1. Introduction
In the control of steel cleanliness, non-metallic inclusion removal is largely accomplished (65–75%) through ladle operations. Non-metallic inclusions directly affect steel products’ fatigue strength, impact toughness, corrosion resistance, and machinability factors such as cutting force, chip/crack formation, power consumption, and tool wear. Non-metallic inclusions can also lead to excessive casting repairs and rejected castings. Cumulatively, these factors negatively impact U.S. energy use and efficiency, infrastructure, economic competitiveness, and national security, because high-quality steel products are keystone parts of fundamental transportation, construction, communications, and defense systems.
In the past few decades, several researchers have investigated inclusion behavior and flow characteristics in a gas-stirred refining ladle. In 1975, Szekely et al. [
1] first modeled and studied the flow characteristic in a ladle based on a simplified water model. Gas was injected from the bottom of the ladle with an assumed constant bubble size, and all other boundary conditions were sourced from the physical experiment. By using Spalding’s k-ω model, the Navier-Stokes equations were solved to predict velocity and turbulence in a water model. In 1978, DebRoy et al. [
2] improved Szekely et al.’s model by revising the bubble model from dispersed bubbles constrained in a single plume with a diameter related to volume fraction to injection based on gas flow rate. Johansen et al. [
3] performed experiments by using a bottom injection water model, discovering key interactions between bubble-generated turbulence and flow velocity in the bubble plume region.
Peranandhantan et al. [
4] conducted experiments to study the formation and size of a slag eye in a simplified water model. Several variables such as gas flow rate, slag thickness, and liquid depth were explored in their work. The slag eye was captured and measured through visual inspection with a camera. Mazumdar et al. [
5] reviewed several studies on physical models and empirical correlations of gas-stirred ladles, covering various simplified expressions of several variables, including gas flow rate and ladle dimensions, among others. Guo et al. [
6] developed a 3-D CFD ladle simulation using a Lagrangian–Eulerian model to predict gas-liquid two-phase flow in the ladle. In this work, more detailed models such as lateral drag force and lift force and so on were added to predict the plume shape. Plume spreading and mass transfer to rising bubbles and the top free surface are included. Zhang et al. [
7] developed a Eulerian–Eulerian two-phase flow system for ladle, with the k-ϵ turbulence model used to simulate turbulence inside of the ladle. Lou et al. [
8] developed a Eulerian–Eulerian numerical model to simulate gas-liquid two-phase flow in the bottom injection gas-stirred system. The effect of the turbulent dispersion force, drag force, and lift force upon the flow field were studied. A bubble-induced turbulence term was developed and included as a source term in order to more accurately predict the bubble plume shape.
For the inclusion removal process, Lou et al. [
9] conducted a review of inclusion behavior mechanisms, including inclusion aggregation and removal. In the past several decades, beginning in the 1970s, researchers have applied numerical methods to calculate and analyze inclusion behavior. In 1975, Szekely et al. [
1] utilized a 2-D model to study inclusion aggregation by inclusion turbulence collision. In 1983, Shirabe et al. [
10] also developed a 2-D model to predict inclusion aggregation by inclusion turbulence collision. However, consideration of all the mechanisms, including inclusion aggregation and inclusion removal was not made. In 2001, Mats Söder [
11] discussed the four mechanisms of inclusion aggregation and all three inclusion removal mechanisms. Four different bubble removal models were compared in this thesis, and more detail was added on inclusion removal due to inclusion-bubble interaction. In this study, inclusion-bubble interactions, including turbulence random collision, turbulence shear collision, laminar shear collision, and stokes collision, were considered. In 2005, Wang et al. [
12] reviewed the inclusion removal process and proposed an expression to represent inclusion removal probability. The expression is related to the probability of adhesion, collision, and detachment. In the study, the author also mentioned that detachment is nearly negligible. In 2013, Lou et al. [
9] performed a more detailed study of inclusion behavior in refining ladles utilizing three inclusion aggregation mechanisms and six inclusion removal mechanisms. In 2019, Chen et al. [
13] first validated simulated inclusion removal results with measured data using the inhomogeneous discrete method of the population balance model.
