# A Review of Physical and Numerical Approaches for the Study of Gas Stirring in Ladle Metallurgy

- 458 Downloads

## Abstract

This article presents a review of the research into gas stirring in ladle metallurgy carried out over the past few decades. Herein, the physical modeling experiments are divided into four major areas: (1) mixing and homogenization in the ladle; (2) gas bubble formation, transformation, and interactions in the plume zone; (3) inclusion behavior at the steel–slag interface and in the molten steel; and (4) open eye formation. Several industrial trials have also been carried out to optimize gas stirring and open eye formation. Approaches for selecting criteria for scaling to guarantee flow similarity between industrial trials and physical modeling experiments are discussed. To describe the bubble behavior and two-phase plume structure, four main mathematical models have been used in different research fields: (1) the quasi-single-phase model, (2) the volume of fluid (VOF) model, (3) the Eulerian multiphase (E–E) model, and (4) the Eulerian–Lagrangian (E–L) model. In recent years, the E–E model has been used to predict gas stirring conditions in the ladle, and specific models in commercial packages, as well as research codes, have been developed gradually to describe the complex physical and chemical phenomena. Furthermore, the coupling of turbulence models with multiphase models is also discussed. For physical modeling, some general empirical rules have not been analyzed sufficiently. Based on a comparison with the available experimental results, it is found that the mathematical models focusing on the mass transfer phenomenon and inclusion behaviors at the steel-slag interface, vacuum degassing at the gas–liquid interface, dissolution rate of the solid alloy at the liquid–solid interface, and the combination of fluid dynamics and thermodynamics need to be improved further. To describe industrial conditions using mathematical methods and improve numerical modeling, the results of physical modeling experiments and industrial trials must offer satisfactory validations for the improvement of numerical modeling.

## Variables

- \( \lambda \), \( \lambda_{{\rho_{\text{l}} }} \), \( \lambda_{{\rho_{\text{g}} }} \), \( \lambda_{{\mu_{\text{l}} }} \), \( \lambda_{\sigma } \)
Geometric ratio of scaling criterion, liquid density ratio of scaling criterion, gas density ratio of scaling criterion, viscosity ratio of scaling criterion, and surface tension ratio of scaling criterion

- \( \beta \)
Fractional submergence of the top lance

- \( d_{\text{b}} \)
Bubble diameter (m)

- \( H \)
Liquid depth (m)

- \( d \)
Inner nozzle diameter (m)

- \( R \)
Ladle radius (m)

- \( R_{\text{av}} \)
Equivalent plume radius (m)

- \( L_{\text{model}} \)
Size of physical modeling ladle (m)

- \( L_{\text{prototype}} \)
Size of prototype ladle (m)

- \( \rho \), \( \rho_{\text{l}} \), \( \rho_{\text{g}} \), \( \rho_{\text{b}} \)
Mixture density, liquid density, gas density, and density of bubble particles (kg m

^{−3})- \( \alpha_{\text{l}} \)\( \alpha_{\text{g}} \)
Liquid volume fraction and gas volume fraction

- \( \vec{u} \), \( \vec{u}_{\text{q}} \), \( \vec{u}_{\text{rel}} \), \( u_{\text{bi}} \), \( U_{\text{p}} \)
Velocity component of mixture fluid, velocity component of liquid phase, relative velocity between gas and liquid, bubble particle velocity, and plume velocity (m s

^{−1})- \( Q \)
Gas flow rates (m

^{3}s^{−1})- \( g \)
Acceleration of gravity (m s

^{−2})- \( p \)
Total pressure (N m

^{−2})- \( \mu \), \( \mu_{\text{t}} \)
Liquid viscosity and turbulent viscosity (kg m

^{−1}s^{−1})- \( F_{\text{s}} \)
Surface tension force (N m

^{−3})- \( \sigma \)
Surface tension coefficient (N m

^{−1})- \( \kappa \)
Curvature (m

^{−2})- \( B_{{{\text{ag}},{\text{i}}}} \), \( B_{{{\text{br}},{\text{i}}}} \)
Birth term due to aggregation and birth term due to breakage

- \( D_{{{\text{ag}},{\text{i}}}} \), \( D_{{{\text{br}},{\text{i}}}} \)
Death term due to aggregation and death term due to breakage

- \( F_{\text{Di}} \)
Drag force (N)

- \( Q_{\text{bi}} \)
Bubble injection mass flow rate (kg/s)

- \( \Delta t \)
Time-step (s)

- \( \tau \) 95 pct
Mixing time (s)

- \( T_{\text{l}} \), \( T_{\text{g}} \)
Liquid temperatures and gas temperatures (K)

- \( {\text{E}}_{\text{o}} \)
Eötvös number

- \( {\text{F}}_{\text{r}} \)
Froude number

## Introduction

Production of clean steel requires strict control of impurity elements, such as O, H, and N, during ladle metallurgy. In addition, the content of nonmetallic inclusions in steel is an important factor affecting the quality of steel. To remove the inclusions, gas bubbling plays an important role in the steel metallurgy. This process is usually applied in the ladle, tundish, and continuous casting processes. Gas bubbling can increase the inclusion removal rate by adhesion or wake flow capture. Moreover, gas stirring is an important means to offer the fluid dynamics and homogenization of the molten steel with respect to alloy content and temperature and to promote reactions at the steel-slag interface.[1, 2, 3, 4, 5, 6, 7] However, gas bubbling can also intensify the fluctuations of the steel–slag interface, and this may cause splashing and exposure of the steel to the atmosphere.

A large number of articles have been published on the study of gas stirring in ladles, and mathematical models and physical models have been used either separately or together according to the research focus. Several reviews have summarized previous studies[1,2,5, 6, 7] involving cold experiments and mathematical modeling carried out several years ago. In Sichen’s[5] review, the understanding of mass transfer and inclusion behaviors, especially the interactions of different types of inclusions, was proposed as the area requiring further study. A good balance between modeling and experimental research was also proposed because experimental studies have become frequent in recent years. Iron *et al*.[6] reported plume dynamics and Froude number similarity in detail. Moreover, the interfacial phenomenon and steel-slag reactions were also highlighted. Based on previous review works, the present article presents a review of the physical and numerical approaches used in the study of gas stirring in ladle metallurgy over the past 3 decades to give some options and find new and meaningful research directions, as well as desired experimental results for simulation validation. Previous contributions to the study of ladle metallurgy have been categorized into four major groups, as covered in the following sections. Section II: physical modeling experiments, Section III: industrial trials, Section IV: criteria for scaling between physical modeling experiments and industrial trials, and Section V: numerical models to study the gas–liquid zone in ladle refining.

## Physical Modeling Experiments

With the aim of improving clean steel’s quality, many researchers[8, 9, 10, 11] have paid attention to either one or several aspects of gas stirring in the ladle. Depending on the research goal, the previous physical modeling experiments in lab scale have been divided into four major groups: (1) mixing and homogenization in the ladle; (2) gas bubbling formation, transformation, and interaction in the plume zone; (3) inclusion behaviors at the steel–slag interface and in the molten steel; and (4) open eye formation.

### Mixing and Homogenization in the Ladle

Physical Modeling Experiments Performed to Study Mixing and Homogenization in Ladles

Author | Experimental Apparatus | Gas Injection Pattern | Gas Injection Position | Scale | Liquid Metal | Gas | Colored Reagent and Injection Position | Remark |
---|---|---|---|---|---|---|---|---|

Joo and Guthrie[8] | cylindrical vessel (UD1000 mm × BD864 mm × H787 mm) | porous plug | central and off-centered bottom blowing | 1/3 | water | air | KCl above the plume | mixing mechanisms as function of porous plug location, tracer injection point, and ladle monitoring point |

Krishnapisharody | cylindrical vessel (D500 mm × H270 mm) | nozzle | central top blowing and off-centered bottom blowing | 1/5 | water | air | KCl above the plume | mixing time for different bottom blowing positions with top blowing |

González-Bernal | cylindrical vessel (D371 mm × H456 mm) | tuyere | off-centered bottom blowing | 1/7 | water | air | vegetal red colorant at the central bottom | effects of locations of single tuyere and dual tuyeres on the mixing time |

Fan | cylindrical vessel (D614 mm × H800 mm) | nozzle | central and off-centered top blowing | 0.25 | water | air | dye NaCl above the water and submerged | optimal position of Ca-Si injection |

Mazumdar | cylindrical vessel (D1120 mm × H930 mm) (D600 mm × H490 mm) (D495 mm × H410 mm) (D300 mm × H250 mm) | submerged lance | central top vertical blowing | 1, 0.53, 0.44, 0.27 | water | air | sulfuric acid above the plume | criteria of mixing time and gas flow rate for dynamic similarity |

Mandal | cylindrical vessel (D600 mm, D450 mm, D300 mm) (0.7 ≤ H/D mm ≤ 1.2) | tuyere/nozzle | ± 0.5R bottom blowing | 0.2 | water | air/N | NaCl or H | mixing time and correlation of liquid depth, vessel radius, and gas flow rate with dual porous plugs stirring |

Mazumdar | cylindrical vessel (D585 mm × H600 mm) | nozzle | central, ± 0.5R, and 0.64R bottom blowing | 0.17 | water | air/N | NaCl or H | mixing time and correlation of flat bottom, tapered cylindrical, step bottom, and funnel- shaped bottom |

Patil | cylindrical vessel (D300 mm, D600 mm) | nozzle | ± 0.5R bottom blowing | water | air | NaCl or KCl or H | effects of slag layer thickness and upper phase physical properties on the mixing time | |

Amaro-Villeda | cylindrical vessel (D537 mm × H410 mm) | nozzle | central and off-centered bottom blowing | 1/6 | water | air | NaOH or HCl | effects of slag properties on mixing time and energy dissipation |

Tang | cylindrical vessel (UD963 mm × BD920 mm × H933 mm) | porous plug | off-centered bottom blowing | 1/3 | water | N | KCl above the exposed eye | effects of dual-plug separation angles and radial locations on mixing time |

Liu | cylindrical vessel (UD676 mm × BD617 mm × H700 mm) | porous plug | central and off-centered bottom blowing | 1/3 | water | N | NaCl above the exposed eye | effects of radial locations and separation angles of single and dual plugs on the mixing time |

Gómez | cylindrical vessel (D335 mm × H391 mm) | nozzle | off-centered bottom blowing | 1/8 | water | air | KCl above the exposed eye | effects of separation angles, radial locations, and slag layer thickness on mixing time |

### Gas Bubble Formation, Transformation, and Interactions in the Plume Zone

_{2}system to simulate the diffusion-controlled decarburization process in liquid steel. More recently, a water-oil-air system was established by Li

*et al*.[39] to predict the bubble size distribution in the plume zone, as shown in Figure 2. Xu

*et al*.[40] studied the effect of the wettability on the formation of separated bubbles using a water model, and the phenomenon of how coaxial bubbles coalesce and how parallel bubbles bounce in one- and two-nozzle systems was shown by Wang

*et al*.[41] in a series of water-based experiments. Ito and co-workers[42,43] studied the behavior of a single rising bubble, and its volumetric mass transfer under vacuum degassing condition was reported.

