Numerical Study on Spreading and Vaporization Process of Liquid Nitrogen Droplet Impinging on Heated Wall

Abstract

Micro-structured surfaces can affect heat transfer mechanisms because of enlarged specific surface areas. However, employing the Leidenfrost effect during liquid nitrogen (LN₂) droplet cooling of a heated micro-structured surface possessing a fin with a spacing much smaller than the diameter of the droplet has not yet been explored. In the present work, a direct numerical simulation (DNS) is carried out to investigate heat transfer mechanisms of the LN₂ droplet, whose diameter is sufficiently larger than the structured spacing of fin, impinging on a micro-structured surface with variable velocities. For a comparative study, a smooth surface is also employed in numerical simulations. The spreading mechanisms and vaporization behavior of the droplet along with liquid film morphology at various conditions are investigated. Results show that a smaller fin size inhibits LN₂ in entering into the grooves between the fins and left the surface untouched by the droplet completely, and eventually, a thinner liquid film is spread out in contrast to the smooth surface. Notably, at a low Weber number, the droplet can be shrunk or even rebounded away from the wall after impinging on the wall. The fastest vaporization behavior for both surfaces, namely smooth and micro-structured, is obtained at a Weber number of 180. Additionally, an effective heat transfer upon the micro-structured surface is observed at a low impinging velocity of the droplet.

1. Introduction

Spray cooling with liquid nitrogen (LN2) has substantial potential for application in the fields of high heat flow electronic equipment cooling [1,2], cryogenic wind tunnel [3], cryogenic surgery [4], and food transportation [5] due to its low evaporation temperature, accessibility, relatively low cost, high safety, and stability. The spray cooling of LN₂ atomizes the jet into tiny droplets that collide with the surface and remove heat through vaporization. Therefore, it is essential to elucidate the mechanisms of spreading and vaporization of the droplets impinging on the surface from a microcosmic view.

Most of the literature on spray cooling, droplet impingement, and vaporization has been focused on room temperature fluids as the working medium [6,7,8]. Liu et al. [9,10,11] demonstrated that increasing pressure difference can increase the uniformity of droplet distribution and reduce particle size, though the far-field spray cone angle was decreased significantly with the increasing variation in pressure. Numerical simulations have been widely used to study jet atomization in order to obtain insight into the internal flow mechanism [12]. Chen et al. [13] established a numerical model coupling the algebraic interfacial area density (AIAD) model and homogeneous relaxation model (HRM). They analyzed the internal mechanisms of LN₂ sprays such as flow pattern, pressure distribution, velocity distribution, and complete core length based on a two-fluid numerical model. Wang et al. [14] developed a numerical model to predict the heating, boiling, and flashing characteristics of LN₂ droplets and the spray patterns during the cooling process to optimize operating conditions. Ruan et al. [15] confirmed that reducing the diameter of the droplet can be a more effective approach than increasing the velocity of the droplet to improve vaporization.

An approach of passive improvement in heat transfer can be a favorable way to enhance heat transfer. The fluid dynamics change with respective surface morphology, and the corresponding heat transfer characteristics can be influenced as well.

Yao et al. [16] analyzed the static and dynamic wettability of the nanoscale hydrophobic surface, and found that the micronano surface can increase the droplet rebound height, which depends on the static contact angle. Yee et al. [17] studied the influence of groove direction, impact velocity, and surface temperature on the shrinkage dynamics of the droplet through high-speed imaging. The oscillating airflow under the droplet caused by grooves resulted in the temperature-induced anisotropic shrinkage. Qin et al. [18] reported that droplets are more likely to splash by hitting a wet surface. Zhan et al. [19] revealed a generation process of droplets when an upward spray impinging at the downward solid surface and the proportion of splashing droplets was increased with the increase in Weber number. The droplet size was proportional to the length of the most unstable Rayleigh–Taylor wave.

In general, most of the current research on LN₂ spray cooling has been based on the analysis of macroscopic flow characteristics of the spray field and the effect of the micro-structured surface on spray cooling based on the experiments of ambient fluids. There are a few literature studies on the cooling effect of a single LN₂ droplet from a microcosmic view.

