Large Eddy Simulation of Multi-Phase Flow and Slag Entrapment in Molds with Different Widths

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Article Large Eddy Simulation of Multi-Phase Flow and Slag Entrapment in Molds with Different Widths

Xianjiu Zhao 1,2,3, Xianglong Li 4,* and Jieyu Zhang 1,2,*

1 School of Materials Science and Engineering, State Key Laboratory of Advanced Special Steel and Shanghai Key Laboratory of Advanced Ferrometallurgy, Shanghai University, Shanghai 200444, China; [email protected] 2 Shanghai University New Materials (TaiZhou) Research Institute Co., Ltd., Taizhou 225500, China 3 Research Institute, Baoshan Iron&Steel Co., Ltd., Shanghai 200941, China 4 School of Iron and Steel, Soochow University, Suzhou 215137, China * Correspondence: [email protected] (X.L.); [email protected] (J.Z.); Tel.: +86-021-5633-7920 (J.Z.)

Abstract: Slag entrapment is a critical problem that affects the quality of steel. In this work, a three- dimensional model is established to simulate the slag entrapment phenomenon, mainly focusing on the slag entrapment phenomenon at the interface between slag and steel in molds with different widths. The large eddy simulation (LES) model and discrete particle model (DPM) are used to simulate the movements of bubbles. The interactions between phases involve two-way coupling. The accuracy of our mathematical model is validated by comparing slag–metal interface fluctuations with practical measurements. The results reveal that the average interface velocity and transverse velocity decrease as the mold width increases, however, they cannot represent the severity of slag entrapment at the interface between slag and steel. Due to the influence of bubble motion behavior, the maximum interface velocity increases with mold width and causes slag entrapment readily, which can reflect the severity of slag entrapment. On this basis, by monitoring the change of impact depths in different molds, a new dimensionless number “C” is found to reveal the severity of slag entrapment at the interface between slag and steel. The results show that the criterion number C increases with mold

Citation: Zhao, X.; Li, X.; Zhang, J. width, which is consistent with the results of ﬂaw detection. Therefore, criterion number C can be Large Eddy Simulation of used to reﬂect the severity of slag entrapment in different molds. Multi-Phase Flow and Slag Entrapment in Molds with Different Keywords: continuous casting; slag entrapment; mold width; large eddy simulation Widths. Metals 2021, 11, 374. https:// doi.org/10.3390/met11020374

Received: 15 January 2021 1. Introduction Accepted: 18 February 2021 The mold is an important part of steel metallurgy and can be called the “heart” of a Published: 23 February 2021 continuous casting machine. During this process, the effect of powder added into the mold can be summarized by three aspects: (1) preventing oxidation of molten steel; (2) absorbing Publisher’s Note: MDPI stays neutral non-metallic inclusions; (3) filling the gap between the mold and the slab surface to improve with regard to jurisdictional claims in heat transfer and lubricating the mold surface. However, due to the instability of level published maps and institutional affil- iations. fluctuation in the mold, the liquid slag is often incorporated into the molten steel, damaging mechanical properties of steel products, especially for automobile steel. Therefore, slag entrapment has become a critical problem that requires serious concern. Due to “black box” operation, the slag entrapment cannot be seen during the real casting process. Therefore, visual models have been developed to reproduce the slag Copyright: © 2021 by the authors. entrapment at the interface between slag and steel. Among these studies, Gguta et al. [1] Licensee MDPI, Basel, Switzerland. found an apparent asymmetric flow near the nozzle. Afterwards, Li et al. [2] captured the This article is an open access article asymmetric vortex distribution of the steel/slag interface by injecting black sesames into distributed under the terms and conditions of the Creative Commons water, and developed a mathematical model to predict the asymmetry flow in the mold. Attribution (CC BY) license (https:// The study of Iguchi et al. [3] showed that the slag entrapment caused by shearing strength creativecommons.org/licenses/by/ is the main mechanism for the slag entrapment during high casting speed operation, and 4.0/). that the interfacial tension has a significant influence on it. Watanabe and Yamashita

Metals 2021, 11, 374. https://doi.org/10.3390/met11020374 https://www.mdpi.com/journal/metals Metals 2021, 11, 374 2 of 13

et al. [4,5] studied slag entrapment by argon blowing, finding that the maximum depth of slag involved would not exceed three times its diameter. In addition, according to Savolainen et al. [6], the effect of the viscosity of slag on the formation and size of slag droplets should be paid attention to take full control of it. Yamada et al. [7] suggested that argon bubbles in a mold become the desirable sites where alumina inclusions are gathered and form large alumina clusters. Despite these studies, some researchers focused on the influence of liquid properties on slag entrapment in the mold, such as liquid metal density and interfacial tension within the slag–metal interface [8–10], in which the critical slag entrapment velocity can be determined. However, this velocity can only predict the slag entrapment caused by shearing stress. Lei Hong et al. [11] proposed another theoretical equation for calculating shear entrapment, taking the viscosity of slag into account. However, Chung and Cramb [12] and others believe that the interfacial tension coefficient should be reduced to about three percent of the original value due to the existence of interfacial reactions. Harman [13] considered nine factors and obtained another formula for calculating critical velocity through non-linear fitting. The above studies are of great significance for understanding the slag entrapment phenomenon. However, the physical models cannot meet all the similarity criteria at the same time, which leads to some limitations for slag entrapment results. With the rapid development of computer science, computational fluid dynamics (CFD) technology has been an important tool in metallurgical process research, and its advantages are increasingly prominent. Saeedipour et al. [14] established a three-phase mathematical model to study the interface wave problem. Liu et al. [15] established a quasi-four-phase model to study the effect of bubbles on the interface fluctuation of the slag–metal interface. Li et al. [16] analyzed three kinds of slag entrapment mechanisms, and gave the transient process of mold slag entrapment in molten steel. Although these studies are of great significance to the study of slag entrapment, the relationship between mold impact depth and velocity, especially the association with slag entrapment, was not explored. There is still a lack of effective evaluation criteria to predict the effect of slag entrapment. In this work, the movement behavior of mold slag in the mold is studied to reveal the influence of mold structure on slag entrapment and a theoretical foundation for mold width adjustment is laid. The innovations of this paper are composed of three parts: Firstly, the slag entrapment between different molds are elaborated in detail. (2) Secondly, a mathematical model of four-phase (slag–metal–gas–air) flow is established to explain differences in slag entrapment in (1). Thirdly, a new dimensionless value is established to characterize the severity of slag entrapment in molds. The research results can act as a guide for the continuous casting process.

