Euler-Euler-Lagrangian Modeling for Two-Phase Flow and Particle Transport in Continuous Casting Mold

Abstract

A mathematical model based on the Euler-Euler-Lagrangian approach has been developed to study the influence of argon gas injection on the molten steel flow and particle transport behaviors in continuous casting mold. The modified k-ε model with extra source term to account for the bubble-induced turbulence is adopted to model the turbulence in this system. The transport of particle is simulated using a Lagrangian approach based on the computed two-phase flow fields. A 1/4th scale water model has been developed to investigate the bubble behavior and fluid flow pattern. Air is injected into the submerged entry nozzle (SEN) through a circumferential inlet chamber which is made using specially-coated samples of mullite porous brick. The predictions of gas bubble distribution and fluid flow pattern are in good agreement with the water model experimental observations. Argon bubbles can change the flow pattern in the upper recirculation zone of the mold, increase the fluctuation of the top surface, and shift the impingement point on the narrow wall. The effect increases with increasing argon gas flow rate, and decreasing casting speed and bubble size. Argon gas injection enhances the removal of particles. The optimum argon gas flow rate between 5 and 10 L/min, casting speed between 0.7 and 0.8 m/min, and argon bubble size around 2.5 mm are obtained using this model to improve the removal of particles.

1. Introduction

Control of fluid flow in the continuous casting mold is important due to its effect on the various phenomena related to the quality of the casting steel product, such as bubble/particle transport and entrapment,^1,2,3,4) transient fluctuations at the top surface,^5,6) and the transport and dissipation of superheat.^7,8) Argon gas is employed in the continuous casting mold to help prevent nozzle clogging, to enhance mixing, to prevent air from entering the submerged entry nozzle (SEN), to promote desired flow patterns, and to promote the floatation of nonmetallic particles from the liquid steel. In addition to these beneficial effects, argon bubbles have some detrimental side effects. They form an exposed eye of steel around the SEN by sweeping off the protective layer of mold flux.^9,10) And small bubbles may be trapped by the solidifying shell and form serious defects just beneath the slab surface.⁸⁾ So it is important for the guidance of quality improvement to study the two-phase flow of molten steel and argon gas in the continuous casting mold.

Experimental works have reported the importance of the two-phase flow in mold when gas is injected. Gupta and Lahiri,¹⁰⁾ and Liu et al.¹¹⁾ studied the asymmetric two-phase flow pattern inside the liquid pool using water experiment. Li et al.¹²⁾ measured the velocity fields at the center section with argon gas injection using a two-dimensional sensor through a laboratory-scale experiment of a continuous casting system using molten tin. Iguchi and Kasai¹³⁾ performed a horizontal water-air two-phase jet experimental study of mean velocity and turbulence components of water flow, which were measured with a Laser Doppler Velocimeter. Esaka et al.¹⁴⁾ showed that the entrapment of argon bubbles depends on the roughness of solid liquid interface. Singh et al.¹⁵⁾ performed a 1/3th water model to study the bubble movement considering the effect of various parameters. Zhang et al.¹⁶⁾ studied the flow pattern in the mold considering the effect of submergence depth, SEN geometry, mold width, water flow rate, and argon gas flow rate. Argon gas disintegrates into many uneven-sized bubbles as it issues out of the SEN. Some researchers^17,18) have studied the bubble size distribution using water model experiments.

