Physical and Mathematical Modeling of Multiphase Flows in a Converter

Abstract

Fluid flow in converter bath, affecting the viability, effectiveness, and efficiency of steelmaking, plays a critical role in the productivity and quality level that can be achieved in the process. Due to a large quantity and diversity of the studies on the characteristics of the multiphase flow, it seems very necessary to make a systematic literature review on state-of-the-art developments in the steelmaking process. This paper presents the recent findings of the characteristics of the multiphase flow in the converter by means of physical and mathematical modeling and the resulting implications for simulating the process. Some representatives include supersonic oxygen jet, stirring and mixing, splashing and droplet generation, and energy transfer. The work summarized in this paper can give an in-depth understanding of the fluid flow in converter and provide references for future modeling of the converter steelmaking process. Future contributions to a fundamentally generalized modeling of the converter steelmaking are still needed. More profoundly, the modeling work can facilitate the real-time data-driven precise BOF process control and be an important component to the realization of intelligent manufacturing in steelmaking process.

1. Introduction

Currently, Basic Oxygen Furnace (BOF) steelmaking is the predominant steelmaking process around the world.¹⁾ The BOF steelmaking process can rapidly refine hot metal and ambient scrap into qualified steel of desired carbon content and temperature with the aid of oxygen-blowing top lance and nonreactive gas-blowing bottom plugs. Figure 1 illustrates the composition of a converter and the blowing information.²⁾ The vessel is lined with basic refractories made from magnesite, dolomite, etc. that provide a relatively inert ambient to the corrosive basic slag prepared by lime dissolution. During the blowing process, highly pure oxygen with high pressure and velocity is injected into the molten bath through nozzles in the form of a gas jet for decarburization, which leads to sophisticated transport phenomena in the bath.²⁾ Besides, the bottom-blowing is also of significant importance, which is aimed to obtain a homogenized melt, to decrease the Fe content in slag and to minimize refractory erosion. Fluid flows in the converter bath work together to affect the viability, effectiveness, and efficiency of converter.³⁾

Fig. 1.

Schematic of a converter of a converter. (a) Characteristics of the vessel and (b) slag-metal-gas interaction and molten steel flow.²⁾

Due to the fairly sophisticated transport phenomena and the difficulty in direct observation, physical water modeling, coupled with mathematical modeling, offers important tools for tackling process development and optimizing the BOF steelmaking process.⁴⁾

In the early days of BOF process development, physical models were considered to be a fantastic tool for investigating the fluid flow phenomena in the converter. Extensive past work has employed water model experiment to successfully visualizing flow patterns, mixing and even some heat transfer phenomena.^{5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30)} R. B. Banks and D. V. Chandrasekhara⁵⁾ investigated the process of a gas jet impinging on and penetrating into a liquid surface. A Plexiglass tank was employed to study the behavior of an air jet penetrating into deep water. N. A. Molloy⁶⁾ identified three types of cavity conditions generated by top blown jet as dimpling mode, splashing mode and penetrating mode depending on the general appearance of the cavity, change in sound of the bath and the reduction of splashing. J. Szekely et al.⁷⁾ determined the flow pattern and quantitatively measured the velocity and turbulence energy fields using hot-film anemometry in a simplified a gas-stirred water model. S. K. Sharma et al.⁸⁾ found that the interaction behavior between top jet and liquid bath can be predicted from a dynamically scaled water model. K. Nakanishi et al.⁹⁾ investigated the mixing rate of molten steel and mass transfer rate between slag and metal in Q-BOP through water model experiments. Extensive studies were also carried out on the stirring behavior andbehavior mixing characteristics in top blown,^{10,11,12,13,14,15)} bottom blown,^{16,17,18,19,20,21,22)} as well as combined blown^{23,24,25,26,27,28,29)} converter at different stages³⁰⁾ during a heat using water models.

However, the transport phenomena in steel bath are so complex that many phenomena cannot be presented by water model experiments such as the compressibility of high speed oxygen, the interaction behavior between different phases, high temperature performance, etc. Mathematical modeling is then an alternative method to visualize fluids flow inside a converter. Moreover, there has been a major growth in the use of mathematical models due to the availability of inexpensive, highly capable computational hardware, the broad range of available software packages and the significant body of accumulated experience in recent years.³¹⁾ J. Szekely et al.³²⁾ first developed the mathematical model for describing the flow field in liquids or melts agitated by a symmetrical placed impinging gas jet. M. Salcudean and R. I. L. Guthrie³³⁾ analysed the fluid flow generated in the course of BOF tapping operation. A mathematical model for the filling process was established and verified by experimental values obtained from a one-tenth scaled-down physical model. Y. Z. Li et al.^34,35) mathematically explored the gas-liquid two-phase fluid flow fields in both top and bottom blown bath. The flow pattern and the distributions of velocity, turbulence viscosity, density and gas void fraction ratio were obtained. S. C. Du et al.³⁶⁾ derived a new velocity boundary condition on the surface of cavity caused by impinging jet. B. K. Li³⁷⁾ proposed a method to predict three-dimensional flow in bottom gas stirred baths. Thereafter, significant progress has been made toward a full simulation^{38,39,40,41,42,43,44,45,46,47,48)} due to the decreasing computational costs and increasing power of commercial modeling packages, which can help us achieve wonderful insights into flow related phenomena without inherent inaccuracies in water model experiments. A comprehensive understanding of the highly coupled phenomena occurring in a converter can be obtained through all these developments of computational fluid dynamics simulations with high fidelity.

Therefore, a clearly systematic knowledge of aforementioned in-furnace highly complex multiphase transport behavior can be conducive to the development of fundamental insights, process optimization and end-point control of BOF operations.^{49,50,51,52,53,54,55,56,57,58)} Therefore, this paper will review the developments in physical modeling and computational simulations of the fluid flow behavior in BOF steelmaking process. Details of fundamental modeling methods, supersonic oxygen jet behavior, stirring and mixing characteristics in the bath, splashing and droplet generation behavior and energy transfer performance will be presented, and finally outlook and future trends will also be discussed.

2. Fundamental Methods and Models

Modeling, which is a well-established scientific technique with powerful capabilities of demonstration and then widely applied in the design, control and optimization of engineering process, implies scientific representation of a process or a phenomenon. The representation can be physical and mathematical.^59,60) A general subdivision of modeling in steelmaking is shown in Fig. 2.⁶⁰⁾ The physical and mathematical modeling work are summarized in the following part.

Fig. 2.

Modeling approaches in steelmaking.⁶⁰⁾

2.1. Establishment of the Physical Modeling

In the context of BOF steelmaking, models made of transparent materials and using liquid like water and mercury as the simulated fluids have most often been used to investigate the hydrodynamics and associated transport phenomena (such as momentum transfer, mass transfer, etc.) in a converter.⁶¹⁾ Amongst water is most often used, and it is because water (at 20°C) and molten steel (at 1600°C) have practically equivalent kinematic viscosities⁶²⁾ as shown in Table 1, making model an excellent tool for investigating various transport phenomena in steelmaking reactor. Physical models, in which water is used to simulate molten steel, are known as “water models” or “aqueous models”. Models can have either reduced scale or full scale. R. I. L. Guthrie and M. Isac⁶³⁾ concluded that a full-scale water modeling can provide a better opportunity of simulating the fluid flow phenomena during real steelmaking operations. Moreover, S. K. Sharma et al.⁸⁾ proved that the metallurgical performance can be predicted from the scaled water models with dynamic similarity to the prototype. The choice of model scale should be made based on dynamic properties and laboratory conditions such as laboratory size, air supply, water supply, etc.⁶⁴⁾ The key objective of the physical modeling is to inexpensively and conveniently measure and visualize the characteristics of an actual converter.