Among the reported simulation methods, direct numerical simulation has been applied to combustion, reaction flow, and spray processes by virtue of its high accuracy and the ability to provide in-depth flow information of each instantaneous flow field [20,21,22,23]. In this work, the direct numerical simulation is used to analyze the spreading process of LN₂ droplets on smooth surface and micro-structured surface. The vaporization rate, residual radius, variation in temperature of LN₂ droplets at the different velocities, and respective surface morphology are quantitatively studied. The effects of Weber number and initial velocity on the collapse and vaporization of the droplets are discussed. Here, the reported mechanisms related to the microscopic flow and laws can ensure the credibility of the accurate establishment of the macroscopic multiphase flow model.

2. Numerical Model

2.1. Physical Model

The geometric model for the LN₂ droplet in free-falling state and impinging on a copper surface is shown in Figure 1. The LN₂ droplet with the initial diameter (d₀) of 2 mm width was considered at a distance (H₀) of 4 mm between the solid surface and center of the LN₂ droplet. The initial temperature (T₀) of LN₂ droplets was kept at 77 K, and the surface temperature (T_w) of copper remained constant at 300 K. The fins with a thickness of 0.4 mm, height of 0.4 mm, and spacing of 0.6 mm were designed.

Some assumptions were considered for the vaporization mechanism of the LN₂ droplet in simulation.

• The effect of temperature on physical properties was disregarded.
• The interface temperature was assumed to be at saturation temperature.
• Heat generation induced by friction was not considered.
• The gas–liquid pair velocity was small, so it could be regarded as incompressible flow.

The analysis was carried out using the dimensionless parameters, namely diameter (d*), velocity (v*), density (ρ*) and time (t*) are defined as follows:

$$d^{*}=d/d_0$$

(1)

$$v^{*}=v/v_0$$

(2)

$${\rho }^{*}=\rho /{\rho }_0$$

(3)

$$t^{*}=d^{*}/v^{*}$$

(4)

The Reynolds number and Weber number of the LN₂ droplet impacting the wall surface are as follows:

$$\mathrm{Re}=\frac{\rho ul}{\mu }=9914$$

(5)

$$\mathrm{We}=\frac{\rho u^{2}l}{\sigma }=180$$

(6)

The residual radius of droplet vaporization can be calculated from the residual liquid phase f_res:

$$r^{*}=\sqrt{f_{res}/\mathsf{\pi }}$$

(7)

where, f_res represents the volume fraction of residual liquid.

2.2. Numerical Model

2.2.1. Governing Equations

In the direct numerical simulation, the governing equations are solved directly by numerical calculation which avoids the limit of the selection of empirical parameters and the accuracy of the turbulence model. The droplet vaporization, a two-phase flow process, is described by a two-dimensional incompressible N–S equation. The momentum equation is as follows:

$$\rho \left({\partial }_{t}u+u·\nabla u\right)=-\nabla p+\nabla ·\left(2\mu D\right)+\sigma \kappa {\delta }_{s}n$$

(8)

where, u is a velocity vector, ρ is the fluid density, p is a pressure, μ is the fluid viscosity, and D is the deformation rate tensor.

$$D=\frac{1}{2}[\left(\nabla u\right)+{\left(\nabla u\right)}^T]$$

(9)

The last term in Equation (8) determines the surface tension term through surface tension coefficient (σ), surface curvature (κ), and Dirac delta function (δ_s) so that the surface tension only acts on the phase interface.

The continuity equation yields:

$$\nabla ·u=0$$

(10)

The gas–liquid continuity equation was established by using the homogeneous flow model and defined as:

$${\partial }_{t}f+\nabla ·\left(fu\right)=0$$

(11)

where, f represents the volume fraction of liquid, then the physical properties in the flow field can be obtained according to f:

$$\rho \left(f\right)\equiv f{\rho }_l+\left(1-f\right){\rho }_v$$

(12)

$$\mu \left(f\right)\equiv f{\mu }_l+\left(1-f\right){\mu }_v$$

(13)

The subscripts l and v represent liquid and gas phase, respectively.

The energy equation yields:

$$\rho c_p\left(\frac{\partial T}{\partial t}+u·\nabla T\right)=\nabla ·\left(\lambda \nabla T\right)+\Phi$$

(14)

where, T, c_p, and λ are termed as the temperature field, heat capacity of the fluid, and thermal conductivity, respectively.