Many mathematical models have been developed to study the two-phase flow in the continuous casting process. Bessho et al.¹⁹⁾ compared the calculated flow pattern, gas volume fraction in a full-scale water model with experimental observations, pointed out that gas created a great change in the flow pattern. Thomas et al.⁸⁾ developed numerical models of fluid, heat, and mass transport, and applied to study the complex inner-related phenomena of two-phase fluid flow and superheat dissipation. Li et al.^12,20) employed a homogeneous fluid model with various densities to track the molten steel-argon gas flow, considering the static magnetic-field application in the mold. Kubo et al.²¹⁾ have demonstrated the two-phase flow inside the mold using different multiphase models, discrete phase model and two-phase model, and it was compared with the plant data of continuous casting mold. Sanchez-perez et al.²²⁾ observed two-way coupled flows in water model experiment and studied the dynamic of coupled and uncoupled two-phase flows using mathematical model. Kwon et al.⁴⁾ represented bubbles distribution and bubble-particle attachment in the continuous casting system by employing the water and CFD model studies. Liu et al.¹¹⁾ developed an Euler-Euler-Large Eddy Simulation model to simulate the transient asymmetric two-phase flow in slab continuous casting mold. Moreover, many researchers^23,24,25) have tracked the trajectories of bubbles inside the liquid pool using a Lagrangian approach. But the effect of argon bubbles on the molten steel flow pattern was neglected, so these results only applied to the low argon flow rates.

However, in spite of the previous mentioned works, very few had been done to fully study the two-phase flow structure inside the continuous casting mold. Understanding the behavior of two-phase flow is essential for the design of effective methods of removing small particles. In the present work, the first objective is to observe and record the liquid flow pattern and bubble distribution inside the liquid pool using water model experiment. The second objective is to develop a mathematical model to study the two-phase flow of molten steel and argon gas, considering the effect of argon flow rate, casting speed, and argon bubble size. Finally, this model is applied to investigate the transport of particle in the mold.

2. Mathematical Model

The Euler-Euler two-fluid model²⁶⁾ is used to simulate the time-averaged flow of argon bubbles in molten steel. Each phase has its own set of continuity and momentum equations. Coupling is achieved through four empirical interfacial forces between liquid steel and argon bubbles. Transport of particles in the mold is simulated using a Lagrangian trajectory-tracking approach based on the two-phase flow fields. However, a relevant summary of the Euler-Euler two-fluid model and the interfacial momentum transfer model have been given in previous work of the authors.¹¹⁾ The following section is the details of the modified k-ε turbulence model and the particle transport model.

2.1. Modified k-ε Turbulence Model

The effective viscosity μ_eff,l of the liquid phase is composed of three contributions: the molecular viscosity μ_l, the turbulent viscosity μ_t,l and the bubble induced turbulence μ_bi,l.   

$${\mu }_{eff,l}={\mu }_l+\mu {}_{t,l}+{\mu }_{bi,l}$$(1)

The calculation of the effective gas viscosity is based on the effective liquid viscosity as was proposed by Jakobsen et al.²⁷⁾   

$${\mu }_{eff,g}=\frac{{\rho }_g}{{\rho }_l}{\mu }_{eff,l}$$(2)

When k-ε model is used, the turbulent viscosity (μ_t,l) is calculated from the turbulent kinetic energy k and turbulent dissipation ε by   

$${\mu }_{t,l}=C_{\mu }{\rho }_l\frac{k^2}{\epsilon }$$(3)

The model proposed by Sato et al.²⁸⁾ is used to take account of the turbulence induced by the movement of the bubbles. The expression is:   

$${\mu }_{bi,l}={\rho }_l{C}_{\mu ,bi}{\alpha }_g{d}_b\left|u_g-u_l\right|$$(4)

with a model constant C_μ,bi which is equal to 0.6.

It has been reported that the bubble induced viscosity in Eq. (4) did not alter the simulation results in case of RANS.²⁹⁾ For this reason, the extra term G in the k and ε proposed by Simonin and Viollet³⁰⁾ is used to represent the migration of the bubbles through the liquid.   

$$\nabla \cdot \left({\alpha }_l{\rho }_l{u}_{l}k\right)=\nabla \left({\alpha }_l\left({\mu }_l+\frac{{\mu }_{t,l}}{{\mathrm{\sigma }}_k}\right)\nabla k\right)+{\alpha }_l\left(G-{\rho }_l\epsilon \right)+{\prod }_{k,l}$$(5)