2. Mathematical Model 2.1. Basic Assumptions To simplify the calculation, the mathematical model used in this work is based on the following assumptions: (i) Liquid steel is regarded as a Newtonian fluid, and its basic parameters, such as density and viscosity, are considered as constants. (ii) The heat transfer and solidification process between molten steel and cooling water are not considered, and the thermal characteristics of slag are ignored. (iii) The discrete phase bubble is assumed to be spherical, and its size change is ignored in the process of floating. (iv) The taper of the mold, as well as heat transfer between the mold and slab, are all ignored.

2.2. Governing Equations There are four phases existing in the mold: molten steel, liquid slag, air, and argon. In this work, we use the volume of ﬂuid (VOF) method to describe interactions between steel, slag, and air and adopt a discrete particle model (DPM) to track trajectories of argon Metals 2021, 11, 374 3 of 13

bubbles. The interactions between these phases involve two-way coupling, following the law of Newton. The continuity equation for these phases can be written as follows:

∂(α ρ ) – k k + ∇ · α ρ u = 0 (1) ∂t k k m – where ρk is the density of the phase, the foot mark k representing the phases um is the velocity of the mixture phase, αk is the volume fraction of each phase, which fulfills the equation αl + αs + αg = 1. Large eddy simulation (LES) is used to solve the Navier–Stokes (N-S) equations of fluid flow: – ∂ ρmum – – h – –T i 6 + ∇ · = −∇ + ∇ · (∇ + ∇ ) − ∇ + + · + ρmumum P µeffect um um τij ρmg 3 F Fγ (2) ∂t πdp

In this equation, ﬂuid motion is solved by a set of equations. The term ρm represents the density of the mixture phase, P is static pressure. The molecular viscosity µm and turbulent viscosity µt in the equation are all weighted average values of the volume fraction of each phase, and the effective viscosity µeffect = µm + µt. F is the forces acting on bubbles. The term Fγ is the interface tension between these phases. The expression of subgrid-scale stress τij is as follows:

1 τ = τ δ − 2µ S (3) ij 3 kk ij t ij

where the term τkk is the isotropic part of the subgrid scale, δij is the Kronecker symbol, Sij 2 is the strain rate, turbulent viscosity µt = ρmLs S . The calculation formula of mixing length is as follows:

Ls = min(κd, ∆Cs) (4)

where κ is the von Karman constant, taken as 0.4, d is the vertical distance from the ﬂuid to the wall, Cs is the Smagorinsky constant, taken as 0.2.

2.3. Discrete Particle Model (DPM) The argon bubbles injected from the nozzle easily ﬂoat up and escape from the upper surface of the mold. In this process, the momentum exchanges between the argon and steel are treated as two-way coupling, which obeys the second law of Newton:

– dup m = F (5) p dt

where up and mp are the velocity and mass of particles, respectively, and F is the resultant force acting on the bubbles, the expression of which can be written as follows:

F = Fg + Fb + Fp + Fd + Fl + Fv−m (6)

where the terms on the right side of the formula are gravity, buoyancy force, pressure gradient force, drag force, lift force, and virtual mass force, respectively. The calculation equations [17–20] can be found in Table1. The bubble size ranges from 0.5–15 mm, following the Rosin–Rammler law:

n −(d/dm) Yd = e (7)

where the variable Yd is the mass fraction of bubbles whose diameters are greater than d. The averaged bubble diameter dm = 5 mm, and the spread parameter n = 2. Metals 2021, 11, 374 4 of 13

Table 1. Forces acting on argon bubbles.

Source Term Formula Annotations The net effect acts on the difference between particle Buoyancy force plus gravity (ρ −ρ )πd3 p m p g and fluid densities. The variable g is gravity force, Fb + Fg 6 acceleration, dp is particle diameter. Where drag coefficient 1 2 24 0.687 |um−up|dp Drag force Fd πd ρCD um − up um − up C = 1 + 0.15Re , Re = ,and u 8 p D Rep p p up p is particle velocity, Rep is particle Reynolds number. Pressure gradient force Fp ρm/ρp up∇um Pressure gradient force is significant when ρ/ρp ≥ 0.1. Where variable G = dum , √ dz |G|up ε = sgn(G) ,Us = u − u 1/2 um−up,l m,l p,l 9 2 h |G| i Lift force Fl − µ d Ussgn(G) J(ε) ε 4π m p up J(ε) = 0.6765 1 + tanh 2.5 log10 +0.191 · (0.667 + tanh(6ε − 1.92))um,1 is stream-wise liquid velocity. 3 ρpπ d du Virtual mass force Fv−m p dum p Where the virtual mass force coefficient Cv−m = 0.5. Cv−m 12 dt − dt

2.4. Boundary Conditions and Numerical Details As shown in Figure1, the whole physical model consists of four parts: nozzle, mold, foot roller zone, and secondary cooling zone. In the production process, the thicknesses of argon and the slag layer are 40 mm and 50 mm, respectively. After entering the mold, the argon gas expands rapidly, resulting in a decrease in the density of argon gas. The bubble density in the molten steel can be calculated through the ideal gas law, as shown in Equation (8). The steel velocity from the nozzle is calculated by the casting speed, and a no-slip condition is used to perform the wall treatment of the mold. In this work, three typical widths of the mold were taken to study the slag entrapment inside the mold, and the cross section are 250 mm × 1100 mm, 250 mm × 1400 mm, 250 mm × 1650 mm, respectively.