In this paper, the numerical simulation software basilisk is used to solve the incompressible N–S equation by employing the geometric volume of fluid (VOF) convective method to capture the phase interface. The detailed numerical method can be seen in Popinet [24].

2.2.2. Vaporization Model

The mass transfer rate is calculated through the heat balance at the interface by using the following relation.

$$\dot{m}=\frac{\left(q_l-q_v\right)\cdot n}{h_{lg}}=\frac{\left({(\lambda \nabla T)}_l-{(\lambda \nabla T)}_v\right)\cdot n}{h_{lg}}$$

(15)

In order to simulate the phase transition process, the surface vaporization flow (Stefan flow) caused by phase change was neglected, and the change in the volume of respective phases during the phase transition process was considered. The specific method to calculate the interface displacement velocity $u_{\mathrm{I}}$ according to the mass transfer rate before employing VOF convection is to obtain the superposition velocity field with the original velocity field and restore the velocity field after completing VOF convection.

$$u_{\mathrm{I}}=\frac{1}{{\rho }_l{h}_{fg}}\left({{\lambda }_v\nabla T|}_v+{{\lambda }_l\nabla T|}_l\right)$$

(16)

The temperature field was solved accurately before calculating the interface displacement velocity. The convection term of the energy equation is handled in the same way as that of the VOF convection equation. Since the phase transition occurs at the gas–liquid interface at the saturation temperature, the Dirichlet boundary condition at the interface is realized by adding a correction source term at the interface to solve the temperature diffusion equation. It is ensured that the interface temperature remains unchanged at the saturation temperature:

$${\partial }_{\mathrm{t}}T=\nabla \cdot \left(D\nabla T\right)-\frac{T-T_{sat}}{{\tau }_c}{\delta }_S\left(x_I\right)$$

(17)

where, ${\tau }_c$ is the control time, and ${\delta }_s$ is the Dirac function $\delta$ centering the interface.

Furthermore, the temperature gradient at the interface is approximately estimated by calculating the temperature gradient on the surface of grid of the vapor and liquid phase adjacent to the interface (taking the two-dimensional case as an example), as shown in Figure 2:

$${\partial }_{\mathrm{x}}T\approx n_x^2{\partial }_{\mathrm{x}}T\left(x+\frac{\mathsf{\Delta }}{2},y\right)+n_y^2{\partial }_{\mathrm{x}}T\left(x+\frac{\mathsf{\Delta }}{2},y+\mathsf{\Delta }\right)$$

(18)

where, $n_x$ and $n_y$ are the component of the interface normal vector along the x- and y-direction, respectively, and $\mathsf{\Delta }$ is the length of the grid side. The source code for this method is visible in Magdelaine’s Sandbox [25].

In the present model, the Peclet number, a ratio of the product of specific heat capacity and superheat to the latent heat of vaporization, is introduced to characterize the weight of convection rate and diffusion rate.

$$Pe=\frac{c_p\theta }{h_{lg}}$$

(19)

2.2.3. Boundary Condition

Initially, the gas is under the static state so the left and right sides of the calculation domain are at the symmetric boundary conditions. The wall surface is defined as the lower boundary at a constant temperature, and the upper boundary is a free flow condition. The domain was considered as 10 times the initial diameter of the droplet. The physical properties of the employed fluids are shown in

Table 1. Physical properties of fluids.

77 K	$\mathit{\rho }$ $\mathbf{k}\mathbf{g}\mathbf{/}{\mathbf{m}}^{\mathbf{3}}$	μ $\mathsf{\mu }\mathbf{P}\mathbf{a}\mathbf{·}\mathbf{s}$	cp $\mathbf{k}\mathbf{J}\mathbf{/}\mathbf{k}\mathbf{g}\mathbf{·}\mathbf{K}$	σ $\mathbf{m}\mathbf{N}\mathbf{/}\mathbf{m}$	α $\mathbf{c}{\mathbf{m}}^{\mathbf{2}}\mathbf{/}\mathbf{s}$	hlg $\mathbf{k}\mathbf{J}\mathbf{/}\mathbf{k}\mathbf{g}$
LN2	807.69	162.94	2.0398	8.9601	0.000883	199.63
N2	4.4367	5.4154	1.1214	/	0.014363	/

, and the physical properties of the solid copper surface are shown in

Table 2. Physical properties of copper.

