The process of tapping steel from the converter to the ladle is a critical step in steelmaking, during which alloying materials are added to ensure rapid homogenization of the molten steel. This study explores the challenges of alloying, particularly the timing of addition and its impact on ladle operation. Improper operation can damage the ladle refractory lining and hinder visual monitoring, while delayed addition reduces mixing efficiency and disrupts process control. To address these issues, a mathematical model was developed to determine the minimum ladle molten steel level required for safe and efficient alloying. This model considers the movement of the alloying material through the feed pipe, its free fall, and its immersion in the molten steel in the ladle.

The tapping of steel from the converter (BOF) to the ladle is a crucial step in the steelmaking process, serving as the initial step in refining molten steel. As shown in Figure 1a, alloying materials are added from an elevated bin and conveyor belt via a movable feed pipe. The agitation energy of the molten steel within the ladle, generated by the impact of the tapping airflow, promotes rapid homogenization. Commonly added materials include synthetic slag, limestone, aluminum (Al), ferrosilicon (FeSi), ferromanganese (FeMn), and carburizing agents.

Figure 1. Basic configuration of the tapping process and the addition of alloy materials (a); Addition of alloy materials during tapping (this generates smoke and subsequently obstructs the operator’s view) (b)

Recent advances in understanding the tapping process have involved thermodynamic modeling and computational fluid dynamics (CFD) simulations. Researchers have used CFD to analyze the converter blowing and electric arc furnace tapping processes, conducting comprehensive parametric studies on alloy type, particle size, and addition time. Their results indicate that smaller alloys (5-20 mm) are more effective than larger particles (up to 80 mm). Building on this, the researchers have further expanded the work by incorporating three-dimensional flow dynamics and entrained gas effects into the CFD method. These studies primarily explore the flow of molten steel within the ladle and the chemical reactions within the ladle during tapping.

This study focuses on the alloying process, specifically the free fall of solid alloy material into a ladle containing partially molten steel. The timing of this process is crucial: improper handling can damage the refractory material at the bottom of the ladle or interfere with the guide sand above the start-up mechanism’s slide plate, potentially preventing the ladle from opening properly for pouring. To avoid these problems, a minimum ladle steel level must be set before alloying begins. However, adding the alloy too late reduces stirring efficiency and mixing performance. The addition of the alloy also generates fumes, which can hinder visual monitoring of the tapping process, as shown in Figure 1b.

To address these challenges, this study developed a physical model to simulate the movement and immersion of alloy materials in molten steel. This model provides valuable insights for optimizing the timing of alloy addition, contributing to a more controllable and efficient process.

Modeling

The model presented in this study aims to simulate the movement of alloy additives through an inclined feed pipe, followed by the free fall of the alloy material into a ladle containing molten steel. The mathematical model consists of two parts, as shown in Figure 2.

Figure 2. Alloy material moves through an inclined feed pipe (a); alloy material falls freely and is immersed in molten steel (b).

In Figure 2a, the first part involves modeling the movement of a single block of alloy material in an inclined feed tube. Here, the interaction between the material block and the tube is simplified by assuming a constant coefficient of friction μ. This assumption allows for the calculation of the velocity vector of the alloy material at the feed tube outlet, thus providing initial conditions for the second part of the model.

In the second part of the model (as shown in Figure 2b), the alloy material block enters a free-fall phase, during which it impacts the surface of the molten steel inside the ladle. As mentioned earlier, the molten steel inside the ladle has a specific height, and the goal of this modeling process is to determine whether the hypothesized spherical alloy material block touches the bottom of the ladle. It is important to note that although this model focuses on the motion of a single spherical block, the simultaneous release of multiple alloy material blocks can create complex interactions between the solid block group and the molten steel, significantly altering the calculation results.

Operation via the inclined feed pipe

The tangential force component that accelerates an alloy additive block of mass m through an inclined material conveying pipe at an angle α can be expressed as:

Friction between alloy block and feed tube surface

The tangential direction is opposite to the direction of motion. In Equation 2, μ represents the dynamic friction coefficient between the alloy block and the steel surface, which, according to Fuller, is approximately 0.3. The constant acceleration a can be determined using the equation of motion in the tangential direction:

Further simplification of equation 3 yields the following: the acceleration is a constant and is independent of the mass of the alloy block.

According to Formula 4, the flow velocity at the end of a pipe of length lp can be calculated as follows:

If the value of μ is too large, such that sin(α) – cos(α)μ ≤ 0, then the alloy block will not move. As the initial conditions for the free fall and immersion process of the alloy block, the velocity components νpx and νpy can be further calculated as:

Free fall and immersion in molten steel

At the end of the feed pipe, an alloy block, assuming it to be a spherical particle with diameter ds and density dp, falls freely into the molten steel in the ladle with velocities νsx and νsy given by Equation 6. As shown in Figure 2b, the gravitational force acting on the sphere in the vertical direction y is:

Furthermore, the magnitude of the buoyant force acting on the sphere is:

Where ρf represents the fluid density, which is the density of air during free fall (0.947 kg/m³ at 100°C), and the density of liquid steel (7000 kg/m³) after the alloy block is fully immersed in the liquid steel. Since air has a lower density, the effect of buoyancy during free fall can be ignored. The resistance acting on the sphere in the direction of its motion in the fluid can be expressed as:

In Equation 9, νs(t) represents the absolute velocity of the sphere, which is calculated from the velocity component, fluid density ρf, and drag coefficient cD, while the drag coefficient cD depends on the flow state defined by the Reynolds number.