The intent of this work is to build upon previous studies on flow characteristics and inclusion behavior conducted by researchers at Purdue University Northwest’s Center for Innovation through Visualization and Simulation (CIVS) to develop a comprehensive CFD ladle model capable of analyzing the effects of gas stirring on a steel-refining ladle, with particular attention to the methods by which inclusions are floated, grow, and are removed from the steel melt. The effects of gas stirring on flow in the ladle, including the impacts of bubble-inclusion interaction, are analyzed using the Euler–Euler approach for multiphase flow modeling. The effects of inclusion aggregation and removal are subsequently modeled via the population balance model (PBM). Through the development of an inclusion removal model and subsequent simulations of flow, inclusion floatation, and inclusion removal within a steel-refining ladle, a better understanding of the mixing and cleaning process can be obtained, and further studies using this modeling approach can be utilized to improve mixing time, enhance steel quality, and minimize potential defects in the downstream steelmaking process.
2. Methodology and CFD Models
The commercially available CFD software ANSYS Fluent 2020R1 was used in this research, with user-defined functions developed specifically for calculating the inclusion aggregation and removal. The simulation process utilized in this study consists of two stages: flow field simulation and inclusion removal simulation. The flow field is simulated using the Euler–Euler multiphase method. In this stage, only two phases are included: liquid steel and injected argon gas. After the multiphase flow simulation in this stage reaches a quasi-steady state, the population balance model will be activated to simulate the inclusion aggregation and removal process. In this second stage, two inclusion phases, inclusion A and inclusion B, are considered. In phase inclusion A, the inclusion aggregation mechanism is activated, allowing smaller inclusions to agglomerate into larger inclusions. This is not the case for inclusion phase B. During the simulation, when inclusion is considered “attached” to a bubble, it will be transferred from inclusion phase A to inclusion phase B. Initial inclusion number density distribution is adopted from published measurements performed by Miyashita et al. [
14]. Here, the initial inclusion number density is assumed as
/m
3 [
13]. The equation of initial number density distribution is shown below:
where
and
are initial inclusion number density with specific size j and initial total inclusion number density.
The balance of mass and momentum of each phase is solved individually. In addition to the standard Navier-Stokes transport equations, population balance modeling based on the PBE is used to handle the transport of particle number density in the fluid, accounting for inclusion aggregation and removal. The turbulence field is predicted by using the standard k-ε model.
2.1. Continuity Equation
The general continuity equation based on Eulerian – Eulerian multiphase model can be expresed as:
represents the phase volume fraction,
is the phase density, and
is the phase local velocity. For the left side,
is source term from phase
m.
2.2. Momentum Equation
The momentum of each phase is solved individually, the general momentum equation can be expressed as:
where
represents phase m stress-strain tensor;
is the summation of rest of forces, including virtual mass force, drag force, and turbulence dispersion force [
8].
2.3. Turbulence Model
In the study, standard k-ε model is used to model the turbulence field in the ladle.
where
and
are constant, and
and
are the turbulent kinetic energy generation due to velocity gradients and buoyancy.
and
represent source terms to turbulent kinetic energy and turbulence dissipation rate. In this study, bubble-induced turbulence [
8] is considered via a source term added into a user-defined function, which was hooked in the simulation.
2.4. Population Balance Model
Population Balance Modeling (PBM) is based on the Population Balance Equation (PBE). In the study, only inclusion aggregation and removal is considered. Inclusion growth and breakup will not be involved. The transport equation for number density of particles can be express as:
where
is the number density of j size inclusions.
represents the aggregation rate of inclusion from size j to size k.
is external source term, and it can be used to calculate inclusion removal.
2.5. Inclusion Aggregation Model
There are three main mechanisms of inclusion aggregation widely studied. They are Brownian motion aggregation, turbulence collision aggregation, and stokes buoyancy collision aggregation. Brownian motion aggregation is neglected since it takes some time for particles to grow [
15]. The overall inclusion aggregation rate can be calculated as follows:
where
represent inclusion the aggregation rate from turbulence collision aggregation and stokes buoyancy collision aggregation.
j and
k represent two different sizes of colliding inclusions.
In a turbulent flow field, the smallest eddy size is called Kolmogorov microscale (
), and it can be expressed as:
In general, in the turbulence field, particle aggregation can occur by viscous mechanism and inertia mechanism. A viscous mechanism occurs when the size of colliding particles is smaller than the Kolmogorov microscale, while an inertia mechanism occurs when the size of colliding particles is greater than the Kolmogorov microscale. Based on Saffman and Turner [
16], the inclusion aggregation due to viscous mechanism can be express as:
where
is the capture efficiency of two colliding inclusions.
is the force coefficient.
is the Hamaker constant and can be assigned to
.