Physical Modeling Experiments Performed to Study Gas Bubbling Behavior in the Plume Zone in Ladles

Author | Experimental Apparatus | Gas Injection Pattern | Liquid Metal | Gas | Remark |
---|---|---|---|---|---|

Sahai and Guthrie[9] | cylindrical vessel (D500 mm × H450 mm) | 2.16-mm nozzle | water | air | velocity pattern and plume structure |

cylindrical vessel (D500 mm × H400 mm) | 4.1-mm, 6.35-mm nozzles | water | air | gas fraction, bubble velocity, bubble frequency, and bubble pierced length in the plume | |

cylindrical vessel (D500 mm × H600 mm) | 6.35-mm nozzle | ||||

Johansen | cylindrical vessel (UD1100 mm × BD930 mm × H1237 mm) | 50-mm porous plug | water | air | radial mean and turbulent velocities |

Taniguchi | cylindrical vessel (D290 mm × H200 mm) | 6-mm nozzle | water | nitrogen | fluid flow, bubble dispersion, and gas-liquid mass transfer |

Anagbo | cylindrical vessel (D500 mm × H400 mm) | 60-mm porous plug | water | air | spatial distributions of properties of the plume above the plug |

cylindrical vessel (D500 mm × H420 mm) | 4-mm nozzle | water | air | variety of bubble size during the floating, velocity pattern, and void fraction of gas along the plume | |

Kishimoto | cylindrical vessel (D500 mm × H420 mm) (D500 mm × H500 mm) | 3-mm nozzle | water | air | location of the interface, propagation velocity, and energy dissipation |

Iguchi | cylindrical vessel (D126 mm × H233 mm) | 2-mm nozzle | water | air | comparison of four regions in the plume zone |

Iguchi | cylindrical vessel (D90 mm × H120mm) | 1-mm nozzle | 1600 °C molten iron | argon | bubble characteristics in a metallurgical reactor |

cylindrical vessel (D400 mm × H370 mm) | 2-mm, 3-mm, 5-mm nozzles | Wood’s metal | nitrogen, argon or helium | gas fraction and bubble frequency, bubble size distribution, mean rising velocity, and physical properties of gas | |

cylindrical vessel (D420 mm × H500 mm) | 1-mm nozzle,10-mm to 50-mm porous plugs | NaOH solution (0.02 mol/L) | CO2 | diffusion-controlled decarburization in molten steel | |

Li | cylindrical vessel with 2.44 deg slope angle (D617 mm × H700 mm) | 43.4-mm porous plug | water | N2 | bubble size distribution in the plume zone |

Xu | cylindrical vessel (D150 mm × H75 mm) | 0.5-mm, 1-mm, 2-mm nozzles | water | air | effect of the wettability on the bubble formation |

Wang | cylindrical vessel (D120 mm × H80 mm) | 1.5-mm, 2-mm, 2.5-mm nozzles | water | air | motion of single bubble and interactions between two bubbles |

cylindrical vessel (D100 mm × H400 mm) | 2-mm orifice | silicon oil | nitrogen | expansion of single bubble rising and its volumetric mass transfer under vacuum degassing condition |

### Inclusion Behavior at the Steel-Slag Interface and in the Molten Steel

*et al*.[50] used a Ga-In-Sn alloy with a melting temperature of 283 K to simulate the steel and MgCl

_{2}-glycerol (87 pct), as well as a hydrochloric acid solution to simulate the ladle slag. The Ga-In-Sn alloy–12 pct hydrochloric acid system showed better applicability for the prediction of slag particle entrainment around the open eye zone. In addition, Dayal

*et al*.[51] studied the effect of the shear force on the particle droplet behavior at the steel-slag interface. Furthermore, an experimental approach was used by Yang

*et al*.[52] to analyze the mechanism of nonmetallic inclusion removal by the wake flow. In their work, the effects of the bubble size, particle concentration, and inclusion particle size on the inclusion capture rate were studied in detail. Liu

*et al*.[53] studied and discussed the forces of nonmetallic inclusions at the steel-slag interface, and the inclusion separation from the molten steel to the slag was analyzed. Zhou

*et al*.[54] also studied the separation of nonmetallic inclusions at the steel-slag interface, and the effects of inclusion geometry and slag properties were investigated in detail.

Physical Modeling Experiments Performed to Study Inclusion Behaviors in Ladles

Author | Experimental Apparatus | Gas Injection | Liquid Metal | Slag | Inclusion | Gas | Remark |
---|---|---|---|---|---|---|---|

Kang | rectangular container (L430 mm × W210 mm × H590 mm) | 10-mm nozzle | water | silicon oil (5 × 10 | charcoal powder | air | inclusion removal around open eye and comparison of inclusion removal contribution by gas plume and buoyancy |

cylindrical vessel (D250 mm × H400 mm) | |||||||

Huang | cylindrical vessel (D1225 mm × H1252 mm) | purging plug | water | mixed oil | alumina hollow balls (0.5, 1, 2 mm) | nitrogen | slag entrapment around open eye |

Thunman | rectangular container (L150 mm × W250 mm × H350 mm) | 5-mm nozzle | Ga-In-Sn alloy (0.34 mm | MnCl2-glycerol (42.11 mm | argon | slag entrainment around open eye | |

HCl solution (1.02 mm | |||||||

Yang | rectangular container (L200 mm × W50 mm × H400 mm) | water | polystyrene particle (15 to 589 | air | inclusion removal by wake flow | ||

Dayal | rectangular container (L950 mm × W150 mm × H400 mm) | 10-mm nozzle | water | oil | air | effect of the shear force on the particle droplet behaviors at the steel-slag interface | |

Liu | water | silicon oil (5 × 10 | hollow aluminum (4 mm) | forces of nonmetallic inclusion at the steel-slag interface and inclusion behavior separated from molten steel to slag | |||

Zhou | water | bean oil kerosene pump oil | paraffin wax (sphere, plate, octahedron) | effects of inclusion geometry and slag properties on the separation process of nonmetallic inclusion at the steel-slag interface |

### Open Eye Formation

_{2}-glycerol (87 pct), and Ga-In-Sn alloy-hydrochloric acid solutions[59] have also been used to model the molten steel. The obtained results show that the spout shape is well described by a Gaussian distribution. Furthermore, the empirical equation for open eye[55,57,58,60,61] has been modified based on parameters such as the density ratios of the bulk and slag phases, the Froude number, and the Reynolds number. Recently, two up-to-date experimental works[19,62] studying the effects of the slag layer thickness and the location and separation angle of dual plugs on the flow pattern and slag eye formation in the ladle have been reported.

Physical Modeling Experiments Performed to Study Open Eye Formation in Ladles

Author | Experimental Apparatus | Gas Injection | Scale | Bulk Phase | Slag Layer | Gas | Colored Reagent | Remark |
---|---|---|---|---|---|---|---|---|

Yonezawa and Schwerdtfeger[10] | cylindrical vessel (D290 mm × H225 mm) | 0.5, 1, 1.5 mm | mercury | silicon oil | high-purity nitrogen | sudan blue | open eye, time average of the free surface area, and time fraction of complete coverage | |

Krishnapisharody and Irons[55] | cylindrical vessel (D420 mm × H500 mm) | 3-mm nozzle | 0.1 | water-paraffin oil, CaCl | air | dimensionless eye size as a function of density ratio and Froude number | ||

Guo and Irons[56] | square vessel (L500 mm × W500 mm × H400 mm) | 1.5-mm nozzle 25-mm porous plug | water | air | spout height | |||

Iguchi | cylindrical vessel (D200 mm × H300 mm) (D500 mm × H750 mm) | 0.5-, 1-, 1.5-mm nozzles | water | silicon oil | air | expression to describe open eye | ||

Peranandhanthan and Mazumdar[58] | cylindrical vessel (D300 mm × H300 mm) | 8-mm nozzle | 0.1 | water | petroleum ether mustard oil soybean oil tetrachloro ethelene perfumed coconut oil | air | modified expression of dimensionless slag eye | |

Wu | cylindrical vessel (D600 mm × H500 mm) | 6-mm nozzle | 0.2 | water-silicon oil (0.050, 0.100, 0.200, 0.515 Pa s) | air | sudan blue | open eye formation | |

cylindrical vessel (D240 mm × H145 mm) | 6-mm nozzle | 1/13 | Ga-In-Sn alloy-hydrochloric acid ((12 pct) (0.006 Pa s) (0.001 Pa s) | argon | sudan yellow | |||

Liu | cylindrical vessel (D617 mm × H700 mm) | 43.4-mm porous plug | 0.33 | water | bean oil | nitrogen | effects of gas flow rate, slag layer thickness, and plug separation angles on slag eye formation | |

Lv | cylindrical vessel (D600 mm, D290 mm) | 6-mm nozzle | water | silicon oil (97 Pa s) | air | sudan blue | size of slag eye | |

cylindrical vessel (D188 mm × H172 mm) | 6-mm nozzle | sodium tungstate (10 Pa s) | ||||||

Amaro-Villeda | cylindrical vessel (D537 mm × H410 mm) | nozzle | 1/6 | water | oil | air | effects of flow rate and slag properties on open eye formation | |

Mazumdar | cylindrical vessel (D600 mm × H705 mm) | 0.28 | water | petroleum ether mustard oil coconut oil | air | optimization of gas bubbling for mixing time and slag eye area | ||

cylindrical vessel (D300 mm × H359 mm) | 0.14 | |||||||

Pérez | cylindrical vessel (D500 mm × H410 mm) | nozzle | 1/6 | water | air | flow pattern measured by PIV and its effect on open eye formation |

In general, using physical modeling, it is possible to investigate the fluid dynamics phenomena, but it is difficult to study the phenomena related to the reaction kinetics in ladle metallurgy. For the physical modeling, required for the optimization of mixing and homogenization in ladles, the general empirical rules have not been analyzed sufficiently.