P = ρPRgT (8)

where the argon density at 20 ◦C is 1.78 kg/m3. The pressure P is standard atmospheric pressure, which can be assumed to be equal to standard atmosphere pressure near the top of the mold. Based on this equation, the bubble density in the molten steel (1556 ◦C) is 0.266 kg/m3. Metals 2021, 10, x FOR PEER REVIEW 5 of 13

FigureFigure 1. 1. Three-dimensionalThree-dimensional model model and and boundary boundary conditions. conditions. (a) Submerged-entry (a) Submerged-entry nozzle nozzle and (b and) (b) continuouscontinuous casting casting slab. slab. In order to obtain a fine vortex structure, the mesh is refined near the slag–metal interface and gas–slag interface. The whole region contains 2.1 × 106 structural grids with good independence. The calculation step is set to 0.01 s and the total calculation time is 100 s. To save calculation time and cost, the k-ε model is used to calculate the steady-state field, and then switched to the LES model to simulate the transient flow field. The specific model parameters are shown in Table 2.

Table 2. Basic model parameters.

Parameters Data Casting speed (m·min−1) 1.10, 0.86, 0.73 Slab size/mm × mm 1100 × 250, 1400 × 250, 1650 × 250 Argon blowing rate (L·min−1) 49.2 (1800 K) Submerged depth of nozzle (mm) 166.5 Mold height (mm) 900 Air layer thickness (mm) 50 Thickness of slag layer (mm) 40 Nozzle angle (°) 15°, downward Dynamic viscosity of molten steel, slag, and air (kg·m−1·s−1) 0.0051, 0.39, 1.7894 × 10−5 Density of molten steel, slag, and argon (1809.15 K) (kg·m−3) 7100, 3000, 0.266 Interfacial tension between slag and liquid metal (N·m−1) 1.2 Interfacial tension between slag and air (N·m−1) 0.31

The movements of argon bubbles consist three key stages, as shown in Figure 2: (1) Entering the mold from the nozzle and being carried downward by the nozzle jet; (2) floating up into the molten steel and passing through the slag–metal interface and entering into the slag; (3) collapsing and subsequently disappearing after floating near the gas/slag interface. The movements of bubbles in the continuous phases and the escape near the gas/slag interface are achieved by coding user-defined functions (UDFs). In addition, the impact point is defined as the position where the vertical shear force equals zero on the center plane of the mold.

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In order to obtain a fine vortex structure, the mesh is refined near the slag–metal interface and gas–slag interface. The whole region contains 2.1 × 106 structural grids with good independence. The calculation step is set to 0.01 s and the total calculation time is 100 s. To save calculation time and cost, the k-ε model is used to calculate the steady-state field, and then switched to the LES model to simulate the transient flow field. The specific model parameters are shown in Table2.

Table 2. Basic model parameters.

Parameters Data Casting speed (m·min−1) 1.10, 0.86, 0.73 Slab size/mm × mm 1100 × 250, 1400 × 250, 1650 × 250 Argon blowing rate (L·min−1) 49.2 (1800 K) Submerged depth of nozzle (mm) 166.5 Mold height (mm) 900 Air layer thickness (mm) 50 Thickness of slag layer (mm) 40 Nozzle angle (◦) 15◦, downward Dynamic viscosity of molten steel, slag, and air (kg·m−1·s−1) 0.0051, 0.39, 1.7894 × 10−5 Density of molten steel, slag, and argon (1809.15 K) (kg·m−3) 7100, 3000, 0.266 Interfacial tension between slag and liquid metal (N·m−1) 1.2 Interfacial tension between slag and air (N·m−1) 0.31

The movements of argon bubbles consist three key stages, as shown in Figure2: (1) Entering the mold from the nozzle and being carried downward by the nozzle jet; (2) floating up into the molten steel and passing through the slag–metal interface and entering into the slag; (3) collapsing and subsequently disappearing after floating near the gas/slag interface. The movements of bubbles in the continuous phases and the escape near the gas/slag interface are achieved by coding user-defined functions (UDFs). In addition, the impact point is defined as the position where the vertical shear force equals zero on the Metals 2021, 10, x FOR PEER REVIEWcenter plane of the mold. 6 of 13

Figure 2. Motion of bubbles in continuous phases. 3. Analysis of Inclusions on Slab 3. Analysis of Inclusions on Slab Figure3 shows the surface defects after the rolling process, which is caused by the large Figure 3 shows the surface defects after the rolling process, which is caused by the inclusions. It is clearly seen that many black lines on the surface of the slabs are stretching large inclusions. It is clearly seen that many black lines on the surface of the slabs are across the whole surface of the slab, the lengths of which are more than 90 millimeters. stretching across the whole surface of the slab, the lengths of which are more than 90 This can degrade the quality of steel signiﬁcantly. The “black line” problem is a signiﬁcant problemmillimeters. that This constrains can degrade the production the quality of automobileof steel significantly. steel. The “black line” problem is a significant problem that constrains the production of automobile steel.

Figure 3. “Black line” defects of hot-rolled sheets of automobile steel with: (a) 1100 mm width, (b) 1650 mm width.

Figure 4 shows the compositions of inclusions inside the black line characterized by scanning electron microscopy (SEM) and energy dispersive spectroscopy (EDS). The results show that Na, Al, Si, Ca, O, and other elements exist in the black lines, whose compositions are similar to that of the liquid slag in the mold. Therefore, it is certain that the black line defect of hot-rolled sheets is caused by slag entrapment.

(a)

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Figure 2. Motion of bubbles in continuous phases. Figure 2. Motion of bubbles in continuous phases.

3. Analysis of Inclusions on Slab Figure 3 shows the surface defects after the rolling process, which is caused by the large inclusions. It is clearly seen that many blackblack lineslines onon thethe surfacesurface ofof thethe slabsslabs areare stretching across the whole surface of the slab, the lengths of which are more than 90 Metals 2021, 11, 374 6 of 13 millimeters. This can degrade the quality of steel significantly. The “black line” problem is a significant problem that constrains the production of automobile steel.

