Modeling and Analysis of Metal Liquid Film Flow Characteristics during Centrifugal Spray Forming

Abstract

Liquid films are an important part of liquid metal granulation in the process of centrifugal spray forming. The size of the granulated particles has an important influence on the density, grain size and microstructure uniformity of the deposited blanks. The particle size is closely related to the flow characteristics of liquid films. Therefore, enhancing our understanding of the flow characteristics of liquid films can provide guidance for forming blanks. In this study, force analysis of a liquid film on the surface of a high-speed rotating centrifugal disc used in centrifugal spray-forming technology was carried out using D’Alembert’s principle and Newton’s law of viscosity. Then, combined with the principle of mass conservation, a theoretical model of the smooth flow of the liquid metal film was established. The experimental values obtained by Leshev were compared with our values to verify the correctness and accuracy of the model. Through the model, the influencing factors of the liquid film flow were obtained, such as the centrifugal disc speed, centrifugal disc radius, inlet volume flow rate and kinematic viscosity. Taking A390 aluminum alloy as the research object, the influence of the process parameters on the thickness, velocity and trajectory of the liquid film was revealed theoretically, and the relationship between the process parameters and the trajectory length and liquid film thickness was clarified. Modeling and analysis can not only help us to understand the flow of a liquid film, but also help us to predict the relevant parameters, which is convenient for the accurate and rapid regulation of the process to obtain the desired flow parameters. Therefore, the research content of this paper is of great significance for the preparation of billets with a uniform microstructure and excellent mechanical properties.

1. Introduction

Centrifugal spray-forming technology uses the centrifugal force provided by a high-speed rotating centrifugal disc to granulate a film-like metal [1,2], and the granulated droplets are deposited on a moving substrate to form a metal blank with a refined microstructure and uniform chemical composition [3,4]. The microstructure and related mechanical properties of the deposited blank [5,6,7] are mainly determined by the size, temperature and speed of the atomized droplets, and the atomization process [8,9,10] is closely related to the flow behavior of the liquid film. Therefore, it is of great significance to study the flow behavior of liquid films for the preparation of high-performance blanks.

It is very difficult to measure the flow behavior of a liquid film on the surface of a high-speed rotating centrifugal disc [11,12,13]. Therefore, research on the flow characteristics of liquid films is mostly carried out using theoretical and simulation methods. Yubin Wang [14] used the method of computational fluid dynamics to analyze the hydrodynamic behavior of a liquid in a rotor–stator spinning disc reactor. The results showed that the average liquid velocity increased with the increase in the rotational speed, volume and radial radius. The average radial velocity increased with the decrease in the rotational speed, volume and radial radius. Zhao [15,16,17] obtained a simplified equation set by analyzing the force of a fluid and combining the dimensionless relationships of the velocity, radius and angular velocity. When solving the equation set, the relationship between the liquid film thickness and velocity and radius was obtained using a discrete method. Based on the order-of-magnitude analysis and the film hypothesis, the Navier–Stokes equation was simplified by Wang Dongxiang [18,19]. A theoretical model of the liquid film flow characteristics in a turntable area after a hydraulic jump was established, and the thickness and velocity of a liquid film after a hydraulic jump were obtained. Sisoev’s study [20] found that air-induced shear force may cause fluctuations in the surface of the film, but that the average liquid film thickness will not change significantly, and the liquid film in the center and edge regions of the turntable has an approximately steady-state flow. In summary, the existing research lacks a detailed exploration of the liquid film flow process. Therefore, the aims of this paper were to study the liquid film flow behavior and reveal the formation mechanism and variation law of the liquid film flow.

It is difficult to obtain the required data using the conventional test equipment for forming liquid films with a high-temperature liquid metal [21], and experimental research encounters problems with high costs and difficulties in traversing a wide range of test parameters. Therefore, a steady-state flow model of a liquid film was established by using D’Alembert’s principle, Newton’s viscosity law and the mass conservation principle. The flow characteristics of the A390 aluminum–silicon alloy were solved and calculated using Matlab software. The influence of the process parameters on the liquid film thickness, velocity and trajectory was revealed, and the relationship between the process parameters and trajectory length and liquid film thickness was clarified. The research results can not only help us to understand the flow behavior of the liquid film, but also predict the relevant parameters. The predicted values can provide a theoretical basis for obtaining the required flow parameters and lay a foundation for the study of liquid film granulation into particles.