Table 1. Physical properties of water at 20°C and steel at 1600°C.⁶²⁾

Property	Water (20°C)	Steel (1600°C)
Molecular viscosity (μ), kg/(m·s)	0.001	0.0064
Density (ρ), kg/m3	1000	7014
Kinematic viscosity (υ=μ/ρ), m2/s	10–6	0.913×10–6
Surface tension (σ), N/m	0.073	1.6

The physical model is established based on the similarity principle. Mainly there are four similarity criteria i.e. geometrical similarity, mechanical similarity, thermal similarity and chemical similarity. Geometrical similarity and mechanical similarity are usually considered during water modeling process. Geometrical similarity represents the size similarity between the model and its prototype. Mechanical similarity implies the similarity of forces and the related entities such as momentum. It is compartmentalized into three categories, namely static similarity, dynamic similarity, and kinematic similarity.⁶⁵⁾ For BOF steelmaking process, equivalence of Froude and Reynolds numbers ensures dynamic similarity between the model and its prototype.⁶⁶⁾ Weber number is chosen to keep interface state.⁶⁷⁾ Moreover, further modifications may also be needed for a specific process. For BOF process, the top inlet fluid is supersonic gas, so the gas is compressible. While the gas used in water modeling experiment is subsonic. If subsonic fluid is taken to simulate the supersonic one, it will lead to a lager impact area simply on account of the geometric similarity of oxygen lance. Correction terms for this error, therefore, should be assumed.⁶⁸⁾ Consequently, dimensions and operating conditions of a converter can be scaled to a model system. Table 2 summarized some water modeling systems published.

Table 2. Details of the various physical modeling investigations.

NO.	Investigators (Year)	Scale factor	System	Focus	Measurement	Refs.
1	Banks et al. (1963)	–	air-water	cavity depth, cavity diameter, lip height, liquid-drop formation	stagnation-pressure analysis, displaced-liquid analysis	5
2	Szekely et al. (1976)	–	air-water, gas stirred	flow pattern, turbulent kinetic energy	flow visualization technique, hot-film anemometry	7
3	Nakanishi et al. (1980)	1/20	air-water	mixing time, mass transfer capacity coefficient	tracer dispersion technique	9
4	He et al. (1989, 1990)	–	air-water, nitrogen- mercury–glycerin	droplet amount, droplet distribution ,droplet generation, emulsion from	nitrogen bath, high speed film	69, 70
5	Koria et al. (1990)	–	air-paraffin oil-aqueous solution	mixing intensity, mass transfer rate, agitation energy	conductivimetric method	26
6	Singh et al. (1990)	1/40	air-benzene-water	mixing time, volumetric mass transfer rate	conductivimetric method	27, 28
7	Duan et al. (1990)	1/11.5	air-water-oil	mixing time, mass transfer rate, stirring energy	conductivimetric method	25
8	Peaslee (1993)	1/2	air-water, air-mercury–water-glycerol	splash type, penetration depth, liquid circulation , cavity formation, drop size	two dimensional slice ,image analysis equipment	71
9	Iguchi et al. (1995, 1996, 1997, 1998)	–	air-water/air-silicone oil-water	velocity distribution, turbulence distribution, bubble characteristics, mixing time	LDV, conductivimetric method	72, 73, 74, 75
10	Qian et al. (1996)	–	air-corn oil-water/air- kerosene oil -water	interface shapes, penetration depth	surface-tracking resistance probe	76
11	Loumala et al. (2002, 2004)	1/7 1/9	air-oil-water	mixing time ,splashing rate, bath oscillation.	conductivimetric method, collecting splashing water	17, 77, 78
12	Ersson et al. (2006, 2014)	1/6	air-water	velocity distribution, mixing time, cavity shape	electroresistivity probe, video recording, PIV system	79, 80, 81
13	Martina et al. (2005)	1/10	air-Vaseline oil-water	mixing time	conductivimetric method	82
14	Nordquist et al. (2006)	–	air-water	Penetration depth	video recording	13
15	Choudhary et al. (2006)	1/6	air-water	mixing time	conductivity measurement technique	18
16	Zhu et al. (2008)	1/8.5	air-water	mixing time, bath oscillation	electroresistivity probe	20
17	Lai et al. (2008)	1/8.5	air-water	mixing time	conductivimetric method	20
18	Conejo et al. (2013, 2015)	1/18 1/8	air-hexane- water, air-motor oil-water	mixing time	electric conductivity sensor	22, 83
19	Li et al. (2015, 2016)	1/10	air-oil-water	cavity depth, cavity diameter	video recording	15, 84
20	Brooks et al. (2014, 2015, 2016, 2017)	1/10, 1/5	air-water, air-motor oil-water	splashing, droplet amount, cavity modes	video recording, FFT technique, sound recorder, waveform and spectrum analysis	85, 86, 87, 88

Abbreviations: Laser Doppler Velocimeter (LDV); Particle Image Velocimetry (PIV); Fast Fourier Transform (FFT)

Many physical modeling studies have been carried out towards the converter steelmaking process in recent years. Results obtained from physical modeling can be supplied to directly evaluate the characteristics of the real system and quantitatively describe the behavior of the multiphase flow in the converter bath during actual production. Besides, physical modeling is always performed to validate a mathematical model, which has become a popular approach in steelmaking process. Hence, physical modeling and mathematical modeling are frequently applied in conjunction, as it is usually difficult to derive validation dataset from industrial operations.⁸⁹⁾ Detailed information will be discussed in the subsequent sections.

2.2. Construction of the Mathematical Model

Mathematical modeling is an alternative method for visualizing the multiphase flow behavior during the converter steelmaking process. A mathematical model is a set of equations, which is used to represent and predict certain phenomena.⁹⁰⁾ The fundamental basis for these differential equations is usually derived from thermodynamics, kinetics, heat flow, mass transfer and other relative phenomena.³¹⁾ Hence, the quantitative mathematical models can play a key role in process control and process optimization, as well as the planning and interpretation of measurements.⁹⁰⁾ The early mathematical models of fluid flow appeared in the early 1970s, and by the early 1980s the availability of computers led to the acceptance of computational fluid mechanics as a standard modeling tool.⁹⁰⁾ Computational Fluid Dynamics (CFD) is an acronym for a combination of physics, numerical mathematics and, to some extent, computer science employed to simulate fluid flows.⁹¹⁾

A modeling program involves the following parts as illustrated in Fig. 3.⁹⁰⁾ It mainly contains problem identification, formulation, simple scoping and scaling calculations, after which machine calculations and experimental work should be done in parallel.

Fig. 3.

The schematic flow chart of a typical mathematical modeling.⁹⁰⁾

The scope of rigorous modeling in steelmaking appears to be somewhat limited due to the complexity of the multiphysics melting process. Therefore, it is practically impossible to build a mathematical model in steelmaking without empiricism and/or idealization. The gross features of multiphase flow in the converter steelmaking process are summarized in Table 3.

Table 3. Fluid characteristics and currently modeling strategy of converter steelmaking.