Figure 3. “Black line” defects of hot-rolled sheets of automobile steel with: (a) 1100 mm width, (b) 1650 mm width. Figure 3. “Black“Black line” defects of hot-rolled sheets of automobile steel with: ( a) 1100 mm width, (b) 1650 mm width. Figure4 shows the compositions of inclusions inside the black line characterized by Figure 4 shows the compositions of inclusions inside the black line characterized by scanning electron microscopy (SEM) and energy dispersive spectroscopy (EDS). The results scanning electron microscopy (SEM) and energy dispersive spectroscopy (EDS). The re- show that Na, Al, Si, Ca, O, and other elements exist in the black lines, whose compositions sults show that Na, Al, Si, Ca, O, and other elements exist in the black lines, whose are similar to that of the liquid slag in the mold. Therefore, it is certain that the black line compositions are similar to that of the liquid slag in the mold. Therefore, it is certain that defect of hot-rolled sheets is caused by slag entrapment. the black line defect of hot-rolled sheets is caused by slag entrapment.

(a)

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(b)

FigureFigure 4.4. MorphologyMorphology andand compositionscompositions ofof “black“black lines”lines” inin hot-rolledhot-rolled sheets:sheets: ((aa)) PositionPosition AA inin FigureFigure3 a;3a; ( (bb)) positionposition BB inin FigureFigure3 b.3b.

FigureFigure5 5 shows shows thethe slag entrapment ratio ratio in in slabs slabs with with different different widths. widths. The The slabs slabs are aretested tested after after the therolling rolling process. process. The Theratio ratio is defined is deﬁned as the as percentage the percentage of slabs of slabs with with slag slagentrapment entrapment problems problems against against totally totally tested tested slabs. slabs. It can It canbe seen be seen from from Figure Figure 5 that5 that the thelarger larger the thewidth width is, is,the the higher higher the the slag slag entr entrapmentapment ratio ratio will will be. be. There There must bebe somesome specialspecial reasonsreasons behindbehind this this phenomenon, phenomenon, however, however, few few works works have have reported reported this this problem prob- andlem tendency.and tendency. So, in So, this in work, this work, the reason the reason for this for phenomenon this phenomenon will be will explained be explained in detail, in throughdetail, through our mathematical our mathematical model. model.

4.0 Grade 1：less than 1050mm； Grade 2：1051mm~1250mm, 3.5 Grade 3：1251mm~1450mm；Grade 4：1451mm~1650mm, Grade 5：more than 1651mm. 3.0 Grade 5 Grade 4 2.5 Grade 3 Grade 2 2.0 Grade 1

1.5

1.0 Slag entrapment Ratio (%) Ratio Slag entrapment

0.5

0.0 Slab width range (mm) Figure 5. The slag entrapment ratio of slabs with different widths.

4. Results and Discussion 4.1. Level Height of Slag–Metal Interface Firstly, the accuracy of the mathematical model is verified through interface-level detections obtained from an eddy current sensor. The current sensor is installed on the top of the mold (Figure 6a), which can reveal the distance between the mold top and slag–metal interface. The installation position is in the middle of the mold, and the distance between the two probes is 0.4 m, as shown in Figure 6a. It can be seen from Figure 6b that the actual distance between the mold top and slag–metal interface is 83–93 mm. By comparison, the simulation results show that the value is 97–100 mm. It is easily seen that the simulated result is a little higher than the experimental result, the reason for which is the fact that the height of slag–metal interface is enhanced by bulging. Generally, the simulated results agree well with the experimental results, implying that the mathematical model established in this work is reliable (Figure 6b).

Metals 2021, 10, x FOR PEER REVIEW 7 of 13

(b)

Figure 4. Morphology and compositions of “black lines” in hot-rolled sheets: (a) Position A in Figure 3a; (b) position B in Figure 3b.

Figure 5 shows the slag entrapment ratio in slabs with different widths. The slabs are tested after the rolling process. The ratio is defined as the percentage of slabs with slag entrapment problems against totally tested slabs. It can be seen from Figure 5 that the larger the width is, the higher the slag entrapment ratio will be. There must be some special reasons behind this phenomenon, however, few works have reported this prob- Metals 2021, 11, 374 7 of 13 lem and tendency. So, in this work, the reason for this phenomenon will be explained in detail, through our mathematical model.

4.0 Grade 1：less than 1050mm； Grade 2：1051mm~1250mm, 3.5 Grade 3：1251mm~1450mm；Grade 4：1451mm~1650mm, Grade 5：more than 1651mm. 3.0 Grade 5 Grade 4 2.5 Grade 3 Grade 2 2.0 Grade 1

1.5

1.0 Slag entrapment Ratio (%) Ratio Slag entrapment

0.5

0.0 Slab width range (mm) FigureFigure 5. 5. TheThe slag slag entrapment entrapment ratio ratio of of slabs slabs with with different different widths. widths.

4.4. Results Results and and Discussion Discussion 4.1. Level Height of Slag–Metal Interface 4.1. Level Height of Slag–Metal Interface Firstly, the accuracy of the mathematical model is veriﬁed through interface-level Firstly, the accuracy of the mathematical model is verified through interface-level detections obtained from an eddy current sensor. The current sensor is installed on the top detections obtained from an eddy current sensor. The current sensor is installed on the of the mold (Figure6a), which can reveal the distance between the mold top and slag–metal top of the mold (Figure 6a), which can reveal the distance between the mold top and interface. The installation position is in the middle of the mold, and the distance between slag–metal interface. The installation position is in the middle of the mold, and the dis- the two probes is 0.4 m, as shown in Figure6a. It can be seen from Figure6b that the actual tance between the two probes is 0.4 m, as shown in Figure 6a. It can be seen from Figure distance between the mold top and slag–metal interface is 83–93 mm. By comparison, the 6b that the actual distance between the mold top and slag–metal interface is 83–93 mm. simulation results show that the value is 97–100 mm. It is easily seen that the simulated By comparison, the simulation results show that the value is 97–100 mm. It is easily seen result is a little higher than the experimental result, the reason for which is the fact that that the simulated result is a little higher than the experimental result, the reason for the height of slag–metal interface is enhanced by bulging. Generally, the simulated results which is the fact that the height of slag–metal interface is enhanced by bulging. Generally, agree well with the experimental results, implying that the mathematical model established Metals 2021, 10, x FOR PEER REVIEWthein thissimulated work is results reliable agree (Figure well6b). with the experimental results, implying that the math-8 of 13 ematical model established in this work is reliable (Figure 6b).

