1. Introduction
Due to the rising need for high-quality steel products, ladle metallurgy, a secondary steelmaking process, has attracted a lot of attention over time. Ladle metallurgy is practiced by adjusting a steel’s composition, cleanliness, and temperature over wide ranges to meet diverse plant and customer requirements. The technique involves a series of different steps after tapping liquid steel into a teeming ladle and before sending it to the caster. These steps include de-oxidation, composition adjustment, temperature control, degassing, cleanliness adjustments, etc. Depending upon the final grade of steel, the steelmaking professional must select different operations wisely. However, all secondary steelmaking operations have one thing in common: the injection of inert gas into the steel melt using one or more porous plugs installed in the bottom brickwork of a ladle. Since argon (Ar) gas has zero solubility in steel and is inert by nature, it is frequently used for purging a melt to ensure temperature and composition homogeneity as well as to encourage reactions that lead to the refining of the steel through slag–metal interactions. Many studies have also confirmed that argon gas injection can aid in inclusion removal.
The rate of injection of argon gas varies, depending upon end requirements, i.e., inclusion removal, floating out the non-metallic inclusions, or desulphurization of steel. The argon gas bubbles become the carrying agents, taking the non-metallic inclusions toward the slag surface. For homogenization or alloy additions, an intermediate flow rate is needed. The dissipation of the buoyant energy of the injected gas primarily causes the homogenization of bath temperature and composition via gas bubbling. On the other hand, a relatively high argon flow rate is used for desulphurization or reoxidation, for which intense mixing conditions are desired. In
Figure 1, a schematic of ladle purging in a Ladle Refining Furnace (LRF) is shown [
1].
The gas rising through the melt induces a turbulent re-circulatory motion in which mass transfer-controlled processes (such as the melting of deoxidizer and alloying additions as well as their dissolution and dispersion) take place. Furthermore, as the injected gas escapes to the surroundings, the redirected bulk flow from the spout region (plume eye) pushes the slag layer radially outwards, exposing the melt surface to the ambient atmosphere. The uncovered area of the melt thus created is typically referred to as an SOE or the “slag open eye”. Note that the slag eye is a potential site for reoxidation, nitrogen pick up, and slag entrainment/entrapment phenomena and hence can profoundly influence the quality of steel. Therefore, during the final stage of ladle refining and immediately before continuous casting, it is customary to practice gentle stirring (commonly termed in the industry as ‘argon rinsing’) to ensure a small “Slag Open Eye” (SOE) area.
Numerical studies of this gas–liquid flow phenomena in a ladle can be categorized into four methods: (a) Quasi-single-phase model, (b) Volume of Fluid (VOF) model, (c) Eulerian multiphase (E–E) model, and (d) Eulerian–Lagrangian (E–L) model. Two good review papers [
2,
3] have discussed all these methods in detail. In this present work, the quasi-single-phase method was used for the optimization study of the porous plug position at the bottom of the ladle. This current research opted for quasi-single-phase modeling to compare mixing times in a steel ladle with a dual-plug configuration at different plug locations and flow rates, which is a less computationally expensive method than multiphase modeling. In quasi-single-phase modeling, the gas–liquid two-phase region is considered a homogenous liquid with a slightly reduced density compared to the surrounding bulk liquid. This allows for a single set of equations of motion to be used to represent the flow in the liquid phase, where the buoyant forces resulting from gas injection are included in the momentum conservation equation in the axial or vertical direction. To understand the state of stirring or agitation efficiency in a ladle, the concept of mixing time has been commonly used. Some previous works [
4,
5] have successfully validated numerically calculated mixing time using the quasi-single-phase method with their corresponding physical model. Several mathematical studies [
4,
5,
6,
7,
8,
9] have used mixing time in the ladle to identify the positions best suited for porous plugs to ensure rapid mixing. Several works [
10,
11,
12] have been conducted to study the effect of differential gas flow rates from a dual plug. Luis et al. [
10] showed that the 3:1 ratio of gas flow rate from a porous plug gives a reduced (i.e., better) mixing time compared to a 1:1 ratio of gas flow rate from the two porous plugs. In this dual-plug design, the net gas flow rate is divided into two regions, resulting in two weakened plumes compared to a single-plug system.
