Numerical Analysis of Effect of Operation Conditions on Bubble Distribution in Steel Continuous Casting Mold with Advanced Bubble Break-up and Coalescence Models

Abstract

Argon bubbles are usually injected into steel continuous casting mold to prevent the clogging of submerged entry nozzle (SEN), but some bubbles may be entrapped to form defects in the final slab. In order to provide a reference for improving the quality of steel, a mathematical model based on the Eulerian-Lagrangian approach with advanced bubble break-up and coalescence models was established to study the effect of operation conditions on bubble distribution in a steel continuous casting mold. A bubble break-up model based on a daughter bubble fraction, which is suitable for the continuous casting system, was considered. The mathematic model was validated by comparing of the size and number of captured bubbles with the plant measurements of previous work. The result shows that argon gas injection has obvious effect on the flow pattern in the upper recirculation zone of the mold. In the upper recirculation zone, the bubbles mean diameter decreases and the bubble number increases with increasing casting speed, and both of the bubble size and number increase with the increase of gas flow rate. From the result, it can be found that the number and diameter of bubbles arriving at the advancing solidified shell region increase with increasing casting speed. In addition, the increase of gas flow rate causes more bubbles arriving at the advancing solidified shell region, but has little effect on the size of bubbles.

1. Introduction

Continuous casting has been widely applied in the steel industry. As the last step of steelmaking, continuous casting plays an important role in controlling the quality of steel. Argon gas is usually injected to prevent nozzle clogging, encourage mixing molten steel and remove inclusions in the process of steel continue casting.¹⁾ The bubble size distribution plays an important role in the gas-liquid flow process. Large bubbles tend to float up to liquid steel surface and may cause the exposed eye of steel around the SEN,²⁾ resulting in slag entrapment at the same time. Small bubbles can be carried to the sidewall and may be captured by solidifying shell, causing defects in the final products, such as pencil pipe, sliver and so on. The operation conditions play a vital role in production efficiency and product quality. Besides, in the turbulent gas-liquid system, the behavior of bubble coalescence and break-up determine the bubble size distribution. Therefore, the research about the effect of operation conditions on bubble distribution with accurate bubble coalescence and break-up models in a steel continuous casting mold is essential to improve the final steel products.

Due to the high operating temperature, the direct investigation of bubble motion in the continuous steel mold is difficult and expensive. Extensive experiment works have been performed to study the gas-liquid flow in the continuous casting mold. Lee et al.³⁾ investigated the initial state of bubble formation through porous MgO refractory under different operating conditions. They quantified the effects of gas flow rate, water velocity and surface contact angle on initial bubble size distribution. Bai and Thomas⁴⁾ investigated the bubble size distribution during the bubble formation using a horizontal hole for injecting gas into a shearing downward flow. Ramos-Banderas et al.⁵⁾ have investigated the bubble interaction behavior in the continuous casting mold by water model experiments. They found that the bubble population and bubble size are closely related to the casting speed and gas flow rate. Esaka et al.⁶⁾ conducted water model experiments to study the interaction between bubbles and solidified shell, and they characterized the effect of bubble size, copper plate angle and the interfacial morphology between solid and liquid on the bubble entrapment.

A lot of mathematical models have been established to simulate the bubble motion in the continuous casting mold. Some of them used the Eulerian-Eulerian (E-E) approach. Both of bubble and liquid phases are treaded as continuum with averaged properties in the Eulerian-Eulerian method. Kubo et al.⁷⁾ studied the gas-liquid flow inside the continuous casting mold using Eulerian-Eulerian model, and the result was compared with the data from water model. Liu et al.⁸⁾ developed an Eulerian-Eulerian Large Eddy model to simulate the two-phase flow in the mold. Liu et al.^9,10,11) applied the population balance model using an Eulerian-Eulerian framework to investigated the bubble distribution in the continuous casting mold considering the bubble interaction. The result showed that the input bubble size distribution affects the molten steel flow field. However, Eulerian-Eulerian method can not get the individual bubbles information, such as path, velocity, location and so on. Others used the Eulerian-Lagrangian method. Most of the simulations with Eulerian-Lagrangian approach^{12,13,14,15,16,17,18)} did not consider the bubbles interaction. Zhang et al.^19,20) established a mathematic model with Eulerian-Lagrangian approach considering the bubbles interaction to simulate the bubbles transport in the continuous casting mold. However, the bubble coalescence model of Zhang is not suitable for large bubbles. Besides, two equal size daughter bubbles were generated from the break-up bubble in his model, which resulted in an inaccurate number of 100-micron-scale bubbles. Those small bubbles are the main reason of slab defects caused by bubble entrapment by solidifying shell. Due to the lack of reasonable bubble coalescence and breakup models, few papers have reported the effect of condition operation conditions on bubble distribution near the advancing solidified shell.