300 K	$\mathit{\rho }$ $\mathbf{k}\mathbf{g}\mathbf{/}{\mathbf{m}}^{\mathbf{3}}$	cp $\mathbf{J}\mathbf{/}\mathbf{k}\mathbf{g}\mathbf{·}\mathbf{K}$	$\mathit{\lambda }$ $\mathbf{W}\mathbf{/}\mathbf{m}\mathbf{·}\mathbf{K}$	α ${\mathbf{m}}^{\mathbf{2}}\mathbf{/}\mathbf{s}$
copper	8930	386	398	0.000115

. The thermophysical properties of the fluids were derived from the software Refprop, which is a physical property database of National Institute of Standards and Technology, NIST. The thermophysical properties of copper are derived from the Thermophysical Properties of Matter Databases, TPMD. which is a manual published by the Center for Information and Numerical Data Analysis and Synthesis, CINDAS.

The open source code software basilisk is used to directly solve the above-mentioned N–S equation. The variable impact of the characteristic spatial scale of the droplet was considered, i.e., some regions remained stable with time and showed no obvious features, while other regions contained details of the impact such as the interaction of the droplet with the wall. In order to reduce the number of mesh, the adaptive mesh refinement (AMR) method was used to reduce computing resource consumption during DNS process. The software basilisk uses quadtree/octree to achieve effective AMR to accommodate details. For each time step, when the discretization error is greater than the set value, the grid is automatically encrypted. The multistage Poisson equation was used to solve the pressure, and the complex solid boundary is represented by the VOF method.

3. Model Validation

3.1. Grid Independence Verification

The process of droplets hitting the heated wall involves vaporization and spreading, and the grid independence of the phase transition model and motion model are verified respectively.

Three kinds of adaptive mesh were used to verify the grid independence. The levels of the mesh were coarse, fine, and refined, which means that the grid encryption levels are level 7, level 8, and level 9, respectively. The corresponding grid sizes are $2\times {10}^{-9}\mathrm{m}$, $2\times {10}^{-10}\mathrm{m}$, $2\times {10}^{-11}\mathrm{m}$, respectively. A schematic of the fine mesh is shown in Figure 3. During the calculation, the software Basilisk will also perform local encryption on the mesh, and the local mesh encryption level is 10, the corresponding mesh size is $2\times {10}^{-11}\mathrm{m}$.

3.1.1. Grid Independence Verification of Phase Transition Model

The process of the LN2 droplet at 77 K impinging on the copper surface at 300 K was simulated. The remaining radius of the vaporized droplet at the different grid encryption levels is shown in Figure 4. All three kinds of mesh possess good convergence. When the encryption grade is coarse, the simulation results of the residual radius of the droplet are not accurate enough, while the results are basically consistent for the fine or refined mesh.

3.1.2. Grid Independence Verification of Motion Model

The process of the oil droplet impinging on the copper surface at room temperature was simulated, The grid of different encryption grades is shown in Figure 5. When the encryption grade is coarse, the edge of the droplet is blurred. When the encryption grade is fine, the edge of the droplet is clearer. When the encryption grade is refined, the edge and inside of the droplet are well encrypted, which can accurately show the shape and contour of the droplet. Therefore, in order to ensure the accuracy of numerical simulation and save computing resources, the mesh encryption grade was considered as the fine one.

3.2. Numerical Model Verification

3.2.1. Comparison with Experimental Results

Verification of Phase Transition Model

Experimental data obtained by Volkov et al. [26] were used to verify the numerical model. The experiment was conducted by taking into account the vaporization of a suspended water droplet with an initial diameter of about 3 mm in a hollow transparent silicon glass cylinder with a radius of 0.1 m. A hot air fan was located below the cylinder that blew wind in an upward direction at 3~4.5 m/s. The initial diameter of the water drop was 3.06 mm at 30℃ temperature and the hot air at 400 ℃ and 800 ℃, respectively, blew at speed of 3 m/s is considered for numerical simulation verification.