FigureFigure 6. Schematic6. Distance diagram from mold for top (a) theto the application slag–metal of currentinterface sensor obtained and through (b) comparison experimental of distance and simulation from mold analysis. top to the slag-metal interface through simulation and industrial measurements. 4.2. Transient Flow Spectrum Analysis of Mold Figure 7 shows the transient molten steel flow patterns of different cross sections. It can be seen that the flow field in the mold is asymmetric and unstable, with multiple vortices of different scales.

Figure 7. Transient flow of molten steel in slabs with different cross sections of (a) 1100 mm × 250 mm; (b) 1400 mm × 250 mm; (c) 1650 mm × 250 mm.

Generally, the liquid steel that flows out of the nozzle and impinges on the narrow surface of mold may be divided into two streams: upper recirculation flow and lower recirculation flow. Obviously, increasing the mold width leads to smaller impacting velocity and deeper injection of liquid steel. Furthermore, the maximum velocity of the slag–metal interface also increases, as shown in Figure 8.

Figure 8. Velocity of slag–metal interfaces in molds with different cross sections of: (a) 1100 mm × 250 mm; (b) 1400 mm × 250 mm; (c) 1650 mm × 250 mm.

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FigureFigure 6. 6. Distance Distance from from mold mold top top to to the the slag–metal slag–metal interface interface ob obtainedtained through through experimental experimental and and simulation simulation analysis. analysis.

4.2.4.2.4.2. Transient TransientTransient Flow Flow Flow Spectrum Spectrum Spectrum Analysis Analysis Analysis of of Mold Mold FigureFigureFigure 7 7shows shows the the the transient transienttransient molten moltenmolten steel steelsteel fl flowflowow patterns patterns of of of different different different cross cross cross sections. sections. sections. It It canItcan can be be beseen seen seen that that that the the the flow flow flow field field field in in the the mold moldmold is isis asymmetric asymmetricasymmetric and andand unstable, unstable,unstable, with withwith multiple multiple vorticesvorticesvortices of of of different different different scales. scales. scales.

FigureFigureFigure 7. 7.7. TransientTransient Transient flow ﬂowflow of ofof molten moltenmolten steel steelsteel in inin slabs slabsslabs with withwith different differentdifferent cross crosscross sections sectionssections of of ( a ()a )1100 1100 mm mm × ×250 250250 mm;mm;mm; ( (bb(b)) )1400 1400 1400 mm mm mm ×× ×250 250 mm; mm; (c ()c )1650 1650 mm mmmm × ×250 250250 mm. mm. mm.

Generally,Generally,Generally, the thethe liquid liquidliquid steel steelsteel that thatthat flows flowsflows out outout of ofof the the the nozzle nozzle nozzle and and and impinges impinges impinges on onon the thethe narrow narrow narrow surfacesurfacesurface of ofof mold moldmold may maymay be bebe divided divideddivided into intointo two twotwo st streams: streams:reams: upper upperupper recirculation recirculationrecirculation flow flowflow and andand lower lowerlower recirculationrecirculationrecirculation flow.flow. flow. Obviously,Obviously, Obviously, increasing increasing increasing the the the mold mold mold width width width leads leads leads to smallerto to smaller smaller impacting impacting impacting velocity ve- ve- locityandlocity deeper and and deeper injectiondeeper injection injection of liquid of of steel. liquid liquid Furthermore, steel. steel. Furthermore, Furthermore, the maximum the the maximum maximum velocity of velocity thevelocity slag–metal of of the the slag–metalinterfaceslag–metal also interface interface increases, also also asincreases, increases, shown in as as Figure shown shown8. in in Figure Figure 8. 8.

FigureFigureFigure 8. 8.8. VelocityVelocity Velocity of of of slag–metal slag–metal slag–metal interfaces interfacesinterfaces in inin molds moldsmolds with withwith different differentdifferent cross crosscross sections sectionssections of: of:of: (a ((a)a )1100) 11001100 mm mmmm × × × 250250250 mm; mm; mm; ( (b b(b)) 1400) 1400 1400 mm mm mm ×× ×250 250 mm; mm; ( c ()c )1650) 16501650 mm mmmm × ×250 250250 mm. mm. mm.

The phenomenon can be attributed to the distribution of argon and the velocity field, as shown in Figure9. The argon bubbles easily float up with a relatively high speed, even higher than the injection velocity of molten steel. When it floats to the gas–slag interface, the bubble collapses and pushes slag firmly into the steel, resulting in an increase in the interface velocity of the slag–metal interface. Therefore, the movements of argon should be strictly controlled in the continuous casting process. Figure 10 shows the impact depth and impact velocity in molds with different widths. Here, the impact depth is defined as the distance between the slag–metal interface and impact point; and the impact velocity is defined as the velocity of the impact point. It can be seen that the flowing strand develops more fully and the impact depth becomes larger with the increasing of mold width. However, due to the flow loss in the flow process, the larger the mold width is, the smaller the impact velocity will be. Metals 2021, 10, x FOR PEER REVIEW 9 of 13

The phenomenon can be attributed to the distribution of argon and the velocity field, as shown in Figure 9. The argon bubbles easily float up with a relatively high speed, even higher than the injection velocity of molten steel. When it floats to the gas–slag interface, the bubble collapses and pushes slag firmly into the steel, resulting in an increase in the interface velocity of the slag–metal interface. Therefore, the movements of argon should Metals 2021, 10, x FOR PEER REVIEWbe strictly controlled in the continuous casting process. 9 of 13

The phenomenon can be attributed to the distribution of argon and the velocity field, as shown in Figure 9. The argon bubbles easily float up with a relatively high speed, even higher than the injection velocity of molten steel. When it floats to the gas–slag interface, the bubble collapses and pushes slag firmly into the steel, resulting in an increase in the Metals 2021, 11, 374 interface velocity of the slag–metal interface. Therefore, the movements of argon should9 of 13 be strictly controlled in the continuous casting process.

Figure 9. The distributions of (a) bubbles in mold and (b) velocity distribution in the slag–metal interface.