To simulate bubble entrapment in the continuous casting process, a suitable capture criterion is necessary. Many studies have focused on the behavior of a particle in front of the solidifying shell of steel during the continuous casting process. Several researchers^21,22) have simulated the entrapment of bubbles in the continuous casting process using a simple capture criterion that bubbles are entrapped when touching the solidification front. This is reasonable if the bubble diameter is smaller than the PDAS (Primary Dendrite Arm Spacing).²¹⁾ Zhang and Wang¹⁸⁾ investigated the particle entrapment considering the solid shell and the solidification heat transfer, and they introduced a capture criterion that particles are entrapped if they go to the region with a liquid fraction smaller than 0.6. However, mushy region must be applied with fine meshes, which needs very computationally demands, and the computation results depends largely on some empirical parameters in this method. Yuan²³⁾ developed a capture criterion considering local force balance on the bubble at the solidification front. It includes the effect of Primary Dendrite Arm Spacing (PDAS), interfacial tension, and other effects.

This present work presents an analysis of the effect of operation conductions on bubble distribution in the continuous casting mold with the Eulerian-Lagrangian method. Advanced bubble break-up and coalescence models are established. The model was validated by the experimental plant measurements. The bubble size and number distributions in the upper recirculation zone and the region near the advancing solidified shell in the mold were studied. Besides, the effect of casting speed and gas flow rate on the bubbles distribution in the mold were investigated.

2. Mathematical Model Formulation

2.1. Fluid Phase Equations

The liquid phase is computed by solving the volume-average Navier-Stokes equations. Equations that express the mass and momentum transfers are:   

$$\frac{\partial }{\partial t}({\alpha }_{\text{l}}{\rho }_{\text{l}})\text{+}\nabla \cdot \left({\alpha }_{\text{l}}{\rho }_{\text{l}}\overrightarrow{u_{\text{l}}}\right)\text{=}0$$(1)

  

$$\frac{\partial }{\partial t}\left({\alpha }_{\text{l}}{\rho }_{\text{l}}\overrightarrow{u_{\text{l}}}\right)+\overrightarrow{v_{\text{l}}}\cdot \nabla \left({\alpha }_{\text{l}}{\rho }_{\text{l}}\overrightarrow{u_{\text{l}}}\right)=-\nabla p+\nabla \cdot [{\alpha }_{\text{l}}({\mu }_{\text{l}}+{\mu }_{\text{t}})]\nabla \overrightarrow{u_{\text{l}}}+F_{\text{gl}}$$(2)

where α_l, ρ_l, $\overrightarrow{u_{\text{l}}}$, μ_l are liquid-phase volume fraction, density, velocity, viscosity respectively. p is pressure, F_gl is the interfacial forces acting on liquid phase by gas. The liquid-phase volume fraction, α_l is calculated as:   

$${\alpha }_l=1-\frac{{\sum }_i{V}_{\text{d},i}}{V_{\text{cell}}}$$(3)

where, V_d,i is the discrete phases volume and V_cell is the grid cell volume. μ_t in Eq. (2) is the turbulent viscosity, which is defined as:   

$${\mu }_{\text{t}}=C_{\mu }{\rho }_{\text{l}}\frac{k^2}{\epsilon }$$(4)

The standard k–ε model is used to model turbulence. The transport equations of k and ε are as follows.   

$${\alpha }_{\mathrm{l}}{\rho }_{\mathrm{l}}\left(\frac{\partial k}{\partial t}+\overrightarrow{u_{\text{l}}}\cdot \nabla k\right)=-\nabla \left({\alpha }_{\mathrm{l}}\frac{{\mu }_{\text{t}}}{{\sigma }_k}\nabla k\right)+{\alpha }_{\mathrm{l}}{G}_k-{\alpha }_{\mathrm{l}}{\rho }_{\mathrm{l}}\epsilon$$(5)

  

$${\alpha }_{\mathrm{l}}{\rho }_{\mathrm{l}}\left(\frac{\partial \epsilon }{\partial t}+\overrightarrow{u_{\text{l}}}\cdot \nabla \epsilon \right)=-\nabla \left({\alpha }_{\mathrm{l}}\frac{{\mu }_{\text{t}}}{{\sigma }_{\epsilon }}\nabla \epsilon \right)+{\alpha }_{\mathrm{l}}{C}_1\frac{\epsilon }{k}{G}_k-{\alpha }_l{C}_2{\rho }_l\frac{{\epsilon }^2}{k}$$(6)

  

$$G_k={\mu }_{\text{t}}\left(\frac{\partial u_{i,j}}{\partial x_j}+\frac{\partial u_{i,j}}{\partial x_i}\right)\frac{\partial u_{i,j}}{\partial x_j}$$(7)

where standard model constants C_μ = 0.99, σ_k = 1.0, σ_ε = 1.3, C₁ = 1.44, C₂ = 1.92.