Co-existing phases	Nature of fluid motion	Currently adopted modeling approach
Supersonic and subsonic gas, liquid metal, molten slag, unmelted solids (with internal dispersions of droplets, particles and bubbles)	Multiphase, compressible, turbulent flow, chemical reactions	Three-phase, turbulent flow models without chemical reactions, solid melting and heat transfer

A typical mathematical model for the converter steelmaking process solves the continuity equation and Navier-Stokes equations considering the compressibility of the supersonic oxygen jet^92,93,94) in a boundary fitted coordinate system. The solution of these equations presents the pressure and velocity distribution in the domain.⁹⁵⁾ As the high flow rates involved in this process, these models must incorporate turbulent fluid flow. Many different turbulence models have been attempted by different researchers to describe the turbulence flow in converter steelmaking, e.g. one equation turbulence models,^32,34) two-equation turbulence models.^{44,96,97,98,99)} In this way, the converter steelmaking can be expressed in terms of some physical variables using partial differential equations with appropriate operating and boundary equations. Subsequently, it is the generation of grid in the domain and discretization of the established partial differential equations into algebraic form using different schemes.¹⁰⁰⁾ However, analytical solution of these equations is rather difficult. Recently, it becomes increasingly possible to model the complex flows and to find optimal solutions¹⁰¹⁾ with the ever increasing power and capabilities of computer hardware, and together with the development of CFD commercial software packages like Fluent, CFX, Phoenics, OpenFOAM, etc.

Furthermore, it is necessary to divide the modeling into manageable subroutines due to the complexity of the BOF process. To date, there is no general CFD model for a BOF, at least in the open literature.⁵⁷⁾ The subroutines are analyzed, described and modeled based on different fluid-dynamic observations. Individual solution is calculated and implemented in the overall model.⁴⁰⁾ Some of the published modeling work are summarized in Table 4.

Table 4. Details of the various mathematical modeling.

No.	Authors (Year)	System	Focus	Turbulence Flow	Multiphase flow	Software	Refs.
1	Szekely et al. (1972)	impinging gas-liquid	fluid flow in liquids	two length scale model102)/PK103,104)	Treating vorticity at the cavity walls	–	32
2	Li et al. (1992)	bottom blown	velocity field, mixing	RANS/sKE	–	FORTRAN	37
3	Tago et al. (2003)	supersonic oxygen jet	jet behavior, fluid flow	RANS/sKE, RSM	–	–	11
4	Odenthal et al. (2006, 2014, 2015)	top-blown bottom blown	fluid flow, mixing, supersonic jet	RANS/sKE	VOF, DPM	FLUENT	40
		Top-bottom blown		URANS/SST-SAS	VOF, DPM	FLUENT, OpenFOAM	105, 106
5	Nguyen et al. (2006)	top-blown	gas-liquid interface deformation	RANS/sKE	VOF	FLUENT	107
6	Jalkanen (2006)	oxygen converter	chemical and thermal evolution	–	–	CONSIM108,119)	110
7	Singh et al. (2007)	bottom blown	mixing	RANS/sKE	Lagrangian approach111)	FLUENT	112
8	Ersson et al. (2008, 2015)	top-blown	gas jet, surface deformation, fluid flow, mixing, thermodynamics	RANS/sKE, rKE, mKE	VOF	–	41
		top-blown		Reactions /RANS/rKE	VOF	Thermo-Calc113) +FLUENT	42
		top-bottom-side blown		RANS/sKE	VOF, DPM	FLUENT	114
9	Asai et al. (2009)	top- blown	impingement interfacial area		MPS		43
10	Alam et al. (2010)	supersonic oxygen jet	jet behavior, fluid flow, temperature, dropt generation	RANS/mKE	–	AVL FIRE115)	116
11	Asahara et al. (2011)	top blown	jet behavior and cavity formation	RANS/SA, KE, KW	VOF	FLUENT	99
12	Wang et al. (2010)	top- blown	compressible Flow	NS/sKW,117) rKE118)	–	FLUENT	44
13	Li et al. (2013)	combined blown	flow field, mixing	RANS/sKE	VOF+DPM	FLUENT	119
14	Doh et al. (2013)	top-blown	deformation of the bath free surface	Filter-based RANS120)/sKE	VOF	PHYSICA121)	45
15	Li et al. (2014)	gas–slag/metal interaction	jet behaviour, cavity evolution	RANS/sKE	VOF	FLUENT	122
16	Chu et al. (2016)	bottom- blown	mixing efficiency, fluid flow	NS/sKE	VOF	FLUENT	48
17	Cao et al. (2016)	top blown	cavity formation, shape, mixing	RANS/sKE	VOF	FLUENT	123
18	Lin et al. (2017)	bottom injection	gas-liquid interaction/heat and mass transfer	RANS+ LES	VOF/DPM	FLUENT	124

Abbreviations: Navier–Stokes equations (NS); Reynolds-averaged Navier–Stokes (RANS); Large Eddy Simulation (LES); Prandtl-Kolmogorov model (PK); k–ε model (KE); standard k–ε model (sKE); realizable k–ε model (rKE); modified k–ε model (mKE); RNG k–ε model (RNGKE); k–ω model (KW); standard k–ω model (sKW); Shear Stress Transport-Scale Adaptive Simulation (SST-SAS); Reynolds Stress Model (RSM); Spalart-Allmaras model (SA); Moving Particle Semi-implicit (MPS); Volume of Fluid (VOF); Discrete Phase Model (DPM).

Accordingly, many numerical attempts have been initiated by many investigators, especially in the near decades with the development of efficient solution algorithms and powerful computational software as well reasonably priced high performance computers. However, numerous idealizations are still adopted to formulate the converter steelmaking process. For example, the reacting turbulent flows, solids, heat transfer, et al. are not included. Moreover, many of the simulations just focus on one specific area as can be seen in Table 4, e.g. the compressible supersonic jet behavior, the interaction between top gas and liquid metals, the mixing in the bath, etc. Much more remains to be done in the development of a generalized CFD model of BOF steelmaking.

In the next section, the fluid flow phenomena inside the BOF, which are widely studied by many investigators, will be discussed.

3. Supersonic Oxygen Flow Behavior

Supersonic oxygen jets impinging on the molten bath are the basis of refining in basic oxygen furnaces, playing a significant role in oxygen supply, bath recirculation and mixing, and chemical reactions. The whole converter steelmaking process is significantly affected by the behavior of the compressible supersonic jet, which is mainly responsible for the phenomena such as oxidization, foaming, splashing, skulling, sloping, converter oscillation and noise.⁴⁰⁾

Detailed work has been done in accurate and efficient modeling of the supersonic oxygen flow in the converter.^10,125) The supersonic oxygen jet, produced by the convergent-divergent (CD) nozzle, is subdivided into potential core, supersonic and subsonic region.¹²⁵⁾ The length of the potential core is one of the significant operating parameters, which represents the attenuation rate of the supersonic flow.⁶⁴⁾ Long potential core length can be obtained by large Mach number,^94,10) high operating pressure,^126,127) incorrect expansion,¹⁰⁾ high ambient pressure,^128,129) high ambient temperature,^116,130,131) etc. Longer potential core will lead to slower attenuation of the oxygen jet, which results in more intensive stirring of the molten bath in the converter. Figure 4 shows some modeling results regarding to the supersonic jets.^40,44)

Fig. 4.

CFD simulation of the supersonic jet under different conditions (a) Mach number distribution under different converter back- pressure;⁴⁰⁾ (b) velocity magnitude maps of the jets with the different nozzle angle.⁴⁴⁾ (Online version in color.)

