample surface is adjusted by a liftable stand to control the initial impact speed. A single water droplet is ejected from a needle and detaches when the gravity is dominant over the surface tension. Finally, the behavior of the water droplet impingement on the hot surface is captured by a high-speed camera. The photographic observations are recorded on the computer. Each experiment under different conditions was performed repeatedly at least three times. The translation distance of the droplet during 1 ms is calculated to obtain the impact velocity. The initial impact velocity ranges from 0.63 m/s to 1.40 m/s, corresponding to the Weber numbers from 31.37 to 92.65. The droplets interaction with the extracted titanium tailings surface is observed by a high-speed camera (IDT Y4-S1) with a 35 mm KOWA macro lens at f/1.4 aperture, which has an image frequency of 2000 fps (frame per second) at the full resolution of 1024 × 786 pixels. The camera is placed horizontally, parallel to the tailings sample surface. The image sequences of the dynamic droplets are processed by Image-Pro plus (version 6.0.0.260, Media Cybernetics Corporation, Rockville, MD, USA) software. The droplet sizes can be obtained by pixel analysis of the captured images.

The uncertainty in the experiments is mainly caused by the measuring errors of impact conditions of the droplet and the surface temperature, as well as uncertainties in the thermocouple location. The thermocouple is installed 1 mm below the middle of the surface. The maximum absolute uncertainty of surface temperature is ±0.6 K. The imaging pixel resolution of the high-speed camera is calculated by observing an object with known length and measuring the length in pixels of the image. According to the error estimation of Negeed et al. [34], the error in the pixel resolution is approximately ±0.021 µm/pixel. The measuring error of the droplets’ diameter is ±0.03 mm. The droplet impact velocity is calculated by using the sequence of two frames of the images, and the measuring error of the droplets’ velocity is probably ±0.01 m/s.

3. CFD Simulation

3.1. The Governing Equations

The VOF method is used for interface tracking, which is applicable to simulating fluid flow between two immiscible fluids. In this model, the volume fraction, $\alpha$, is expressed as the volume percent covered by the liquid phase. The $\alpha =0$ indicates that the cell is covered by air, and $\alpha =1$ means a cell filled with liquid. The conservation of momentum and mass is satisfied by solving the Navier–Stokes equation for an incompressible flow.

The volume fraction advection, mass conservation equation, and momentum conservation equation can be written as follows:

$$\frac{\partial \alpha }{\partial t}+\nabla \cdot \left(\overrightarrow{v}\alpha \right)=0$$

(1)

$$\frac{\partial }{\partial t}\left(\rho \right)+\nabla \cdot \left(\rho \overrightarrow{v}\right)=0$$

(2)

$$\frac{\partial }{\partial t}\left(\rho \overrightarrow{v}\right)+\nabla \cdot \left(\rho \overrightarrow{v}\overrightarrow{v}\right)=-\nabla p+\nabla \cdot \left[\mu \left(\nabla \overrightarrow{v}+\nabla {\overrightarrow{v}}^T\right)\right]+\rho g+{\overrightarrow{F}}_s$$

(3)

where $\overrightarrow{v}$ is the velocity vector; p is the pressure; T is the temperature; ${\overrightarrow{F}}_s$ is the surface-tension term; $\rho$ and $\mu$ are the density and viscosity, respectively; and g is the gravitational acceleration.

The fluid properties are volume-fraction-averaged, which can be further defined as follows based on the volume fraction value of the cell:

$$\rho =\alpha {\rho }_l+\left(1-\alpha \right){\rho }_g$$

(4)

$$\mu =\alpha {\mu }_l+\left(1-\alpha \right){\mu }_g$$

(5)

where ${\rho }_l$ is the liquid density, ${\rho }_g$ is the air density, ${\mu }_l$ is the liquid viscosity, and ${\mu }_g$ is the air viscosity.

Surface tension plays an important role in the dynamic behaviors of the droplet impact on the solid surface. Brackbill [35] used the continuum surface force model (CSF) to transform the surface tension into the body force; the effect of surface tension is regarded as the source item of the momentum equation:

$${\overrightarrow{F}}_s=\sigma \frac{\rho \kappa \nabla \alpha }{\left({\rho }_l+{\rho }_g\right)/2}$$

(6)

where σ and $\kappa$ represent the surface tension value and surface curvature, respectively.