2.2. Bubble Transport Model

In the present work, a coupled Lagrangian approach is used to calculate the bubble transport. The governing equations for translational motion of each bubble can be calculated as   

$$m_{\text{b}}\frac{d{\overrightarrow{u}}_{\text{b}}}{dt}=-F_{\text{gl}}=F_{lg}=\overrightarrow{F}{}_{lg}^{\text{d}}+\overrightarrow{F}{}_{lg}^{\text{p}}+\overrightarrow{F}{}_{lg}^{\text{buo}}+\overrightarrow{F}{}_{lg}^{\text{vm}}+\overrightarrow{F}{}_{lg}^{\text{gra}}+\overrightarrow{F}{}_{lg}^{\text{L}}$$(8)

where ${\overrightarrow{u}}_{\text{b}}$ and m_b are the bubble velocity and mass respectively, F_lg represents the interfacial forces acting on gas by liquid phase. The forces considered in the model are drag force, pressure gradient force, buoyancy force, virtual mass force, gravitational force and lift force as shown in the following equations.   

$$\overrightarrow{F}{}_{lg}^{\text{d}}=C_{\text{D}}\frac{{\rho }_{\text{l}}\left|\overrightarrow{u}-{\overrightarrow{u}}_{\text{b}}\right|\left(\overrightarrow{u}-{\overrightarrow{u}}_{\text{b}}\right)}{2}\frac{\pi d_{\text{b}}^2}{4}$$(9)

  

$$\overrightarrow{F}{}_{lg}^{\text{p}}=\frac{1}{6}\pi d_{\text{b}}^3{\rho }_{\text{l}}\frac{d\overrightarrow{u}}{dt}$$(10)

  

$$\overrightarrow{F}{}_{lg}^{\text{buo}}=-\frac{\text{1}}{\text{6}}\pi d_{\text{b}}^{\text{3}}{\rho }_{\text{l}}\cdot \overrightarrow{g}$$(11)

  

$$\overrightarrow{F}{}_{lg}^{\text{vm}}=\frac{1}{6}\pi d_{\text{b}}^3{C}_{\text{vm}}{\rho }_{\text{l}}\frac{d}{dt}\left(\overrightarrow{u}-{\overrightarrow{u}}_{\text{b}}\right)$$(12)

  

$$\overrightarrow{F}{}_{lg}^{\text{gra}}=\frac{1}{6}\pi d_{\text{b}}^3{\rho }_{\text{b}}\cdot \overrightarrow{g}$$(13)

  

$$\overrightarrow{F}{}_{lg}^{\text{L}}=-\frac{1}{6}\pi d_{\text{b}}^3{C}_{\text{L}}{\rho }_{\text{l}}\left({\overrightarrow{u}}_{\text{b}}-\overrightarrow{u}\right)\left(\nabla \times \overrightarrow{u}\right)$$(14)

Random walk model was used to consider the effect of turbulent velocity fluctuation on the transport of bubbles.   

$${\overrightarrow{u}}_{\text{b}}={\overrightarrow{u}}_{\text{b}}+\xi \sqrt{2k/3}$$(15)

where k is the local turbulent kinetic energy and ξ is a Gaussian-distributed rand number ranging from 0–1.

2.3. Bubble Breakup Model

In a turbulent dispersion system, the bubble break-up occurs through the bubble interaction with turbulent eddies. The eddies have an uneven pressure distribution on a bubble surface, thus causing bubble break-up. The bubble break-up criterion and breakup fraction are two important parameters in the bubble breakup model. The result of Sevik and Park²⁴⁾ shows that the breakup of bubble is related with the turbulent kinetic energy dissipation rate. The maximum stable bubble diameter, d_max, is calculated as   

$$d_{\text{max}}=W{e}_{\text{crit}}^{3/5}\frac{{\sigma }^{3/5}}{{\rho }_{\text{l}}^{3/5}}{\epsilon }^{-2/5}$$(17)

where ε is the turbulent kinetic energy dissipation rate, σ is the gas-liquid surface tension. Zhang et al. showed that the critical Weber number, We_crit, for bubble break-up is 0.53 in the argon-steel melt systems.

Only the binary breakup into two daughter bubbles is considered. The break-up volume fraction of daughter bubble f_bv is an essential parameter in the bubble break-up process. The break-up volume fraction is a random variable, which is obtained by empirical observations or statistical distribution. By categorizing the basis of the shape of daughter size distribution, the daughter size distribution can be classified into three categories,²⁵⁾ as shown in Fig. 1.

Fig. 1.

Shapes of daughter bubble size distribution. (a) Bell-shape (b) M-shape (c) U-shape (d) shape of this work.

Bell-shape: The mother bubble breaks into two daughter bubbles with the same size has the highest probability, while the probability of generating one small and one large daughter bubble is lowest.

M-shape: Neither the probability that the formation of two same size daughter nor a small daughter bubble occurs is high, but the probability of breakup volume fraction between these two extremes is highest.

U-shape: the probability that the formation of one small daughter bubble is high and two daughter bubbles of the same size break off from the mother bubble has low probability.