These studies explored the fundamental behavior of the supersonic jets and greatly helped in propagating the knowledge on oxygen lance performance and its design. R. Sambasivam et al.¹³²⁾ designed a new oxygen lance with a central subsonic nozzle as described in Fig. 5. The augment of droplet generation can be controlled through the subsonic nozzle. The results obtained from simulations and water model experiments show that the droplet generation rate was significantly improved in the presence of the central subsonic jet. The modeling of co-jet technology for BOF converters was reported by A. R. Naji Meidani et al.¹³³⁾ The experimental results demonstrated that a better performance in terms of greater penetration depths and reduced mixing times can be obtained with a co-axial, low density subsonic jet and a main supersonic jet. H. J. Odenthal et al.¹⁰⁵⁾ proposed an adaptive BOF top lance nozzle displayed in Fig. 6. It reacts to the changes in the oxygen pressure and controls the surface ratio between throat and outlet section of each nozzle. The service life of the Laval nozzles can be increased and the blowing becomes more efficient. Therefore, the BOF process can be improved to a large extent by operating the lance from its designed point. However, more detailed knowledge still needs to be investigated on the supersonic jet behavior under the actual conditions of the complex converter steelmaking process. For instance, the study on supersonic jet behavior in hot and dense surroundings is rarely considered,¹³⁴⁾ even though many of the studies were conducted under hot gaseous surroundings. During the steelmaking process, the supersonic jets expand into the surroundings formed by hot and dense slag foam. Moreover, chemical reactions are also taking place in the emulsion slag.

Fig. 5.

Schematic representation of the lance design with a central subsonic nozzle.¹³²⁾

Fig. 6.

CAD model of the adaptive BOF top lance nozzle, (1) Impeller, (2) Generator, (3) Accumulator, (4) Pressure and temperature sensor, (5) Micro-controller, (6) Transmitter, (7) Drive unit, (8) Spindle gear.¹⁰⁵⁾ (Online version in color.)

4. Stirring and Mixing Behavior

The stirring and mixing of molten bath is the most important and basic unit process and phenomena of modern metallurgical process. A proper stirring and mixing is a key factor determining the working status and reaction rate of production process.¹³⁵⁾ For the converter steelmaking process, the interaction between the injected gases and molten liquid is of decisive importance. Therefore, intensive work have been implemented on the stirring and mixing behavior in the converter caused by top blowing, bottom blowing and combined blowing interaction by many investigators.

The interaction between gas and liquid metal/slag, as described in Fig. 7, is the primary determinant of the complicated phenomena during the converter process.²⁾ The formation of cavity is a typical character of the interaction and has been largely investigated. The cavity dimension, which is closely related to the interfacial area in actual production,¹³⁶⁾ is influenced by nozzle diameter and angle, lance height, oxygen pressure, flow rate, slag properties, etc.^{13,14,15,41,43,76,99,137)} The operating parameters such as oxygen pressure and lance height have a much bigger impact on the cavity formation compared to the slag properties.^46,138) Moreover, a reasonable cavity profile can attain a better mixing effect in the converter steelmaking process.¹²³⁾ M. Ersson et al.⁴²⁾ developed a novel approach where the computational fluid dynamics software is coupled with the thermodynamic databases to obtain dynamic simulations of metallurgical process phenomena. This modeling approach has been applied to a fundamental model for a top-blown converter. Reactions between gas–steel, gas–slag, steel–slag and gas–steel–slag have been considered. Figure 8 shows the mass fraction of carbon in the steel and CO in the gas phase. Results show that turbulent diffusion of species cannot be neglected when considering the species transport in the surface area. A large amount of CO produced during the decarburization process might slow down the rate of decarburization in droplets ejected from the bath. This work provides us the sights that it is possible to develop a dynamic coupling of the Thermo-Calc databases and a CFD software to carry out the dynamic simulations. Moreover, the high temperature physical property values such as viscosity, surface tension, density and thermal conductivity of molten slag and metal during steelmaking process are greatly affected by the intensive stirring caused by the coexisting multiphase flows.¹³⁹⁾ Nevertheless, the detailed information on a plurality of fundamental physical property values involving multiphase fluids are still lost. Further studies are still needed.

Fig. 7.

Interaction between top blown jet and molten liquid obtained from (a) high speed imaging⁸⁶⁾ and (b) CFD simulation.⁴⁶⁾ (Online version in color.)

Fig. 8.

Simulated mass fraction of carbon in steel (a) and CO in the gas phase (b).⁴²⁾

The high mixing efficiency in the converter is largely attributed to stirring effects of the bottom injected gas. Therefore, the effects of various bottom nozzles conditions,^{16,17,18,19,24,80,83)} gas flow rates and gas supply schemes,^{21,22,25,48,140)} and combination schemed of top gas and bottom gas^{20,26,27,30,78,79,119,141)} are studied to optimize the mixing effects in the converter. The velocity distribution inside a 335-ton converter bath as well as the velocity and bubble size distribution in a six plugs 190-ton bottom blown converter with standard bottom stirring are presented in Fig. 9.¹⁰⁵⁾ The determined optimizing results implemented in the actual steelmaking vessel at Tata Steel indicated that the average phosphorus partition ratio was improved by 10 to 12 points, and the average %Fe total in the slag dropped by about 1–2%, and a record vessel life of more than 2000 heats was achieved in the first campaign itself. Gradual improvement in the approach to equilibrium was also observed.¹⁸⁾ Moreover, the reduced splashing to the knuckle areas can be obtained with certain lance gaps by positioning bottom nozzles directly between the cavity and knuckle area with remarkable (approximately 30–40%) overlap.¹⁴²⁾ However, the best performance scheme varies from different production conditions. For example, the bottom tuyere configuration in a symmetric non-equiangular position was found to be the best arrangement with respect to mixing in the vessel studied by S. K. Choudhary et al.¹⁸⁾ However, a new bottom tuyere scheme with an asymmetrical configuration was found to be one of the best cases with respect to a decreased mixing time in the bath in the work of X. B. Zhou et al.⁸⁰⁾ Thus, detailed and specific investigations are required to gain an excellent stirring and mixing performance for different production conditions such as asymmetrical and symmetrical bottom blowing.

Fig. 9.

The velocity distribution inside the 335-ton converter and the velocity (a) and bubble size distribution in 190-ton bottom blown converter (b).¹⁰⁵⁾ (Online version in color.)

5. Splashing and Droplet Generation Behavior

“Splashing” refers to the liquid projected from the bath including the material ejected from the vessel, whereas “droplet generation” refers to the part of splash that ends up in the emulsion as distinct droplets.⁸⁶⁾ Figure 10 shows the droplet generation process in a transparent vessel.¹⁴³⁾ The reasons for the droplets generation can be pressure fluctuations, horizontal and vertical cavity oscillations,¹⁴⁴⁾ shearing by the gas stream, breakup of large droplets, entrainment into a gas stream,⁶⁹⁾ and by the growth of the ripples.¹⁴⁵⁾ M. Iguchi¹⁴⁶⁾ exhaustively reviewed the mechanisms of the bubble and droplet generation in a bath subjected to bottom gas injection through a single-hole nozzle using many examples. Various influential factors like lance height and flowrate,^{6,67,85,86,87,147,148,149)} bottom blowing nozzles and flowrate^{69,78,142,150,151,152)} were elaborately investigated to illustrate the splashing and droplet generation behavior by means of water modeling and CFD simulations. The splashing and droplet generation under different flowrate can be seen in Fig. 11. On the one hand, droplet generation has both beneficial and detrimental effects.^116,148) The droplet increases the interfacial area, which in turn, increases the refining rate.¹⁴⁸⁾ On the other hand, the splashing and droplet generation may cause a wearing of refractories or a skulling on the mouth of the vessels and lances, which can result in a loss of production.^145,152) Foamy slag, decreasing top gas flow rate and increasing lance height reduces metal losses and the skulling of the upper converter cone and converter mouth.⁷⁸⁾ However, none of the predictive models for predicting droplet generation in Oxygen steelmaking have yet incorporated the effect of bottom gas injection into their predictions and there is evidence that it likely to be significant and is worthy of further research.⁸⁸⁾ Whilst, there is a clear need for further high temperature trials at industrial scale to fully quantify the splashing and droplet generation phenomena.

