Constant Flow Rate Pouring of the Steel Ladle: Analytical Model, Simulation Model, and Experimental Verification

Abstract

To realize accurate ladle pouring, an analytical model of the constant flow rate pouring was established. By integrating a user-defined function (UDF), a CFD simulation model of the constant flow rate pouring was established to investigate the liquid steel pouring behavior under different inner wall inclination angle α, initial liquid volume V_c, and target flow rate q. Finally, the accuracy of the analytical model and the simulation model was verified through experiments. The results show that the experimental results agree well with the theoretical and simulation results, which verify the accuracy of the analytical model and the simulation model. Moreover, the simulation results indicate that increasing both α and V_c leads to an increase in the pouring flow rate. To achieve a stable pouring process and a constant flow rate value, a proper α, V_c and q_t should be selected. In this study α = 7.5° V_c = 70% V_capacity and q in the range of 0.10–0.12 m³/s are proper. To realize constant flow rate pouring, a time-variant ladle angular velocity is obtained and it can be adjusted by the motor speed. Therefore, different constant flow rates could be acquired by adjusting the motor speed, which provide guidance to the casting technology.

1. Introduction

An accurate pouring process plays a crucial role in the casting technology. Conversely, improper pouring, such as the fluctuation in flow rate or the presence of residual liquid steel, could result in defects like inclusions and porosity, which, in turn, results in an increased scrap rate and an increased production cost. However, traditional pouring methods rely heavily on the experience and skill of operators, which would result in a high defect rate. Consequently, to ensure high-quality casting production and reduce the influence of human factors, the constant flow rate pouring technology has become a key focus in casting research [1,2,3]. This technology encompasses various control methods, including quality control, electrode control, volume control, time control, demonstration-based reproduction control, and image processing control [4,5,6,7,8].

Wei et al. [9] investigated the liquid steel pouring process using a computational fluid dynamics (CFD) simulation model. The results showed that the two-phase flow model could effectively simulate the liquid steel pouring process, which confirmed the feasibility of CFD simulation. Han et al. [10] further utilized a CFD simulation model to analyze the dynamic changes in the flow field, temperature field, and phase interfaces during ladle pouring at a constant rotational speed in two-dimensional space. Jin et al. [11] found that operating the tilting pouring machine at a fixed angular velocity could not realize a constant flow rate pouring. However, continuously adjusting the angular velocity of the ladle could achieve it.

Nobutoshi [12] implemented a flow rate control system based on differential flatness in an automatic pouring machine and experimentally validated its effectiveness in mitigating disturbances. Cheng [13] experimentally investigated a pneumatic constant flow rate pouring system. The results showed that the pouring flow rate increases with the air flow rate, and it can be controlled by adjusting the pouring duration. Gao [14] introduced a machine vision-based image processing technology for automatic pouring and proposed an edge extraction method for metal level detection in pouring images.

However, stable and accurate pouring has not yet been realized by controlling the ladle angular velocity. Thus, to realize accurate ladle pouring, an analytical model of the constant flow rate pouring was established, and a control curve of the ladle angular velocity was acquired. By integrating a user-defined function (UDF), a CFD simulation model of the constant flow rate pouring was established to investigate the liquid steel pouring behavior under different inner wall inclination angles, initial liquid volumes, and target flow rates. Finally, the accuracy of the analytical model and the simulation model was verified through experiments.

2. Analytical Model of Pouring Process of the Steel Ladle

After the steel ladle is designed, the structural parameters remain constant except for the initial liquid volume, as shown in Figure 1.

The ladle pouring process can be divided into three stages. During these three stages, the residual liquid volume in the ladle changes over time. In the first stage, the residual liquid volume remains unchanged, and no liquid flows out of the ladle, as shown in Figure 2a. In the second stage, the liquid begins to flow out of the ladle and the residual liquid volume decreases, and the ladle bottom is still fully submerged in liquid steel, as shown in Figure 2b. In the third stage, the ladle bottom is exposed. Meanwhile, the liquid keeps flowing out of the ladle, and the residual liquid volume continues decreasing, as shown in Figure 2c.