  

$$\begin{array}{l}\nabla \cdot \left({\alpha }_l{\rho }_l{u}_l\epsilon \right)=\nabla \left({\alpha }_l\left({\mu }_l+\frac{{\mu }_{t,l}}{{\mathrm{\sigma }}_{\epsilon }}\right)\nabla \epsilon \right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }+{\alpha }_l\left(C_1\frac{\epsilon }{k}G-C_2{\rho }_l\frac{{\epsilon }^2}{k}\right)+{\prod }_{\epsilon ,l}\end{array}$$(6)

where standard model constants C₁ = 1.44, C₂ = 1.92, C_μ = 0.09, σ_k = 1.00, σ_ε = 1.30.   

$$G=-{\mu }_{eff,l}\left(\nabla u_l+{\left(\nabla u_l\right)}^T\right)\cdot \nabla u_l$$(7)

The extra source terms are written for this model as:   

$${\prod }_{k,l}=C_k{C}_f{\rho }_l{\alpha }_g{\alpha }_{l}k$$(8)

  

$${\prod }_{\epsilon ,l}=C_{\epsilon }{C}_f{\rho }_l{\alpha }_g{\alpha }_l\epsilon$$(9)

where C_f is the interphase friction coefficient given by   

$$C_f=\frac{3}{4}\cdot \frac{C_D}{d_b}\left|u_g-u_l\right|$$(10)

where the coefficients of this model are C_k = 0.75 and C_ε = 0.6.

2.2. Particle Transport Model

In order to analyze the particle transport behavior in the mold, a mathematical model of the three-dimensional motion of particle is applied. Assumptions for the calculation are as follows: (1) the particle is spherical and small, and its motion does not affect the two-phase flow field; (2) coalescence and breakup of particles are not considered; (3) top surface fluctuation and slag entrainment is not considered. Due to the low volume fraction of impurity particles in continuous casting mold, transport of particles can be computed using the Lagrangian approach. The equation of motion for such a particle was derived:   

$$m_P\frac{d{u}_P}{dt}=F_g+F_b+F_d+F_{vm}+F_p$$(11)

The terms on the right hand side of Eq. (11) are gravitational force of the particle, buoyancy force due to gravity, drag force, virtual mass force, and pressure gradient force. Each of the five hydrodynamic forces has been discussed in previous work.³¹⁾

2.3. Numerical Details

Due to the symmetry of the structure and reducing the computational cost, only a quarter of the mold was modeled, as shown in Fig. 1(b). The fluid properties and operating conditions used in numerical simulation are given in Table 1. The computational domain is discretized with a structured mesh. The whole grid for the domain consists of around 600000 cells. A constant velocity boundary condition for two phases is prescribed at the inlet of the SEN based on the casting speed, the ratios of liquid and gas phase at the inlet are defined based on their respective flow rate. A constant pressure condition at the bottom of the calculation domain is applied. The top surface of the mold is treated as a flat surface and modeled as degassing boundary condition. Along the walls, no-slip boundary conditions are adopted. Argon gas is injected into the SEN at a room temperature. It expands descending in the nozzle due to heat transfer. Thus, the argon gas injection rate used in the model is the hot argon flow rate. Gas injection may be characterized by the average gas volume fraction, which can be found from Thomas et al.⁸⁾

Fig. 1.

Schematic of the water model system (a) and the calculation model (b).

Table 1. The geometrical, physical properties and operating conditions used in numerical simulation model.

Parameter	Value	Parameter	Value
Diameter of SEN, mm	80	Length of mold, mm	800
Length of SEN, mm	1000	Length of domain, mm	3500
Exit angle of nozzle, °	15	Casting speed, m/min	0.5–1.0
Height of SEN port, mm	80	Density of molten steel, kg/m3	7020
Width of SEN port, mm	70	Density of argon, kg/m3	0.56
Submergence depth of SEN, mm	300	Viscosity of molten steel, kg·m–1·s–1	0.0056
Width of mold, mm	2200	Viscosity of argon, kg·m–1·s–1	7.42 × 10–5
Thickness of mold, mm	300	Diameter of argon bubble, mm	0.5–3

Five classes of particles with diameters of 2, 5, 10, 20, and 50 μm are defined. Each class is injected from the same position. And the same number for each class is 2000. The initial locations are uniformly distributed at the inlet surface. An escape boundary condition is defined at the top surface and the bottom of the mold. A reflecting boundary for the walls inside the SEN is defined. The particles are modeled as to be captured at the side walls of the mold and outside of the SEN.