Fig. 10.

Schematic illustration of droplet generation process.¹⁴³⁾ (Online version in color.)

Fig. 11.

Droplet generation at lance height 0.120 m and (a) flow rate 50 L/min; (b) flow rate 70 L/min.⁸⁵⁾

6. Energy Transfer Behavior

The main characteristic phenomena during the converter steelmaking process are summarized in the previous sections. In fact, these phenomena are essentially the result of the inherent energy transfer between the jets and the molten bath. The transfer efficiency of the fluids determines the representative parameters of steelmaking process. However, there are few studies on the energy transfer behavior in relation to the multiphase flow in the converter. The stirring and mixing of the bath is achieved by the transferring of agitation energy generated by the injected gas to the molten liquid. Water modeling experiments were used to evaluate the agitation caused by top jet,²⁷⁾ bottom gas,^27,153) and combination gases.⁶⁶⁾ The main reasons for circulation and mixing of the liquid bath are energy transferred from the bottom gas and the buoyancy caused by bubbles rising.²⁷⁾ However, only 10% of the agitation energy stemmed from top jet is transferred to stirring and mixing the bath.⁶⁶⁾ The relation between the agitation energy and mixing intensity in an 80 t converter can be seen in Fig. 12.¹⁵⁴⁾

Fig. 12.

The effect of agitation energy on mixing behavior in an 80 t converter.¹⁵⁴⁾ (Online version in color.)

H. Y. Hwang and G. A. Irons¹⁴⁾ numerically explored the energy transfer efficiency of a top-blown jet into the bath. An energy transfer index was defined to quantify the transfer of kinetic energy. The index showed little dependency on the gas flow rate but increased as the lance height increased. The fast Fourier transform technique (FFT) and power spectral density function (PSD) were found useful to analyze the energy associated with variations in surface profile, cavity depth, cavity width, and the horizontal position of the cavity tip. S. Sabah et al.¹⁵⁵⁾ estimated the amount of energy consumption by using an energy balance approach as displayed in Fig. 13. Calculations showed that dissipation and splashing are the dominant processes for which most of the power of the jet is used, and cavity formation consumes the least amount. This simplified approach provides an improved understanding of the gas injection process and may be used for developing models of the injection process of steelmaking. Besides, the energy transfer phenomena in top blowing¹⁵⁶⁾ and combined blowing¹⁵⁷⁾ converter were numerically investigated. For the top blown system, the efficiency of the energy transfer from the jets to the molten bath is very low. Decreasing lance height or increasing operation pressure promotes the efficiency of the momentum transfer from the jets to the molten bath but lowers the efficiency of the kinetic energy transfer.¹⁵⁶⁾ However, the energy transfer for the bottom blowing is much more efficient than that of the top blowing operations.¹⁵⁷⁾ And the kinetic energy transfer is closely related to the parameters like flowrate, radial position and the configuration of the bottom tuyeres but indistinctively sensitive to the viscosity of the upper slag or the foaming phases. These studies can provides us more fundamental insights into the multiphase flow behavior in the converter.

Fig. 13.

The control volume between the nozzle exit and the bath surface.¹⁵⁵⁾ (Online version in color.)

7. Conclusions and Outlook

This article has shown the significant progress achieved in regard to the multiphase flow behavior in the converter steelmaking process. It can be concluded that modeling plays an increasing role in augmenting traditional methods to achieve future advance in BOF practice with the aid of increasing development in computational power and simulation tools. Future contributions toward a generalized computational fluid dynamics model should include the full dimensional transient flow simulation, complex chemical reactions, time-varying foamy slag behavior, enhanced coupling with other phenomena such as the solid materials behavior, data - driven online quality prediction and control.

However, many challenges still need to be addressed. For example, even higher-performance supercomputers should be made accessible due to the really time-intensive modeling task. Numerical stability and big data management need to be enhanced for the establishment of a collaborated program. High-order accurate methods are still limited. Methods for massive data transfer, storage, extraction and visualization from large datasets need to be developed. Nonetheless, with sustained efforts and improvements in our knowledge base, we will eventually realize fundamentally generalized modeling of the converter steelmaking. It is an important part of a solid understanding of the fundamentals towards BOF and which will in turn facilitate the real-time data-driven precise process control of the converter steelmaking process. More profoundly, the fundamentally knowledge-based understanding of this process obtained from the modeling work could be an important component to the realization of intelligent steelmaking. Meanwhile, the implementation of intelligent steelmaking is one of the core sectors in the intelligent manufacturing for steel industry, which is pushed by the intelligent technology development nowadays, since several national strategies for intelligent manufacturing have been launched such as Industry internet (USA, 2009), Industry 4.0 (Germany, 2013), Made in China 2025 (China, 2015), Society 5.0 (Japan, 2016), etc. Therefore, it is envisaged that the intelligent manufacturing in steelmaking process will witness even more intense application of modeling in the coming year.

Acknowledgements

The financial support from Doctoral Fund of Ministry of Educations of China (No. 20120006110036) and Jiangxi Provincial Department of Science and Technology (20171 ACE50020) are highly acknowledged. The authors would like to give their thanks to Dr. Muxing Guo (KU Leuven), Prof. Bart Blanpain (KU Leuven), Dr. Dongsheng Liao (ArcelorMittal Dofasco Inc.), and Prof. Hongbiao Dong (University of Leicester) for their inspiring talking to this work. Lingling Cao and Yannan Wang also want to thank the support of the China Scholarship Council (CSC).