During the pouring process, the flow characteristics and the liquid distribution vary at each stage. Typically, the liquid steel flows in the shape of an obliquely truncated cone or an irregular horseshoe as it exits the ladle. Due to the absence of an accurate volume calculation formula, liquid steel is often modeled as a cylinder [15]. The diameter D of the cylinder is calculated as D = (D₁ + D₂)/2, where D₁ and D₂ represent the diameters at the top and bottom of the ladle, respectively. In ladle design, this assumption is reasonable due to the generally small taper. Additionally, the initial liquid volume is typically assumed to be 70% of the total ladle capacity. Using the relationship between the ladle rotation angle θ and the residual liquid volume, the residual liquid volume at any given θ can be calculated after establishing an analytical model.

2.1. Analytical Model of the Residual Liquid Volume During the Pouring Process

In the first stage (0° < θ < θ₁), the ladle starts to rotate, but no liquid steel flows out, as the surface of the liquid steel has a certain distance from the outlet. θ₁ is the critical angle of the first stage. When the ladle rotates to θ₁, the liquid steel begins to flow out of the ladle. The residual liquid volume in the first stage is set to V₀, as shown in Figure 2a.

The critical value of the ladle rotation angle θ₁ is:

$${\theta }_1=\arctan \left(2\left(H-H_1\right)/D\right)$$

(1)

At this time, the residual liquid volume V₀ in the ladle is:

$$V_0={\displaystyle \frac{1}{4}}\pi D^2\left(H_1-T\right)$$

(2)

In the second stage (θ₁ ≤ θ < θ₂), the liquid steel in the ladle is in the shape of an obliquely truncated cone, set the residual liquid volume in the second stage to V₁, as shown in Figure 2b.

The critical value of the ladle rotation angle θ₂ is:

$${\theta }_2=\arctan \left(h/D\right)$$

(3)

At this time, the residual liquid volume V₁ in the ladle is:

$$V_1={\displaystyle \frac{1}{8}}\pi D^2\left(2h-D\tan \theta \right)$$

(4)

In the third stage (θ₂ ≤ θ < 90°), the liquid steel continues flowing out of the ladle, and the steel in the ladle is in the shape of an irregular horseshoe until the pouring completion, as shown in Figure 2c.

To simplify the calculation, the shape is modeled as a regular cylinder obliquely truncated by the residual diagonal tangent. The coordinate axes are established, as shown in Figure 3. The residual volume of the ladle is determined through calculus.

There is a liquid surface plane 1, which passes through points E(0, r, h) and point F(0, r − htanθ, h). Set the equation of plane 1 to Ax + By + Cz = D. This equation of plane 1 can be solved by finding the normal vector of plane 1 as follows:

$$\mathrm{y}+{\displaystyle \frac{a-r}{h}}z-a=0$$

(5)

where r = D/2, a = D/2 − h/tanθ.

The residual liquid volume V₂ in the ladle is as follows:

$$\begin{array}{r}{V}_2={\displaystyle \underset{D}{\iint }z}dxdy=2{\displaystyle {\int }_a^r{\displaystyle {\int }_0^{\sqrt{r^2-y^2}}\frac{h(a-y)}{a-r}}}dxdy={\displaystyle \frac{h}{a-r}}({\displaystyle \frac{\pi a{r}^2}{2}}-a{r}^2\cdot \arcsin (2a/D)\\ -a^2{(r^2-a^2)}^{0.5}-{\displaystyle \frac{2}{3}}{(r^2-a^2)}^{1.5})\end{array}$$

(6)

Summarizing the above three stages, the analytical model of the residual liquid volume V in the ladle at any ladle rotation angle θ can be obtained.

$$V=\left\{\begin{array}{cc}\pi r^2\left(H_1-T\right)& 0°<\theta <{\theta }_1\\ {\displaystyle \frac{1}{2}}\pi r^2\left(2h-D\tan \theta \right)& {\theta }_1<\theta <{\theta }_2\\ {\displaystyle \frac{h}{a-r}}({\displaystyle \frac{a\pi r^2}{2}}-a{r}^2\cdot \arcsin (2a/D)-a^2{(r^2-a^2)}^{0.5}-{\displaystyle \frac{2}{3}}{(r^2-a^2)}^{1.5})& {\theta }_2<\theta <90°\end{array}\right.$$

(7)

2.2. Analytical Model of the Flow Rate During the Pouring Process

By taking the first-order derivative of the residual liquid volume with respect to time and then taking the absolute value of this derivative, the flow rate q is obtained. Similarly, the flow rate changes over time during the pouring process, and the flow rates at the three stages q₀, q₁, and q₂ are also different.