## Industrial Trials

*et al*.[50] and Dayal

*et al*.[51] Moreover, in the work of Wu et al.,[59,63,64] the open eye area was measured in a ladle at Saarstahl AG. Specifically, the influence of the flow pattern on the mixing conditions and open eye formation in the industrial plant was evaluated. Furthermore, the analyses of the temperature distribution and heat transfer on the ladle lining during the preheating process[65] and the teeming process[66] were carried out based on a comparison of the industrial data and the data calculated by Glaser

*et al.*A study of the influence of the stirring rate on the inclusion characteristics was carried out by Malmberg

*et al*.[45] based on tool steel ladle data from Uddeholm AB. Recently, experiments were carried out at the SFIL Steelworks[67] to study hydrogen degassing in the industrial process.

Industrial Trials for Gas Bubbling in Ladles

Company | Steel Grade | Experimental Apparatus | Stirring Condition | Capacity | Alloy Component | Slag Layer Thickness | Remark |
---|---|---|---|---|---|---|---|

SSAB AB[131] | cylindrical vessel (D2.6 m × H2.9 m) | 107t | thermal stratification during holding | ||||

Yawata steelworks of Nippon Steel Corporation[10] | cylindrical vessel (D4.4 m × H3.5 m) | 100 to 500 NL/min | 350t | 0.02 pct C, 0.01 pct Si, 0.20 pct Mn, 0.015 pct P, 0.010 pct S | 50 mm | open eye | |

Uddeholm AB[45] | AISI H13 tool steel | 700 A + 10 L/min Ar 900 A + 100 L/min Ar | 65t | 0.39 pct C, 1.0 pct Si, 0.4 pct Mn, 5.3 pct Cr, 1.3 pct Mo, 0.9 pct V, N, S | optimization of gas stirring for decreasing inclusion content | ||

tool steel | cylindrical vessel (D2.95 m × H1.36 m) | 300, 600, 750, 900 A | 65t | Cr, Mo, Mn, Si, V, Ni, S | 100 mm | slag droplets generated at the steel-slag interface | |

special steels | cylindrical vessel (D2.97 m × H3.18 m) | 20, 30 STP m | 170t | Al, C, Mn, Si | mixing phenomena and open eye formation | ||

cylindrical vessel (D3.59 m × H4.75 m) | 215t | analysis on the ladle lining during the preheating process and teeming process | |||||

SFIL Steelworks[67] | cylindrical vessel (D2.8 m × H2.79 m) | 10.8 Nm | 100t | hydrogen degassing |

## Criteria for Scaling Between Physical Modeling Experiments and Industrial Trials

Scaling Criteria in Ladles

Author | Froude Number | Gas Flow Rate (Model and Prototype) | Gas Flow Rate (Prototype and Industrial Scale) | Void Fraction | Plume Radius | Plume Velocity | Mixing Time |
---|---|---|---|---|---|---|---|

\( \frac{{\rho_{\text{g}} }}{{\rho_{\text{l}} }}\frac{{Q^{2} }}{{gd^{5} }} \) | \( \frac{{{\text{Q}}_{\text{model}} }}{{Q_{\text{prototype}} }} = \lambda^{5/2} \) | \( \frac{{\alpha_{\text{model}} }}{{\alpha_{\text{prototype}} }} = \lambda^{0} \) | \( \frac{{R_{{{\text{av}}_{\text{model}} }} }}{{R_{{{\text{av}}_{\text{prototype}} }} }} = \lambda \) | \( \frac{{U_{{p_{\text{model}} }} }}{{U_{{p_{\text{prototype}} }} }} = \lambda^{1/2} \) | |||

Yu | \( \frac{{Q_{\text{model}} }}{{{\text{Q}}_{\text{prototype}} }} = \left( {\frac{{\lambda_{\sigma } }}{{\lambda_{{\rho_{\text{l}} }} }}} \right)^{1/4} \lambda^{2} \) | ||||||

Fan and Hwang[21] | \( \frac{{Q_{\text{model}} }}{{Q_{\text{prototype}} }} = \frac{{\lambda_{\sigma } }}{{\lambda_{{\mu_{\text{l}} }} }}\lambda^{2} \) | \( \frac{{Q_{\text{prototype}} }}{{Q_{\text{industrial scale}} }} = \frac{1873}{293}\frac{{P_{\text{atm}} }}{{P_{\text{atm}} + \rho_{\text{steel}} gH_{\text{real}} }} \) | |||||

Mazumdar | \( \frac{{Q_{\text{model}} }}{{Q_{\text{industrial scale}} }} = \lambda^{5/2} \) | \( \frac{{\tau_{\text{model}} }}{{\tau_{\text{full scale}} }} = \lambda^{1/2} \) | |||||

Mazumdar[69] | \( \frac{{U^{2} }}{gH} \) | \( \frac{{Q_{\text{model}} }}{{Q_{\text{industrial scale}} }} = \lambda^{3/2} \) | \( \frac{{U_{{p_{\text{model}} }} }}{{U_{{p_{\text{full scale}} }} }} = \lambda^{1/6} \) | \( \frac{{\tau_{\text{model}} }}{{\tau_{\text{full scale}} }} = \lambda^{5/6} \) | |||

Pan | \( \frac{{Q_{\text{model}} }}{{Q_{\text{prototype}} }} = \frac{{\lambda_{\sigma } }}{{\lambda_{{\mu_{\text{l}} }} }}\lambda^{2} \) | \( \frac{{Q_{\text{prototype}} }}{{Q_{\text{industrial scale}} }} = \frac{1873}{293}\frac{{P_{\text{atm}} }}{{P_{\text{atm}} + \rho_{\text{steel}} gH_{\text{real}} }} \) |

## Numerical Models to Study the Gas–Liquid Zone in Ladle Refining

### Multiphase Models Applied to Study Ladle Refining

To describe gas–liquid two-phase flow, there are four main mathematical methods: (1) the quasi-single-phase model, (2) the volume of fluid (VOF) model, (3) the Eulerian multiphase (E–E) model, and (4) the Eulerian–Lagrangian (E–L) model. In early works, the plume zone mixed with gas and liquid was treated as a quasi-single-phase. With the increase in computational capabilities, the VOF and the E–E models have become widely used for the simulation of the interphase interactions of the gas and liquid phases. In comparison to the predictive quasi-single-phase model, the VOF and the E–E models are more computationally expensive. Recently, commercial codes coupled with user-defined functions (UDFs) have been widely employed for the study of gas bubbling in ladles. In recent years, a new approach[39] to calculate the gas bubble size distribution within the E–E model based on the population balance model (PBM) has been proposed. In the E–L model,[71, 72, 73] the VOF model is used to track the free surface of the melt coupled in Eulerian coordinates, whereas the discrete phase model (DPM) is used to describe the stirring generated by the bubbles under a Lagrangian reference frame. Some previous works[74, 75, 76] have compared the quasi-single-phase model, VOF model, E–E model, and E–L model, and these comparisons reveal that the advanced models have gradually improved in the last few years.

#### Quasi-single-phase model

The quasi-single-phase model is the simplest of the four models discussed previously. The quasi-single-phase model avoids the need to compute the motion of the bubbles. The key principal in this model is that the characteristics of the plume, such as the gas fraction, velocity pattern, and plume diameter, are set using empirical equations. Thus, the volume fraction equation is not coupled to the equation group to be solved. A buoyancy term generated by the gas bubbling is added into the momentum conservation equation. In this model, the plume is treated as a quasi-single phase in which the volume fraction of gas in each control volume is affected by various parameters during calculation.

In the quasi-single-phase model, the equations of continuity and momentum are written as follows.

The density in the plume is \( \rho = \alpha_{\text{g}} \rho_{\text{g}} + \alpha_{\text{l}} \rho_{\text{l}} \), and the recirculation zone is treated as a liquid phase\( \rho = \rho_{\text{l}} \).

*et al*.[81] proposed a new equation to estimate the average velocity of the rising plume of the gas–liquid mixture and reported a model with slip (rather than with no slip) between the gas phase and the liquid phase to provide a more realistic description of the actual physical phenomenon. Furthermore, Madan

*et al*.[75] compared the mixing conditions in a dual-plug ladle calculated by the quasi-single-phase model and the E–L model. In contrast to previous quasi-single-phase models in cylindrical coordinates, Goldschmit and Owen[82] also constructed their calculation system in Cartesian coordinates to study the effect of different separation angles and positions of the porous plugs on the flow pattern. In addition, Ganguly and Chakraborty[83] coupled the thermal energy equation with the quasi-single-phase model to study the fluid flow and heat transfer in the gas-stirred ladle. The combined model aimed to control the thermal stratification in the molten steel. A quasi-single-phase model based on slippage and no slippage between the rising bubbles and the surrounding liquid was also set up by the same research group.[84] Compared with the experimental velocities and mixing time data, the numerical calculation with slippage showed a higher accuracy than that without a slippage for predicting the flow pattern.