3. Model Validation

3.1. Comparison with Bubble Distribution Observations

In order to visualize the flow pattern and bubble distribution inside the mold, and validate the mathematical model, a 1/4th scale water model was established, the schematic of experiment system is shown in Fig. 1(a). Water was circulated in the circuit through a buffer cylinder to the simplified tundish. Water level in the tundish and mold was maintained constant with the help of the slide gate nozzle and outlet of the buffer tank. Air from the gas cylinder was fed to the SEN through a circumferential inlet chamber which was made using specially-coated samples of mullite porous brick; the material is used for the upper nozzle of the actual continuous casting mold. Once the water and air flow reached a steady state, the pictures of bubble dispersion were taken using a high speed camera with 4500 frames per second.

The instantaneous distributions of gas bubbles in the mold at different times are shown in Figs. 2(a) to 2(c), which are obtained at a given water flow rate of 16 L/min and gas flow rate of 0.833 L/min. The upward bending of the jet is quite evident. Bubbles are found to be floating around the SEN, and are subsequently removed from the mold. The measured mean air bubble diameter through the image analyzing method is 1.49 mm. Although it is very important to consider the local bubble size distribution for the industrial molten steel–argon flow, it is difficult and almost impossible to measure the argon bubble diameter in the molten steel. So the mean bubble diameter obtained from the water model experiment is used as the initial bubble diameter in the simulation. The streamline of gas bubble is given in Fig. 2(d). It can be seen that the distribution and penetrating length of argon bubbles agree well with the transient experimental results.

Fig. 2.

Instantaneous bubble distribution in water model (a–c) and streamline obtain from the calculation model (d).

3.2. Comparison with Flow Pattern Observations

Figures 3(a) and 3(b) show the fluid flow pattern in the water model without and with gas injection, respectively. These photographs were obtained by black-colored dye injection against a white background. The flow pattern in the upper recirculation zone with gas injection is very different from that without gas injection. The fluid left the SEN port as a strong jet impinge on the narrow face of the mold, then split vertically to create upper and lower recirculation, as shown in Fig. 3(a). The influence of the gas bubbles on the fluid flow pattern can be seen in Fig. 3(b), part of fluid moved up toward the top surface after leaving the SEN port, another part of fluid continue as a jet impinge at a slightly higher location on the narrow wall. Figures 4(a) and 4(b) illustrate the typical molten steel flow pattern predicted numerically without and with argon gas injection, respectively. The flow pattern in the upper recirculation zone is very similar to that one observed in the experiment (Fig. 3). The upward motion due to the bubbles was predicted to shift the location of the impingement point on the narrow wall and the lower recirculation eye. And the special flow pattern with gas injection is important to the particle transport inside the liquid pool.

Fig. 3.

Fluid flow pattern obtain from the water model experiment without (a) and with (b) gas injection.

Fig. 4.

Streamlines of molten steel obtain from the calculation model without (a) and with (b) argon gas injection.

4. Typical Flow Results

4.1. Two Phase Velocity Fields

Figures 5(a) and 5(b) show the argon volume fraction and the velocity field of argon gas at the central plane of the mold, with a casting speed of 0.7 m/min and a gas flow rate of 10 L/min, and the diameter of the argon gas bubble is 1.49 mm. Argon gas bubbles are driven by the molten steel flow inside the calculation domain. After entering the mold with the molten steel jet, most of the bubbles are crowded together close to the SEN and float upward through the upper recirculation zone, and then escape from the top surface. This result agrees well with the water model experiment result (Fig. 2(a)). From the velocity field of argon gas, in Fig. 5(b), it can be seen that the velocity is stronger at the position of larger argon volume fraction than other positions; the possible reason is the effect of the bubble buoyancy.