The compared results are shown In Figure 6. It can be seen that the vaporization rate obtained by simulation is faster than that of the experiment, resulting in a sharp decrease in diameter of the droplet. However, the variation pattern of the diameter obtained from simulation is in good agreement with experiment results, i.e., an average error of 13.08% at 400 ℃ and 1.49% at 800 ℃. The ISM algorithm used in this paper does not consider the influence of disturbance speed on the movement speed of the phase interface. Compared with the working condition of 400 °C, the temperature gradient at 800 °C is larger, and the influence of the disturbance speed is small, so the numerical calculation results are more consistent with the test results.

Verification of Motion Model

Figure 7 shows the morphological comparison of the simulation and experiment [27] when the water droplets hit the wall. In the environment of 0.1 mPa, the water droplets with an initial diameter of 2.6 mm hit a hydrophobic wall with a contact angle of 160° vertically at an initial velocity of 0.5 m/s at a distance of 3 mm from the wall. The simulation results show that the water droplets have successively experienced spreading, retraction, rebound, and peristaltic rise after hitting the wall, which is consistent with the experimental results.

Figure 8 shows the spreading coefficient trend comparison of the simulation and experiment [28] when the water droplets hit the wall, where the abscissa t* represents dimensionless time, and the ordinate d/d0 is the ratio of the spreading diameter of the liquid film to the initial diameter of the droplet. In the experiment, a single droplet with a diameter of 0.5 mm hits the surface of the iron-nickel alloy at an initial velocity of 2.32 m/s, and the droplet diffusion diameter with a static contact angle of 57° is measured. The trend of the two curves is highly consistent, and the maximum deviation does not exceed 10%, which proves the reliability of the model.

3.2.2. Comparison with Analytical Results

The vaporization of an n-decane droplet in stationary air was simulated, and the numerical model was validated according to the reported $d^2$ law [29]. By considering the assumption of uniformity and quasi-steady state, the analytic solution for the behavior of droplets can be obtained by solving the conservation equations of mass and energy. The variation in the diameter can be expressed by $d^2$ law.

$$d^2=d_0^2-Kt$$

(20)

where K is the constant determined by the physical properties of gas and liquid phase.

Figure 9 compares the numerical and analytical solutions for the $d^2$ value of the vaporized droplet. The numerical simulation calculates the diameter of the droplet based on liquid volume. It can be seen that the numerical results are quite consistent with the analytical results, and the vaporization rate is slightly higher than that of the quasi-steady state. At the end of the simulation, the $d^2$ value of droplet decreases from 3.98 to 1.4 mm².

4. Results and Discussion

4.1. LN₂ Droplet Spreading Process

4.1.1. Effect of the Micro-Structured Surface

After impinging on the wall, the droplet’s behavior is mainly affected by inertia force, surface tension, viscous force, inner wall pressure, and boundary layer resistance. The liquid viscosity and surface tension can be the critical parameter to impact the collision behavior [29]. At an appropriate initial velocity, the droplets can impinge the wall and experience the reciprocating movement of the spreading and contraction under the combination of the forces mentioned above. Under the consumption of viscous force and boundary layer resistance, the amplitude of spreading radius as well as velocity is decreased gradually and can reach the equilibrium by the combined effect of the internal pressure and surface intension.

Firstly, the process of droplet impinging on the smooth surface was simulated. The initial impact velocity of LN₂ was 1 m/s. The low surface tension and high impact velocity resulted in the fine deformation of the droplet during falling down. After hitting the bottom wall, a strong deformation occurred at the bottom surface of the droplet, while the upper part of the droplet remained in perfect shape. The liquid splashed at both ends of the droplet, and droplets started to flatten, break up, and spread out over a film that is much thinner than the initial radius of the droplet. The simulated Weber number and Reynolds number were 180 and 9914, respectively. According to the morphological determination formula given by Sommerfeld [30], i.e.,$K=\sqrt{We\sqrt{Re}}$, the calculated value of K was 133.87 that is consistent with the mode of crushing and splashing.