References

• 1)  World Steel Association: Steel Statistical Yearbook 2016, Worldsteel Committee on Economic Studies, Brussels, (2016), 15.
• 2)   Y.  Higuchi and  Y.  Tago: Tetsu-to-Hagané, 86 (2000), 654.
• 3)   R. I. L.  Guthrie: Metall. Mater. Trans. B, 35B (2004), 417.
• 4)   J.  Szekely,  J. W.  Evans and  J. K.  Brimacombe: The Mathematical and Physical Modeling of Primary Metals Processing Operations, John Wiley & Sons, Inc., New York, (1988), 261.
• 5)   R. B.  Banks and  D. V.  Chandrasekhara: J. Fluid Mech., 15 (1963), 13.
• 6)   N. A.  Molloy: J. Iron Steel Inst., 208 (1970), 943.
• 7)   J.  Szekely,  H. J.  Wang and  K. M.  Kiser: Metall. Trans. B, 7B (1976), 287.
• 8)   S. K.  Sharma,  J. W.  Hlinka and  D. W.  Kern: Iron Steelmaker, 4 (1977), 7.
• 9)   K.  Nakanishi,  Y.  Kato,  T.  Nozaki and  T.  Emi: Tetsu-to-Hagané, 66 (1980), 1307.
• 10)   K. I.  Naito,  Y. J.  Ogawa and  T.  Inomoto: ISIJ Int., 40 (2000), 23.
• 11)   Y.  Tago and  Y.  Higuchi: ISIJ Int., 43 (2003), 209.
• 12)   W.  Yang,  Y.  Ding,  M.  Wang,  H.  Shi,  T.  Wang and  L.  Zhu: Iron Steel, 39 (2004), 16.
• 13)   A.  Nordquist,  N.  Kumbhat,  L.  Jonsson and  P.  Jönsson: Steel Res. Int., 77 (2006), 82.
• 14)   H. Y.  Hwang and  G. A.  Irons: Metall. Mater. Trans. B, 43B (2012), 302.
• 15)   M.  Li,  Q.  Li,  S.  Kuang and  Z.  Zou: Metall. Mater. Trans. B, 47B (2016), 116.
• 16)   C.  Roth,  M.  Peter,  M.  Schindler and  K.  Koch: Steel Res. Int., 66 (1995), 325.
• 17)   M. J.  Luomala,  T. M. J.  Faritius and  J. J.  Harkki: ISIJ Int., 44 (2004), 809.
• 18)   S. K.  Choudhary and  S. K.  Ajmani: ISIJ Int., 46 (2006), 1171.
• 19)   V.  Singh,  J.  Kumar,  C.  Bhanu,  S. K.  Ajmani and  S. K.  Dash: ISIJ Int., 47 (2007), 1605.
• 20)   Z. Y.  Lai,  Z.  Xie and  L. C.  Zhong: ISIJ Int., 48 (2008), 793.
• 21)   K.  Nakatani,  F.  Ogasawara,  T.  Yamauchi,  G.  Okuyama and  Y.  Uchida: Iron Steel Technol., 10 (2013), 180.
• 22)   A. N.  Conejo,  S.  Kitamura,  N.  Maruoka and  S. J.  Kim: Metall. Mater. Trans. B, 44B (2013), 914.
• 23)   R. S.  Zhang,  L. G.  Chen and  Y. S.  Zhou: Cent. Iron Steel Res. Inst. Tech., 3 (1983), 14.
• 24)   S.  Paul and  D. N.  Ghosh: Metall. Trans. B, 17 (1986), 461.
• 25)   L.  Duan and  X. W.  Chen: J. Univ. Sci. Technol. Beijing, 12 (1990), 510.
• 26)   S. C.  Koria and  S.  Pal: Ironmaking Steelmaking, 17 (1990), 325.
• 27)   R. P.  Singh and  D. N.  Ghosh: Ironmaking Steelmaking, 17 (1990), 333.
• 28)   R. P.  Singh and  D. N.  Ghosh: ISIJ Int., 30 (1990), 955.
• 29)   M.  Martína,  M.  Renduelesb and  M.  Díaza: Chem. Eng. Sci., 60 (2005), 5781.
• 30)   L. Y.  Wen,  D. F.  Chen,  L. Y.  Dong,  C. G.  Bai,  Y. H.  Zhou,  M.  Liao and  J. N.  Zhang: J. Iron Steel Res., 18 (2006), 14.
• 31)   J.  Szekely: JOM, 42 (1990), 16.
• 32)   J.  Szekely and  S.  Asai: Metall. Trans., 5 (1974), 463.
• 33)   M.  Salcudean and  R. I. L.  Guthrie: Metall. Trans. B, 9 (1978), 181.
• 34)   Y. Z.  Li and  D. Y.  Li: J. Beijing Iron Steel Inst., 4 (1983), 48.
• 35)   Y. Z.  Li and  H. Z.  Guo: J. Beijing Iron Steel Inst., 4 (1984), 75.
• 36)   S. C.  Du,  J. Y.  Zhang and  S. K.  Wei: Acta Metall. Sin., 22 (1986), 237.
• 37)   B. K.  Li,  J. C.  He and  Z. W.  Lu: Acta Metall. Sin., 28 (1992), 475.
• 38)   P.  Gittler,  R.  Kickinger,  S.  Pirker,  E.  Fuhrmann,  J.  Lehner and  J.  Steins: Scand. J. Metall., 29 (2000), 166.
• 39)   H.  Laux,  S. T.  Johansen,  H.  Berg and  O. S.  Klevan: Scand. J. Metall., 29 (2000), 71.
• 40)   H. J.  Odenthal,  U.  Falkenreck and  J.  Schlüter: Proc. European Conf. on Computational Fluid Dynamics, TU Delft, Netherlands, (2006), 1.
• 41)   M.  Ersson,  A.  Tilliander,  L.  Jonsson and  P.  Jönsson: ISIJ Int., 48 (2008), 377.
• 42)   M.  Ersson,  L.  Höglund,  A.  Tilliander,  L.  Jonsson and  P.  Jönsson: ISIJ Int., 48 (2008), 147.
• 43)   M.  Asai,  H.  Nijo and  K.  Ito: ISIJ Int., 48 (2009), 178.
• 44)   W. J.  Wang,  Z. F.  Yuan,  H.  Matsuura,  H. X.  Zhao,  C.  Dai and  F.  Tsukihashi: ISIJ Int., 50 (2010), 491.
• 45)   Y.  Doh,  P.  Chapelle,  A.  Jardy,  G.  Djambazov,  K.  Pericleous,  G.  Ghazal and  P.  Gardin: Metall. Mater. Trans. B, 44B (2013), 653.
• 46)   Q.  Li,  M. M.  Li,  S. B.  Kuang and  Z. S.  Zou: Metall. Mater. Trans. B, 46B (2015), 1494.
• 47)   F. H.  Liu,  R.  Zhu,  Q. G.  Wang and  R. G.  Bai: ISIJ Int., 55 (2015), 1633.
• 48)   K. Y.  Chu,  H. H.  Chen,  P. H.  Lai,  H. C.  Wu,  Y. C.  Liu,  C. C.  Lin and  M. J.  Lu: Metall. Mater. Trans. B, 47B (2016), 948.
• 49)   R. Y.  Yin: ISIJ Int., 42 (2002), 1061.
• 50)   R. J.  Fruehan: Ironmaking Steelmaking, 32 (2005), 3.
• 51)   A.  McLean: Metall. Mater. Trans. B, 37B (2006), 319.
• 52)   G. H.  Li,  B.  Wang,  Q.  Liu,  X. Z.  Tian,  R.  Zhu,  L. N.  Hu and  G. G.  Cheng: Int. J. Min. Metall. Mater., 17 (2010), 715.
• 53)   K.  Chattopadhyay,  M.  Isac and  R. I. L.  Guthrie: Ironmaking Steelmaking, 37 (2010), 562.
• 54)   Z.  Wang,  Q.  Liu,  F. M.  Xie,  B.  Wang,  B.  Wang and  G. Q.  Fu: Ironmaking Steelmaking, 39 (2012), 228.
• 55)   Z.  Wang,  J.  Chang,  Q. P.  Ju,  F. M.  Xie,  B.  Wang,  H. W.  Li,  B.  Wang,  X. C.  Lu,  G. Q.  Fu and  Q.  Liu: ISIJ Int., 52 (2012), 1585.
• 56)   Q.  Liu,  B.  Wang,  Z.  Wang,  B.  Wang and  X. F.  Zhang: China Sci. Technol. Achiev., 11 (2014), 78.
• 57)   K.  Chattopadhyay: AIST Trans., 11 (2014), 277.