Moreover, the prescribed dynamic contact angle, $\theta$, is determined as the boundary condition, and a method is applied to express the surface unit vector, $\widehat{n}$, at the wall boundary as follows:

$$\kappa =\nabla \cdot \widehat{n}$$

(7)

$$\widehat{n}={\widehat{n}}_w\mathit{cos}\theta +{\widehat{n}}_t\mathit{sin}\theta$$

(8)

where ${\widehat{n}}_w$ and ${\widehat{n}}_t$ are the unit vectors to the wall normal and tangent, respectively.

3.2. Numerical Model and Boundary Conditions

The finite volume method is carried out to simulate the droplet impact process in the 2D axisymmetric computational domain by using the commercial software ANSYS Fluent 2021 R1 [36]. The structural and uniform staggered grid is generated in the mesh-generating tool ICEM. The computational domains are set to be 20 × 15 mm. The domain consists of 30,000 structured computational cells with a refined mesh. Figure 2 represents the regional mesh refinement of the gas–liquid interface. The pressure-outlet conditions are used in other boundaries. The droplet velocity is defined as V. The direction of gravitational acceleration is vertically toward the X axis. The transient pressure and velocity are coupled by executing the PISO (Pressure-Implicit with Splitting of Operators) algorithm, and the body force-weighted discretization was imposed on the pressure equation. The QUICK (Quadratic Upstream Interpolation for Convective Kinematics) scheme is used for the discretization of the momentum equation, and the Geo-reconstruction scheme is applied to discretize the VOF function. The time step is set to be 0.05 ms in the numerical simulation. The convergence criterion is that the residuals of all equations are less than 10⁻⁴.

In the simulation, the fluid domain around the droplet is air, and water is chosen as the material of the droplet. The thermal properties of the surface material are assumed to be the same as the extracted titanium tailings. The physical properties of materials in the simulation are listed in

Table 1. Physical parameters of materials at 298 K.

Materials	Air	Water (Liquid)	Extracted Titanium Tailings
Density, kg·m−3	1.225	998.2	2600
specific heat, kJ·kg−1·K−1	1.006	4.182	0.74
thermal conductivity, kw·m−1·K−1	0.024	0.600	1.38
Viscosity, Pa·s−1	1.790 × 10−5	1.005 × 10−3	-

. The temperature of the solid surface and air is set at 573 K. The ambient pressure, $P$, is 1 atm. The conditions of Weber numbers (We) are chosen similar to the experiments, ranging from 30 to 130. Considering the impingement of a water droplet onto both hydrophilic and hydrophobic surfaces, the range of contact angles is between 0° and 180°. The volume fraction of the droplet, $\alpha$, before impact is 1. The initial diameter of droplets, D₀, increases from 1 mm to 4 mm. The surface-tension value is $\sigma$ = 0.07 N/m, and gravity is g = 9.8 m/s². The droplet density, ${\rho }_l$, is 998.2 kg/m³; dynamic liquid viscosity, ${\mu }_l$, is 1.005 × 10⁻³ P·s; gas density, ${\rho }_g$, is 1.225 kg/m³; and dynamic gas viscosity, ${\mu }_g$, is 1.79 × 10⁻⁵ P·s.

4. Results and Discussion

4.1. Comparison between Experiments and Simulation

Figure 3 displays the experimental observations and simulated morphology of a water droplet with a diameter of 1 mm impact on the extracted titanium tailings’ surface at T_s = 573 K. The corresponding Weber number is 100. We observed a good agreement in the impact process between the results of the experiments and simulation. At the beginning of a water-droplet impact on the hot surface, the droplet spreads out quickly and reaches the maximum spreading factor at t = 2 ms. Then secondary droplets split out from the rim of the droplet and splash out at t = 3.5 ms. At t = 7.5 ms, the droplet is shattered into several smaller droplets under the effect of Rayleigh–Plateau instability [37].

The spreading factor, β, is formulated as D/D₀, where D is the diameter of the interface between the liquid droplet and solid surface, and D₀ is the initial droplet diameter. The spreading and retraction process of a water droplet on the hot surface is successfully simulated. Upon considering all the simulated cases, the droplet diameter is 1 mm; the impact velocities are 1.71 m/s, 2.09 m/s, and 2.42 m/s; and the corresponding Weber numbers are 40, 60, and 80, respectively. The temperature of surface is 573 K, and the contact angle is 100°. Figure 4 illustrates a comparison between the measured and simulated of droplet-spreading factors at different Weber numbers. It can be seen that the time evolutions of spreading factors present a similar tendency. As the droplet impact velocity increases, the maximum spreading factor becomes larger, and the time for the droplet to reach the maximum spreading factor is shorter. The spreading factor of the droplet is determined by the combination of inertial force, surface tension, liquid viscosity, and kinetic energy loss. In the initial stage of droplet spreading, the maximum spreading factor observably increases with the impact velocity due to the inertial force; as a result, the kinetic energy is lost. Since the droplet reaches the maximum spread factor, the spreading factor gradually decreases, and the effect of impact velocity is small. Above all, the time evolution of droplet-spreading factors is qualitatively consistent with the experimental results, thus verifying the accuracy of the numerical model.