Nambiar et al.²⁶⁾ noted that the U-shape is the most reasonable daughter size distribution for that generating a small daughter bubble requires the least energy than the other models. However, in the model of U-shape, only the energy constraint is considered, and the capillary pressure constraint of bubble themselves is ignored, as known from Wang et al.²⁷⁾ Wang et al.²⁷⁾ indicated that the mother bubble size and energy dissipation rate both affect the daughter bubbles fraction. In this paper, the maximum energy dissipation rate is about 1 m²/s³near the SEN ports, and the smallest break-up bubble is about 6 mm. So as known from the model of Wang et al.,²⁷⁾ the daughter bubble size distribution tends to U-shape, but it can not be infinitely small due to the capillary pressure constraint of bubble themselves. Thorsten and Tacke²⁸⁾ have reported that the smallest blister in the casting slab is about 60 μm. Therefore, f_bv is calculated with a tanh function, which has U-shape distribution with lower limits.   

$$f_{\text{bv}}=\frac{1}{2}[1\text{+}tanh(14.4x-7.2)]$$(18)

where x is a random variable from 0 to 1.

Because of the nature of the Lagrangian approach, the information of the daughter bubble location is required after a bubble break-up event. According to Lau et al.,²⁵⁾ the location of larger daughter bubble is at the original position of mother bubble. The smaller daughter bubble is placed randomly around the larger daughter bubble. The center-to-center line distance between two daughter bubbles is 1.1 times the sum of the radii of the two daughter bubbles to avoid the immediate coalescence in a subsequent simulation time step. The velocities of the two daughter bubbles are consistent with the mother bubble.

2.4. Bubble Collision Model

The bubbles collision coalescence is divided into three steps: Two bubbles collide then forming a liquid film; the liquid film drains out to the critical thickness during the contact of two bubbles; bubbles coalesce at the moment of rupture of liquid film. If liquid film does not drain out to a critical thickness during the bubble contact time, two bubble will bounce off. Chesters and Hofman²⁹⁾ studied the bubble coalescence process by developed a model of bubble liquid film thinning process, which predicted that the bubbles coalesce if the relative Weber number is less than a critical value. Duineveld³⁰⁾ got a critical We between coalescence and bouncing of 0.18 with the water experiment of side by side collision. In recent years, Yang et al.³¹⁾ indicated that the spherical bubble collision result is also related to the off-center degree when collision. The result shows that bubbles coalesce when We(1 - B²) < 0.16 + B/24. The relative Weber is calculated as:   

$$We=\frac{r_{\text{e}}{v}_{\text{r}}^2{\rho }_{\text{l}}}{\sigma }$$(19)

where r_e is the equivalent radius of two bubbles, calculated as: r_e = 2r₁r₂/(r₁ + r₂); v_r is the relative velocity; ρ_l is the liquid density and σ is the liquid surface tension.

The parameter B, which characterizes the off-center degree, is a dimensionless number from 0 to 1. It is head-on collision when B is 0, and it is edge collision when B is 1. The impact parameter B is expressed as:   

$$B=\frac{b}{r_1+r_2}$$(20)

where r₁ and r₂ are the radii of the two bubbles, b is the distance from the center of the one bubble to the relative velocity vector originating from the center of the other bubble at contact, as shown in Fig. 2(a).

Fig. 2.

(a) Bubble bounce off process; (b) Bubble coalescence process. (Online version in color.)

However, the model proposed by Yang et al.³¹⁾ is established based on spherical bubble. It is not suitable for ellipsoid bubbles with large size. The result got by Ribeiro and Mewes^31,32) shows that the critical relative velocity is a decreasing function of the equivalent diameter of two bubbles when smaller than 2.30 mm, and it becomes constant if the equivalent diameter is larger than 2.30 mm. In the present work, the model of Yang et al.³¹⁾ is used when the equivalent diameter is below 2.30 mm. When the equivalent diameter is larger than 2.3 mm, two bubbles bounce off if the relative approach velocity exceeds 0.11 m/s.^32,33)

In this work, the bubbles collision occurs when the center-to-center line distance of two bubbles is less than the sum of their radii. Two bubbles bounce off (Fig. 2(a)) with a completely elastic collision according to the momentum conservation. The velocities of two bubbles after bouncing off can be calculated as   

$$u_{\text{ra}}^{\text{'}}=2\frac{m_a{u}_{\text{ra}}+m_b{u}_{\text{rb}}}{m_a+m_b}-u_{\text{ra}}$$(21)

  

$$u_{\text{rb}}^{\text{'}}=2\frac{m_a{u}_{\text{ra}}+m_b{u}_{\text{rb}}}{m_a+m_b}-u_{\text{rb}}$$(22)

where $u_{\text{ra}}^{\text{'}}$ and $u_{\text{rb}}^{\text{'}}$ are the velocities projected on the line connecting bubble centers after collision. m_a and m_b are the masses of bubbles.