Quasi-Single-Phase Model to Study Ladle Metallurgy

Author | Dimension | Position of Gas Injection | Volume Fraction | Plume Velocity | Slip Velocity | Plume Shape | Bubble Diameter | Remark |
---|---|---|---|---|---|---|---|---|

Joo and Guthrie[8] | 2 | one off-centered two off-centered | \( \frac{Q}{{\pi R_{\text{av}}^{2} U_{\text{p}} }} \) | \( 4.17Q^{0.333} H^{0.25} R^{ - 0.33} \) | mixing mechanisms with single or dual bubbling of different positions | |||

Goldschmit and Owen[82] | 3 | one central two off-centered | \( \frac{{Q_{1} - \pi R_{\text{av}}^{2} \alpha (1 - \alpha )U_{\text{s}} }}{{2\pi \mathop \smallint \nolimits_{0}^{{R_{av} }} U_{\text{p}} r{\text{d}}r}} \) | \( 4.5Q^{0.333} H^{0.25} R^{ - 0.25} \) | \( 1.08*\left( {\frac{{gd_{\text{b}} }}{2}} \right)^{0.5} \) | \( 0.291\left( {\frac{{Q_{1}^{2} }}{g}} \right)^{0.2} {\text{Fr}}_{\text{m}}^{ - 0.129} \left( {\frac{z}{{d_{\text{o}} }}} \right)^{0.43} \) | \( 0.35*(Q_{\text{g}}^{2} /g)^{0.2} \) | position of Ar injection |

2 | central vertical submerged lance | \( \frac{{Q\frac{{T_{\text{l}} }}{{T_{\text{g}} }}\frac{{P_{0} }}{P}}}{{2\pi R_{\text{av}}^{2} U_{p} }} \) | \( 4.17Q^{0.333} H^{0.25} R^{ - 0.33} \) | \( 1.08*\left( {\frac{{gd_{\text{b}} }}{2}} \right)^{0.5} \) | \( k\left( {\frac{\sigma }{{\rho_{\text{l}} }}} \right)^{1/2} \) | flow pattern | ||

Mazumdar and Guthrie[80] | 2 | central vertical submerged lance | \( \frac{Q}{{\pi R_{\text{av}}^{2} U_{\text{p}} }} \) | \( 4.19\beta^{0.333} Q^{0.333} H^{0.25} R^{ - 0.33} \) | with or without tapered side walls and surface baffles | |||

Mazumdar | 2 | one bottom center | \( \frac{Q}{{\pi R_{\text{av}}^{2} U_{\text{p}} }} \) | \( 4.5Q^{0.333} H^{0.25} R^{ - 0.25} \) | average rise velocity in the plume zone | |||

Ganguly and Chakraborty[83] | 2 | one center | \( \frac{Q}{{\pi R_{\text{av}}^{2} U_{\text{p}} }} \) | \( 4.17Q^{0.333} H^{0.25} R^{ - 0.333} \) | \( \left( {\frac{1}{\sqrt 3 }} \right){\text{radius at surface}} \) \( \frac{Q}{{\pi R_{\text{av}}^{2} U_{\text{p}} }}\left( {\text{no slip}} \right) \) \( \frac{{Q - \pi R_{\text{av}}^{2} \alpha \left( {1 - \alpha } \right)u_{\text{rel}} }}{{2\pi \mathop \smallint \nolimits_{0}^{{R_{\text{av}} }} rU_{\text{p}} {\text{d}}r}}(slip) \) | thermal stratification | ||

Ganguly and Chakraborty[84] | 3 | one center | \( \frac{Q}{{\pi R_{\text{av}}^{2} U_{\text{p}} }} \) | \( 4.17Q^{0.333} H^{0.25} R^{ - 0.333} \) | \( 1.08*\left( {\frac{{gd_{\text{b}} }}{2}} \right)^{0.5} \) | \( \left( {\frac{1}{\sqrt 3 }} \right){\text{radius at surface}} \) | effect of gas flow rate, bottom nozzle configurations, and tracer addition locations on mixing time |

#### VOF model

The VOF model is widely used to track the interfaces of different phases and is a type of Eulerian method. When the flow rate is low, separate bubbles are generated and the interfaces of the different phases are sharp. As the flow rate increases, the gas injection leads to the plume transferring from a bubble regime to a jetting regime.[40] In the VOF model, one set of equations for continuity, momentum, and phase volume fraction is calculated. The conservation formulas are shown as follows.

*e.g.*, liquid and gas)

*et al.*[40,85] considered the surface force in the VOF model to investigate a single bubble rising in molten steel and bursting at the interface. Furthermore, the effect of the wettability on single bubble formation was reported based on VOF model predictions and water model experiments. Wang

*et al.*[41] extended single bubble formation and rising to coaxial bubble coalescence and parallel bubble bouncing. Li

*et al*.[86] used the multiphase VOF model to simulate the flow pattern and the interface behavior of the molten steel and slag layer. The effects of the gas flow rate and nozzle arrangement were the focus in their work. The VOF model was employed by Llanos

*et al*.[11] to study the influence of various gas injection arrangements on the mixing time, the wall skin friction coefficient, and the open eye area. Compared with the use of one argon injection tuyere, the use of two argon injection tuyeres showed no obvious reduction in the mixing time. However, the slag layer opening and the skin friction coefficient value decreased. In a recent work by Ramasetti

*et al.*,[87] the same method was also used to predict open eye formation. In the work of Huang

*et al.*,[49] the VOF model and the large eddy simulation (LES) model were used to analyze slag droplet entrainment in the open eye zone. The model was then modified using a UDF by Li,[88] yielding a new method to track the number of droplets, as well as the volume and location of every droplet in the domain. A similar model was also developed by Sulasalmi

*et al*.[89,90] to analyze the effect of the interfacial velocity on droplet distribution and slag emulsifications at the steel-slag interface. In the research of Senguttuvan[91] and Senguttuvan and Irons,[92] the entrainment of slag into molten metal and vice versa was modeled by using a coupled VOF-LES model. The relationship between the amounts of entrained slag and the interfacial mass transfer rate was discussed. Depending on the research characteristics, not only the VOF model could be used to track the interface of different phases, but also other improved models could be used to study the inclusion behavior in molten steel. Based on the preliminary work, a coupled model was employed by Xu

*et al*.[93] to study the effect of wake flow on inclusion removal. In their work, the VOF model was used to calculate the fluid dynamics induced by single bubbles and the DPM was used to track the motion of the particles. Some researchers have also made efforts to combine fluid dynamics and thermodynamics calculations. Ersson

*et al*.[94] developed a coupled model, combining the VOF model and the Thermo-Calc software database, to compute the fluid dynamics and thermodynamics simultaneously. They investigated the interfacial decarburization in a top blown converter. Later, Singh

*et al*.[95] used the same method to model the desulfurization process around the open eye area, as well as at the steel–slag interface. However, concerning bubble behavior under vacuum conditions, only a few works using mathematical modeling have been reported.

VOF Model to Study Ladle Metallurgy

Author | Model | Turbulence | Dimension | Gas Injection | Inclusion | Slag | Code | Remark |
---|---|---|---|---|---|---|---|---|

Llanos | VOF |
| 3 | one off-centered two off-centered | no | yes | fluent | mixing time, wall skin friction coefficient, open eye of various gas injection arrangements |

VOF | laminar | 3 | one center | no | no | fluent | single bubble rising and bursting, effect of the wettability on bubble formation | |

Xu | VOF (interface) DPM (inclusion) | laminar | 3 | one center | yes | no | fluent | effect of wake flow on inclusion removal |

Wang | VOF | laminar | 3 | one center two off-centered | no | no | fluent | coaxial bubbles coalescence and parallel bubbles bounce with one and two nozzles |

Huang | VOF | LES model | 3 | one off-centered | no | yes | fluent | slag droplet entrainment at the open eye area |

Li | VOF |
| 3 | one off-centered two symmetric | yes | no | fluent | flow and interface behavior of steel and slag |

Ramasetti | VOF |
| 3 | one off-centered | no | yes | Fluent | open eye formation |

VOF | LES model | 3 | one center | no | yes | Fluent | effect of interfacial velocity on droplet distributions and slag emulsification at the steel-slag interface | |

VOF | LES model | 3 | one center | no | yes | entrainment of slag into molten metal, vice versa, and slag-metal interfacial mass transfer rates | ||

Ersson | VOF + Thermo-Calc |
| 3 | top blowing | no | yes | Fluent + Thermo-Calc | interfacial reactions and decarburization |

Singh | VOF + Thermo-Calc |
| 3 | one off-centered two off-centered | no | yes | Fluent + Thermo-Calc | steel-slag interfacial reaction and desulfurization |

#### E–E model

In the E–E model, multiple sets of equations for the continuity, momentum, turbulent energy, and dissipation rate are calculated for each phase. This increased complexity affects the convergence behavior.[96,97] E–E models can be used to include the effects of forces, such as the virtual mass force, drag force, lift force, and turbulent dissipation force, on the flow pattern. In practice, the relatively complex E–E model is mostly used. Based on this model,[47,48,98, 99, 100] a coupling with the PBM has been developed to simulate the subsize bubble behavior generated from a porous plug in the plume zone.[39] In this model, conservation equations are solved.

PBM for the description of bubble behavior:

*et al*.[46,101,102] and Söder[103] developed a model to investigate the inclusion behavior in gas-stirred ladles with the PHOENICS code. In this code, the interphase slip algorithm, originally developed by Spalding

*et al.*,[104,105] was used to solve the two-phase problem. Based on the same fluid dynamics code, they[106, 107, 108] also combined fluid dynamics and thermodynamics to study the flow pattern and chemical reactions around the steel-slag interface. At the same time, several forces in the E–E model[109,110] were analyzed by using the CFX code to predict the flow pattern in ladle furnaces. In addition, Wang

*et al*.[111,112] used the CFX code to investigate three mechanisms of inclusion collisions, that is, Brownian collision, turbulent collision, and Stokes collision, and two main mechanisms of inclusion removal, that is, Stokes flotation and bubble adhesion. Geng

*et al*.[113] also used the CFX code to study the effect of the dual-plug separation angle and axial distance on the mixing time. Moreover, Maldonado-Parra

*et al*.[114] studied the effects of the radial position of single plug and dual plugs on mixing time using the PHOENICS code. In recent works by De Felice

*et al*.[99] and Bellot

*et al.*,[100] the PBM coupled with the traditional E–E model was employed to study the inclusion transport, aggregation, and surface entrapment in the gas-stirred ladle. Huang

*et al*.[115] also used the E–E model to calculate two-phase flow and the DPM to predict inclusion trajectories and to describe the effects of purging plug arrangement and gas flow rate on the erosion of the lining of the refining ladle. Lou and Zhu[47,48,96,116,117] have made step-by-step contributions to the numerical simulation of the ladle process. In the first step, they[96] investigated the effects of the turbulent dissipation force, bubble-induced turbulence, drag force, lift force, and bubble size on the profile of the plume. In the next step, they combined the PBM and the E–E model developed in their previous work[96] and used to investigate the various mechanisms of inclusion growth and removal under different tuyere conditions.[47,48] In the last step,[116,117] the calculations of the thermodynamics and fluid dynamics in a gas-stirred ladle were used to describe the efficiencies of desulfurization, dealumination, desilication, and demanganization. Based on previous work on the E–E model, the simultaneous reaction model (SRM) coded by these researchers was added to investigate the metal-slag reactions. In contrast, Yu

*et al*.[118, 119, 120, 121, 122] mainly focused on investigating the dehydrogenation and denitrogenation behaviors in an industrial vacuum tank degasser with different operating conditions. A recent work by Li

*et al*.[39] used the PBM to calculate the bubble size distribution affected by the coalescence and breakage in the plume. Based on the E–E model, studies on the combination of fluid dynamics and thermodynamics, such as desulfurization, need to be developed further.