Fig. 5.

Argon volume fraction distribution and argon velocity vector plot in central plane of the mold.

Figure 6 shows the corresponding velocity field of molten steel with the same flow conditions. The molten steel jet from the SEN port is divided into two parts. One part flows to the top surface and another part flows to the narrow wall of the mold. Comparing the flow pattern of the two phases, it can be found that the flow pattern of the molten steel which flows to the top surface is similar to the flow pattern of argon gas, implying that the upward molten steel flow is induced by the argon bubble.

Fig. 6.

Molten steel velocity vector plot in the central plane of the mold.

Argon gas injection affects the casting process through its influence on the molten steel flow pattern. The extent of this effect depends on the argon gas flow rate, casting speed and the bubble size. Figures 7, 8, 9, 10, 11, 12 show the effects of these important parameters on the motion of argon bubble, velocity fluctuation on the top surface and impingement point on the narrow wall.

Fig. 7.

Argon volume fraction distribution in the upper roll with different argon gas flow rates.

Fig. 8.

Argon volume fraction (a), vertical velocity of steel (b) along the centerline of 0.001 m below the top surface and velocity of steel(c) on the narrow wall with different argon gas flow rate.

Fig. 9.

Argon volume fraction distribution in the upper roll with different casting speeds.

Fig. 10.

Argon volume fraction (a), vertical velocity of steel (b) along the centerline of 0.001 m below the top surface and velocity of steel(c) on the narrow wall with different casting speed.

Fig. 11.

Argon volume fraction distribution in the upper roll with different bubble sizes.

Fig. 12.

Argon volume fraction (a), vertical velocity of steel (b) along the centerline of 0.001 m below the top surface and velocity of steel (c) on the narrow wall with different bubble size.

4.2. Effect of Argon Gas Flow Rate

Figure 7 shows the effect of argon gas flow rate on the argon volume fraction distribution in the mold. For all cases, the casting speed is 0.7 m/min, and the diameter of the argon gas bubble is 1.49 mm. Due to the injection of more bubbles, stronger buoyancy make the bubbles themselves float more easily and more dispersed in the upper roll, when the gas flow rate increases from 5 to 15 and 25 L/min. Figure 8(a) shows the argon volume fraction distribution at the centerline of the top surface with different argon gas flow rates. This result illustrates that the argon volume fraction in the mold naturally increases with increasing argon gas flow rate at the inlet. Most of bubbles escape from the top surface around the region of 0.06 to 0.7 m distance from the SEN. The peak values of argon volume fraction are located at 0.1 m distance from the SEN when the argon gas flow rate is ≥ 15 L/min, otherwise, the location is at the region of 0.4 and 0.5 m distance from the SEN when the argon gas flow rate is 10 and 5 L/min, respectively.

The top-surface velocity greatly influences the harmful entrainment of liquid slag, which causes surface defects or sub-layer defects in the slab.⁵⁾ Figure 8(b) shows the velocity fluctuation of molten steel at the centerline of 0.001 m below the top surface with different argon gas flow rates. The velocity from the liquid pool to the top surface is defined with a positive value in this plot. The figure shows that the vertical velocity is a negative value near the SEN(at the region of 0.1 m to 0.6 m distance from the SEN) and is a positive value near the narrow wall (0.6 m to 1.1 m distance from the SEN) when there is no argon gas injection. However, when the argon gas is injected into the liquid pool, the velocity fluctuation is obvious, with a positive value near the SEN, especially at the edge of SEN. The liquid slag in this region has a big opportunity to be pushed away and generate an open “eye” of the molten steel, inducing serious reoxidation from the air and worsening steel cleanliness.^9,11)

The upward motion due to the bubbles was predicted to shift the location of the impingement point on the narrow wall. The velocity of molten steel at 5 mm from the narrow wall is used to study this effect; it becomes stagnation at the impingement point, as shown in Fig. 8(c). The stagnation point falls between adjacent velocity peaks, where the jet splits to flow upward and downward along the narrow wall. The figure shows that the impingement points gradually move toward the top surface with increasing argon gas flow rate. This upward motion is predicted to shift the location of the upper recirculation eye toward the mold center. More jet could flow upward into the upper recirculation zone, and more particles would be taken and removed from the top surface.