After bubbles coalesce (Fig. 2(b)), the velocity of the newly formed bubble is given by   

$$u_{\text{c}}=\frac{m_{\text{a}}{u}_{\text{a}}+m_{\text{b}}{u}_{\text{b}}}{m_{\text{a}}+m_{\text{b}}}$$(23)

where u_c is the velocity of newly bubble formed by coalescence.

2.5. Bubble Entrapment Model

There are several models to predict the capture of bubbles during the continuous casting in the previous studies.^18,34) Recently, an advanced capture criterion, which considers the local force balance on the bubble at the solidification front, was proposed by Yuan and Thomas.^17,23) The effects of bubble size, local flow field, PDAS and some special forces are considered in this model. The flow chart of the model is shown in Fig. 3(a). Bubbles smaller than PDAS can move between the dendrite arms easily to be captured when the bubbles touch the wall of the mold. If the bubbles are larger than PDAS, forces acting on the spherical bubble when touching dendrites are shown in Fig. 3(b). The additional forces, including lubrication force F_lub, surface tension gradient force F_Grad, Van der Waals force F_IV, are also considered.   

$$F_{lub}=6\pi V_{\text{sol}}\frac{R_{\text{b}}^2}{h_{\text{o}}}{\left(\frac{r_{\text{d}}}{r_{\text{d}}+R_{\text{b}}}\right)}^2$$(24)

  

$$\begin{array}{l}{F}_{\text{Grad}}=-{\displaystyle \frac{m\beta \pi R_{\text{b}}}{{\xi }^2}}\\ \{{\displaystyle \frac{({\xi }^2-R_{\text{b}}^2)}{\beta }}ln\left[{\displaystyle \frac{(\xi +R_{\text{b}})[\alpha (\xi -R_{\text{b}})+\beta ]}{(\xi -R_{\text{b}})[\alpha (\xi +R_{\text{b}})+\beta ]}}\right]+{\displaystyle \frac{2R_{\text{b}}}{\alpha }}-{\displaystyle \frac{\beta }{{\alpha }^2}}ln\left[{\displaystyle \frac{\alpha (\xi +R_{\text{b}})+\beta }{\alpha (\xi -R_{\text{b}})+\beta }}\right]\}\end{array}$$(25)

  

$$F_{\text{IV}}=2\pi (E_{\text{sb}}-E_{\text{sl}}-E_{\text{bl}})\frac{r_{\text{d}}{R}_{\text{b}}}{r_{\text{d}}+R_{\text{b}}}\frac{a_{\text{o}}^2}{h_{\text{o}}^2}$$(26)

where R_b is the bubble radius; r_b is the radius of dendrite tip; h_o is the distance between the dendrite tip and bubble; a_o is the diameter of the liquid steel atomic; E is the surface energy and the subscripts b, l, and s denote the bubble, liquid steel solid steel respectively; ξ = R_b+ + r_d + h_o, α = 1 + nC₀ and β = (C^* − C₀)nr_d where m, n, C^* and C₀ are model parameters. More details about parameters in the force expression can be found in a previous work.²³⁾

Fig. 3.

(a) Flow chart of bubble capture criterion by shell; (b) Schematic diagram of bubble touching dendrite tips. (Online version in color.)

2.6. Simulation Details

The work is based on the No.4 continuous slab caster of Baosteel Shanghai¹²⁾ for validating the mathematic model. The geometric schematic and boundary conditions are shown in Fig. 4. The bubble model with a Lagrange approach is a developed code written in C langue. The liquid phase is dealt with the commercial package ANSYS FLUENT coupling with the discrete phase realizing with the help of extensive user-defined subroutines developed by the authors. The parameters of geometry and operation conditions are given in Table 1. Three different grids with different quantities of cells (500000, 650000 and 750000) where each grid has a similar meshing scheme were tested. There were no significant differences in the flow field and bubble distribution between the three grids. Therefore, the mesh consisting of 500000 was used. The simulation time step is 0.001 s.

Fig. 4.

Schematic diagram of the mold and boundary conditions. (Online version in color.)

Table 1. Process Parameters.

Parameters	value
Height of SEN port (mm)	83
Width of SEN port (mm)	65
Nozzle port downward angle (deg)	15
SEN submergence depth (mm)	160
Width of slab (mm)	1300
Thickness of slab (mm)	230
Length of slab (mm)	2000
Casting speed (m/min)	1.5
Argon volume fraction (vol pct)	8.2
Liquid density (kg/mm3)	7000
Gas density (kg/mm3)	0.5
Liquid viscosity (kg/(m·s))	0.0063
Gas viscosity (kg/(m·s))	0.0000212
Initial bubble diameter (mm)	1 (30%), 1.5 (45%), 2 (25%)

The conditions for molten steel are as follows: (I) the top inlet of the SEN is set with a constant velocity inlet boundary condition; (II) the bottom of the mold is applied as the outflow outlet boundary condition; (III) the liquid top surface is modeled as the free-slip condition; (IV) no-slip condition boundary is set at the SEN wall and the liquid pool wall. For the bubbles, the escape boundary is applied for the top surface of the mold. The bubbles touching the walls of SEN are modeled to be reflected. The criteria for bubbles that touching the side walls of liquid pool to be caught or reflected is based on the bubble entrapment model.