2. Principle and Condition Assumption of Centrifugal Spray-Forming Technology

2.1. Formation Principle of Liquid Metal Film

Figure 1 is a schematic diagram of centrifugal spray deposition for blank preparation [22]. The centrifugal force provided by the liquid metal centrifugal disc atomizes the film-like metal, and the atomized droplets are deposited on a moving substrate in order to prepare a blank with a certain size. For the convenience of expression and subsequent calculation, the reference frame is fixed on the centrifugal disc, and the cylindrical coordinate system $r-\theta -z$ is introduced [23,24,25]. The coordinate origin of the cylindrical coordinate system coincides with the center of the centrifugal disc, as shown in Figure 1. The liquid flows vertically along the negative direction of the z-axis to the surface of the centrifugal disc rotating around itself at an angular velocity and rapidly spreads into the liquid film under the action of centrifugal force. In a specific flow state, there will be a hydraulic jump phenomenon. Because the liquid flow near the hydraulic jump is in a strong turbulent state, it is difficult to theoretically characterize the liquid film flow in the hydraulic jump region. Therefore, this paper aimed to conduct a theoretical analysis of liquid film flow with a steady-state flow.

2.2. Condition Hypothesis

The molten metal is regarded as a Newtonian, incompressible, isothermal fluid [26], and the spreading process of the liquid film is assumed to be as follows [17,18,27]:

(a)

The temperature of the liquid film on the surface of the centrifugal disc does not change.

(b)

Considering that the liquid film is thin, the internal pressure of the liquid film and the pressure on the surface of the liquid film are regarded as the environmental pressure.

(c)

The liquid film has a steady-state axisymmetric flow, so the velocity and pressure do not change with $\theta$.

(d)

The axial velocity is much smaller than the radial and circumferential velocities, so the influence of the axial velocity of the liquid film is ignored.

(e)

There is no slip between the liquid film and the surface of the centrifugal disc, that is, the contact part of the liquid film and the centrifugal disc rotates with the centrifugal disc.

(f)

The gravity and other inertial forces on the liquid film on the centrifugal plate are neglected compared with the centrifugal force and viscous force.

3. Modeling of Liquid Film Flow Characteristics

The main forms of motion of the liquid film on the centrifugal disc are circular motion along the disc surface and linear motion sliding from the center to the edge of the disc. A fluid element is taken at any radial coordinate $r$, circumferential coordinate $\theta$ and axial coordinate $z$. The radial size of the fluid element is $dr$, the circumferential size is $rd\theta$ and the thickness is $dz$. The volume of the fluid element is as follows:

$$dV=rd\theta drdz$$

(1)

Figure 2 is a schematic diagram of the fluid element motion on the surface of the centrifugal disc. The coordinate system $r-\theta -z$ rotates with the centrifugal disc, and the motion and force of the liquid film correspond to the dynamic coordinate system (non-inertial system). According to D’Alembert’s principle, the centrifugal force $d\overrightarrow{F}$ on the fluid element is as follows:

$$d\overrightarrow{F}=-\mathsf{\rho }[\overrightarrow{w}\times (\overrightarrow{w}\times \overrightarrow{r})+2\overrightarrow{w}\times \overrightarrow{u_{\theta }}]dV$$

(2)

where $\overrightarrow{F}$ is the centrifugal force (vector), $\mathrm{N}$; $\mathsf{\rho }$ is the density of the liquid metal, $\mathrm{kg}/{\mathrm{m}}^3$; $\overrightarrow{w}$ is the rotation angular velocity (vector), $\mathrm{rad}/\mathrm{s}$; $\overrightarrow{u_{\theta }}$ is the relative circumferential velocity (vector), $\mathrm{m}/\mathrm{s}$; $\overrightarrow{r}$ is the radial position (vector), $\mathrm{m}$; $\overrightarrow{w}\times (\overrightarrow{w}\times \overrightarrow{r})$ is the implicated acceleration; and $2\overrightarrow{w}\times \overrightarrow{u_{\theta }}$ is the Coriolis acceleration.