4.3. Effect of Casting Speed

Figure 9 shows the argon volume fraction distribution with different casting speeds inside the mold. For all cases, the argon gas flow rate is 10 L/min, and the diameter of the argon gas bubble is 1.49 mm. The results illustrate that the bubbles dispersed more widely in the upper recirculation zone, when the casting speed increases from 0.5 to 0.7 and 1.0 m/min.

Figure 10(a) shows the argon volume fraction distribution at the centerline of the top surface with different casting speeds. It can be found that, when the casting speed is ≤ 0.6 m/min, the peak values of argon volume fraction are located at 0.1 m distance from the SEN, most bubbles escape from the top surface at the region of 0.06 to 0.4 m distance from the SEN. Otherwise, the peak values located at about 0.4 m distance from the SEN and most bubbles escape at the region of 0.06 to 0.7 m distance from the SEN when the casting speed is ≥ 0.7 m/min.

The velocity fluctuation at the centerline of 0.001 m below the top surface with different casting speeds is shown in Fig. 10(b). The result shows that the vertical velocity near the narrow wall at 0.7 to 1.1 m distance from the SEN is negative close to zero. When the casting speed is ≤ 0.7 m/min, the velocity fluctuation near the SEN is obvious especially at the edge of SEN. Otherwise, when the casting speed is ≥ 0.8 m/min, the velocity fluctuation near the SEN becomes weaker and the velocity value is negative at the edge of the SEN, moreover, the peak values of the vertical velocity are at 0.4 to 0.5 m distance from the SEN and the vertical velocity value decreases with increasing casting speed.

Figure 10(c) shows the velocity of molten steel with different casting speeds at the centerline. The figure shows that the velocity of molten steel increases with increasing casting speed. But there is no obvious effect on changing the impingement points on the narrow wall with increasing casting speed.

4.4. Effect of Argon Bubble Size

Figure 11 shows the effect of bubble size on the argon volume fraction distribution inside the domain. For all cases, the casting speed is 0.7 m/min and the argon flow rate is 10 L/min. The bubble size is 0.5, 1.5 and 3.0 mm, respectively. The results indicate that stronger buoyancy due to the larger bubble makes the bubbles themselves float more easily and less dispersed.

The argon volume fraction with injection of different bubble sizes at the centerline of the top surface is shown in Fig. 12(a). The figure shows that most bubbles escape from the top surface near the SEN when the bubble size is ≥ 2.0 mm, otherwise, most bubbles escape at the region of 0.06 to 0.7 m distance from the SEN when the bubble size is from 0.5 to 1.5 mm.

Figure 12(b) shows the velocity fluctuation with injection of different bubble sizes at the centerline of 0.001 m below the top surface. The figure shows that, when the bubble size is ≥ 2.0 mm, the velocity fluctuates at the region of 0.06 to 0.3 m distance from the SEN, the velocity values are positive near the SEN (0.06 to 0.15 m distance from the SEN), but are negative at the region of 0.15 to 0.7 m distance from the SEN. When the bubble size is ≤ 1.5 mm, the vertical velocity values at the region of 0.06 to 0.7 m distance from the SEN are mostly positive.

The velocity of molten steel with injection of different bubble sizes at the centerlinex is shown in Fig. 12(c). The velocity of molten steel in the upper recirculation zone increases with increasing bubble size. But the change of the impingement point at the narrow wall is complex with injection of different bubble sizes, the location is the lowest (about 0.58 m below the top surface) when the bubble size is 2 mm, and is the highest (about 0.43 m below the top surface) when the bubble size is 0.5 mm. However, there are two impingement points when the bubble size is 0.5 mm; the possible reason is that there are two recirculation zones in the upper roll near the narrow wall.