Figure 10 shows the impact depth and impact velocity in molds with different widths. Here, the impact depth is defined as the distance between the slag–metal interface and impact point; and the impact velocity is defined as the velocity of the impact point. It can be seen that the flowing strand develops more fully and the impact depth becomesFigureFigure 9. Thelarger distributions distributions with the increasingof of (a (a) )bubbles bubbles of in mold in mold mold width. and and (b ( )bHowever, velocity) velocity distribution distributiondue to the in inflowthe the slag–metal loss slag–metal in the flowinterface.interface. process, the larger the mold width is, the smaller the impact velocity will be.

Figure 10 shows the impact depth and impact velocity in molds with different widths. Here, the impact depth is defined as the distance between the slag–metal interface and impact point; and the impact velocity is defined as the velocity of the impact point. It can be seen that the flowing strand develops more fully and the impact depth becomes larger with the increasing of mold width. However, due to the flow loss in the flow process, the larger the mold width is, the smaller the impact velocity will be.

FigureFigure 10. 10. ImpactImpact depth depth and and velocity velocity of of molten molten steel steel under under different different mold mold widths. widths.

4.3.4.3. Analysis Analysis of of Velocity Velocity Characteristics Characteristics of of Slag–Metal Slag–Metal Interface Interface TheThe average average velocity velocity fluctuation fluctuation of of the the slag–metal interface interface is is monitored and the resultsresults are are shown shown in in Figure Figure 11.11. It It can can be be seen seen that that the the fluctuations fluctuations are are quite different in · −1 differentdifferent molds. On the one hand, thethe average velocityvelocity isis 0.07320.0732 mm·ss −1 withwith a a small small width width · −1 ofof the the mold mold and decreases to 0.0693 mm·ss−1 withwith a a medium medium width width of of the the mold. mold. On On the the other other hand,hand,Figure the 10. fluctuationImpactfluctuation depth ofof and thethe velocity velocity velocity of with moltenwith a largea steel large widthunder width different of theof the mold mold mold is widths. more is more significant, significant, and the velocity continues to decrease with an average value of 0.0647 m·s−1. This phenomenon and the velocity continues to decrease with an average value of 0.0647 m·s−1. This phe- is mainly attributed to the loss of molten steel flow. In a word, with the increase in the nomenon4.3. Analysis is mainly of Velocity attributed Characteristics to the loss of Slag–Metal of molten steelInterface flow. In a word, with the increase mold section, the velocity of the slag–metal interface decreases. However, there is little in theThe mold average section, velocity the velocity fluctuation of the slagof the–metal slag–metal interface interface decreases. is monitored However, thereand the is Metals 2021, 10, x FOR PEER REVIEWdifference in the interface velocity between slag and liquid metal in these three kinds10 of of13 littleresults difference are shown in inthe Figure interface 11. Itvelocity can be betweenseen that slagthe fluctuationsand liquid metalare quite in thesedifferent three in molds, which cannot explain the why the slag entrapment increases with different widths. kindsdifferent of molds, molds. whichOn the cannot one hand, explain the averagethe why velocity the slag is entrapment 0.0732 m·s− 1 increaseswith a small with width dif- ferentTherefore, widths. the averageTherefore, velocity the average of the slag–metalvelocity of interfacethe slag–metal cannot interface be used tocannot evaluate be used slag of the mold and decreases to 0.0693 m·s−1 with a medium width of the mold. On the other toentrapment evaluate slag severity. entrapment severity. hand, the fluctuation of the velocity with a large width of the mold is more significant, and the velocity continues to decrease with an average value of 0.0647 m·s−1. This phenomenon is mainly attributed to the loss of molten steel flow. In a word, with the increase in the mold section, the velocity of the slag–metal interface decreases. However, there is little difference in the interface velocity between slag and liquid metal in these three kinds of molds, which cannot explain the why the slag entrapment increases with dif-

Figure 11. AverageAverage velocity velocity fluctuations ﬂuctuations of of slag–metal slag–metal interface interface under under different different sections: sections: (a) (1100a) 1100 mm mm × 250× mm;250 mm;(b) 1400 (b) 1400mm × mm 250× mm;250 ( mm;c) 1650 (c) mm 1650 × mm250 ×mm.250 mm.

The horizontal velocity fluctuation of the slag–metal interface is shown in Figure 12. The velocity is signed: The negative value of velocity points to the narrow side of the mold, and the positive value points to the direction of the submerged nozzle. In the mold with a small width, most of the velocity value is positive, indicating that the up-flow is strong, corresponding to a flow pattern of double-roll flow (see Figure 7a). However, with the medium width of the mold, the value is mostly negative, indicating that the flow has been transformed into single-roll flow and the flow rate is reduced. When the width of the mold reaches 1650 mm, the flow is completely transformed into single-roll flow, while the flow rate increases. Therefore, the horizontal velocity cannot explain the reasons for the differences of slag entrapment as well.

Figure 12. Horizontal shear velocity fluctuation of slag/metal interface under different cross sections: (a) 1100 mm × 250 mm; (b) 1400 mm × 250 mm; (c) 1650 mm × 250 mm.

Figure 13 shows the variation of the maximum interface velocity with time. It can be seen that with the rise in mold width, the maximum velocity of the slag–metal interface also increases, so the probability of slag entrapment increases. This explains the phenomenon that the number of slag droplets entrapped in the mold increases with the increase in the cross section width of the mold, as shown in Figure 3. In addition, it can be found that with the increase in cross section width, the amplitude of velocity fluctuation also increases, that is, a large cross section width easily causes slag entrapment, which is consistent with the changing trend of slag drop number (see Figure 10). Therefore, the maximum interface velocity can reflect the severity of slag entrapment in a different mold, which should be paid more attention to control slag entrapment in steel.

Metals 2021, 10, x FOR PEER REVIEW 10 of 13

ferent widths. Therefore, the average velocity of the slag–metal interface cannot be used to evaluate slag entrapment severity.

Metals 2021, 11, 374 10 of 13 Figure 11. Average velocity fluctuations of slag–metal interface under different sections: (a) 1100 mm × 250 mm; (b) 1400 mm × 250 mm; (c) 1650 mm × 250 mm.