The centrifugal force of the fluid element is decomposed along the radial and circumferential directions, and the formulas of the radial component $d{F}_r$ and circumferential component $d{F}_{\theta }$ are obtained as follows:

$$d{F}_r=\mathsf{\rho }(r{w}^2+2w{u}_{\theta })dV$$

(3)

$$d{F}_{\theta }=-2\mathsf{\rho }w{u}_{r}dV$$

(4)

where $u_r$ and $u_{\theta }$ are the radial velocity and circumferential velocity of the fluid element relative to the centrifugal disc, $\mathrm{m}/\mathrm{s}$; $w$ is the rotation angular velocity of the centrifugal disc, $\mathrm{rad}/\mathrm{s}$; and $F_r$ and $F_{\theta }$ are the radial component and circumferential component of the centrifugal force, $\mathrm{N}$.

In practice, the circumferential velocity of the fluid element is much smaller than that of the centrifugal disc [24]. Therefore, the influence of $u_{\theta }$ can be neglected, and $d{F}_r$ is approximately as follows:

$$d{F}_r\approx \mathsf{\rho }r{w}^{2}dV$$

(5)

According to Newton’s law of viscosity and assumptions, the following formulas can be obtained:

$$d{P}_{\theta }=d{P}_r=0$$

(6)

$$\mathsf{\mu }\frac{\partial u_r}{\partial \theta }={\tau }_{\theta r}$$

(7)

$$\mathsf{\mu }\frac{\partial u_r}{\partial z}={\tau }_{zr}$$

(8)

$$\mathsf{\mu }\frac{\partial u_{\theta }}{\partial z}={\tau }_{z\theta }$$

(9)

In Formulas (7)–(9), $\tau$ is the shear stress, N, where the first subscript letter indicates the direction perpendicular to the surface, and the second subscript letter indicates the stress direction, where ${\tau }_{\theta r}=0$; $\mathsf{\mu }$ is the dynamic viscosity of the liquid, $\mathrm{Pa}·\mathrm{s}$.

The force analysis of the fluid element in the radial and tangential directions is shown in Figure 3. The force balance equation of the fluid element can be obtained through analysis.

$$d{F}_r+({\tau }_{zr}+d{\tau }_{zr})rdrd\theta -{\tau }_{zr}rdrd\theta =0$$

(10)

$$d{F}_{\theta }+({\tau }_{z\theta }+d{\tau }_{z\theta })rdrd\theta -{\tau }_{z\theta }rdrd\theta =0$$

(11)

By substituting $F_r$, $F_{\theta }$, ${\tau }_{zr}$ and ${\tau }_{z\theta }$ into Formulas (10) and (11), the equations of the radial velocity and circumferential velocity can be obtained.

$$\mathsf{\mu }\frac{{\partial }^2{u}_r}{\partial z^2}+\mathsf{\rho }r{w}^2=0$$

(12)

$$\mathsf{\mu }\frac{{\partial }^2{u}_{\theta }}{\partial z^2}-2\mathsf{\rho }w{u}_r=0$$

(13)

On the surface of the centrifugal disc, that is, $z=0$, there is no slip between the centrifugal disc and the liquid film. The boundary conditions are as follows:

$$u_r=0 ,u_{\theta }=0$$

(14)

On the surface of the liquid film, that is, $z=h$, the shear force between the liquid film and the air is zero, and the boundary conditions are as follows:

$$\frac{\partial u_r}{\partial z}=0,\frac{\partial u_{\theta }}{\partial z}=0$$

(15)

According to Equations (12)–(15), the radial velocity and circumferential velocity of the liquid film on the surface of the centrifugal disc are as follows:

$$u_r=\frac{\mathsf{\rho }r{w}^2}{\mathsf{\mu }}(hz-\frac{z^2}{2})$$

(16)

$$u_{\theta }=-\frac{{\mathsf{\rho }}^{2}r{w}^3}{12{\mathsf{\mu }}^2}(8h^{3}z-4h{z}^3+z^4)$$

(17)

The negative sign of the circumferential velocity in Formula (17) indicates that the direction of motion is opposite to the direction of rotation of the centrifugal disc.

