numerical simulation of in-mold electromagnetic stirring
TRANSCRIPT
ISIJ Int. 61(6): 1860-1871 (2021)© 2021 ISIJ 1860
* Corresponding author: E-mail: [email protected]
© 2021 The Iron and Steel Institute of Japan. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs license (https://creativecommons.org/licenses/by-nc-nd/4.0/).
ISIJ International, Vol. 61 (2021), No. 6, pp. 1860–1871
https://doi.org/10.2355/isijinternational.ISIJINT-2020-745
1. Introduction
During the continuous casting, molten steel enters the casting mold through a Submerged Entry Nozzle (SEN), the flow rate of the molten steel is commonly controlled by either slide gate or stopper rod. The off-center slide gate may generate an asymmetric or bias flow in the mold, which may induce asymmetrical level fluctuations and possibly to enhance the risks of superficial defects, such as hook formation, longitudinal cracks.1,2) Therefore, understanding quantitatively the molten steel flow affected by the slide gate and optimizing the asymmetric flow in the mold are of great importance to improve steel quality.
Extensive and elaborate past work were conducted on the different aspects of slide gate such as slide-gate open- ing rate,3–7) slide gate orientation,3) slide gate design8,9) and the influence of these parameters on the mold flow structure.10,11) Bai et al.3,4) simulated two phase flow under different gate opening fractions and orientations, the results were verified using experimental data and found that the off-
Numerical Simulation of In-mold Electromagnetic Stirring on Slide Gate Caused Bias Flow and Solidification in Slab Continuous Casting
Haibiao LU,1) Bin LI,1) Jiaxun LI,1) Yunbo ZHONG,1) Zhongming REN1) and Zuosheng LEI1,2)*
1) State Key Laboratory of Advanced Special Steel & Shanghai Key Laboratory of Advanced Ferrometallurgy, Shanghai Univer- sity, Shanghai, 200444 China. 2) Sino-European School of Technology, Shanghai University, Shanghai, 200444 China.
(Received on December 9, 2020; accepted on February 26, 2021)
A mathematical model coupled with electromagnetic field, flow, heat transfer and solidification has been developed to simulate the bias flow caused by slide gate under different slide-gate opening rates, EMS currents and casting speeds. Through comparing the magnetic flux density and flow field with measured results, the reliability of mathematical model is proved. The symmetric index, variance of solidified shell thickness have been introduced to judge the symmetry of flow field and uniformity of solidified shell, respectively. The results show that bias flow phenomenon has happened when the slide-gate opening rate is less than 100%, the deviated direction is opposite to the opening direction of slide gate. As the slide- gate opening rate decreases, the symmetry of flow field and uniformity of solidified shell decrease. Increasing the EMS current and decreasing the casting speed, the symmetry of flow field caused by slide gate increases, but it can not eliminate completely, while the uniformity of solidified shell increases firstly and then decreases. There exists an optimal EMS to balance the symmetry of flow field and uniformity of solidified shell.
KEY WORDS: continuous casting; slide gate; bias flow; electromagnetic stirring; uniformity of solidified shell.
center blocking effect of the slide gate generated asymmetric flow and the 0-deg gate orientations the worst bias flow between the left and right ports. Gursoy et al.6) developed mathematical model for different slide gate and stopper rod opening rates in a slab continuous casting, the flow structure and meniscus velocity were discussed. Cedillo et al.7) used the water model and mathematical model to investigate the effects of slide gate opening on the flow field in the mold and demonstrated that the slide gate opening will cause the bias flow in the mold. Kononov et al.8) designed new slide gate systems for ingot and shaped casting. However, although great advances have been made in optimizing the flow pattern in the mold and submerged entry nozzle with slide gate, there still exist distinct bias flow in the mold.