In the first stage (0° < θ < θ₁), V₀ = V_c. The flow rate q₀ can be calculated as

$$q_0=\left|{\displaystyle \frac{dV}{dt}}\right|=0$$

(8)

In the second stage (θ₁ ≤ θ < θ₂), the flow rate q₁ can be calculated as follows:

$$q_1=\left|{\displaystyle \frac{dV}{dt}}\right|=\left({\displaystyle \frac{1}{8}}\pi D^3\omega \right){\cot }^2\theta$$

(9)

In the third stage (θ₂ ≤ θ < 90°), the flow rate q₂ can be calculated as follows:

$$\begin{array}{l}{q}_2=\left|{\displaystyle \frac{dV}{dt}}\right|=\omega e\left[{\left(2g\right)}^{1.5}/3+g^{0.5}{f}^2-\pi r^{2}f/2+r^{2}fa\sin \left(f/r\right)\right]-\\ \omega \tan \theta \left[\begin{array}{l}{\displaystyle \frac{1}{2}}\pi r^{2}eh/\left(e-1\right)-r^{2}eha\sin \left(f/r\right)\\ /\left(e-1\right)+eh{f}^2/\left(\left(e-1\right)g^{0.5}\right)-erhf\theta \end{array}\right]/\left[\left(e-1\right){\left(1-f^2/r^2\right)}^{0.5}\right]\end{array}$$

(10)

where $e={\tan }^2\theta + 1, f=(r-h/\tan \theta ) \mathrm{and} g=r^2-{(r-h/\tan \theta )}^2$.

Summarizing the above three stages, the analytical model of the flow rate q at any ladle rotation angle θ as well as the ladle angular velocity ω can be obtained.

$$q=\left\{\begin{array}{cc}0& 0°<\theta <{\theta }_1\\ \left({\displaystyle \frac{1}{8}}\pi D^3\omega \right){\cot }^2\theta & {\theta }_1<\theta <{\theta }_2\\ \omega e\left[{\left(2g\right)}^{1.5}/3+g^{0.5}{f}^2-\pi r^{2}f/2+r^{2}fa\sin \left(f/r\right)\right]-& \begin{array}{l}\\ \end{array}\\ \omega \tan \theta \left[\begin{array}{l}{\displaystyle \frac{1}{2}}\pi r^{2}eh/\left(e-1\right)-r^{2}eha\sin \left(f/r\right)/\\ \left(e-1\right)+eh{f}^2/\left(\left(e-1\right)g^{0.5}\right)-erhf\theta \end{array}\right]/\left[\left(e-1\right){\left(1-f^2/r^2\right)}^{0.5}\right]& {\theta }_2<\theta <90°\end{array}\right.$$

(11)

To realize constant flow rate pouring, a time-invariant ladle angular velocity ω is required. Therefore, the ladle angular velocity ω becomes a function of the ladle rotation angle θ, in which ω can be calculated according to any ladle rotation angle θ.

3. Simulation Model of Pouring Process of the Steel Ladle

3.1. CFD Simulation Setting

The simulation was conducted using ANSYS Fluent 2022 R1 on a workstation equipped with dual Intel Xeon Gold 6248R CPUs (48 cores) and 256 GB of RAM. A User-Defined Function (UDF) for the ladle rotation speed was written using the compiler within Fluent. The convergence criterion is set as the residual being less than 10⁻⁴.

To study the flow behavior during the pouring process, a geometrical model of the ladle was constructed (in it, D₁ = 2305 mm, D₂ = 2985 mm, H = 3740 mm, H₁ = 2927 mm, T = 350 mm, and the inner wall inclination angle α = 7.5°), as shown in Figure 4. The model consists of the following three domains: the liquid domain inside the ladle, the air domain inside the ladle, and the air domain outside the ladle, as shown in Figure 5. During the meshing process, a tetrahedral mesh is generated using the patch-conformal algorithm. For the ladle domain, the mesh size is 70 mm, while for the air domain, it is 200 mm. The total number of nodes is 173,699, and the total number of cells is 956,763. Meanwhile, the initial liquid volume V_c is set to 14,226 L, and the total pouring time t is set to 110 s. The liquid steel does not flow out of the ladle in the range of 0–20 s, and it begins to flow out at t = 20 s and completely flows out at t = 110 s.