Because the actual liquid film is thin, the liquid film on the centrifugal disc can be treated as a small cylinder. According to the principle of mass conservation, for any radius $r$ on the surface of the disk, the volume flow rate $Q$ can be expressed as follows:

$${\displaystyle {\int }_0^{h}2}\pi r{u}_{r}dz=Q$$

(18)

where $Q$ is the volume flow rate of the liquid, ${\mathrm{m}}^3/\mathrm{s}$, and $h$ is the liquid film thickness, $\mathrm{m}$.

Combined with Formulas (16) and (18), the calculation formula for the liquid film thickness can be obtained as follows:

$$h={(\frac{3\mathsf{\mu }Q}{2\pi \mathsf{\rho }{r}^2{w}^2})}^{\frac{1}{3}}$$

(19)

From the relationship between the dynamic viscosity and kinematic viscosity, the following can be obtained:

$$h={(\frac{3\nu Q}{2\pi r^2{w}^2})}^{\frac{1}{3}}$$

(20)

$$\nu =\frac{\mathsf{\mu }}{\mathsf{\rho }}$$

(21)

where $\nu$ is the kinematic viscosity of the liquid, ${\mathrm{m}}^2/\mathrm{s}$.

The overall flow trend of the liquid metal on the centrifugal disc is described by the average radial velocity and circumferential slip. The average radial velocity and average circumferential velocity of the liquid film are as follows:

$$u_{r-a}=\frac{Q}{2\pi rh}={(\frac{w^2{Q}^2}{12{\pi }^2\nu r})}^{\frac{1}{3}}$$

(22)

$$u_{\theta -a}=\frac{{\displaystyle {\int }_0^h{u}_{\theta }}dz}{h}=-\frac{3rw}{5}{(\frac{2Q^2}{3{\pi }^2\nu w{r}^4})}^{\frac{2}{3}}$$

(23)

where $u_{r-a}$ is the average radial velocity of the liquid film, $m/\mathrm{s}$, and $u_{\theta -a}$ is the average circumferential velocity of the liquid film, $m/\mathrm{s}$.

The circumferential slip is expressed as the ratio of the circumferential velocity of the liquid film to the circumferential velocity of the centrifugal disc.

$$\varphi =\frac{\left|u_{\theta -a}\right|}{rw}=\frac{3}{5}{(\frac{2Q^2}{3{\pi }^2\nu w{r}^4})}^{\frac{2}{3}}$$

(24)

where $\varphi$ is the circumferential slip.

The change in the vector diameter with time is the velocity, and the time required for the fluid to move from the center of the centrifugal disc to any position can be obtained by using the velocity as an intermediate variable. Furthermore, the rotation angle $\theta$ and the trajectory length $S$ can be obtained.

$$u_{r-a}=\frac{dr}{dt}={(\frac{w^2{Q}^2}{12{\pi }^2\nu r})}^{\frac{1}{3}}$$

(25)

$$t_L={\displaystyle {\int }_o^r{(\frac{12{\pi }^2\nu r}{w^2{Q}^2})}^{\frac{1}{3}}}dr=\frac{3}{4}{(\frac{12{\pi }^2\nu }{w^2{Q}^2})}^{\frac{1}{3}}{r}^{\frac{4}{3}}$$

(26)

$$\theta =t_{L}w=\frac{3}{4}{(\frac{12{\pi }^2\nu w}{Q^2})}^{\frac{1}{3}}{r}^{\frac{4}{3}}$$

(27)

$$S={\displaystyle {\int }_o^{r}rw{t}_{L}dr}=\frac{9}{40}{(\frac{12{\pi }^2\nu w}{Q^2})}^{\frac{1}{3}}{r}^{\frac{10}{3}}$$

(28)

where $t_L$ is the time it takes the liquid film to spread from the center to the edge, $s$; $\theta$ is the rotation angle of the liquid film spreading from the center to the edge, $\mathrm{rad}$; and $S$ is the length of the liquid film trajectory, m.

4. Numerical Experiment

4.1. Model Validation

In the process of centrifugal granulation, the thickness of the liquid film on the surface of the centrifugal disc is generally about several hundred microns, and it is difficult to accurately measure it using conventional methods. Leshev [25] used the probe method to measure the thickness of a liquid film on the surface of an entire turntable. The experimental conditions are shown in

Table 1. Experimental conditions used by Leshev [25].