TheThe horizontal horizontal velocity velocity fluctuation fluctuation ofof the slag–metal interface interface is is shown shown in inFigureFigure 12. 12 . TheThe velocity velocity is signed:is signed: The The negative negative value value of velocityof velocity points points to theto the narrow narrow side side of the of mold,the andmold, the positiveand the positive value points value topoints the direction to the direction of the submergedof the submerged nozzle. nozzle. In the In mold the mold with a smallwith width, a small most width, of themost velocity of the velocity value is value positive, is positive, indicating indicating that the that up-flow the up-flow is strong, is correspondingstrong, corresponding to a flow to pattern a flow of pattern double-roll of double-roll flow (see flow Figure (see7 a).Figure However, 7a). However, with the mediumwith the width medium of the width mold, of the mold, value the is mostly value is negative, mostly negative, indicating indicating that the that flow the has flow been transformedhas been transformed into single-roll into single-roll flow and flow the flowand the rate flow is reduced.rate is reduced. When When the width the width of the moldof the reaches mold 1650reaches mm, 1650 the mm, flow the is completely flow is completely transformed transformed into single-roll into single-roll flow, while flow, the flowwhile rate the increases. flow rate Therefore, increases. theTherefore, horizontal the horizontal velocity cannot velocity explain cannot the explain reasons the for rea- the differencessons for the of differences slag entrapment of slag as entrapment well. as well.

FigureFigure 12. 12.Horizontal Horizontal shear shear velocity velocity ﬂuctuation fluctuation ofof slag/metalslag/metal interface interface under under different different cross cross sections: sections: (a) 1100 (a) 1100mm × mm 250× 250 mm; (b) 1400 mm mm ×× 250250 mm; mm; (c) ( c1650) 1650 mm mm × 250× 250mm. mm.

FigureFigure 13 13 shows shows the the variation variation ofof thethe maximummaximum interface velocity velocity with with time. time. It Itcan can be be seenseen that that with with the the rise rise in moldin mold width, width, the the maximum maximum velocity velocity of theof the slag–metal slag–metal interface interface also increases,also increases, so the probabilityso the probab ofility slag of entrapment slag entrapment increases. incr Thiseases. explains This explains the phenomenon the phe- thatnomenon the number that the of slagnumber droplets of slag entrapped droplets inentrapped the mold in increases the mold withincreases the increase with the in in- the crosscrease section in the width cross of section the mold, width as of shown the mold, in Figure as shown3. In addition,in Figure it3. can In addition, be found it that can with be thefound increase that in with cross the section increase width, in cross the section amplitude width, of velocitythe amplitude ﬂuctuation of velocity also increases, fluctuation that is,also a large increases, cross section that is, widtha large easilycross section causes width slag entrapment, easily causes which slag entrapment, is consistent which with is the changingconsistent trend with of the slag changing drop number trend of (see slag Figure drop 10number). Therefore, (see Figure the maximum10). Therefore, interface the velocitymaximum can reﬂectinterface the velocity severity can of slagreflect entrapment the severity in of a differentslag entrapment mold, which in a different should be mold, which should be paid more attention to control slag entrapment in steel. Metals 2021, 10, x FOR PEER REVIEWpaid more attention to control slag entrapment in steel. 11 of 13

FigureFigure 13. 13.Maximum Maximum velocity velocity distribution distribution of of slag/metal slag/metal interfaceinterface underunder different cross sections sections of: of: (a (a) )1100 1100 mm mm × ×250250 mm; mm; (b) 1400 mm × 250 mm; (c) 1650 mm × 250 mm. (b) 1400 mm × 250 mm; (c) 1650 mm × 250 mm. 4.4. Prediction of Slag Entrapment in Mold 4.4. Prediction of Slag Entrapment in Mold Figure 14 reveals the number of slag drops in molds of different widths. It can be seenFigure that the 14 numberreveals theof slag number drops of increase slag dropss with in the molds mold of width. different The widths. reason for It canthis be seenphenomenon that the number is related of to slag argon drops blowing, increases that withis, a more the mold considerable width. Theamount reason of argon for this phenomenongas retained isin related the wide-faced to argon mold blowing, and thata longer is, a morefloating considerable time lead to amount more argon of argon bub- gas retainedbles retained in the in wide-faced the slag layer, mold accompanie and a longerd by floating the enhanced time lead emulsification to more argon effect. bubbles The retainedvelocity in of the bubbles slag layer, is so high accompanied that they bymay the significantly enhanced emulsificationincrease the velocity effect. of The the velocity steel ofaround bubbles them. is so Therefore, high that a they large may mold significantly is more likely increase to cause the slag velocity entrapment. of the steel around them. Therefore, a large mold is more likely to cause slag entrapment.

Figure 14. Monitoring of slag droplets in molds with different widths of (a) 1100 mm × 250 mm; (b) 1400 mm × 250 mm; (c) 1650 mm × 250 mm; (d) average value.

5. Slag Entrapment Evaluation Criteria The floatation of bubbles significantly affects the slag–metal interface, and increases the slag entrapment when mold width gets larger. Simultaneously, the floatation of bubbles can also lift the molten steel that is injected from submerged entry nozzle (SEN). Therefore, a dimensionless criterion number can be defined to reflect the effect of jet rigidity on slag entrapment in steel. As shown in Equation (9): α C = θ (9)

Metals 2021, 10, x FOR PEER REVIEW 11 of 13

Figure 13. Maximum velocity distribution of slag/metal interface under different cross sections of: (a) 1100 mm × 250 mm; (b) 1400 mm × 250 mm; (c) 1650 mm × 250 mm.

4.4. Prediction of Slag Entrapment in Mold Figure 14 reveals the number of slag drops in molds of different widths. It can be seen that the number of slag drops increases with the mold width. The reason for this phenomenon is related to argon blowing, that is, a more considerable amount of argon gas retained in the wide-faced mold and a longer floating time lead to more argon bubbles retained in the slag layer, accompanied by the enhanced emulsification effect. The Metals 2021, 11, 374 11 of 13 velocity of bubbles is so high that they may significantly increase the velocity of the steel around them. Therefore, a large mold is more likely to cause slag entrapment.