The spreading morphology of a droplet with a dimensionless diameter of 1 impacted the micro-structured surface with the grooves’ spacing of 0.2 mm, the height of 0.4 mm, and a thickness of 0.4 mm, as shown in Figure 10a. The small spacing between fins caused the gas to become trapped between the adjacent fins when the droplet hits the wall, and trapped gas pushed the droplet upward at the micro-structured surface. The droplets cannot fully spread out smoothly between the adjacent structures, conforming to the Cassie-Baxter model, as shown in Figure 10b. The solid area and liquid phase showed a very small contact area and failed to achieve the large contact area. Figure 11 shows the comparison of the spreading morphology between LN₂ droplets impinging on the smooth surface and the micro-structured surface, respectively. At the dimensionless time, t* = 5, both ends of the droplet retracted and curled up on the micro-structured surface, impeding the continuous spreading of the droplet. However, at t* = 9, the droplet dispersed into several large liquid films at the micro-structured surface, while the liquid films on the smooth surface remained nearly uniform and intact. The diameter of the liquid film on the smooth and the micro-structured surface varies with time, as shown in Figure 12. Before t* = 5, the dimensionless diameter of the liquid film on smooth surface and the micro-structured surface is almost the same, and the spreading diameter on the smooth surface is slightly larger than that on the micro-structured surface. Beyond t* = 5, the spreading speed of the droplet on the micro-structured surface is significantly reduced due to the contraction and breakup of the droplets into multiple liquid films, as shown in Figure 11.

When the droplet impinges the liquid film downward, the wall will exert a reaction force on the droplet, forming a certain dynamic pressure inside the droplet. The raised microstructure surface brings greater diffusion resistance, increases kinetic energy loss, and when the droplet impinges the microstructure surface, the diffusion speed of the liquid film is affected. From the analysis of the micro-structured surface, the maximum diameter and spreading speed of the LN₂ droplets is smaller than that of a smooth surface. In addition, the existence of air gaps between fins causes the LN₂ droplet to wet the surface incompletely.

4.1.2. Effect of the Thermophysical Properties

The viscosity and surface tension of the fluid have a great influence on the movement of the liquid film, and the particularity of the cryogenic fluid causes its spread to be different from that of the normal temperature fluid. The LN2 droplet and oil droplet impinging the smooth wall under the same initial conditions are simulated. In the numerical simulation, the initial diameter is 2 mm, the initial speed is 1 m/s, and the drop height is 1 mm. Figure 13 and Figure 14 show the spreading process of oil droplets and liquid nitrogen droplets impinging the smooth wall, respectively.

Different from the LN2 droplet, the oil droplet maintains a good shape during the impact of the wall. At the moment t* = 1.8, the oil droplet reaches its maximum spreading diameter, after which its edge warps and begins to contract towards the center. Until t* = 4.2, it returns to its initial shape and then rebounds off the wall. Under the same initial conditions, the surface tension and viscosity of the LN2 droplet are much smaller than that of the oil droplet. After the initial impact on the wall, the LN2 droplet continues to spread. At the moment of t* = 1.8, the LN2 droplet shows a trend of high center and low on both sides. The spreading speed is significantly higher than that of the oil droplet, and the rebound phenomenon does not occur.

Figure 15 is the comparison of the spreading coefficient of the nitrogen droplet and the oil droplet. For the oil droplet, it first spread and then retracted after impinging the wall, and the spreading diameter shows a trend of first increasing and then decreasing, reaching the maximum diffusion value at the time of t* = 1.8. For the LN2 droplet, it continues to spread after hitting the wall, the diameter of the liquid film continues to expand, the liquid film becomes thinner and thinner, and the diffusion rate gradually slows down. Viscosity and surface tension together determine the morphological change process of the droplet after impinging the wall, of which, the LN2 droplets are much smaller than the oil droplets. The viscosity of oil and LN2n is 7.15 × 10⁻², 1.6294 × 10⁻⁴ Pa·s, and the surface tension is 3.4 × 10⁻² and 8.9601 × 10⁻³ N/m, respectively.

Under the same initial conditions, the surface energy of the oil droplet is sufficient to overcome its kinetic energy, and the wall has a greater resistance to it. Therefore, after the impact of the oil droplet, the liquid film first spreads, the spreading speed is gradually smaller, and then begins to contract under the action of surface tension. In contrast, after the impact of the LN2, no contraction occurs, the kinetic energy has been dominant, and the liquid film continues to spread, but the spreading speed decreases under the action of the surface tension.