4.2. Dynamic Behaviors Analysis

The simulated morphology of a water droplet at different Weber numbers during the impact process can be seen in Figure 5. The numerical results show that the droplet begins to spread after impacting on the hot surface when the Weber number is below 80 (Figure 5a, 0.5 ms), and its kinetic energy is converted to surface energy. At t = 2.5 ms, the droplet reaches the maximum spreading diameter and is in a transient stationary state. At the same time, the surface energy of the droplet also reaches a maximum. The droplet starts to retract at 3.5 ms (Figure 5a, 3.5 ms), and the three-phase contact line moves toward the center during the retraction process. The droplet completely retracts and reaches the minimum spreading factor at 5.5 ms. At t = 7.5 ms, the droplet bounces off from the surface. However, the droplet in Figure 5b appears to break up when the Weber number is above 80. Afterward, the broken secondary droplets bounce off the surface. It can be concluded that the broken limit of the Weber number for the droplet on a hot surface is We ∼ 80.

To investigate the fluid dynamics associated with a droplet impact on the hot surface, the volume fraction, velocity distribution, static pressure, and temperature of the liquid and air were analyzed at different time instants, as shown in Figure 6. In the simulation, D₀ = 1 mm, T_s = 573 K, and V = 1 m/s. At the early stage of the impact process (2 ms), the droplet begins to spread rapidly, and the velocity of the three-phase junction is the largest. Meanwhile, pressure at the bottom of the droplet is relatively large due to the extrusion of the surface, and a negative pressure zone appears in the upper part of the droplet. It can be seen from the distribution of the velocity contour that small swirls appear on either side above the droplet and further disturb the three-phase junction. Moreover, the temperature distribution of the droplet is in the equilibrium stage, while the temperature of the vapor on both sides of the droplet increases on account of the specific heat capacity less than that of water. At t = 6 ms, the droplet reaches its maximum spreading factor; as shown in Figure 6b, the high-pressure zone transfers from the center to the rim of the droplet. The droplet will undergo a recoiling motion under the influence of surface tension. At t = 8.5 ms (Figure 6c), the droplet breaks up into two secondary droplets during the recoiling stage. A lateral pressure difference is developed between the center of two secondary droplets. This pressure difference is the reason for the recoiling regime of the droplet. Simultaneously, the main velocity contour between secondary droplets is in the up direction. However, there is a backflow near the hot surface due to the uneven distribution of the secondary droplet velocity contour. In Figure 6d, two secondary droplets recombine into one main droplet due to retraction at 10 ms. Meanwhile, a pressure gradient occurs in the radial orientation of the main droplet. It can be seen from the temperature distribution that a temperature gradient is observed in the vapor layer. This is caused by backflow motion, which can promote heat transfer between the droplet and the hot surface.

4.3. Spreading Characteristics Analysis

4.3.1. Effect of Droplet Diameters

Figure 7 shows the time evolutions of the spreading factor, β, of a droplet with different diameters of 1 mm, 2 mm, 3 mm, and 4 mm. The corresponding Weber numbers based on the droplet diameters are 31.3, 62.6, 93.9, and 125.2. In all simulated cases, the impact velocity of the droplet is 1 m/s, and the contact angle is 90°. The results show that, with the increase of droplet diameters, the maximum spreading factors increase. When the droplet diameter is 1 mm, the spreading factor reduces to 0 during the droplet spreading process, indicating that the droplets can rebound with low Weber numbers because the inertia force of the droplet is insufficient to overcome the surface tension. It also can be seen from Figure 5 that the time needed for reaching the maximum spreading factor grows with the increased diameters. The droplet diameter is proportional to the inertial force. The viscous dissipation becomes more dominant with the droplet diameter increasing, and the droplet spread is counteracted. As a result, the spreading speed of the droplet decreases with the droplet diameters.