Working Fluid	$\mathit{R}\mathbf{/}(\mathbf{m})$	$\mathit{\omega }\mathbf{/}(\mathbf{rad}\mathbf{/}\mathbf{s})$	$\mathit{Q}\mathbf{/}(\times {10}^{-6}{\mathbf{m}}^3\mathbf{/}\mathbf{s})$	$\mathit{\nu }\mathbf{/}(\times {10}^{-7}{\mathbf{m}}^2\mathbf{/}\mathbf{s})$
Water	0.2	10.47	5.33	1.01
30% glycerol	0.2	10.47	5.33	6.25
50% glycerol	0.2	10.47	5.33	10.63

. Figure 4 shows a comparison between the liquid film thickness calculated using Formula (20) and Leshev’s measurement results [25]. At the edge of the centrifugal disc, the prediction result of the liquid film thickness has high accuracy, which is in good agreement with the experimental measurement, and the error is large near the center of the rotating disc. The model established in this paper is mainly used to predict the liquid film parameters at the edge of the centrifugal disc, so it is considered that the model can be used to predict the flow characteristics of the liquid film.

4.2. Numerical Calculation

Due to the high silicon content of the Al–Si alloy, under the conventional casting process, the primary silicon exhibits a variety of morphologies, such as star, polygon, plate and feather morphologies, and the eutectic silicon is coarse and needle-like [28,29,30], which easily leads to premature crack initiation and tensile fracture, resulting in a decrease in the mechanical properties. In order to overcome the above shortcomings, the centrifugal spray-forming process of the Al–Si alloy was explored. Therefore, the A390 aluminum–silicon alloy was taken as the research object. The chemical composition of A390 is shown in

Table 2. A390 chemical constituents (wt%).

Element Name	Si	Fe	Cu	Mn	Mg	Zn	Ti	Al
Context	17.0	0.5	4.5	0.1	0.55	0.1	0.2	Balance

. When the temperature is in the range of 660 °C to 1000 °C, the kinematic viscosity of the molten A390 changes between 3.8 × 10⁻⁷ m²/s and 11.4 × 10⁻⁷ m²/s.

The parameters used for the numerical calculation are shown in

Table 3. Parameters of the numerical calculation.

$\mathit{R}\mathbf{/}(\mathbf{m})$	$\mathit{\omega }\mathbf{/}(\mathit{rad}\mathbf{/}\mathbf{s})$	$\mathit{Q}\mathbf{/}(\times {10}^{-5}{\mathbf{m}}^3\mathbf{/}\mathbf{s})$	$\mathit{\nu }\mathbf{/}(\times {10}^{-7}{\mathbf{m}}^2\mathbf{/}\mathbf{s})$
0.1	600	3.77	3.80
0.1	800	6.28	5.70
0.1	1000	8.80	7.60
0.1	1200	12.6	9.47
0.1	1400	16.3	11.4

. In the calculation of Formulas (20), (22) and (24), each formula calculates 15 sets of data, that is, two of the three variables are kept unchanged, with only one of them changed, and the influence of this changed parameter on the liquid film characteristics is analyzed.

In this paper, MATLAB software was used to solve the steady-state model. MATLAB is a commercial mathematics software produced by MathWorks Company in the United States. The version used in this article was MATLAB 9.11. In the calculation, the relevant parameters were input first, then the liquid film thickness was calculated, and finally the calculation speed, trajectory length, etc., were calculated.