4.2. LN₂ Droplet Vaporization Process

4.2.1. Effect of the Micro-Structured Surface

The vaporization of the LN₂ droplet with an initial diameter of 2 mm and a velocity of 1 m/s on the smooth and micro-structured hot wall, respectively, was simulated by considering influences of gravity during falling and impinging. Initially, the dimensionless diameter and velocity are kept at 1, and the initial residual r* is 0.5. The vaporization process of the LN₂ droplet is studied within t*=10. Figure 16 shows the morphology of the droplet during the vaporization on smooth surface and micro-structured surface at t* = 2. The temperature of the solid wall surface is much higher than the boiling temperature of LN₂ that causes an obvious Leidenfrost phenomenon.

The presence of the fins and trapped gas inhibits the droplet to get in contact with the surface, resulted in the detached droplet that doesn’t encroach into grooves.

As the residual radius decrease with time being (Figure 17), it can be seen that the droplet vaporizes more slowly when adding fins to the wall. The pressure field marked by contour lines at t* = 2.5 is shown in Figure 18. At the micro-structured surface, there is a negative pressure area under the droplet, which makes the droplet lifetime longer during vaporization.

At t* = 5, the temperature of the smooth surface is significantly higher than that of the micro-structured surface, and the temperature curve shows violent fluctuation as shown in Figure 19. The gas produced during droplet vaporization is responsible for fluctuation due to a lower thermal conductivity than that of LN₂. The temperature of the area covered by gas shows a high level while the temperature of the wall that is exposed to LN₂ shows a significant drop. The overall temperature curve shows a trend of high on both sides and low in the middle, confirming the fact that cooling capacity is transferred to both sides from the area where the droplet falls and hits.

Specifically, the droplet boils upon the film adjacent to the wall at high temperature, and a clear vapor layer can be seen between the droplet and hot wall. In the case of the micro-structured surface, when the droplet is about to impact the wall surface, the entrapped gas further hinders the droplet to come into contact with the wall surface. Thus, a negative pressure is formed between the fins, and the vaporization rate is lower than that of the smooth surface.

4.2.2. Effect of Weber Number

The Weber number, a decisive parameter of droplet breakage, is determined by the ratio of inertial force to surface tension. Therefore, the spreading of the droplet after impinging on the wall was simulated with various Weber numbers for the smooth wall and micro-structured wall. The simulated Weber numbers were 30, 180, 360, and 450.

The spreading morphology of droplet on the smooth surface at Weber number of 30 is shown in Figure 20. At the Weber number of 30, the spreading morphology of LN₂ droplets on the smooth and micro-structured surface was quite different from that of other cases of different Weber numbers. At the smooth surface, the droplet did not splash at the Weber number of 30, and the droplet’s motor pattern changed from the vertical motion to horizontal motion due to the contact with the solid wall. The symmetrical shape of liquid crowns appeared at both ends of the droplet with inward contraction, and the droplet spread out from the middle to form a thin film. The thickness of the film decreased continuously. After reaching the maximum spreading coefficient (d/d₀) of 4.39, the droplet split from the middle and contracted with the shape of liquid crowns on both sides, and the dimensionless height (h/d₀) of 1.55 was obtained for the liquid crowns after the contraction. As the Weber number became greater than 180, the dominant effect of the surface tension weakened, the droplet struck the surface of the wall, broke, and splashed, and the droplet spread out to a maximum radius in comparison to the initial value, obtaining a larger contact area.

At the different Weber numbers, the width of the liquid film decreased on the micro-structured surface, because a few droplets entered the grooves, which reduces certain droplet volume fractions that can spread out on the surface. In addition, the available fins promote the breaking up of the droplet. At t* = 6, a large number of tiny droplets break away from the initial droplet and get spattered. The existed fins can change the contact angle of the droplet. In addition, simulation results show that spreading on the smooth surface provides a relatively stable liquid film surface while spreading on the micro-structured surface causes a certain degree of fluctuation in the liquid film surface. This may be attributed to the stronger pressure wave inside the transmission of the liquid. The collision of the droplet with the solid surface formed a pressure wave that interacts with the surface tension, resulting in fluctuations, and eventually affecting the flow and motion form of the droplet.

Furthermore, the vaporization of the droplet impacting the smooth surface and micro-structured surface under the different Weber number was simulated, and the residual radius of the droplet was quantified.