4.3.2. Effect of Contact Angles

The droplet with a diameter of 2 mm impacting on a hot surface with a velocity of 1 m/s at different contact angles was simulated. The surface temperature is 573 K. Figure 8 illustrates the evolution of the spreading factor at different contact angles. The maximum spreading factors are 8.01, 5.45, 3.89, 3, and 2.34 for surface contact angles of 0°, 60°, 90°, 120°, and 180°, respectively. It can be seen that the maximum spreading factors and the times needed for reaching it increase with the reduction of the surface contact angles. When a droplet impacts the hydrophobic surface with a contact angle above 90°, it will leave the hot surface, and the spreading factor is 0. The hydrophobicity of the surface enhances as the contact angle increases. Therefore, the spreading of droplets on the surface will be suppressed, which manifests the decrease in the spreading factor. If the contact angle is less than 90°, the kinetic-energy loss of the droplet on the hydrophilic surface is less than it is on the hydrophobic surface. The droplets will spread out further and then retract to reach the equilibrium stage.

4.3.3. Effect of Surface-Tension Value

Figure 9 displays the time variation of the spreading factors at different surface-tension values. At the initial spreading stage (t < 1 ms), the variations of spreading factors for different surface-tension values are basically consistent. As the spread time increases, with σ = 0.04 N/m, the maximum spreading factor is 5.09 at 3.1 ms; with σ = 0.07 N/m, the maximum spreading factor is 3.55 at 2.7 ms, which means that the maximum spreading factor, and the time needed for reaching it tends to decrease with increasing surface tension value. After the droplet impacts the hot surface with a higher surface-tension value of 0.06 and 0.07 N/m, it will undergo the retraction process rapidly and oscillate repeatedly. This phenomenon can be explained that the surface tension becoming a promoting force of the droplet movement during the retracting stage.

4.4. Scaling Analysis of Maximum Spreading Factor

The maximum spreading factors obtained from the experiments are compared with the scaling laws of the maximum spreading factors in previous research studies, as shown in Figure 10. A correction to predict the maximum spreading factors, β_max, is determined as ${\mathsf{\beta }}_{\max }=cW{e}^a$ [38], where c and a are obtained by regressing experiments’ data. The scaling laws of the simulated and experimental results for the extracted titanium tailings surface are displayed and correspond to the following correction:

$${ \mathsf{\beta }}_{\max }=0.724W{e}^{0.396}$$

(9)

$${ \mathsf{\beta }}_{\max }=0.532W{e}^{0.426}$$

(10)

The predicted values of Clanet et al. [8], Wang et al. [39], and Jowkar [40] are applied for comparison. The predictions of maximum spreading factors by Clanet, Wang, and Jowkar are all lower than those of the present work. Moreover, the simulation of the maximum spreading factors is slightly higher than the experimental data. This can be explained by the intensive capillary action of the tailings’ surface, which absorbs a quantity of kinetic energy to restrain the droplet spreading on the extracted titanium tailings’ surface [41].

5. Conclusions

Impingement of a water droplet on the hot tailings’ surfaces was experimentally investigated and simulated by applying the VOF method. The effects of droplet diameters, impact velocity, contact angles, and surface tension coefficient on the spreading characteristics were numerically studied. The main conclusions are as follows:

(1)

Regarding the dynamic behaviors of droplets on the hot surface, the numerical model was validated as compared to the experimental results, using a high-speed camera. A good agreement of spreading factors with different impact velocities between the simulation and experiments was obtained.

(2)

The Weber number has an important effect on the fluid dynamics of droplets. When We is below 80, the main behaviors of droplets on the hot surface after impact include spread, retraction, and rebound. In addition, the larger Weber number (We > 80) results in the breakup and splash of the droplets after reaching the maximum spreading factor. The spreading motion and the dynamic characteristic of the droplet are mainly influenced by the pressure distribution inside the droplet. The distribution of velocity contour promotes heat transfer between the droplet and the surface.

(3)

Contact angles and surface tension coefficients are important parameters to influence the spreading characteristics. The hydrophobic property of the surface is enhanced with the increase of the contact angle, resulting in the reduction of droplet spreading factors. In addition, the maximum spreading factor decreases with the surface tension value rising at the spreading stage, and the droplet undergoes oscillation after retraction due to the larger surface tension.

The findings of the research not only give an insight into the significant effect of the contact angle and surface tension on droplet-spreading characteristics but also provide a novel pathway for engineers to establish a precise prediction method for controlling hydrodynamic interactions on the extracted titanium tailings surfaces.