• 58)   T.  Emi: ISIJ Int., 55 (2015), 36.
• 59)   R. N.  Giere: Using Models to Represent Reality, Springer US, New York, (1999), 41.
• 60)   D.  Mazumdar and  J. W.  Evans: Modeling of Steelmaking Processes, CRC Press, Boca Raton, (2009), 100.
• 61)   G. H.  Geiger and  D. R.  Poirier: Transport Phenomena in Metallurgy, Addison-Wesley Publishing Co., Reading, MA, (1973), 76.
• 62)   F.  Kemeny,  D. J.  Harris,  A.  McLean,  T. R.  Meadowcroft and  J. D.  Young: Proc. 2nd Process Technology Conf., TMS, ISS-AIME, Warrendale, PA, (1981), 232.
• 63)   R. I. L.  Guthrie and  M.  Isac: Iron Steelmaker, 30 (2003), 27.
• 64)   Z. P.  Cai,  Y.  Liang and  C. X.  Zhang: Experimental Research and Testing Technology of Chemical Metallurgy Model, Metallurgical Industry Press, Beijing, (2001), 41.
• 65)   L. I.  Sedov: Similarity and Dimensional Methods in Mechanics, CRC Press, Boca Raton, (1993), 57.
• 66)   K.  Nakanishi,  K.  Saito,  T.  Nozaki,  Y.  Kato,  K. I.  Suzuki and  T.  Emi: Steelmaking Conf. Proc., Iron & Steel Society of AIME, US, (1982), 101.
• 67)   Q. L.  He and  N.  Standish: ISIJ Int., 30 (1990), 305.
• 68)   C. K.  Liu,  Y. M.  Yan and  X. W.  Li: Proc. 3rd Academic Conf. Metallurgical Process Dynamic Reaction Engineering, The Chinese Society for Metals, Beijing, (1986), 13.
• 69)   N.  Standish and  Q. L.  He: ISIJ Int., 29 (1989), 455.
• 70)   Q. L.  He and  N.  Standish: ISIJ Int., 30 (1990), 305.
• 71)   K. D.  Peaslee: Electric Furnace Conf. Proc., The Iron and Steel Society, London, (1993), 403.
• 72)   M.  Iguchi,  T.  Kondoh,  Z.-I.  Morita,  K.  Nakajima,  K.  Hanazaki,  T.  Uemura and  F.  Yamamoto: Metall. Mater. Trans. B, 26B (1995), 241.
• 73)   M.  Iguchi,  H.  Ueda,  T.  Chihara and  Z.-I.  Morita: Metall. Mater. Trans. B, 27B (1996), 765.
• 74)   M.  Iguchi,  T.  Kondoh and  K.  Nakajima: Metall. Mater. Trans. B, 28B (1997), 605.
• 75)   M.  Iguchi,  K.-I.  Nakamura and  R.  Tsujino: Metall. Mater. Trans. B, 29B (1998), 569.
• 76)   F.  Qian,  R.  Mutharasan and  B.  Farouk: Metall. Mater. Trans. B, 27B (1996), 911.
• 77)   M. J.  Luomala,  E. O.  Virtanen,  P. T.  Mure,  T. P.  Siivola,  T. M. J.  Fabritius and  J. J.  Härkki: Steel Res., 73 (2002), 9.
• 78)   M. J.  Luomala,  T. M. J.  Fabritius,  E. O.  Virtanen,  T. P.  Siivola and  J. J.  Härkki: ISIJ Int., 42 (2002), 944.
• 79)   M.  Ersson,  A.  Tilliander,  M.  Iguchi,  L.  Jonsson and  P.  Jönsson: ISIJ Int., 46 (2006), 1137.
• 80)   X. B.  Zhou,  M.  Ersson,  L. C.  Zhong and  P.  Jönsson: ISIJ Int., 54 (2014), 2255.
• 81)   X. B.  Zhou,  M.  Ersson,  L. C.  Zhong,  J.  Yu and  P.  Jönsson: Steel Res. Int., 85 (2014), 273.
• 82)   M.  Martína,  M.  Renduelesb and  M.  Díaza: Chem. Eng. Sci., 60 (2005), 5781.
• 83)   M. C.  Terrazas and  A. N.  Conejo: Proc. Int. Cong. on the Science and Technology of Steelmaking (ICS 2015), The Chinese Society for Metals, Beijing, (2015), 175.
• 84)   Q.  Li,  M. M.  Li and  Z. S.  Zou: Proc. Int. Cong. on the Science and Technology of Steelmaking (ICS 2015), The Chinese Society for Metals, Beijing, (2015), 187.
• 85)   S.  Sabah and  G.  Brooks: ISIJ Int., 54 (2014), 836.
• 86)   S.  Sabah and  G.  Brooks: Metall. Mater. Trans. B, 46B (2015), 863.
• 87)   S.  Sabah and  G.  Brooks: Ironmaking Steelmaking, 43 (2016), 473.
• 88)   G.  Brooks,  S.  Sabah,  B.  Rout and  Z.  Li: Proc. Iron & Steel Technology Conf. and Exposition (AISTech 2017), AIST, Warrendale, PA, (2017), 1253.
• 89)   K.  Chattopadhyay,  M.  Isac and  R. I. L.  Guthrie: ISIJ Int., 50 (2010), 331.
• 90)   J.  Szekely: Metall. Trans. B, 19 (1988), 525.
• 91)   J.  Blazek: Computational Fluid Dynamics: Principles and Applications, Butterworth-Heinemann, Oxford, UK, (2005), 1.
• 92)   J.  Laufer: On Turbulent Shear Flows of Variable Density, Rand Corp, Santa Monica, CA, (1968), 1.
• 93)   J. H.  Ferziger and  M.  Peric: Computational Methods for Fluid Dynamics, Springer Science & Business Media, Berlin, (2002), 309.
• 94)   Z. P.  Cai,  Y. S.  Xie and  A. W.  Xia: Iron Steel, 15 (1980), 14.
• 95)   S. V.  Patankar and  D. B.  Spalding: Int. J. Heat Mass Transf., 15 (1972), 1787.
• 96)   B. E.  Launder and  D. B.  Spalding: Mathematical Models of Turbulence, Academic Press, London, (1972), 27.
• 97)   D.  Mazumdar and  R. I. L.  Guthrie: Appl. Math. Model., 10 (1986), 25.
• 98)   M. P.  Schwarz and  W. J.  Turner: Appl. Math. Model., 12 (1988), 273.
• 99)   N.  Asahara,  K.  Naito,  I.  Kitagawa,  M.  Matsuo,  M.  Kumakura and  M.  Iwasaki: Steel Res. Int., 82 (2011), 587.
• 100)   O. C.  Zienkiewicz and  R. L.  Taylor: The Finite Element Method, McGraw-hill, London, (1977), 90.
• 101)   R. I. L.  Guthrie: ISIJ Int., 49 (2009), 1453.
• 102)   M.  Wolfshtein: Int. J. Heat Mass Transf., 12 (1969), 301.
• 103)   A. N.  Kolmogorov: Dokl. Akad. Nauk SSSR, 30 (1941), 299.
• 104)   L.  Prandtl: Festschr. zum, 60 (1945), 134.
• 105)   H. J.  Odenthal,  J.  Schlüter and  N.  Uebber: Proc. 7th European Oxygen Steelmaking Conf. (EOSC), Česká Hutnická Společnost, Czech Republic, (2014), 1.
• 106)   H. J.  Odenthal: Proc. Int. Cong. on the Science and Technology of Steelmaking (ICS 2015), The Chinese Society for Metals, Beijing, (2015), 183.
• 107)   A. V.  Nguyen and  G. M.  Evans: Appl. Math. Model., 30 (2006), 1472.
• 108)   H.  Jalkanen and  E.  Andersin: Proc. Modern Technologies for Making and Casting of Steel Conf., Varna, Bulgaria, (1989), 1.