Argon Bubbles were injected from the walls of a top portion of the SEN. The bubbles are assumed to be spherical. The diameter of initial bubble ranges from 1 to 2 mm based on the research by Lee et al.,³⁾ as seen in Table 1. For studying the bubble distribution in the continuous casting mold in the following sections, the domain in the upper recirculation zone and the region within 15 mm near the advancing solidified shell were divided into twenty equal zones, as shown in Fig. 4.

3. Model Validation

3.1. Comparison of Bubbles Behavior

A simplified water model recorded by a high speed camera in the mold was established to observe the bubbles behavior. Figure 5(a) shows the process of bubble coalescence in the simulation and water experiment. It can be seen that two bubbles may coalesce into a large bubble when they collide. The process that the large bubble break up into two small bubbles with different diameters is shown in Fig. 5(b). From the figures, it can be seen that the both of the phenomena bubble coalescence and breakup in the simulation agree well with the water model observation.

Fig. 5.

Comparison of bubble coalescence (a) and breakup (b) between simulation and experiment. (Online version in color.)

3.2. Comparison with Bubble Entrapment

Whether the entrapment bubbles can be observed in the examined layers in the simulation is illustrated in Fig. 6. The entrapment of a bubble j with radius r_j at the distance s_j below the strand surface of final slab can be observed on the examined surface of sample k at distance s_k blow the slab surface if it satisfies the following conditions:   

$$\{s_j<s_k\mathrm{\hspace{0.17em} }and\mathrm{\hspace{0.17em} }{s}_j+r_j>s_k\}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{o}\mathrm{r}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\{s_j>s_k\mathrm{\hspace{0.17em} }and\mathrm{\hspace{0.17em} }{s}_j-r_j<s_k\}$$(27)

The shell thickness is calculated as $s=3\sqrt{t}$ mm,¹²⁾ where s is the solidification shell thickness and t is the solidification time. In the continuous casting process, the solidification time can be determined by $t=L/v_{\text{c}}=40L$, where L is the distance below meniscus and v_c is continuous casting speed. Therefore, $s_j=3\sqrt{40L_j}$, where L_j is the distance between the entrapped bubble and the meniscus. The comparison of bubble entrapped in the examined layers between the simulation and the plant measures is shown in Fig. 7. It can be found that the simulation results with advanced bubble break-up and coalescence models agree better with the plant measurement than that without modification. This indicates that the advanced bubble break-up and coalescence models in the present work are reliable.

Fig. 6.

Determination whether the entrapment bubbles can be observed on the surface. (Online version in color.)

Fig. 7.

(a) Locations of samples and examined layers. Predicated bubble distribution in every layer on the wide face: (b) Bubble number per cm²; (c) Average bubble diameter; Predicated bubble distribution in every layer on the narrow face: (d) Bubble number per cm²; (e) Average bubble diameter. (Online version in color.)

4. Result and Discussions

4.1. Liquid Flow and Bubbles Motion in the Mold

Figure 8 shows the contours of liquid velocity and its vectors with and without gas injection at the center cross-sectional plane. Figure 8(a) shows the liquid flow without gas injection. One can notice that a jet flow impinges on the narrow face of the mold when the liquid gets out of the SEN port. Then the liquid splits vertically to create an upper and a lower recirculation zones, forming a typical double-roll flow pattern. Figure 8(b) shows the flow pattern with gas injection. From the figure, one can notice that the gas injection has obvious effect on the flow pattern. Part of liquid goes up toward to the top surface after out of the SEN exit, while another part of fluid continues impinging on the narrow face. In addition, because the molten steel is affected by the drag from the bubble, the flow velocity in the upper recirculation with gas injection is significantly smaller than that without gas injection.

Fig. 8.

Velocity magnitude and streamlines in the mold without argon injection (a) and with argon injection (b). (Online version in color.)

Figure 9 shows the bubble motion in the continuous casting mold. If the diameters of bubbles are smaller than 0.4 mm, they are difficult to be observed with original size scale. Therefore, different bubble sizes are represented by different colors in the figure. From Fig. 9(a), it can be seen that the diameter of bubbles ranges from 1 mm to 2 mm at the beginning of gas injection. As the bubbles move down with the steel in the SEN, a large number of bubble coalescence phenomena occur, which can be known from the change of bubble color in Fig. 9(b). As the continued movement of bubbles, a lot of bubble break-up phenomena occur when the bubbles get out the SEN ports due to the high turbulent stress near the port, which can be reflected by the blue bubbles. From the Figs. 9(c)–9(e), it can be notice that the large bubbles float up quickly after out of the ports due to the large buoyancy. And the small bubbles have the tendency to follow the main stream to the narrow face. These fine bubbles are easy to be entrained by the stream of molten steel, and thus some of them go down to the depth of the mold, and others go to the upper recirculation zone, as seen in Figs. 9(f)–9(h).