5. Particle Transport inside the Mold

With the development of the secondary steelmaking, the technique of removing particles has been significantly improved in ladle and tundish. However, some small particles (usually ≤ 50 μm) would enter into the casting mold through the SEN. Most of these particles gradually float to the top surface and are trapped by the slag layer, but some of them move downward into the lower roll of the mold and are captured by the solidified shell. In the subsequent rolling process, these particles could lead to the formation of defects in the final products. Until now, these defects have not been successfully and completely eliminated from the continuous casting process. Therefore, it is important to clarify the behavior of particles and chose the appropriate operating conditions (casting speed and argon gas flow rate) in the continuous casting process for improving steel plate quality.

The purpose of this section is to develop a numerical model for calculating the particle transport in the two-phase flow of the mold, considering the effects of the argon gas flow rate, casting speed and bubble size. Five classes of particles with diameters of 2, 5, 10, 20, and 50 μm were injected into the SEN, and then the destinations of the particles, containing remove from the top surface (define “removal”), escape from the bottom of the domain and are trapped by the walls (define “captured”) and “midway” inside the liquid pool, were recorded.

Some of the particles rise to the top surface and are trapped by the slag, the trajectories of these particles (50 μm) without and with argon gas injection are significantly different, as shown in Figs. 13(a) and 13(b). Argon gas injection changes the particle trajectory; most of these particles (trapped by the slag layer) would enter the upper recirculation zone close to the SEN, and finally float out of the domain, as shown in Fig. 13(b). This result confirms the importance of the two-phase flow in affecting the particle transport in the mold. If argon bubbles accumulate more and escape from a small region, the top slag in this region would be pushed away and generate an open eye, and result in the exposure of the molten steel. Due to the low pressure of this region, more particles and bubbles would escape from here. But this is not good due to absorbing oxygen and entraining slag.¹⁶⁾

Fig. 13.

Predicted particle (d = 50 μm) trajectories inside the mold without (a) and with (b) argon gas injection.

Figures 14 and 15 show the predicted particle final removal location on the top surface without and with argon gas injection, respectively. The particles distribute at the top surface more dispersed without argon gas injection (Fig. 14(a)). And most these particles are removed from 0.2 m to 0.8 m distance from the SEN (Fig. 15(a)). With argon gas injection, most these particles are removed from 0.3 m to 0.6 m distance from the SEN, as shown in Figs. 14(b) and 15(b). More particles would move to the narrow wall with increasing particle diameter. The possible reason is that these particles can travel a long distance with the molten steel jet, stronger buoyancy of the larger particles makes floating the particles themselves more easily and more dispersed in the upper roll, but smaller ones with weaker buoyancy would move downward into the lower roll of the mold with the steel jet and cause the defects of the slabs. The results also show that more particles would be removed from the top surface with argon gas injection.

Fig. 14.

Particle final removal locations on the top surface without (a) and with (b) argon gas injection.

Fig. 15.

Distribution of removal particle on the top surface without (a) and with (b) argon gas injection.

Figures 16, 17, 18 show the removal ratio of particles from the molten steel to the top surface and the captured ratio of particles by the solidifying shell and escaping from the bottom of the domain, considering the effects of the argon gas flow rate, casting speed and the bubble size, respectively. The results show that larger particles are easier removed than smaller particles. Figures 16(a) and 16(b) show that gas injection helps to remove particles, in the same conditions, with increasing argon gas flow rate, the removal ratio increases first and then decreases, so that an optimum gas flow rate for this mold is between 5 and 10 L/min.

Fig. 16.

Influence of argon gas flow rate on the removal and captured ratio of particles.

Fig. 17.

Influence of casting speed on the removal and captured ratio of particles.

Fig. 18.