For a sphere, according to White’s formula, the drag coefficient cD(t) can be expressed as:

The kinematic viscosity of air, νf, is 2.08 × 10⁻⁵ m²/s, and the kinematic viscosity of liquid steel is 5 × 10⁻⁷ m²/s. The equation of motion of the alloy block in the x-direction can be written as:

Where θ(t) represents the angle of the spherical alloy block relative to the y-axis. In the numerical simulation, the velocity and position in the x-direction can be updated using a discrete time step Δt, as follows:

The equation of motion in the y-direction is:

The velocity and position in the y-direction can be calculated using Equation 16:

When a spherical object impacts the surface of liquid steel (y < Hs), the properties of the fluid (density and kinetic viscosity) change from those of air to those of molten steel.

Given that the free fall of the alloy block and its entry into the molten steel pool takes approximately 2 seconds, this method requires 15,000 time steps per simulation. Figure 3 shows the results of the numerical calculations; Figure 3a shows the change of the y-coordinate over time, and Figure 3b shows the trajectory of an alloy block with a diameter of 60 mm and a density of 4000 kg/m³. In Figure 3, the liquid steel level is indicated in red, with a set value of 0.45 meters, representing the standard ladle level height prior to this study.

Figure 3. Y-coordinate as a function of time (a) and the trajectory of the alloyed block with a diameter of 60 mm and a density of 4000 kg/m³ (b). The liquid steel surface (marked in red) reading is 0.45 meters.

Once the alloy block is fully immersed in the molten steel and reaches its maximum depth (i.e., the lowest y-coordinate position), it will float upwards, exhibiting slight fluctuations. The key calculation result is information about the lowest immersion point, as this information determines whether the alloy block has touched the bottom of the ladle.

The model was applied using a set of parameters related to the density and size of typical alloy materials, listed in Table 1. According to the data in Table 1, the density ranged from 700 kg/m³ for coke to 4500 kg/m³ for metallic manganese, while the size of the additive blocks ranged from 4 mm to 80 mm. In each simulation run, the lowest y-coordinate value was recorded, representing the maximum immersion depth of the alloy material block in molten steel. These results are summarized for all parameters in Figure 4.

Table 1. Density and particle size analysis of various additives

Figure 4. Relationship between maximum immersion depth of alloy materials in molten steel and density and size.

Figure 5 shows the maximum size of an alloy material flow with a specific density in molten steel at different levels within a ladle, which allows the alloy block to remain suspended without contacting the bottom of the ladle. The maximum size that can remain suspended without contacting the bottom of the ladle increases significantly when the molten steel level increases from 10% to 15%.

The data shown in Figure 4 clearly demonstrates that the greater the density and size of the alloy ingot, the greater its maximum immersion depth. The smallest ingot, coke, had an immersion depth of only 3 mm, while the largest ingot, metallic manganese, reached a maximum immersion depth of 375 mm. This result is significant because, considering that the standard molten steel level before this study was 450 mm (equivalent to 10% of the total ladle volume), this maximum immersion depth is close to the bottom of the ladle.

Given that the model only considers a single addition of alloying material, and considering the previously mentioned potential for severe disturbance to the molten steel surface from releasing multiple alloying materials simultaneously, it can be inferred that in actual operation, alloying materials with densities higher than 4000 kg/m³ (such as FeSiMn, FeMnHC, FeMnMC, and metallic manganese) may impact the bottom of the ladle. Based on these findings, the minimum ladle molten steel level required to begin adding alloying materials increases to 675 mm, equivalent to 15% of the total ladle volume.

So far, this model has focused solely on the immersion process of a single alloy bulk in molten steel. However, in practical applications, the alloy addition process is better represented by particle flows. These bulk flows interact more extensively with the liquid surface, potentially inducing turbulence and surface waves. The next step is to approximate the aggregate of alloy material as a single bulk with the same diameter as the individual pieces, thus providing a simplified approach for further analysis.

Figure 4 illustrates the relationship between the maximum size of the alloy material and its density under different molten steel levels. The graph shows the relationship between the alloy material density (kg/m³) and the maximum immersion depth, with the curves corresponding to different molten steel levels (5%, 10%, 15%, 20%, and 25%). As the alloy material density increases, the maximum permissible size decreases, while higher molten steel levels allow for deeper immersion. This effect is particularly pronounced for lower-density materials, where the increase in immersion depth is significantly greater than for higher-density materials. Notably, Figure 4 indicates that, as recommended in this study, increasing the molten steel volume from 10% to 15% (equivalent to an increase in molten steel level of 0.22 meters) significantly increases the size of the largest alloy block capable of suspending without touching the bottom of the ladle.

A developed model for determining the minimum molten steel level required for the safe addition of alloying materials during converter tapping shows that the size and density of the added material significantly affect its immersion depth. Calculations indicate that denser materials and larger addition amounts are more likely to reach the bottom of the ladle. This necessitates a higher minimum molten steel level to ensure safe operation and prevent damage to the refractory material or disturbance of the ladle guide sand, thus avoiding situations where the ladle cannot be opened for pouring. The model’s findings suggest that in this specific application, when using denser materials such as ferrosilicon, ferromanganese, or metallic manganese, the ladle level should be increased.