• 109)   M.-L.  Suomi,  H.  Jalkanen,  E.  Andersin and  M.  Wallgren: Proc. 2nd Colloquium on Process Simulation (CONSIM), Helsinki University of Technology, Espoo, Finland, (1995), 289.
• 110)   H.  Jalkanen: Advanced Processing of Metals and Materials, TMS, Warrendale, PA, (2006), 541.
• 111)   C.  Crowe,  M.  Sommerfeld and  Y.  Tsuji: Multiphase Flows with Droplets and Particles, CRC Press, Boca Raton, (1998), 191.
• 112)   V.  Singh,  J.  Kumar,  C.  Bhanu,  S. K.  Ajmani and  S. K.  Dash: ISIJ Int., 47 (2007), 1605.
• 113)   J. O.  Andersson,  T.  Helander,  L.  Höglund,  P. F.  Shi and  S.  Bo: Calphad, 26 (2002), 273.
• 114)   X. B.  Zhou,  M.  Ersson,  L. C.  Zhong and  P.  Jönsson: Steel Res. Int., 86 (2015), 1328.
• 115)  AVL FIRE Inc.: CFD Solver v8.5 Manual, AVL Fire, Graz, Austria, (2006), 1.
• 116)   M.  Alam,  J.  Naser and  G.  Brooks: Metall. Mater. Trans. B, 41B (2010), 636.
• 117)   D. C.  Wilcox: AIAA J., 30 (1992), 2639.
• 118)   T. H.  Shih,  W. W.  Liou,  A.  Shabbir,  Z.  Yang and  J.  Zhu: Comput. Fluids, 24 (1995), 227.
• 119)   Y.  Li,  W. T.  Lou and  M. Y.  Zhu: Ironmaking Steelmaking, 40 (2013), 505.
• 120)   S. T.  Johansen,  J.  Wu and  W.  Shyy: Int. J. Heat Fluid Flow, 25 (2004), 10.
• 121)   N.  Croft,  K.  Pericleous and  M.  Cross: Numerical Methods in Laminar and Turbulent Flows, Vol. 9, ed. by C. Tayler and P. Durbetaki, Pineridge Press, Swansea, UK, (1995), 1269.
• 122)   Q.  Li,  M. M.  Li,  S. B.  Kuang and  Z. S.  Zou: Can. Metall. Q., 53 (2014), 340.
• 123)   L. L.  Cao,  Q.  Liu,  Z.  Wang and  N.  Li: Ironmaking Steelmaking, (2016), DOI: 10.1080/03019233.2016.1255373.
• 124)   L. K.  Lin,  Z. H.  Yang,  S. Y.  Hsu,  J. S.  Wang,  Y. C.  Liu,  Y. L.  Huang and  C. H.  Tsai: Proc. 7th Int. Conf. on Modelling and Simulation of Metallurgical Processes in Steelmaking (STEELSIM2017), The Chinese Society for Metals, Beijing, (2017), 204.
• 125)   A.  Chatterjee: Iron Steel, 45 (1972), 627.
• 126)   F. L.  Wu: Energy Metall. Ind., 8 (1989), 47.
• 127)   K.  Liu: Ph.D. thesis, Northestern University, (2008), http://cdmd.cnki.com.cn/Article/CDMD-10145-2010259808.htm, (accessed 2008-02-25).
• 128)   I.  Sumi,  G.  Okuyama,  S.  Nabeshima,  H.  Matsuno and  Y.  Kishimoto: ISIJ Int., 47 (2007), 73.
• 129)   Z.  Song,  M.  Ersson and  P. G.  Jönsson: ISIJ Int., 51 (2011), 1637.
• 130)   I.  Sumi,  Y.  Kishimoto,  Y.  Kikuchi and  H.  Igarashi: ISIJ Int., 46 (2006), 1312.
• 131)   R.  Sambasivam and  F.  Durst: J. Heat Transf., 132 (2010), 031701.
• 132)   R.  Sambasivam,  S. N.  Lenka,  F.  Durst,  M.  Bock,  S.  Chandra and  S. K.  Ajmani: Metall. Mater. Trans. B, 38B (2007), 45.
• 133)   A. R.  Naji Meidani,  M.  Isac,  A.  Richardson,  A.  Cameron and  R. I. L.  Guthrie: ISIJ Int., 44 (2004), 1639.
• 134)   R.  Sambasivam and  F.  Durst: Ironmaking Steelmaking, 37 (2010), 195.
• 135)   Z. Q.  Xiao: Studies on Unit Processes and Unit Phenomena in Metallurgy, Metallurgical Industry Press, Beijing, (2006), 17.
• 136)   H. W.  Meyer,  W. F.  Porter,  G. C.  Smith and  J.  Szekely: JOM, 20 (1968), 35.
• 137)   O.  Olivares,  A.  Elias,  R.  Sánchez,  M.  Díaz-Cruz and  R. D.  Morales: Steel Res., 73 (2002), 44.
• 138)   F.  Memoli,  C.  Mapelli,  P.  Ravanelli and  M.  Corbella: ISIJ Int., 44 (2004), 1342.
• 139)   Y.  Kishimoto and  N.  Saito: Tetsu-to-Hagané, 100 (2014), 445.
• 140)   V.  Singh,  S. N.  Lenka,  S. K.  Ajmani,  C.  Bhanu and  S.  Pathak: ISIJ Int., 49 (2009), 1889.
• 141)   C.  Roth,  M.  Peter,  M.  Juhart and  K.  Koch: Steel Res., 70 (1999), 502.
• 142)   T. M. J.  Fabritius,  M. J.  Luomala,  E. O.  Virtanen,  H.  Tenkku,  T. L. J.  Fabritius,  T. P.  Siivola and  J. J.  Härkki: ISIJ Int., 42 (2002), 861.
• 143)   M. A.  Ehteram,  H. B.  Tabrizi,  G.  Ahmadi,  M.  Safari and  M. A.  Mirsalim: J. Aerosol Sci., 65 (2013), 49.
• 144)   R. C.  Urquhart and  W. G.  Davenport: Can. Metall. Q., 12 (1973), 507.
• 145)   K. D.  Peaslee and  D. G.  Robertson: Extraction & Processing Division Cong. (EPD) Proc., TMS, Warrendale, PA, (1994), 1129.
• 146)   M.  Iguchi: Tetsu-to-Hagané, 103 (2017), 119.
• 147)   S. C.  Koria and  K. W.  Lange: Metall. Trans. B, 15 (1984), 109.
• 148)   Subagyo,  G. A. Brooks, K. S. Coley and G. A. Irons: ISIJ Int., 43 (2003), 983.
• 149)   B. K.  Rout,  G.  Brooks,  Subagyo,  M. A. Rhamdhani and Z. Li: Metall. Mater. Trans. B, 47 (2016), 3350.
• 150)   G.  Turner and  S.  Jahanshahi: Trans. Iron Steel Inst. Jpn., 27 (1987), 734.
• 151)   T.  Fabritius,  P.  Mure,  E.  Virtanen,  P.  Hannula,  M.  Luomala and  J.  Härkki: Ironmaking Steelmaking, 29 (2002), 29.
• 152)   B. T.  Maia,  R. K.  Imagawa and  R. P.  Turner: Proc. Iron & Steel Technology Conf. and Exposition (AISTech 2016), AIST, Warrendale, PA, (2016), 1083.
• 153)   Y.  Sundberg: Scand. J. Metall., 7 (1978), 81.
• 154)   L. L.  Cao,  Z.  Wang,  Q.  Liu,  X. C.  Lu and  F. M.  Xie: Proc. Int. Cong. on the Science and Technology of Steelmaking (ICS 2015), The Chinese Society for Metals, Beijing, (2015), 171.
• 155)   S.  Sabah and  G.  Brooks: Metall. Mater. Trans. B, 47B (2016), 458.
• 156)   M. M.  Li,  Q.  Li,  S.  Kuang and  Z. S.  Zou: Steel Res. Int., 87 (2016), 288.
• 157)   X. B.  Zhou,  M.  Ersson,  L. C.  Zhong and  P.  Jönsson: Metall. Mater. Trans. B, 47B (2016), 434.