Influence of bubble sizes on the removal and captured ratio of particles.

Faster casting speed removes less large particles (≥ 50 μm) than slower casting speed, as shown in Figs. 17(a) and 17(b). But for small particles (≤ 20 μm), with the increasing of casting speeds from 0.5 to 1.0 m/min, the removal ratio decreases first, and then increases, and then decreases again. As previous studied, however, slower casting speed is not suitable for improving productivity, and faster casting speed would induce many defects. So that there should be an optimal casting speed that gives not only high productivity but also less defects. This result suggests the optimal casting speed for this mold might be from 0.7 to 0.8 m/min.

Figures 18(a) and 18(b) show that decreasing bubble size is more efficient for removing particles. However, small bubbles, such as those smaller than 1 mm, may be trapped by the solidifying shell during moving through the lower recirculation zone. Thus, there should be an optimal bubble size that gives not only high removal ratio but also low captured ratio of bubbles. This result suggests that the optimal bubble size might be about 2.5 mm. Of course, it is very difficult to control the bubble size in the continuous casting mold. Until now, it is failed to make this vision a reality in the industry. The present work mainly explains the optimal bubble size injection for removing particles from the mold in theory.

6. Conclusions

Eulerian-Eulerian simulations of the two-phase flow inside the continuous casting mold using the modified k-ε model predictions have been compared with the water model experiment. A Lagrangian approach is performed to calculate the transport of particles inside the liquid pool based on the two-phase flow field. From the obtained results, the following conclusions can be drawn.

(1) Good qualitative agreement has been obtained between the model predictions and the experimental observations of two-phase flow pattern characteristics.

(2) Argon gas injection changes the flow pattern in the upper recirculation zone of the mold. The fluid jet from the SEN port is divided into two parts: one part flows upward to the top surface close to the SEN; another part continues as a jet and impinges at a slightly higher location on the narrow wall.

(3) Argon gas injection increases the fluctuation of the top surface and shifts the impingement point on the narrow wall and the recirculation eyes in both the upper roll and lower roll of the domain. The effect increases with increasing argon gas flow rate, and decreasing casting speed and bubble size.

(4) Argon gas injection increases the removal ratio of inclusion particles. In order to improve the removal of particles, optimum argon gas flow rate between 5 and 10 L/min, casting speed between 0.7 and 0.8 m/min, and argon bubble size around 2.5 mm are obtained using this model.

Acknowledgements

Authors are grateful to the National Natural Science Foundation of China for support of this research, Grant NO. 51210007.

Nomenclature

C_f Interphase friction coefficient, dimensionless

d_b Bubble diameter, [m]

F_b Buoyancy force of the particle, [N·m^–3]

F_d Drag force of the particle, [N·m^–3]

F_g Gravitational force of the particle, [N·m^–3]

F_p Pressure gradient force, [ N·m^–3]

F_vm Virtual mass force of the particle, [N·m^–3]

g Acceleration of gravity, [m·s^–2]

G Production of turbulent kinetic energy, [kg·m^–1·s^–3]

l Liquid phase, dimensionless

m_p Particle mass, [kg]

u_g Velocity of fluid phase, [m·s^–1]

u_l Velocity of gas phase, [m·s^–1]

u_p Velocity of particle, [m·s^–1]

ρ_l Density of fluid phase, [kg·m^–1·s^–2]

ρ_g Density of gas phase, [kg·m^–1·s^–2]

μ_eff Effective viscosity, [N·s·m^–2]

μ_l Molecular viscosity of liquid phase, [N·s·m^–2]

μ_t Turbulent viscosity, [N·s·m^–2]

μ_bl Bubble induced viscosity, [N·s·m^–2]

α_l Liquid volume fraction, dimensionless

α_g Gas volume fraction, dimensionless

ε Turbulent dissipation rate, [m²·s^–3]

${\prod }_{k,l},{\prod }_{\epsilon ,l}$  Influence of the dispersed phases turbulence on the continuous phase turbulence

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