To verify mesh independence and eliminate the influence of mesh number on calculation accuracy, four sets of meshes were generated. The number of meshes in each set was 420,235, 550,378, 956,763, and 1,131,109, corresponding to Mesh No. 1, No. 2, No. 3, and No. 4, respectively. Error analysis was then carried out on the pouring flow rates calculated using these four sets of meshes, and the results are shown in

Table 1. Mesh independence test results.

	Number of Meshes	Flow Rate/m3·s−1
No.1	420,235	0.1649
No.2	550,378	0.1636
No.3	956,763	0.1621
No.4	1,131,109	0.1620

.

The results show that the calculation error of the s flow rate between Mesh No. 1 and No. 2 is 0.79%; between Mesh No. 2 and No. 3 is 0.95%; and between Mesh No. 3 and No. 4 is 0.025%. Through comparing the calculation errors, it is evident that the calculation error between Mesh No. 3 and No. 4 is extremely small, indicating that the impact of mesh number on the solution results can be ignored. Given the limitations of computing capacity, Mesh No. 3 which has a mesh number of 956,763 was selected for subsequent simulation calculations.

The viscosity and the density of the liquid steel are set to 0.0062  Pa/s and 7138  kg/m³, respectively [10]. As an isothermal model was adopted, thermal properties were not relevant to the flow simulation. The initial temperature of 1600 K was only used for phase initialization (liquid steel). The air is treated as an incompressible gas, and its viscosity, thermal conductivity, specific heat capacity, and molar mass are set to 1.7894 × 10⁻⁵ Pa/s, 0.0242  W/m K, 1006.4  J/kg K and 28.9  g/mol, respectively. The boundary conditions are defined as follows: (1) the upper surface boundary of the air domain is specified as a pressure outlet, with its value set to the atmospheric pressure; (2) the boundary between the liquid domain inside the ladle and the air domain is defined as an internal boundary. Additionally, the ladle outlet surface in contact with the air is also defined as an internal boundary; (3) all the other boundaries are defined as static, no-slip, adiabatic wall surfaces. The realizable k-ε model and volume of fluid (VOF) model are used to track the phase interface between the liquid steel and the air.

The liquid domain inside the ladle is initialized at 1600 K (the temperature of the liquid steel), while all the other domains are set to 300 K. The external pressure is set to the standard atmospheric pressure of 101,325 Pa. Gravity is applied in the negative direction along the Y-axis, with the value of 9.8 m/s². The pressure-velocity coupling is handled using the SIMPLE algorithm. For investigating the flow behavior, a dynamic mesh technology was used. The number of time steps is 250 s, the time step size is 0.08 s, the maximum number of iterations is 5, and the turbulence model is adopted. A diffusion smoothing method is applied, and the meshes are periodically re-meshed to maintain high quality. The geometrical parameters are substituted into Equation (7) to establish a relationship between the ladle rotation angle θ and time, as shown in Figure 6. Simultaneously, the relationship between the ladle angular velocity ω and time is obtained, as shown in Figure 7.

In the range of 0–20 s, the ladle angular velocity remains constant, and the liquid steel has not yet begun to flow out of the ladle. In the range of 20–110 s, the angular velocity first decreases, then gradually increases, and finally sharply increases. To accurately define the ladle angular velocity over time, a user-defined function (UDF) was used. By defining the boundary as a rotating body around a fixed axis, the dynamic behavior during the pouring process can be precisely captured.

3.2. Transient Flow Field During the Constant Flow Rate Pouring Process

During the constant flow rate pouring process, the change in the phase interface is a crucial factor affecting the flow characteristics. The simulation results indicate that the whole pouring process can be divided into four stages. Each stage reflects different physical phenomena and flow characteristics of the liquid steel. Next, the phases are analyzed in detail at the following four stages, as shown in Figure 8.

(1)

The first stage of the pouring process (0~20 s). The ladle begins to rotate, and the angular velocity remains constant. The liquid steel starts to flow slowly on the upper surface, with the flow mainly concentrated in a small area inside the ladle.

(2)

The second stage of the pouring process (20~56.8 s). The liquid steel begins to flow out of the ladle, with the flow concentrated around the ladle outlet.