Fig. 9.

Bubble movement in the mold at different moments. (a) 0 s, (b) 0.15 s, (c) 0.65 s, (d)1 s, (e) 2 s, (f) 3 s, (g) 8 s, (h) 16 s. (Online version in color.)

4.2. Effect of the Casting Speed

Casting speed is an important production index in continuous casting process. A higher casting speed means a higher productivity. However, a high casting speed always results in more slag entrapment and internal defects. For investigating the effect of casting speed on bubble distribution in the mold, three different casting speeds with other process parameters unchanged were simulated. Figure 10 shows the whole bubble distribution in the upper recirculation zone of continuous casting model. It can be seen that the higher casting speed results a more widely bubble distribution in the upper recirculation zone. More bubbles are carried to the region far away from the SEN with the increase of casting speed. The bubble size decreases with the increase of casting speed.

Fig. 10.

The bubble distribution inside the mold with different casting speeds. (a) 1.3 m/min (b) 1.5 m/min (c) 1.7 m/min. (Online version in color.)

Figure 11 presents the size and number distribution of bubbles in the upper recirculation zone with three casting speeds. The results indicate that the mean radius and number of bubbles under all casting speeds decrease with the increase of the distance from SEN. The reason is that most of the large bubbles float up with a short moving distance after escaping from the SEN port due to the large buoyancy. Figure 11(a) shows that the bubble mean radius in the region within a distance of 0.15 m from the SEN with high casting speed is smaller than that with low casting speeds. The bubbles number increases with increasing casting speed. Because the critical break-up size of bubbles decreases due to the higher turbulent with the higher casting speed, and more bubbles break up while out the SEN port, which results in that the quantity of small bubbles increases in the mold and the size of bubbles decrease near the SEN port. In addition, the mean diameter of bubble with the casting speed of 1.7 m/s tends to be constant, while that with 1.3 and 1.5 m/s decreases with the distance from the SEN. The reason is that critical break-up size of bubbles decreases due to the higher turbulent with the higher casting speed, resulting in more uniform bubble size with the casting speed of 1.7 m/s. Besides, greater liquid velocity has a stronger drag force on bubbles, carrying large bubbles to farther region. So, the mean diameter of bubble with 1.7 m/s the casting speed of tends to be constant. In the region farther than 0.15 m from the SEN, the bubble mean radius at high casting speed is larger than with low casting speed. Because the bubble horizontal velocity is mainly determined by the liquid steel velocity. A higher steel flow rate has a stronger driving force on bubbles, resulting in a lager bubble horizontal velocity. This means more large bubbles are carried to the farther region from the SEN under a higher casting speed.

Fig. 11.

(a) Bubble size distribution and (b) bubble number distribution under different casting speeds in upper recirculation zone. (Online version in color.)

Figure 12 presents the size and number distributions of bubbles at the region near the narrow face under different casting speeds. The results indicate that the mean radius of bubble decreases with increasing depth from the meniscus within 0.5 m, and it tends to small and constant in the deeper region. Because larger bubbles tend to float up along the narrow face, the smaller bubbles can reach the deeps of the mold along with the downward flow. From Fig. 12(a), one can notice that the mean radius of bubble near the narrow face increases with the increase of casting speed. Because a higher steel flow rate has a stronger driving force on the bubbles and carries more larger bubbles to the narrow face, which is mentioned in the upper recirculation zone. This means more larger bubbles may be entrapped by the advancing solidified shell in the narrow face when the casting speed increases, resulting in larger internal defects of the slab. It can be seen from Fig. 12(b) that the bubble number decreases with the increase of the distance within 0.7 from meniscus except a jump region 0.4 m below the meniscus. Because many smaller bubbles reach the region 0.4 m below the meniscus with the jet liquid flow, as seen Fig. 8(b). Meanwhile, the number of bubbles at the region near the narrow face increases with increasing the casting speed. Because a higher casting speed causes a higher turbulent kinetic energy dissipation rate, decreasing the critical breakup size of bubbles. So more small bubbles were generated by breakup, resulting in more bubbles getting to the narrow face. This means high casting speed may cause more inner defects in slab at the narrow face.

Fig. 12.

(a) Bubble size distribution and (b) bubble number distribution under different casting speeds in the region near the narrow face. (Online version in color.)