5. Results and Discussion

5.1. Analysis and Discussion of Liquid Film Thickness

Figure 5a–c shows the distribution of the liquid film thickness along the radial direction under different kinematic viscosities, different centrifugal disc speeds and different volume flow rates. It can be seen in the figure that the liquid film is thicker at a position closer to the rotation axis, which means that the liquid film has not yet fully spread and is present on the surface of the centrifugal disc in the form of a liquid column. As the liquid film flows outwards along the radial direction, the thickness of the liquid film decreases rapidly and then slowly, which means that the fluid is in a stable flow state. For any specified radius, the liquid film thickness increases with the increase in the volume flow rate and kinematic viscosity, and decreases with the increase in the centrifugal disc rotation speed. When the radius is 0.1 m, as the kinematic viscosity increases from 3.80 × 10⁻⁷ m²/s to 11.4 × 10⁻⁷ m²/s, the liquid film thickness increases from 11.7 μm to 16.9 μm; as the rotation speed of the centrifugal disc increases from 600 rad/s to 1400 rad/s, the liquid film thickness decreases from 20.7 μm to 11.8 μm; and as the volume flow rate increases from 3.77 × 10 ⁻⁵ m³/s to 16.3 × 10⁻⁵ m³/s, the liquid film thickness increases from 11.1 μm to 18.1 μm. By quantifying the thickness of the liquid film at the edge, it can be concluded that the change in the rotational speed has a more significant effect on the thickness of the liquid film, which indicates that the centrifugal field provided by the centrifugal force at the edge plays a leading role in the spreading of the liquid film.

5.2. Analysis and Discussion of Average Radial Velocity of the Liquid Film

Figure 6a–c shows the distribution of the average radial velocity along the radial direction under different kinematic viscosities, different centrifugal disc speeds and different volume flow rates. It can be seen in the diagram that the average radial velocity value is higher at positions closer to the rotation axis. When the liquid metal hits the center of the centrifugal disc from the inlet, the direction of the fluid’s motion changes rapidly in a short period of time, making the average radial velocity value relatively high. As the liquid film spreads along the radial direction, the average radial velocity first decreases rapidly and then slowly. For any specified radius, the average radial velocity increases with the increase in the volume flow rate, and decreases with the increase in the kinematic viscosity and rotation speed of the centrifugal disc. When the radius is 0.1 m, as the kinematic viscosity increases from 3.80 × 10⁻⁷ m²/s to 11.4 × 10⁻⁷ m²/s, the average radial velocity decreases from 11.9 m/s to 8.3 m/s; as the rotation speed of the centrifugal disc increases from 600 rad/s to 1400 rad/s, the average radial velocity increases from 6.8 m/s to 11.9 m/s; and as the volume flow rate increases from 3.77 × 10⁻⁵ m³/s to 16.3 × 10⁻⁵ m³/s, the average radial velocity increases from 5.4 m/s to 14.4 m/s. By quantifying the thickness of the liquid film at the edge, it can be concluded that the change in the volume flow rate has a more significant effect on the average radial velocity, which indicates that the overflow rate on the surface of the centrifugal disc increases at a large volume flow rate, the energy obtained by the liquid film from the centrifugal disc increases and the spreading to the edge is accelerated.

5.3. Analysis and Discussion of Circumferential Slip

Figure 7 a–c shows the distribution of the circumferential velocity slip along the radial direction under different kinematic viscosities, different centrifugal disc speeds and different volume flow rates. It can be seen in the diagram that the closer to the rotation axis, the higher the degree of slip; this is because when the liquid metal hits the center of the centrifugal disc from the inlet, the direction of motion of the fluid changes to the radial direction, and the circumferential component is almost zero. As the liquid film spreads outwards along the radial direction, the sliding degree first decreases rapidly and then slowly to a certain value and remains unchanged, which means that the force in the circumferential direction of the liquid film reaches the viscous force limit. For any specified radius, the degree of slip increases with the increase in the volume flow rate, and decreases with the increase in the rotational speed and kinematic viscosity of the centrifugal disc. When the radius is 0.1 m, as the kinematic viscosity increases from 3.80 × 10⁻⁷ m²/s to 11.4 × 10⁻⁷ m²/s, the slip decreases from 3.44% to 1.66%; as the rotational speed of the centrifugal disc increases from 600 rad/s to 1400 rad/s, the slip decreases from 3.05% to 1.73%; and as the volume flow rate increases from 3.77 × 10⁻⁵ m³/s to 16.3 × 10⁻⁵ m³/s, the circumferential slip increases from 0.7% to 4.9%. By quantifying the thickness of the liquid film at the edge, it can be concluded that the change in the volume flow rate has a more significant effect on the circumferential slip. At the same time, according to the small slip at the edge, the tangential velocity of the liquid film can be approximately considered to be consistent with the tangential velocity of the centrifugal disc.