E–E Model to Study Ladle Metallurgy

Author | Model | Dimension | Inclusion | Code | Virtual Mass Coefficient | Drag Coefficient | Lift Coefficient | Turbulence Dissipation Coefficient | Remark |
---|---|---|---|---|---|---|---|---|---|

Xia | E–E | 2 | no | CFX | 0.44, 4/3, \( \frac{2}{3}E_{\text{o}}^{1/2} \) | 0.1, 0.15, 0.3 | 0.1 | drag coefficient and lift force coefficient for different bubble shapes | |

Mendez | E–E | 2 | no | CFX | 0 to 0.06 | \( \frac{24}{\text{Re}}(1 + 0.15{\text{Re}}^{0.687} ) \) | 0.05 | 0 to 1 | drag force and nondrag force |

Lou and Zhu[96] | E–E | 3 | no | FLUENT | universal drag | − 0.05, 0, 0.5 | Simonin | interaction forces between gas-liquid two-phase | |

E–E + PBM (inclusion) | 3 | yes | FLUENT | inclusion behavior and mixing phenomena with different arrangements of tuyeres | |||||

E–E + PBM (inclusion) | 3 | yes | FLUENT | 0.5 | 8/3 | Tomiyama | Simonin | transport, aggregation, and surface entrapment of inclusions | |

E–E | 3 | yes | CFX | \( \frac{24}{\text{Re}}\left( {1 + 0.15{\text{Re}}^{0.687} } \right) + \frac{0.42}{{1 + \frac{{4.25 \times 10^{ - 4} }}{{{\text{Re}}^{1.16} }}}} \) | mechanisms of inclusion growth and inclusion removal | ||||

Geng | E–E | 3 | no | CFX | \( { \hbox{max} }\left( {\frac{24}{\text{Re}}\left( {1 + 0.15{\text{Re}}^{0.687} } \right), 0.44} \right) \) | effects of dual-plug separation angle and axial distance on mixing time | |||

Maldonado-Parra | E–E | 3 | no | PHOENICS | effects of radial position of single plug and dual plugs on mixing time | ||||

Huang | E–E + DPM (inclusion) | 3 | yes | CFX | \( \frac{24}{\text{Re}}(1 + 0.15{\text{Re}}^{0.687} ) \) | influences of purging plug arrangement and gas flow rate on the erosion of the ladle lining | |||

E–E | 2 | yes | PHOENICS | \( \frac{24}{\text{Re}}\left( {1 + 0.15{\text{Re}}^{0.687} } \right) + \frac{0.42}{{1 + \frac{{4.25 \times 10^{ - 4} }}{{{\text{Re}}^{1.16} }}}} \) | microinclusion growth and separation and removal | ||||

E-E | 2 | no | PHOENICS | flow pattern and chemical reaction around the steel-slag interface | |||||

E–E + SRM | 3 | no | FLUENT | universal drag | Simonin | thermodynamics and fluid dynamics of desulfurization, dealumination, desilication, and demanganization | |||

E–E | 3 | no | FLUENT | universal drag | 0.1 | dehydrogenation and denitrogenation in industrial vacuum tank degassers | |||

Li | E–E + PBM (bubble) | 3 | no | FLUENT | 0.5 | Schiller–Naumann | Tomiyama | Sato | bubble size distribution affected by the coalescence and the breakage in the plume |

#### E–L model

^{®}by Aoki

*et al*.[123] They used this model to study how the bubble morphology affects the probability of inclusion attachment. The interaction forces on the liquid–gas plume were taken into consideration in the calculations. In addition, Singh

*et al*.[124] used a similar model and improved the grid resolution near the wall to investigate the wall shear stress distribution in a gas agitated vessel. Because of the limitations of the E–E model, particle tracking using the DPM in the E–E model showed flow field interacts only with the primary phase. Therefore, Cloete

*et al*.[71] replaced the E–E model with the VOF model for tracking of the melt interface, wherein the DPM was used to calculate the trajectory of each bubble. In addition, Liu

*et al*.[72] used a similar model to predict the interface behaviors and the mixing times of one-plug and dual-plug systems with the plugs placed 90 and 180 deg apart. As a result, an arrangement of the dual-plug system having a separation angle of 180 deg at low gas flow rates was recommended for inclusion removal. The same method was also employed by Li

*et al*.[73] for studying alloy dispersions. Using the E–L model with the consideration of bubble aggregation, Li

*et al.*[125,126] added the LES model to calculate the multiscale eddies and study the unsteady state of the open eye.

Eulerian–Lagrangian Model to Study Ladle Metallurgy

Author | Model | Turbulence | Dimension | Position of Porous Plug | Slag | Virtual Mass Coefficient | Drag Coefficient | Buoyancy Force | Lift Coefficient | Pressure Gradient force | Remark |
---|---|---|---|---|---|---|---|---|---|---|---|

Aoki | E–E + DPM (bubble) |
| 3 | off-centered | no | 0.5 | Kuo and Wallis[132] | yes | \( 0.00165 \alpha_{\text{g}}^{ - 0.78} \)[133] | yes | inclusion removal |

Singh | E–E + DPM (bubble) |
| 3 | one center | no | Morsi and Alexander[134] | wall shear stress distribution in gas agitated vessels | ||||

Cloete | VOF + DPM (bubble) |
| 3 | one center | no | Xia[109] | yes | bubble growth and period of swirl motion | |||

Liu | VOF (interface) + DPM (bubble) |
| 3 | one-off-centered two symmetric two off-centered (90 deg) | yes | 0.5 | nonspherical drag law | yes | yes | interface behavior and mixing time of one plug with dual-plug system | |

Li | VOF (interface) + DPM (bubble) |
| 3 | one off-centered | yes | 0.5 | Kuo and Wallis[132] | yes | 0.1 | alloy dispersion | |

VOF (interface) + DPM (bubble) |
| 3 | one off- center | yes | 0.5 | Ishii–Zuber | yes | yes | unsteady state of open eye |

### Turbulence Models Applied in Ladle Refining

Concerning turbulence models applied in the mathematical modeling of ladle metallurgy, two kinds of turbulence models, namely, the *k*–*ε* model and LES model, have been widely used. The standard *k*–*ε* model is most commonly used for the calculation of the flow pattern in industrial ladles. In some works, the standard *k*–*ε* model[96] modified with bubble-induced turbulence and the renormalization (RNG) *k*–*ε* model[39] have also been employed. Compared with the standard *k*–*ε* model, the form of the RNG *k*–*ε* model is similar but has an additional term that improves the accuracy for rapidly strained flows and enhances the accuracy for swirling flows. The *k*–*ε* model is a Reynolds averaged numerical simulation (RANS), which averages the numerical information of eddies with various sizes. However, in the LES model, the large and small eddies are treated separately in the calculations. The large eddies are resolved directly and the small eddies are calculated by the Smagorinsky–Lilly subgrid-scale model. Here, the LES model can be seen as a compromise between direct numerical simulation and the RANS model in terms of accuracy and computational cost. With the development of computational abilities, the LES model[49,125,127] has become increasingly used to predict the slag entrapment and bubble distribution in the modeling of metallurgical systems.

### Comparison of Calculation Systems

*L*/

*H*= 0.5 from the bottom under the same conditions in the quasi-single-phase model,[82] E–E model,[96] and E–L model[31] are compared with the measured data[31] in Figure 5 (for detailed experimental conditions, see Reference 31). The quasi-single-phase model is the simplest one of the three models. In this model, the shape of the two-phase zone is predicted in advance, so this model cannot predict the flow pattern of circulation in the ladle accurately. However, on reviewing the principles of this model, the relationship between various parameters is deeply distinct. The E–E model is most consistent with the experimental results, and the E–L model is the second-best option. Because of the high volume fraction in the primary zone, the E–E model is more accurate in describing the phenomena in this zone. With respect to the axial distance from the nozzle exit and the transformation of bubbles in the plume, the DPM is more suitable for the prediction of the bubble behavior at the bubble buoyancy zone and spout zone. Modifying the drag coefficient and adding nondrag forces improves the E–E model considerably. In recent studies of ladle metallurgy, based on the modified E–E model of the gas stirring conditions, a large number of special models, either in commercial packages or research codes, have been coupled together to describe the complex physical and chemical phenomena in the ladle. This includes inclusion behavior, degassing, refractory erosion, and thermodynamic reactions. The different research directions can be classified by the different locations in the ladle and the preferred combinations of the numerical models, as shown in Figure 6. In general, several types of mathematical model have been studied in ladle refining: (1) velocity distribution and turbulent dissipation; (2) alloy and temperature homogenization (mixing time); (3) number, location, and pattern of plugs; (4) bubble behavior in the plume zone; (5) open eye formation; (6) slag entrapment and inclusion behavior at the steel-slag interface; (7) inclusion removal in molten steel; (8) vacuum degassing; and (9) the combination of fluid dynamics and thermodynamics for interface reactions. Furthermore, based on the analysis of previous studies, both the PBM and DPM have been used to describe the movement and interaction inclusion particles and bubbles in advanced models. In conclusion, many physical and mathematical models have been developed to investigate alloy homogenization and open eye formation. However, in terms of the physical and mathematical modeling of the mass transfer phenomenon, there are only a few works studying the steel-slag interface reactions,[76,92] vacuum degassing at the gas–liquid interface,[28,37,43,128] and dissolution rate of the solid alloy at the liquid–solid interface.[73,123,129] For the inclusion behavior at the steel-slag interface, physical[50,51,53,54,130] and mathematical models[49,89, 90, 91] have been developed, but the interactions of the slag layer phase and inclusion particles need to be improved to study the droplet behavior at the interface. Moreover, the existing combined analysis of the combination of fluid dynamics and thermodynamics[94,95,107,108,116] is not sufficient and needs to be improved further. On the whole, none of these mathematical models functions well in all research aspects and each has its own limitations. To better describe industrial conditions using mathematical methods and to improve numerical modeling, the results of physical modeling experiments and industrial trials must offer satisfactory validations.