As for the inertia mechanism, the aggregation rate is expressed as: [
17]
where
is capture efficiency of two colliding inclusion.
is inclusion velocity with size
.
Another inclusion aggregation mechanism is Stokes buoyancy collision aggregation. Because of density difference, alumina inclusion can rise up. When inclusions rise up, the flow field can influence the inclusion rising path. The inclusion aggregation rate due to Stokes buoyancy collision can be expressed as [
13]:
where
represents the capture efficiency of two colliding inclusions due to the stokes buoyancy collision aggregation, and
is the stokes velocity calculated from stokes law.
2.6. Inclusion Removal Model
There three different inclusion removal mechanisms: inclusion removal due to wall adhesion, inclusion removal due to slag absorption and inclusion removal due to bubble attachment. The overall inclusion removal rate is the summation of the rate from these mechanisms:
The first part of inclusion removal will be wall adhesion. Based on Zhang et al.’s [
18] model, the turbulence dissipation rate is introduced into calculation. The removal rate can be expressed as follows:
where
represents the current time size j inclusion number density, and the unit for this is n/m
3.
and
represent local slag cell effective area and volume.
is the density of inclusion.
is the volume of inclusion with size j.
The second part of the inclusion removal calculation is inclusion removal due to slag absorption. In the current model, when inclusion reaches the slag-steel interface under the influence of the flow field, it is assumed ideally removed by slag. However, when inclusions reach the liquid-free surface in the slag eye zone, they will not be removed, and instead, they will continue to circulate within the steel melt. The removal rate due to slag absorption can be described as:
For inclusions attached to bubbles, when the rupture of the bubble occurs at the slag-steel interface, any inclusions attached to the bubble will be considered totally removed. However, similarly to inclusions floating in the melt, when a bubble ruptures at the slag eye zone, any inclusions attached to that bubble will be considered re-released back to steel. The inclusion removal due to bubble attachment can be described as:
where
is the relative velocity between the gas bubble and inclusion particles;
is the volume fraction of gas bubble in the computational cell;
is the efficiency of inclusion attached to bubble [
19]. P consists of two factors:
, which represents the probability of inclusion collision with bubbles, and
, describing the probability of inclusion attaching to bubbles through adhesion. In this study, based on Trahar’s [
20] work, the detachment rate of inclusions from bubbles is assumed to be zero. This is an idealized assumption to reduce computational complexity, meaning that when an inclusion attaches to a bubble, it will never detach.
4. Conclusions
An unsteady, three-dimensional, isothermal, multiphase computational fluid dynamics (CFD) model was developed for simulating fluid flow and inclusion agglomeration and removal in a steel-refining ladle. The inclusion aggregation and removal model is validated by comparing simulated inclusion number density distribution with experimental measurements. With this model, a comprehensive study of inclusion aggregation and removal in different bottom gas-stirred ladles was conducted. Based on a simplified Nucor Steel two-plug bottom injection ladle, two different separation angle ladle cases, 180° and 90°, were developed. Three argon gas flow rate combinations (5/5 SCFM, 5/20 SCFM, and 20/20 SCFM) were employed across these ladle designs. Inclusion aggregation and removal in different ladles have been investigated.
A study of the impacts of ladle design and flow rate on inclusion aggregation and removal has been conducted utilizing the aforementioned CFD model. Key findings include notes on the impacts of gas flow rates on slag eye formation and the further impacts of these conditions on inclusion aggregation and removal. Without considering re-oxidation at slag eye, for a fixed gas flow rate, an increase in plug separation angle will result in an increase in the inclusion removal rate. This was demonstrated by comparing 180° and 90° separation angles. Likewise, when the plug separation angle is fixed, and the gas flow rate increases, the inclusion removal rate will also increase.
Further research may be necessary to explore the potential impacts of these design modifications. A comparison of potential erosion on the ladle wall due to shear stress may be necessary to ascertain whether the increased flow rate near the ladle walls would result in shorter lifespans for refining ladles, and correspondingly whether adjusting the position of porous plugs would present a beneficial design modification for industrial facilities.