(3)

The third stage of the pouring process (56.8~106.4 s). As the ladle rotation angle increases, the residual liquid volume decreases and the bottom of the ladle becomes exposed, so the ladle angular velocity gradually increases.

(4)

The fourth stage of the pouring process (106.4~110 s). The residual liquid volume further decreases until the pouring is complete, and little liquid steel is left in the ladle.

In addition, a cross-section of the ladle is presented to observe the velocity change in the liquid steel during the constant flow rate pouring process, as shown in Figure 9.

(1)

The first stage of the pouring process (0~20 s). Initially, the liquid velocity was zero. When the ladle begins to rotate, liquid velocity occurs inside the ladle, while no liquid velocity happens outside the ladle.

(2)

The second stage of the pouring process (20~56.8 s). The liquid steel begins to flow out of the ladle, so liquid velocity occurs outside the ladle. The liquid velocity gradually increases with the increase in the falling height due to gravity. Meanwhile, the liquid distribution area increases with the increase in the falling height due to the liquid diffusion. To realize the constant flow rate pouring, the maximum liquid velocity gradually decreases during this stage.

(3)

The third stage of the pouring process (56.8~106.4 s). Similarly, both the liquid velocity and the liquid distribution area increase with the increase in the falling height. Conversely, to realize constant flow rate pouring, the maximum liquid velocity gradually increases during this stage.

(4)

The fourth stage of the pouring process (106.4~110 s). When the ladle rotation angle approaches 90°, the residual liquid steel is very little, so the maximum liquid velocity continues decreasing, although the ladle angular velocity increases rapidly.

3.3. Effect of Inner Wall Inclination Angle α

During the constant flow rate pouring process, the inner wall inclination angle α and the initial liquid volume V_c would significantly influence the instantaneous flow behavior and the residual liquid volume, according to Equation (11). Constant flow rate pouring at different ladle inner wall inclination angles α are simulated and change curves of the residual liquid volume and the instantaneous flow rate are obtained, are shown in Figure 10.

No matter how large α is, the residual liquid volume and the flow rate have the same trend. The residual liquid volume first remains unchanged and then decreases to zero, while the flow rate first is zero and then sharply increases to a nearly constant value. As α increases, the initial liquid volume V_c also increases, which directly influences the change curve of the residual liquid volume. Both the initial volume and the flow rate of the liquid steel increase with the increase in α. When α is in the range of 2.5° and 7.5°, the flow rate fluctuates, and when α is in the range of 7.5° and 12.5°, the flow rate slightly decreases. However, when α = 7.5°, the flow rate curve is constant and stable. Therefore, to achieve a stable pouring process and a constant flow rate value, a proper inner wall inclination angle α should be selected in actual production.

3.4. Influence of Initial Liquid Volume V_c

Constant flow rate pouring at different initial liquid volumes V_c is simulated and change curves of the residual liquid volume and the instantaneous flow rate are obtained, as shown in Figure 11. No matter how large V_c is, the residual liquid volume and the flow rate have the same trend. The residual liquid volume first remains unchanged and then decreases to zero, while the flow rate first is zero and then sharply increases to a nearly constant value. As V_c increases, the initial volume, the residual liquid volume, and the flow rate of the steel liquid increase. Except for when V_c = 70% V_capacity, the flow rate first slightly decreases and then obviously decreases before pouring completion. However, when V_c = 70% V_capacity, the flow rate curve is constant and stable. Therefore, to achieve a stable pouring process and a constant flow rate value, a proper initial liquid volume V_c should be selected in actual production.

3.5. Influence of Target Flow Rate q_t

As different target flow rate q_t requires different ladle angular velocity, constant flow rate pouring at different target flow rate q_t is simulated and change curves of the residual liquid volume and the instantaneous flow rate are obtained, as shown in Figure 12.

No matter how large q_t is, the residual liquid volume and the flow rate have the same trend. The residual liquid volume first remains unchanged and then decreases to zero, while the flow rate first is zero and then sharply increases to a nearly constant value. As q_t increases, the initial volume remains unchanged, the flow rate of the steel liquid increases, the residual liquid volume decreases more sharply, and the total pouring duration decreases. When q_t is in the range of 0.10 m³/s and 0.12 m³/s, the flow rate curve is constant and stable. However, when q_t is in the range of 0.16 m³/s and 0.24 m³/s, the flow rate slightly decreases. Therefore, to achieve a stable pouring process and a constant flow rate value, a proper target flow rate q_t should be selected in actual production.