Comparison of Momentum Equation and Turbulence Model Used in Diffenent Models

Model | Momentum Equation | Main Turbulence Model Used |
---|---|---|

Quasi-single-phase model | \( \frac{\partial }{\partial t}\left( {\rho \vec{u}} \right) + \rho \vec{u} \cdot \nabla \vec{u} = - \nabla p + \nabla \cdot \left[ {\left( {\mu + \mu_{\text{t}} } \right)\left( {\nabla \vec{u} + \nabla \vec{u}^{T} } \right)} \right] +\varvec{\rho}_{{\mathbf{L}}} \varvec{\vec{g}\alpha }_{{\mathbf{g}}} \) | standard |

VOF model | \( \frac{\partial }{\partial t}\left( {\rho \vec{u}} \right) + \nabla \cdot \left( {\rho \vec{u}\vec{u}} \right) = - \nabla p + \nabla \cdot \left[ {\left( {\mu + \mu_{\text{t}} } \right)\left( {\nabla \vec{u} + \nabla \vec{u}^{T} } \right)} \right] + \varvec{\rho \vec{g}} + \varvec{F}_{{\mathbf{s}}} \) | standard |

E–E model | \( \frac{\partial }{\partial t}\left( {\alpha_{q} \rho_{q} \vec{u}_{q} } \right) + \nabla \cdot \left( {\alpha_{q} \rho_{q} \vec{u}_{q} \vec{u}_{q} } \right) = - \alpha_{q} \nabla p + \nabla \cdot \left[ {\alpha_{q} \left( {\mu + \mu_{\text{t}} } \right)\left( {\nabla \vec{u} + \nabla \vec{u}^{T} } \right)} \right] +\varvec{\alpha}_{\varvec{q}}\varvec{\rho}_{\varvec{q}} \vec{\varvec{g}} + \vec{\varvec{F}}_{{{\mathbf{drag}},\;\varvec{q}}} + \vec{\varvec{F}}_{{{\mathbf{lift}},\;\varvec{q}}} + \vec{\varvec{F}}_{{{\mathbf{VM}},\;\varvec{q}}} + \vec{\varvec{F}}_{{{\mathbf{TD}},\;\varvec{q}}} \) | standard |

Eulerian–Lagrangian model | \( \frac{\partial }{\partial t}\left( {\rho \vec{u}} \right) + \nabla \cdot \left( {\rho \vec{u}\vec{u}} \right) = - \nabla p + \nabla \cdot \left[ {\left( {\mu + \mu_{\text{t}} } \right)\left( {\nabla \vec{u} + \nabla \vec{u}^{T} } \right)} \right] + \varvec{\rho \vec{g}} + \varvec{F}_{{\mathbf{s}}} + \varvec{F}_{{{\mathbf{bi}}}} \) \( \varvec{F}_{{{\mathbf{bi}}}} = \mathop \sum \limits_{1}^{{\varvec{N}_{{\mathbf{b}}} }} \left( {\vec{\varvec{F}}_{{{\mathbf{drag}},\varvec{ }{\mathbf{b}}}} + \vec{\varvec{F}}_{{{\mathbf{buoyancy}},\varvec{ }{\mathbf{b}}}} + \vec{\varvec{F}}_{{{\mathbf{VM}},\varvec{ }{\mathbf{b}}}} + \vec{\varvec{F}}_{{{\mathbf{pressure gradient}},\varvec{ }{\mathbf{b}}}} } \right)\varvec{\rho}_{{\mathbf{b}}} \varvec{Q}_{{{\mathbf{bi}}}} \Delta \varvec{t} \) | standard |

## Concluding Remarks

- 1.
Depending on the research goal, previous physical modeling experiments in the lab scale have been divided based on four major research focuses: (a) mixing and homogenization in the ladle, (b) gas bubble formation, transformation, and interactions in the plume zone; (c) inclusion behavior; and (d) steel-slag interface and open eye formation. Several industrial trials have focused on open eye formation and the optimization of gas stirring. Concerning physical modeling, such as mixing and homogenization in ladles, the general empirical rules have not been analyzed sufficiently.

- 2.
The connection between industrial trials and physical modeling experiments is important for determining scaling criteria. Froude similarity and modified Froude similarity seem to be the most common dimensionless numbers used in gas-stirred metallurgical reactors. The parameters of the water model, prototype, and industrial ladles (that is, the gas flow rate, void fraction, plume radius, plume velocity, and mixing time) are directly linked by the ratios of geometric factors, or transferred to dimensionless patterns first and then mathematically related.

- 3.
According to the basic fundamental theory and applicability of each model, four kinds of multiphase models coupled with three kinds of turbulence models, particularly paying attention to different research directions, have been discussed. The VOF model is mainly used to track the sharp interfaces of different phases for the analysis of bubble separation and slag eye opening. The quasi-single-phase model, E–E model, and Eulerian–Lagrangian model have been widely used to calculate the flow patterns in industrial metallurgical ladles. In the three mathematical models, the set of equations for the main liquid phase are solved using a Eulerian algorithm. The difference between the three models is the method used to describe the liquid–gas two-phase zone. Based on the modified E–E model for the gas stirring conditions, a large number of special models in commercial packages or research codes have been coupled to describe the complex physical and chemical phenomena in the ladle. This includes inclusion behavior, degassing, refractory erosion, alloy homogenization, and thermodynamic reactions. For the turbulence models, the

*k*–*ε*model is commonly used for the calculation of the flow pattern in industrial ladles, and the LES model has become increasingly used in the study of metallurgical systems. - 4.
Based on the present review, the following recommendations regarding model combinations are suggested: (a) for physical modeling, such as mixing and homogenization in ladles, the general empirical rules have not been analyzed sufficiently; (b) the mathematical models focusing on inclusion behaviors at the steel-slag interface need to be improved; and (c) the phenomena governing the transfer of elements, vacuum degassing, and the combination of fluid dynamics and thermodynamics, such as in desulfurization, need to be developed further.

## Notes

### Acknowledgments

One of the authors (YL) extends his sincere appreciation to the China Scholarship Council for financial support of his study at the KTH-Royal Institute of Technology (Stockholm).