ElectroMagnetic (EM) systems have been validated as an attractive method to control relevant phenomena related to fluid flow in continuous casting. Recently many research- ers12–17) have previously reported that electromagnetic brake (EMBr) can improve the asymmetric flow either in SEN or mold. Garcia-Hernandez12) et al. studied the effects of mold curvature, slide gate and EMBr fields on steel flow in a slab mold with a mathematical model, the results showed
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that the symmetry of flow field increases with the increase of magnetic flux density. Jin et al.13) constructed a math- ematical model to study the effect of electromagnetic brake on the bias flow in the mold together with slide gate, the results show that EMBr can reduce the extent of asymmetric flow caused by slide gate. Additionally, Electromagnetic stirring (EMS), as another electromagnetic controller, has shown the ability to change the flow pattern18–21) and opti- mize the adverse mold flow such as the bias flow caused by SEN clogging.22,23) Li et al.22,23) reported that M-EMS can greatly improve the asymmetry of flow field in the slab mold with SEN clogging, and the stirring intensity should be controlled below a safe level to avoid slag entrapment. However, relatively little study has reported on the influence of EMS on flow field in the mold under slide gate system. Geng et al.24) investigated the effect of rotating magnetic field on the flow field in the SEN and the billet mold under different slide-gate opening rates, and showed that the EMS current can improve the bias flow in the mold, but the elec- tromagnetic stirrer was equipped around the SEN but not the mold, furthermore, most of research did not consider the solidified shell in their work, in addition, some other process parameters, such as the casting speed and SEN depth, are also lack of consideration.
The current work presents a mathematical model to inves- tigate the influence of EMS on bias flow, heat transfer and solidification behavior in an actual slab mold under different slide-gate opening rates and casting speeds. By comparisons with the magnetic flux density data and flow field, the reli- ability of our mathematical model is proved. The symmetric index and variance of solidified shell have been introduced to judge the symmetry of flow field and uniformity of solidi- fied shell thickness, respectively.
2. Mathematical Modeling
In this mathematical model, the Maxwell’s equation was solved to calculate the electromagnetic field and electromag- netic force, and the latter was acted as the momentum source term into the Navier-Stokes equation. The electromagnetic field, fluid flow, heat transfer and solidification have been coupled to investigate the phenomenon occurring in con- tinuous casting under different slide-gate opening rates and casting speeds.
2.1. Assumption In order to simplify the numerical simulation, the present
work includes the following assumptions: (1) The influence of flow field on the electromagnetic field
is ignored due to the small magnetic Reynolds number,25) and the electromagnetic field is assumed to be quasi-static.
(2) The influence of Joule heat generated by currents is ignored in simulation of heat transfer and solidification due to its low frequency.
(3) The liquid steel and the liquid slag behave as incom- pressible Newtonian fluids, and each phase is considered as a homogeneous phase.
(4) The effects of mold oscillation and mold curvature are not taken into account.
(5) The mushy zone is modeled as a porous medium, where the flow follows Darcy’s law.
(6) The sintering layers and the raw material layers of the mold flux are not considered.
2.2. Governing Equation 2.2.1. Electromagnetic Model
Electromagnetic field was determined by solving Maxwell’s equations:
where
H is magnetic field strength, A/m. The time averaged electromagnetic force can be calcu-
lated by:
B* is the complex conjugate of
B; and Re denotes the real part of the complex quantity.
2.2.2. Fluid Flow Model Continuity equation:
u 0 ................................ (6)
Momentum equation (N-S):
t uu p u S Fm ... (7)
Where u is the fluid velocity, m/s; p is the pressure, Pa; ρ is the fluid density, kg/m3; and μeff is the effective viscos- ity, Pa·s, namely:
eff t pc 0 0 2
................. (8)
The standard κ-ε turbulence model is used to determine the effective viscosity, μeff.
Energy equation:
T
................... (10)
Where λ is the thermal conductivity of the fluid, W/ (m·K); cp is the specific heat capacity of the fluid, J/(kg·K), Prt is the turbulence Prandtl number, 0.85, H is enthalpy, J/kg; href is reference enthalpy relative to temperature, Tref, J/kg; fl is liquid fraction of liquid steel, %; and Ls is latent heat of solidification, J/kg.