4. Experimental Setup and Verification

4.1. Experimental Setup

Liquid steel is of high cost and safety risk, so water is selected as an alternative fluid to verify the pouring process. The water model experiment is based on the Fraude similarity criterion to ensure that the gravity-dominated flow is similar to the dynamics of molten steel. The dynamic viscosity difference is evaluated using the Reynolds number Re. In this study, the Reynolds number Re > 10⁵. Under turbulent conditions, the influence of viscosity can be ignored. Therefore, in the steel liquid pouring simulation, the water pouring experiment can be used to verify the flow behavior of the steel liquid instead of the steel liquid pouring experiment [16]. To verify the accuracy of the analytical model and the simulation model, an experimental setup is constructed, as shown in Figure 13. It consists of many components, including profiles, motors, PLCs, binocular cameras, pressure sensors, circuit breakers, fixed pulleys, pouring ladles, liquid collection buckets, and wire ropes.

The liquid collection bucket is used to collect the poured liquid, and the residual liquid volume is used to verify the analytical model and the simulation model.

A ladle with an inner wall inclination angle of 7.5° is selected. In the constant flow rate pouring experiments, the motor operates at a pre-set speed and then drives the ladle rotating at a time-variant angular velocity, so precise control of the motor speed is crucial to ensure the constant flow rate.

4.2. Different Initial Liquid Volumes

Three sets of constant flow rate pouring experiments are conducted at the initial liquid volume V_c = 3.90 L, 4.50 L, and 5.10 L, which are about 60%, 70%, and 80% of the total liquid volume, respectively. The target flow rate q_t is set to 50 cm³/s. The experimental residual liquid volume is acquired and then compared with that calculated by the analytical model and the simulation model, as shown in Figure 14.

It is observed that the experimental results agree well with the theoretical and simulation results at different initial liquid volumes, which verifies the accuracy of the analytical model and the simulation model. Furthermore, the experimental residual liquid volume changes approximately linearly, indicating that the constant flow rate pouring is realized and a constant flow rate can be acquired by adjusting the motor speed.

4.3. Different Target Flow Rates

Three sets of constant flow rate pouring experiments were conducted at the target flow rates q_t = 50 cm³/s, 45 cm³/s, and 40 cm³/s. The initial liquid volume V_c is set to 4.5 L, which is about 70% of the total ladle volume. The experimental residual liquid volume is acquired and then compared with that calculated by the analytical model and the simulation model, as shown in Figure 15.

It is observed that the experimental results agree well with the theoretical and simulation results at different target flow rates, which verifies the accuracy of the analytical model and the simulation model. Furthermore, the experimental residual liquid volume changes approximately linearly, indicating that the constant flow rate pouring is realized and different constant flow rates can be acquired by adjusting the motor speed.

5. Conclusions

To realize accurate ladle pouring, an analytical model of the constant flow rate pouring was established. By integrating a user-defined function (UDF), a CFD simulation model of the constant flow rate pouring was conducted to investigate the liquid steel pouring behavior under different inner wall inclination angles α, initial liquid volumes V_c, and target flow rates q_t. Finally, the accuracy of the analytical model and the simulation model was verified through experiments. The results are as follows:

(1)

The experimental results agree well with the theoretical and simulation results, which verify the accuracy of the analytical model and the simulation model.

(2)

During the pouring process, the residual liquid volume and the flow rate have the same trend. The residual liquid volume first remains unchanged and then decreases to zero, while the flow rate first is zero and then sharply increases to a nearly constant value.

(3)

Increasing both α and V_c leads to an increase in the flow rate. To achieve a stable pouring process and a constant flow rate value, proper α, V_c, and q_t should be selected. In this study α = 7.5°, V_c = 70% V_capacity and q in the range of 0.10–0.12 m³/s are proper.

(4)

To realize constant flow rate pouring, a time-variant ladle angular velocity is obtained, and it can be adjusted by the motor speed. Therefore, different constant flow rates could be acquired by adjusting the motor speed, which provides guidance to the engineering application of casting technology.