## References

- 1.D. Mazumdar and R.I.L. Guthrie:
*ISIJ Int.*, 1995, vol. 35, pp. 1–20.Google Scholar - 2.L.F. Zhang and S. Taniguchi:
*Int. Mater. Rev.*, 2000, vol. 45, pp. 59–82.Google Scholar - 3.K. Krishnapisharody and G.A. Irons:
*Metall. Mater. Trans. B*, 2013, vol. 44B, pp. 1486–98.Google Scholar - 4.S. Yu, Z.S. Zou, L. Shao, and S. Louhenkilpi:
*Steel Res. Int.*, 2017, vol. 88, pp. 1–5.Google Scholar - 5.D. Sichen:
*Steel Res. Int.*, 2012, vol. 83, pp. 825–41.Google Scholar - 6.G. Irons, A. Senguttuvan, and K. Krishnapisharody:
*ISIJ Int.*, 2015, vol. 55, pp. 1–6.Google Scholar - 7.D. Mazumdar and J.W. Evans:
*ISIJ Int.*, 2004, vol. 44, pp. 447–61.Google Scholar - 8.S. Joo and R.I.L. Guthrie:
*Metall. Trans. B*, 1992, vol. 23B, pp. 765–78.Google Scholar - 9.Y. Sahai and R.I.L. Guthrie:
*Metall. Trans. B*, 1982, vol. 13B, pp. 193–202.Google Scholar - 10.K. Yonezawa and K. Schwerdtfeger:
*Metall. Mater. Trans. B*, 1999, vol. 30B, pp. 411–18.Google Scholar - 11.C. A. Llanos, S. Garcia-Hernandez, J.A. Ramos-Banderas, J.D. Barret, and G. Solorio-Diaz:
*ISIJ Int.*, 2010, vol. 50, pp. 396-402.Google Scholar - 12.K. Krishnapisharody, N.B. Ballal, P.K. Sinha, M.K. Sardar, and K.N. Jha:
*ISIJ Int.*, 1999, vol. 39, pp. 419–25.Google Scholar - 13.R. González-Bernal, G. Solorio-Diaz, A. Ramos-Banderas, E. Torres-Alonso, C.A. Hernández-Bocanegra, and R. Zenit:
*Steel Res. Int.*, 2017, vol. 89, art. no. 1700281.Google Scholar - 14.J. Mandal, S. Patil, M. Madan, and D. Mazumdar:
*Metall. Mater. Trans. B*, 2005, vol. 36B, pp. 479–87.Google Scholar - 15.N. Mazumdar, A. Mahadevan, M. Madan, and D. Mazumdar:
*ISIJ Int.*, 2005, vol. 45, pp. 1940–42.Google Scholar - 16.S.P. Patil, D. Satish, M. Peranandhanathan, and D. Mazumdar:
*ISIJ Int.*, 2010, vol. 50, pp. 1117–24.Google Scholar - 17.A.M. Amaro-Villeda, M.A. Ramirez-Argaez, and A.N. Conejo:
*ISIJ Int.*, 2014, vol. 54, pp. 1–8.Google Scholar - 18.H.Y. Tang, X.C. Guo, G.H. Wu, and Y. Wang:
*ISIJ Int.*, 2016, vol. 56, pp. 2161–70.Google Scholar - 19.Z.Q. Liu, L.M. Li, and B.K. Li:
*ISIJ Int.*, 2017, vol. 57, pp. 1971–79.Google Scholar - 20.A.S. Gómez, A.N. Conejo, and R. Zenit:
*J. Appl. Fluid Mech.*, 2018, vol. 11, pp. 11–20.Google Scholar - 21.C.M. Fan and W.S. Hwang:
*Ironmak. Steelmak.*, 2002, vol. 29, pp. 415–26.Google Scholar - 22.S. Asai, T. Okamoto, J.C. He, and I. Muchi:
*Trans. Jpn Inst. Met.*, 1983, vol. 23, pp. 43–50.Google Scholar - 23.D. Mazumdar and R.I.L. Guthrie:
*Metall. Trans. B*, 1986, vol. 17B, pp. 725–33.Google Scholar - 24.D. Mazumdar, H.B. Kim, and R.I.L. Guthrie:
*Ironmak. Steelmak.*, 2000, vol. 27, pp. 302–09.Google Scholar - 25.A.H. Castillejos and J.K. Brimacombe:
*Metall. Trans. B*, 1987, vol. 18B, pp. 649–58.Google Scholar - 26.A.H. Castillejos and J.K. Brimacombe:
*Metall. Trans. B*, 1987, vol. 18B, pp. 659–71.Google Scholar - 27.S.T. Johansen, D.G. C. Robertson, K. Woje, and T.A. Engh:
*Metall. Trans. B*, 1988, vol. 19B, pp. 745–54.Google Scholar - 28.S. Taniguchi, S. Kawaguchi, and A. Kikuchi:
*Appl. Math. Modell.*, 2002, vol. 26, pp. 249–62.Google Scholar - 29.P.E. Anagbo and J.K. Brimacombe:
*Metall. Trans. B*, 1990, vol. 21B, pp. 637-48.Google Scholar - 30.Y.Y. Sheng and G.A. Irons:
*Int. J. Multiphase Flow*, 1991, vol. 17, pp. 585–98.Google Scholar - 31.Y.Y. Sheng and G.A. Irons:
*Metall. Mater. Trans. B*, 1995, vol. 26B, pp. 625–35.Google Scholar - 32.Y. Kishimoto, Y.Y. Sheng, G.A. Irons, and J.S. Chang.
*ISIJ Int.*, 1999, vol. 39, pp. 113–22.Google Scholar - 33.M. Iguchi, H. Takeuchi, and Z. Morita.
*ISIJ Int.*, 1991, vol. 31, pp. 246–53.Google Scholar - 34.M. Iguchi, H. Kawabata, K. Nakajima, and Z. Morita.
*Metall. Mater. Trans. B*, 1995, vol. 26B, pp. 67–74.Google Scholar - 35.Y.K. Xie and F. Oeters:
*Steel Res. Int.*, 1992, vol. 63, pp. 93–104.Google Scholar - 36.Y.K. Xie, S. Orsten, and F. Oeters:
*ISIJ Int.*, 1992, vol. 32, pp. 66–75.Google Scholar - 37.D. Guo and G.A. Irons:
*Metall. Mater. Trans. B*, 2000, vol. 31B, pp. 1447–55.Google Scholar - 38.D. Guo and G.A. Irons:
*Metall. Mater. Trans. B*, 2000, vol. 31B, pp. 1457–64.Google Scholar - 39.L.M. Li, Z.Q. Liu, B.K. Li, H. Matsuura, and F. Tsukihashi:
*ISIJ Int.*, 2015, vol. 55, pp. 1337–46.Google Scholar - 40.Y.G. Xu, M. Ersson, and P.G. Jonsson:
*Metall. Mater. Trans. B*, 2015, vol. 46B, pp. 2628–38.Google Scholar - 41.G.C. Wang, H.C. Zhou, Q.R. Tian, X.G. Ai, and L.F. Zhang:
*ISIJ Int.*, 2017, vol. 57, pp. 805–13.Google Scholar - 42.T. Tatsuoka, C. Kamata, and K. Ito:
*ISIJ Int.*, 1997, vol. 37, pp. 557–61.Google Scholar - 43.K. Sakaguchi and K. Ito:
*ISIJ Int.*, 1995, vol. 35, pp. 1348–53.Google Scholar - 44.Y.J. Kang, L. Yu, and D. Sichen:
*Ironmak. Steelmak.*, 2007, vol. 34, pp. 253–61.Google Scholar - 45.K. Malmberg, M. Nzotta, A. Karasev, and P.G. Jönsson:
*Ironmak. Steelmak.*, 2013, vol. 40, pp. 231–37.Google Scholar - 46.M. Söder, P.G. Jönsson, and L. Jonsson:
*Steel Res. Int.*, 2004, vol. 75, pp. 128–38.Google Scholar - 47.W.T. Lou and M.Y. Zhu:
*Metall. Mater. Trans. B*, 2013, vol. 44B, pp. 762–82.Google Scholar - 48.W.T. Lou and M.Y. Zhu:
*ISIJ Int.*, 2014, vol. 54, pp. 9–18.Google Scholar - 49.A. Huang, H. Harmuth, M. Doletschek, S. Vollmann, and X.Z. Feng:
*Steel Res. Int.*, 2015, vol. 86, pp. 1447–54.Google Scholar - 50.M. Thunman, S. Eckert, O. Hennig, J. Bjorkvall, and D. Sichen:
*Steel Res. Int.*, 2007, vol. 78, pp. 849–56.Google Scholar - 51.P. Dayal, K. Beskow, J. Björkvall, and D. Sichen:
*Ironmak. Steelmak.*, 2006, vol. 33, pp. 454–64.Google Scholar - 52.H.L. Yang, P. He, and Y.C. Zhai:
*ISIJ Int.*, 2014, vol. 54, pp. 578–81.Google Scholar - 53.C. Liu, S.F. Yang, J.S. Li, L.B. Zhu, and X.G. Li:
*Metall. Mater. Trans. B*, 2016, vol. 47B, pp. 1882–92.Google Scholar - 54.Y.L. Zhou, Z.Y. Deng, and M.Y. Zhu:
*Int. J. Miner. Metall. Mater.*, 2017, vol. 24, pp. 627–37.Google Scholar - 55.K. Krishnapisharody and G.A. Irons:
*Metall. Mater. Trans. B*, 2006, vol. 37B, pp. 763–72.Google Scholar - 56.D. Guo and G.A. Irons:
*Metall. Mater. Trans. B*, 2002, vol. 33B, pp. 377–84.Google Scholar - 57.M. Iguchi, K. Miyamoto, S. Yamashita, D. Iguchi, and M. Zeze:
*ISIJ Int.*, 2004, vol. 44, pp. 636–38.Google Scholar - 58.M. Peranandhanthan and D. Mazumdar:
*ISIJ Int.*, 2010, vol. 50, pp. 1622–31.Google Scholar - 59.L. Wu, P. Valentin, and D. Sichen:
*Steel Res. Int.*, 2010, vol. 81, pp. 508–15.Google Scholar - 60.N. Lv, L. Wu, H. Wang, Y. Dong, and C. Su:
*Int. J. Iron Steel Res.*, 2017, vol. 24, pp. 243–50.Google Scholar - 61.D. Mazumdar, P. Dhandapani, and R. Sarvanakumar:
*ISIJ Int.*, 2017, vol. 57, pp. 286–95.Google Scholar - 62.L.E.J. Pérez, A. Amaro-Villeda, A.N. Conejo, C. González-Rivera, and M.A. Ramirez-Argáez:
*Mater. Manuf. Process*, 2018, vol. 33, pp. 882–90.Google Scholar - 63.P. Valentin, C. Bruch, Y. Kyrylenko, H. Kochner, and C. Dannert:
*Steel Res. Int.*, 2009, vol. 80, pp. 552–58.Google Scholar - 64.M. Ek, L. Wu, P. Valentin, and D. Sichen:
*Steel Res. Int.*, 2010, vol. 81, pp. 1056–63.Google Scholar - 65.B. Glaser, M. Görnerup, and D. Sichen:
*Steel Res. Int.*, 2011, vol. 82, pp. 1425–34.Google Scholar - 66.B. Glaser, M. Görnerup, and D. Sichen:
*Steel Res. Int.*, 2011, vol. 82, pp. 827–35.Google Scholar - 67.F. Karouni, B.P. Wynne, J. Talamantes-Silva, and S. Phillips:
*Steel Res. Int.*, 2018, vol. 89, art. no. 1700551.Google Scholar - 68.K. Krishnapisharody and G.A. Irons:
*ISIJ Int.*, 2010, vol. 50, pp. 1413–21.Google Scholar - 69.D. Mazumdar:
*Metall. Trans. B*, 1990, vol. 21B, pp. 925–28.Google Scholar - 70.Y.H. Pan, D.C. Guo, J.J. Ma, W.Z. Wang, F.P. Tang, and C. Li:
*ISIJ Int.*, 1994, vol. 34, pp. 794–801.Google Scholar - 71.S.W.P. Cloete, J.J. Eksteen, and S.M. Bradshaw:
*Progr. Comput. Fluid Dynam.*, 2009, vol. 9, pp. 345–56.Google Scholar - 72.H.P. Liu, Z.Y. Qi, and M.G. Xu:
*Steel Res. Int.*, 2011, vol. 82, pp. 440–58.Google Scholar - 73.Y.L. Li, L.F. Zhang, and Y. Ren:
*Proc. Extraction and Processing Division Symp. on Pyrometallurgy in Honor of David G.C. Robertson*, San Diego, CA, 2014, pp. 659–66.Google Scholar - 74.D. Mazumdar and R.I.L. Guthrie:
*Metall. Mater. Trans. B*, 1994, vol. 25B, pp. 308–12.Google Scholar - 75.M. Madan, D. Satish, and D. Mazumdar:
*ISIJ Int.*, 2005, vol. 45, pp. 677–85.Google Scholar - 76.Q. Cao and L. Nastac:
*Metall. Mater. Trans. B*, 2018, vol. 49B, pp. 1388–1404.Google Scholar - 77.Y. Sahai and R.I.L. Guthrie:
*Metall. Trans. B*, 1982, vol. 13B, pp. 125–27.Google Scholar - 78.D. Mazumdar:
*Metall. Trans. B*, 1989, vol. 20B, pp. 967–69.Google Scholar - 79.Y. Sahai and R.I.L. Guthrie:
*Metall. Trans. B*, 1982, vol. 13B, pp. 203–11.Google Scholar - 80.D. Mazumdar and R.I.L. Guthrie:
*Metall. Trans. B*, 1985, vol. 16B, pp. 83–90.Google Scholar - 81.D. Mazumdar, R.I.L. Guthrie, and Y. Sahai:
*Appl. Math. Modell.*, 1993, vol. 17, pp. 255–62.Google Scholar - 82.M.B. Goldschmit and A.H.C. Owen:
*Ironmak. Steelmak.*, 2001, vol. 28, pp. 337–41.Google Scholar - 83.S. Ganguly and S. Chakraborty:
*ISIJ Int.*, 2004, vol. 44, pp. 537–46.Google Scholar - 84.S. Ganguly and S. Chakraborty:
*Ironmak. Steelmak.*, 2008, vol. 35, pp. 524–30.Google Scholar - 85.Y.G. Xu, M. Ersson, and P. Jonsson:
*Steel Res. Int.*, 2015, vol. 86, pp. 1289–97.Google Scholar - 86.B.K. Li, H.B. Yin, C.Q. Zhou, and F. Tsukihashi:
*ISIJ Int.*, 2008, vol. 48, pp. 1704–11.Google Scholar - 87.E.K. Ramasetti, V.-V. Visuri, P. Sulasalmi, and T. Fabritius: A CFD and Experimental Investigation of Slag Eye in Gas Stirred Ladle.
*Proc. 5th Int. Conf. on Fluid Flow*, Canada, 2018.Google Scholar - 88.G.N. Li: UDF to Count the Number of Droplets in a VOF Simulation. http://www.eureka.im/1249.html, 2007.
- 89.P. Sulasalmi, V.-V. Visuri, A. Kärnä, and T. Fabritius:
*Steel Res. Int.*, 2015, vol. 86, pp. 212–22.Google Scholar - 90.P. Sulasalmi, A. Kärnä, T. Fabritius, and J. Savolainen:
*ISIJ Int.*, 2009, vol. 49, pp. 1661–67.Google Scholar - 91.A. Senguttuvan: Doctoral Thesis, McMaster University, Hamilton, ON, Canada, 2016.Google Scholar
- 92.A. Senguttuvan and G.A. Irons:
*ISIJ Int.*, 2017, vol. 57, pp. 1962–70.Google Scholar - 93.Y.G. Xu, M. Ersson, and P.G. Jönsson:
*ISIJ Int.*, 2016, vol. 56, pp. 1982–88.Google Scholar - 94.M. Ersson, L. Höglund, A. Tilliander, L. Jonsson, and P.G. Jönsson:
*ISIJ Int.*, 2008, vol. 48, pp. 147–53.Google Scholar - 95.U. Singh, R. Anapagaddi, S. Mangal, K.A. Padmanabhan, and A.K. Singh:
*Metall. Mater. Trans. B*, 2016, vol. 47B, pp. 1804–16.Google Scholar - 96.W.T. Lou and M.Y. Zhu:
*Metall. Mater. Trans. B*, 2013, vol. 44B, pp. 1251–63.Google Scholar - 97.O.J. Ilegbusi, M. Iguchi, K. Nakajima, M. Sano, and M. Sakamoto:
*Metall. Mater. Trans. B*, 1998, vol. 29B, pp. 211–22.Google Scholar - 98.L.F. Zhang.
*Modelling Simul. Mater. Sci. Eng.*, 2000, vol. 8, pp. 463–76.Google Scholar - 99.V. De Felice, I.L.A. Daoud, B. Dussoubs, A. Jardy, and J.P. Bellot:
*ISIJ Int.*, 2012, vol. 52, pp. 1273–80.Google Scholar - 100.J.P. Bellot, V. De Felice, B. Dussoubs, A. Jardy, and S. Hans:
*Metall. Mater. Trans. B*, 2014, vol. 45B, pp. 13–21.Google Scholar - 101.D.Y. Sheng, M. Söder, P. Jönsson, and L. Jonsson:
*Scand. J. Metall.*, 2002, vol. 31, pp. 134–47.Google Scholar - 102.M. Söder, P. Jönsson, and J. Alexis:
*Scand. J. Metall.*, 2002, vol. 31, pp. 210–20.Google Scholar - 103.M. Söder: Doctoral Thesis, KTH Royal Institute of Technology, Stockholm, 2004, 91-7283-886-8.Google Scholar
- 104.D.B. Spalding:
*Math. Comput. Simul.*, 1981, vol. 23, pp. 267–76.Google Scholar - 105.D.B. Spalding:
*Numerical Computation of Multi-Phase Fluid Flow and Heat Transfer*, Pineridge Press, London, 1980, pp. 139–67.Google Scholar - 106.L. Jonsson and P. Jönsson:
*ISIJ Int.*, 1996, vol. 36, pp. 1127–34.Google Scholar - 107.P.G. Jönsson, L. Jonsson, and D. Sichen:
*ISIJ Int.*, 1997, vol. 37, pp. 484–91.Google Scholar - 108.L. Jonsson, D. Sichen, and P. Jönsson:
*ISIJ Int.*, 1998, vol. 38, pp. 260–67.Google Scholar - 109.J.L. Xia, T. Ahokainen, and L. Holappa:
*Scand. J. Metall.*, 2001, vol. 30, pp. 69–76.Google Scholar - 110.C.G. Mendez, N. Nigro, and A. Cardona:
*J. Mater. Process. Technol.*, 2005, vol. 160, pp. 296–305.Google Scholar - 111.L.T. Wang, Q.Y. Zhang, S.H. Peng, and Z.B. Li:
*ISIJ Int.*, 2005, vol. 45, pp. 331–37.Google Scholar - 112.L.T. Wang, S.H. Peng, Q.Y. Zhang, and Z.B. Li.
*Steel Res. Int.*, 2006, vol. 77, pp. 25–31.Google Scholar - 113.D.Q. Geng, H. Lei, and J.C. He:
*Int. J. Miner. Metall. Mater.*, 2010, vol. 17, pp. 709–14.Google Scholar - 114.F.D. Maldonado-Parra, M.A. Ramirez-Argáez, A.N. Conejo, and C. González:
*ISIJ Int.*, 2011, vol. 51, pp. 1110–18.Google Scholar - 115.A. Huang, H.Z. Gu, M.J. Zhang, N. Wang, T. Wang, and Y. Zou:
*Metall. Mater. Trans. B*, 2013, vol. 44B, pp. 744–49.Google Scholar - 116.W.T. Lou and M.Y. Zhu:
*Metall. Mater. Trans. B*, 2014, vol. 45B, pp. 1706–22.Google Scholar - 117.W.T. Lou and M.Y. Zhu:
*ISIJ Int.*, 2015, vol. 55, pp. 961–69.Google Scholar - 118.S. Yu and S. Louhenkilpi:
*Metall. Mater. Trans. B*, 2013, vol. 44B, pp. 459–68.Google Scholar - 119.S. Yu, J. Miettinen, and S. Louhenkilpi:
*Mater. Sci. Forum*, 2013, vol. 762, pp. 253–59.Google Scholar - 120.S. Yu, J. Miettinen, and S. Louhenkilpi:
*Steel Res. Int.*, 2014, vol. 85, pp. 1393–1402.Google Scholar - 121.S. Yu, J. Miettinen, L. Shao, and S. Louhenkilpi:
*Steel Res. Int.*, 2015, vol. 86, pp. 466–77.Google Scholar - 122.S. Yu: Doctoral Thesis, Aalto University, Helsinki, 2014.Google Scholar
- 123.J. Aoki, L. Zhang, and B.G. Thomas:
*3rd Int. Congr. on Science & Technology of Steelmaking*, Warrendale, PA, 2005, pp. 319–32.Google Scholar - 124.R. Singh, D. Mazumdar, and A.K. Ray:
*ISIJ Int.*, 2008, vol. 48, pp. 1033–35.Google Scholar - 125.L.M. Li, B.K. Li, and Z.Q. Liu:
*ISIJ Int.*, 2017, vol. 57, pp. 1–10.Google Scholar - 126.L.M. Li, Z.Q. Liu, M.X. Cao, and B.K. Li:
*JOM*, 2015, vol. 67, pp. 1459–67.Google Scholar - 127.T. Kulju, S. Ollila, R.L. Keiski, and E. Muurinen:
*IFAC-PapersOnLine*, 2015, vol. 48, pp. 1–5.Google Scholar - 128.T. Stapurewicz and N.J. Themelis:
*Can. Metall. Q.*, 1987, vol. 26, pp. 123–28.Google Scholar - 129.A.K. Singh and D. Mazumdar:
*Metall. Mater. Trans. B*, 1997, vol. 28B, pp. 95–102.Google Scholar - 130.J. Savolainen, T. Fabritius, and O. Mattila:
*ISIJ Int.*, 2009, vol. 49, pp. 29–36.Google Scholar - 131.C.E. Grip, L. Jonsson, P.G. Jönsson, and K.O. Jonsson:
*ISIJ Int.*, 1999, vol. 39, pp. 715–21.Google Scholar - 132.J.T. Kuo and G.B. Wallis:
*Int. J. Multiphase Flow*, 1988, vol. 14, pp. 547–64.Google Scholar - 133.S.W. Beyerlein, R.K. Cossmann, and H.J. Richter:
*Int. J. Multiphase Flow*, 1985, vol. 11, pp. 629–41.Google Scholar - 134.S.A. Morsi and A.J. Alexander:
*J. Fluid Mech.*, 1972, vol. 55, pp. 193–208.Google Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.