FigureFigure 14. 14. MonitoringMonitoring of ofslag slag droplets droplets in molds in molds with with different different widths widths of (a of) 1100 (a) 1100 mm mm× 250× mm;250 ( mm;b) (b) 14001400 mm mm × ×250250 mm; mm; (c) (1650c) 1650 mm mm × 250× mm;250 mm; (d) average (d) average value. value.

5.5. Slag Slag Entrapment Entrapment Evaluation Evaluation Criteria Criteria TheThe floatation floatation of ofbubbles bubbles significantly significantly affects affects the theslag–metal slag–metal interface, interface, and increases and increases thethe slag slag entrapment whenwhen moldmold widthwidth gets gets larger. larger. Simultaneously, Simultaneously, the the floatation floatation of bubblesof bubblescan also can lift also the moltenlift the molten steel that steel is injectedthat is injected from submerged from submerged entry nozzleentry nozzle (SEN). (SEN). Therefore, Therefore,a dimensionless a dimensionless criterion numbercriterion cannumber be defined can be todefined reflect to the reflect effect the of effect jet rigidity of jet ri- on slag gidityentrapment on slag inentrapment steel. As shownin steel. in As Equation shown in (9): Equation (9): α Metals 2021, 10, x FOR PEER REVIEW α 12 of 13 CC== (9) (9) θ θ where α is the angle of the jet and θ is the angle of the nozzle (see Figure 2). There- fore,where theα physicalis the angle meaning of the of jet Equation and θ is (9) the can angle be interpreted of the nozzle as the (see ratio Figure of jet2). angle Therefore, to the the originalphysical nozzle meaning angle. of EquationWithout bubble (9) can injection, be interpreted the rigid as jet the is ratioequal of to jet C = angle 1. Considering to the original thenozzle effect angle. of argon, Without C < bubble1 when injection,argon floats. the rigidFigure jet 15 is shows equal tovariationsC = 1. Considering of C value with the effect differentof argon, moldC < 1widths. when It argon can be floats. seen from Figure Figure 15 shows 15 that variations the C value of increasesC value with with mold different

width.mold widths.This trend It canis similar be seen to that from of Figure slag drop 15 thatvariations the C shownvalue increasesin Figure with14, indicating mold width. thatThis the trend C value is similar can be to thatusedof as slag a guide drop to variations reflect the shown slag entrapment in Figure 14 in, indicatingmolds with that dif- the C ferentvalue widths. can be used as a guide to reﬂect the slag entrapment in molds with different widths.

FigureFigure 15. 15. PredictionPrediction of of slag slag severity severity through through dimensionless dimensionless value. value.

6. Conclusions In this work, a three-dimensional model is established to simulate the slag entrapment phenomenon, mainly focusing on the slag entrapment phenomenon in different molds. The large eddy simulation (LES) model is applied for the calculation of the turbulence of molten steel, and the Smagorinsky–Lilly model is used to describe the sub-grid scale vortices. Based on the obtained study results, the following conclusions can be drawn: (1) The amount of slag entrapment increases with mold width, which is mainly due to the bubble pulsion. There exists a transformation of flow pattern in the mold when mold width increases. The double-roll flow pattern produces less slag entrapment than a single-roll flow pattern. (2) The impact depth of molten steel decreases with mold width, while the impact velocity increases with mold width. The larger the mold width, the weaker the rigidity of the jet. Thus, the bubble injection significantly affects the flow field in the mold. (3) Both of the average velocity and horizontal velocity of slag/metal interface cannot reflect the severity of slag entrapment in the mold. By comparison, the maximum velocity at the interface shows good advantages in predicting the severity of slag entrapment in the mold. (4) A dimensionless criterion number C with the physical meaning of the ratio of jet angle to nozzle angle is successfully established to predict slag entrapment in different molds.

Author Contributions: X.Z. contributed to writing the original draft and compiled the manuscript and the data collection; X.L. contributed to the outline of the paper, writing, drawing and validation; J.Z. contributed to the conceptualization, reviewing and editing the manuscript. All authors have read and agreed to the published version of the manuscript. Funding: This work is financially supported by the Baoshan Iron&Steel Company (Grant No. R17ECP0310), and Special funding from Jiangyan district in Taizhou City (No. SDXCL(2020)-12). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable.

Metals 2021, 11, 374 12 of 13

6. Conclusions In this work, a three-dimensional model is established to simulate the slag entrapment phenomenon, mainly focusing on the slag entrapment phenomenon in different molds. The large eddy simulation (LES) model is applied for the calculation of the turbulence of molten steel, and the Smagorinsky–Lilly model is used to describe the sub-grid scale vortices. Based on the obtained study results, the following conclusions can be drawn: (1) The amount of slag entrapment increases with mold width, which is mainly due to the bubble pulsion. There exists a transformation of flow pattern in the mold when mold width increases. The double-roll flow pattern produces less slag entrapment than a single-roll flow pattern. (2) The impact depth of molten steel decreases with mold width, while the impact velocity increases with mold width. The larger the mold width, the weaker the rigidity of the jet. Thus, the bubble injection significantly affects the flow field in the mold. (3) Both of the average velocity and horizontal velocity of slag/metal interface cannot reflect the severity of slag entrapment in the mold. By comparison, the maximum velocity at the interface shows good advantages in predicting the severity of slag entrapment in the mold. (4) A dimensionless criterion number C with the physical meaning of the ratio of jet angle to nozzle angle is successfully established to predict slag entrapment in different molds.

Author Contributions: X.Z. contributed to writing the original draft and compiled the manuscript and the data collection; X.L. contributed to the outline of the paper, writing, drawing and validation; J.Z. contributed to the conceptualization, reviewing and editing the manuscript. All authors have read and agreed to the published version of the manuscript. Funding: This work is financially supported by the Baoshan Iron&Steel Company (Grant No. R17ECP0310), and Special funding from Jiangyan district in Taizhou City (No. SDXCL(2020)-12). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy. Conflicts of Interest: The authors declare no conflict of interest.

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