The predictions of the mean radius and number of bubbles at the region near the wide face under different casting speeds are presented in Fig. 13. From Fig. 13(a), one can notice that the mean radius of bubbles in the region near the meniscus at the wide face is larger than that at the narrow face. The reason is that the wide face is closer to the SEN, so more large bubbles can reach the wide face. But there is no difference in the deeper region. Because the bubbles that can follow the downward flow and reach the depths near the wide face are small bubbles with similar size and good ability of following. Similar to the narrow face, the mean radius and number of bubbles in the region near the wide face increase with increasing casting speed, which means that there may be more and larger internal defects at the wide face with the increase of casting speed.

Fig. 13.

(a) Bubble size distribution and (b) bubble number distribution under different casting speeds in the region near the wide face. (Online version in color.)

Fig. 14.

The bubble distribution under different argon volume fractions. (a) 4% (b) 8.2% (c) 12%. (Online version in color.)

4.3. Effect of Gas Flow Rate

The gas flow rate plays an important role in the continuous casting process. Low gas flow rate may not achieve the function of protecting SEN clogging by inclusions. High gas flow rate may cause slag entrapment, resulting in product defects. For investigating the effect of gas flow rate, three different argon volume fractions were simulated with other process parameters being the same as the No. 4 continuous slab caster of Baosteel Shanghai in this work. Figure 10 presents the distribution of bubbles in the mold. The results show that the gas flow rate has no obvious effect on the distribution range of bubbles, but the bubble size and number increase with increasing the gas flow rate.

The number and size distributions of bubbles in the upper recirculation zone with three argon volume fractions are shown in Fig. 15. The result shows that a high gas flow rate leads to the increase of the bubble size near the SEN. Because a higher argon volume fraction means that more bubbles coalesce into larger ones, which may cause the slag entrapment. However, the farthest distance from the SEN that the bubbles with large size can reach are mainly determined by the steel flow rate, so there is no significant difference in bubble size when far away from the SEN when the casting speed is the same. One can notice that the number of bubbles increases with increasing gas volume fraction. Higher argon volume fraction means more big bubbles, which can break up into more small bubbles, resulting in the increase of bubble number in the mold.

Fig. 15.

(a) Bubble size distribution and (b) bubble number distribution under different argon volume fractions in upper recirculation zone. (Online version in color.)

The effects of gas flow rate on the number and size distribution of bubbles in the region near the narrow face are present in Fig. 16. The results show that the gas flow rate has no significant effect on the bubble size distribution. The reason has been stated in the analysis of upper recirculation that the farthest distance that bubbles with the same size can reach is mainly determined by the steel flow rate. So, there is almost no difference in the size of bubbles that can reach the narrow face with different gas flow rate under the same casting speed. From Fig. 16(b), one can notice that the bubble number at the narrow face increases with gas flow rate. Because, under the higher gas flow rate, more large bubbles birth due to the bubbles coalescence, resulting in more small bubbles generated by the break-up of large bubbles. Near the wide face, the size and number distributions of bubbles are similar with that at narrow face, as shown in Fig. 17. This means a higher gas flow rate may cause more bubbles entrapped by the advancing solidification shell, resulting in more internal defects of the slab but not affecting the defect size.

Fig. 16.

(a) Bubble size distribution and (b) bubble number distribution under different argon volume fractions in the region near the narrow face. (Online version in color.)

Fig. 17.

(a) Bubble size distribution and (b) bubble number distribution under different argon volume fractions in the region near the wide face. (Online version in color.)

5. Conclusions

Eulerian-Lagrangian simulations considering the coalescence and break-up of bubbles have been conducted to study the bubble transport in the steel continuous casting mold. Modified bubble coalescence model has been established to accommodate the off-center collision of sphere and non-sphere bubbles. A bubble break-up model based on a daughter bubble volume fraction following a U-shape profile, which is suitable for the continuous casting system, was considered. The effects of casting speed and argon flow rate on the distribution of bubbles in the mold were investigated. The conclusions from the study can be summarized as:

(1) Argon gas injection has obvious effect on the flow pattern in the upper recirculation zone of the mold. The result shows that part of liquid floats up to the top surface leaving the SEN exit, and another part rushes to the narrow surface.

(2) Large bubbles float up quickly leaving the SEN ports, while small ones are carried to the narrow face or even the deeps of the mold by the main flow. The bubble size and number in the region near the narrow and wide faces both decrease within a range with increasing depth from the meniscus, and they get almost to be constant in the deep region.

(3) In the region of upper recirculation: the number of bubbles increases with increasing casting speed; the size of bubbles near the SEN decreases with the increase of the casting speed but it increases in the region far away the SEN. In the region near the narrow and wide faces, both the number and size of bubbles increase with the increase of casting speed.

(4) The number of bubbles in the upper recirculation increases with the increasing of gas flow rate. A high argon volume fraction leads to the increase of the bubble radius near the SEN, but there is no significant difference when far away from the SEN. There are more bubbles near the narrow and wide faces with the increase of gas flow rate, but the gas flow rate has no significant effect on the bubble size distribution at the narrow and wide faces.

Acknowledgements

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. The authors wish to express thanks to Yang You and Weiqiang Liu for the academic communication.

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