In addition, previous studies have explored the formation of slag open eyes using multiphase modeling techniques [
13,
14,
15,
16,
17,
18,
19]. Ramasetti et al. [
13] investigated the impact of top layer thickness and density on open-eye formation in a gas-stirred ladle. Mantripragada et al. [
14] used the Coupled Level Set Volume of Fluid (CLSVOF) model to investigate inlet-gas-purging rate, melt height, slag layer thickness, and angular and radial positions of gas inlets affecting slag opening area. Liu et al. [
15] simulated a four-phase flow consisting of bubble–steel–slag–top gas in a bottom-blown argon-stirred ladle. Results showed that at low gas flow rates, small open eyes formed and collapsed alternately, while at high gas flow rates, the size of the slag eye increased and its shape changed from circular to oval.
Apart from those studies, Liu et al. [
7] have shown that a ladle with a dual-plug configuration can perform better in reducing wear on refractory linings compared to a single-plug system for an equal net flow rate of argon gas injection. The two plugs can reduce the values of interfacial velocity, reducing the potential for slag entrainment and erosion of the upper sections of the refractory wall. Similar observations are reported in some other literature [
20,
21].
While numerous studies have focused on simulating steel flow fields and optimizing the positions of porous plugs, these investigations have customarily been limited to cylindrical ladles. In contrast, this study aims to analyze and simulate mixing behavior within an elliptical ladle. The primary objective is to conduct a numerical investigation to determine how the arrangement of plugs and their varying flow rates would affect the mixing times in the ladle. To achieve this, a quasi-single-phase modeling approach is employed. Furthermore, this study compares the mixing behavior between a single-plug configuration and a dual-plug configuration in an argon-stirred ladle using a single-phase modeling technique for computational efficiency. Additionally, a transient multiphase model incorporating a volume of fluid (VOF) model was developed to simulate the formation of slag open eyes (SOE) in the single- and dual-plug ladle configurations.
2. Mathematical Modeling
Mathematical modeling was carried out using the Computational Fluid Dynamics (CFD) software ANSYS Fluent. In this present work, the simulation process was performed in two parts. In the first part, a quasi-single phase, isothermal, three-dimensional, incompressible, turbulent flow model was developed to simulate and understand flow dynamics inside an elliptical ladle and to calculate its mixing times numerically. In this case, slag and air phases in the ladle were ignored. Only the effects of argon bubbling into the ladle were considered. In the quasi-single-phase modeling, the gas–liquid two-phase region is treated as a homogeneous liquid with a slightly reduced density compared to the surrounding bulk liquid. This modeling approach ignores the interactions and exchanges between different phases and assumes constant fluid properties, such as density and viscosity, throughout the entire domain. Consequently, it fails to capture the complexity and dynamics of multiphase flows accurately, including interfacial phenomena. The objective was to numerically investigate the fluid flow behavior and mixing phenomena between a single-plug configuration and a dual-plug configuration in the argon-stirred ladle. Additionally, the effect of the different positional arrangements of an additional plug along with the existing plug is studied in terms of mixing times. Additionally, the effect of differential gas flow rates through the two plugs was studied. As such, the relevant governing equations used in the simulation are as follows.
2.1. Governing Equations
2.1.1. Mass Conservation Equation
In Equation (1), terms and denote density (kg/m3) and velocity (m/s), respectively at a point .
2.1.2. Momentum Conservation Equation
In Equation (2), the term P represents pressure (Pa), g is acceleration due to gravity (m·s−2), and is the effective viscosity, representing the summation of molecular viscosity and turbulent viscosity (μ + μt).
2.1.3. Transport Equations for k and ε in the k-ε Model
The kinetic energy of turbulence,
k, is given as follows:
The rate of dissipation of kinetic energy, ε is given as follows:
where
C1,
C2,
and
are empirical constants, whose values are 1.38, 1.92, 1.0, and 1.3, respectively. Moreover,
G represents the generation of kinetic energy of turbulence due to mean velocity gradients.