The effect of Weber number on droplet vaporization on the smooth and micro-structured surface is shown in Figure 21 and Figure 22, respectively. In the case of smooth surface, the variation in Weber number showed no obvious effect on the vaporization process of the droplet. However, at the micro-structured surface, the vaporization rate is the slowest at 30, and the highest vaporization rate is observed at the Weber number of 180. However, beyond the 180, i.e., Weber number of 360 and 450, the vaporization rate showed no significant increment. The smooth surface offered a better vaporization effect than that of the micro-structured surface.

4.2.3. Effects of Impinging Velocity

The spreading and vaporization process of the droplet impinging on both smooth and the micro-structured surfaces were simulated at a falling velocity of 0.5 m/s, 1 m/s, 2 m/s, and 3 m/s, respectively. A higher velocity upon a smooth surface provides more residual energy of the initial kinetic energy under certain surface tension and viscous force as well as higher the spreading velocity resulted in a short time required to reach the maximum spreading diameter. At the Weber number of 180, the surface tension is not enough to maintain the complete shape of the droplet so that the droplet hit the wall, broke up, and spattered into some small secondary droplet. Figure 23a shows the variation curve of the residual radius of LN₂ droplet vaporizing on the smooth surface. Since the initial velocity of the droplet is different, the beginning of the vaporization is also different so the time origin to the moment when the lower end of the droplet touches the wall is translated, and the translated variation curves of the residual radius are shown in Figure 23b. It can be seen that the initial vaporization rate of the droplet at the different velocities is consistent after the droplet contacts the wall surface. At t* > 3, the droplet with the initial velocity of 0.5 m/s maintains a higher vaporization rate and obtains the minimum residual radius. With the increasing impinging velocity, the vaporization rate gradually decreases, and the value of the residual radius is positively correlated with the value of the initial velocity.

On the other hand, for the micro-structured surface, the variation curve of the residual radius is shown in Figure 24b, which is translated at the time origin to the moment when the lower end of the droplet touches the wall (refer to Figure 24a). The effect of impinging velocity and the droplet vaporization on the micro-structured surface is similar to the above-mentioned smooth surface. At the velocity of 0.5 m/s, the droplet showed the highest vaporization rate after contacting the surface. With the increasing velocity, the droplet maintained a high vaporization rate for a shorter time, and the remaining radius became larger. At the falling velocity of 0.5 m/s, the droplet maintained a complete shape before touching the wall surface. After slowly contacting the wall surface, it spread out into a liquid film about twice the size of the initial diameter and sufficiently transferred heat to the micro-structured surface, resulting in a fast vaporization effect. With the increasing velocity, the droplet quickly broke up and spattered after impinging on the surface with fins. The small secondary droplets produced rolled onto the micro-structured surface and failed to transfer heat. At t* = 10, no vaporization was observed in the droplet.

5. Conclusions

In summary, a direct numerical simulation was carried out to investigate the heat transfer performance of the LN₂ droplet impinging a micro-structured surface composed of fins that were designed with a spacing much smaller than the diameter of the droplet. For a comparable study, numerical simulations with a smooth surface were also conducted. The structural parameters of the micro-structured surface affect the heat transfer. Accordingly, the droplet’s behavior by means of the motion and liquid film morphology under the various conditions were investigated. The major findings are as follows.

(1)

The Leidenfrost phenomenon occurs on both surfaces, namely micro-structured and smooth surfaces owing to a large temperature difference. The produced vapor layer hindered the heat transfer between the hot surfaces and the LN₂ droplet. The droplet cannot completely wet the micro-structured surface, resulting in a small spreading diameter in the counterpart of the droplet at the smooth surface. The slow vaporization process is observed at the micro-structured surface rather than that of the smooth surface.

(2)

The Weber number affects the heat transfer of liquid nitrogen droplets on both smooth and micro-structured surfaces. On the smooth surface, the Weber number greater than 180 showed a minor effect on the droplet’s spreading morphology, and no obvious effect was observed at the vaporization rate. On the other hand, the micro-structured surface exhibited the highest vaporization rate at the Weber number of 180.

(3)

Impinging velocity showed a noticeable impact on the heat transfer behavior. At the micro-structured surface, a large impinging velocity can lead to severe crushing and splashing of the droplet. The secondary droplet rolls over on the micro-structured surface, and the heat transfer performance becomes worse than that of the smooth surface.