5.4. Analysis and Discussion of Liquid Film Trajectory

Figure 8a–c shows the trajectory of the liquid film on the surface of the centrifugal disc under different kinematic viscosities, different centrifugal disc speeds and different volume flow rates. In the trajectory diagram, it can be seen that the trajectory of the liquid film is a spiral line. The length of the liquid film trajectory increases with the increase in the kinematic viscosity and rotational speed of the centrifugal disc, and decreases with the increase in the volume flow rate. When the radius is 0.1 m, as the kinematic viscosity increases from 3.80 × 10⁻⁷ m²/s to11.4 × 10⁻⁷ m²/s, the rotation angle increases from 6.26 rad to 9.03 rad; as the rotation speed of the centrifugal disc increases from 600 rad/s to 1400 rad/s, the rotation angle increases from 6.65 rad to 8.82 rad; and as the volume flow rate increases from 3.77 × 10⁻⁵ m³/s to 16.3 × 10⁻⁵ m³/s, the rotation angle decreases from 13.87 rad to 5.22 rad. By quantifying the rotation angle at the edge, it can be concluded that the change in the volume flow has a more significant effect on the rotation angle; this is because the larger the volume flow, the greater the radial velocity and circumferential slip, and the rotation angle is reflected in the trajectory.

5.5. The Relationship between Liquid Film Thickness and Trajectory Length

Formula (29) can be obtained through the nonlinear fitting of the liquid film thickness and trajectory length.

$$h=8.7924\times {10}^{-6}(\frac{Q^2}{{\pi }^2\nu \omega }{)}^{\frac{1}{15}}{(\frac{1}{r})}^{\frac{2}{3}}$$

(29)

Figure 9 shows the relationship between the liquid film thickness and the trajectory length. The thickness of the liquid film decreases with the increase in the length of the trajectory, and the error between the fitted value and the model value is controlled within ±15%. The formula can be used for the design and optimization of the centrifugal disc structure. In the design and optimization, the trajectory length of the liquid film on the surface of the centrifugal disc can be increased to obtain a thinner liquid film, and then a uniform atomized droplet can be obtained.

6. Conclusions

According to the principle of liquid film formation, the cylindrical coordinate system $r-\theta -z$ was introduced. Force analysis of the fluid element was carried out using D’Alembert’s principle and Newton’s viscosity law, and the force balance equation was established. The model of the flow characteristics of the liquid film was obtained through combination with the principle of mass conservation. The model can not only help us to understand how the liquid film flows in the centrifugal disc, but can also help us to predict the relevant parameters. The predicted values provide a theoretical basis for obtaining the required flow parameters and lay a foundation for the subsequent study of liquid film atomization into particles. On this basis, molten A390 aluminum alloy was modeled and analyzed, and the results showed were as follows:

1. The trajectory of the liquid film formed by the liquid metal on the surface of the rotating centrifugal disc is a spiral line. When moving to the edge, the thickness of the liquid film along the radial direction first quickly and then slowly decreases. After the flow enters a stable state, the thickness of the liquid film remains basically unchanged. At the same time, the tangential velocity of the liquid film at the edge can be approximately considered to be the same as the tangential velocity of the centrifugal disc.

2. The effects of the centrifugal disc speed, kinematic viscosity and volume flow rate on the flow characteristics of the liquid film are different. The liquid film thickness increases with an increase in the volume flow rate and kinematic viscosity, and decreases with an increase in the centrifugal disc rotation speed. The average radial velocity and circumferential slip increase with an increase in the volume flow rate, and decrease with an increase in the kinematic viscosity and rotation speed of the centrifugal disc. The trajectory length increases with an increase in the kinematic viscosity and centrifugal disc speed, and decreases with an increase in the volume flow rate.

3. The nonlinear relationship between the thickness of the liquid film and the length of the trajectory was obtained through data fitting. The relationship is as follows:

$$h=8.7924\times {10}^{-6}(\frac{Q^2}{{\pi }^2\nu \omega }{)}^{\frac{1}{15}}{(\frac{1}{r})}^{\frac{2}{3}}.$$