2.1.4. Species Transport Equation
Once the flow fields have converged during simulation, the transient state is switched on for the species transport equation to visualize mass flow rate within the domain.
In Equation (5), is the effective mass diffusion coefficient given by . and are the laminar and turbulent Schmidt numbers, respectively.
In the second part of the simulation, a transient, isothermal, multiphase phase (steel–slag–argon–air) model was developed. The volume of fluid (VOF) modeling technique was used to investigate slag open eye (SOE) formation. To conduct this simulation, the mass and momentum equations were solved as shown in Equations (1) and (2), respectively. Additionally, the k-ε model was used to incorporate turbulence phenomena, as shown in Equations (3) and (4). It is important to note that the transient term was added to all four equations to account for the system’s transient behavior.
2.1.5. Volume of Fluid Model
The VOF method is widely used for simulating multiphase flows. It assumes incompressible fluid phases with no mass transfer between them. The VOF method tracks the interface between phases with a sharp boundary assumption, facilitating clear interface location tracking during the simulation. However, this assumption can lead to inaccuracies in capturing small-scale features or sharp gradients at the interface. In this work, it was used to track the liquid steel/slag/air interface behavior. The finite volume equation of the VOF model can be written in the following form:
where
and
represent the mass transfer from phase
p to phase
q and
q to
p, respectively, in unit time and volume;
is the volume fraction of phase
q;
is the density of phase
q;
is a source term (=0). When the volume fractions are summed, the following equation must be satisfied:
2.2. Characteristics of Gas-Liquid Plume
In quasi-single-phase modeling, the gas–liquid mixture is modeled as a homogeneous fluid. This region is referred to as a plume; the volume fraction of gas inside the plume and the dimension of the plume region play a significant role in modeling. They are predicted with experimental results or equations available in the literature. The volume fraction of gas is estimated from the principle of volume continuity using the average rise velocity of the gas–liquid mixture and plume dimensions. Assuming a no-slip condition for this current study average gas volume fraction is given as follows:
where
is the average velocity of plume, and
is plume rising velocity; Sahai and Guthrie [
22] provided the following equation to calculate plume rising velocity:
Using gas volume fraction value, density of the plume can be estimated as follows:
Similarly, Goldschmit and Owen [
23] have estimated top surface radius of plume for a ladle with height, H containing liquid steel as
, where b is radius of plume in which gas fraction is half of centerline gas fraction, given by the following:
where value of
is given by the following:
where
is equivalent to gas flow rate a bottom (
z = 0) at steel melting temperature. It is expressed as follows:
and
is the equivalent gas flow rate a top (
z =
H) at steel melting temperature.
2.3. Geometry and Mesh Setup
The geometry of an elliptical ladle operating in an industrial set-up was developed using ANSYS SpaceClaim software.
Figure 2 shows the three-dimensional geometry of the elliptical ladle. For the quasi-single-phase simulation, the air region and slag layer were ignored, and the gas–liquid plume region was introduced to incorporate the effect of argon bubbling. Plume dimension calculations are discussed in the previous section. The constructed geometry was then exported to the ANSYS Meshing tool, in which a tetrahedral mesh was generated.
Figure 2a presents the geometry created for the ladle with a single plug, and
Figure 2b presents a dual-plug configuration. The generated mesh for the ladle with a single plug contained 217,812 nodes and 136,267 elements. Similarly, the mesh of a ladle with a dual plug had 589,903 nodes and 363,293 elements. The geometrical parameters of the ladle for the quasi-single-phase modeling are shown in
Table 1.
For the VOF multiphase model, all four phases (liquid steel, slag, argon, and air) were considered. The same geometries and meshes were used, as shown in
Figure 2, after ignoring the plume region. The height of the air region was considered as 609 mm, and the slag layer thickness was kept at 106 mm as per the industrial setup. The details of material properties used in the mathematical model are shown in
Table 2. To replicate the actual plant process, the density of argon gas is taken, corresponding to a temperature around 1100 °C [
24].