The enthalpy-porous model is used to simulate the solidi- fication of steel in a slab continuous casting mold, the liq- uid-solid mushy zone is treated as a porous zone. The sink
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term (Sm) is added to the turbulence equations to account for reduced porosity in the mushy region, as follows:
S
f
2
3 .
............. (11)
Where Amush is a mushy zone constant which was set as 108 in current study;26) and uc is casting speed, m/s.
t u 0 ...................... (12)
Where t is time, s; u is the viscosity of the molten steel, Pa·s; and αsteel is the volume fraction of the molten steel in the mold, where αsteel = 0 indicates liquid slag, 0 < αsteel < 1 indicates steel slag interface, and αsteel = 1 means molten steel.
2.3. Geometry and Boundary Condition 2.3.1. Geometry Model
A three-dimensional mathematical model coupled with electromagnetic stirring has been established in continuous casting with slide gate. The flow rate of the molten steel through the SEN into the mold is controlled by a slide gate that moves between the geometric center and the Outside Radius (OR) of the mold, as shown in Fig. 1. The linear fraction of opening distance is defined as slide-gate opening rate (FL).4) Figure 2(a) shows the geometry for electromag- netic simulation, two electromagnetic stirrers are installed in both side of the mold wide face, the center of stirrer is
located at 75 mm below the meniscus. Figure 2(b) indicates the geometry for fluid simulation with slide gate system, in order to simulate the behavior in steel/slag interface, a liq- uid slag with 15 mm-thickness is involved in computation domain. The model mesh type was a hexahedral structure mesh, the grid was encrypted in special regions and grid- independent verification was performed. The number of grids are about 1.6 million, and the effective height of the mold is 800 mm, considering the influence of reflow on the flow field of molten steel in the upper mold, we extended the mold model to 3 000 mm. The process conditions and physical properties of numerical simulation are given in Tables 1 and 2.
Fig. 1. Schematic of slide gate configuration. (Online version in color.)
Fig. 2. Schematics of the calculation model. (a) model used for electromagnetic simulation, (b) model used for fluid simu- lation. (Online version in color.)
Table 1. Process parameters used in mathematical simulation.
Parameters Value Unit Parameters Value Unit
Section size of mold 1 800× 230 mm2 Casting speed 0.8, 1.0, 1.2, 1.4 m/min
Size of SEN port 80×60 mm2 Slide-gate opening rate, FL 40, 50, 60, 70, 100 %
Outer diameter of SEN 120 mm Coil number of each stirrer 36 –
Inner diameter of SEN 80 mm Turn number of each coil 20 –
Inclination angle 15 ° Stirrer center from meniscus 75 mm
Submerged entry depth 170 mm EMS frequency 4.5 Hz
Total nozzle length 905 mm EMS current 0, 400, 500, 600, 700 A
Table 2. Physical parameters used in mathematical simulation.
Molten steel parameters Value Unit Liquid slag
parameters Value Unit
Density 7 000 kg/m3 Density 2 600 kg/m3
Specific heat 720 J/(kg·K) Specific heat 1 200 J/(kg·K)
Thermal conductivity 31 W(m·K) Thermal
conductivity 3 W(m·K)
Latent heat 275 000 J/kg Solidus temperature 1 307 K
Solidus temperature 1 724 K Liquidus
temperature 1 486 K
tension 1.2 N/m
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2.3.2. Boundary Conditions for Electromagnetic Simula- tion
The whole geometry model was taken to be surrounded by an air cuboid (2.95 m × 1.2 m × 3.6 m) in which most of the magnetic flux lines are closed. Boundary conditions are applied on the external surface of this cuboid with mag- netically flux parallel boundary.27)
2.3.3. Boundary Conditions for Fluid Field and Tempera- ture Field Simulation
(1) The inlet velocity of SEN was calculated based on the mass conservation, and turbulent kinetic energy and the energy dissipation rate are estimated by the semiempirical equations,28) the casting temperature is set as 1 815 K.
(2) The outlet boundary in the mold bottom is a fully developed outflow condition.
(3) The mold wall is treated with the nonslip boundary condition and the heat flux on wide and narrow face of the mold is a function of distance down the mold, as shown in Eq. (13), which is similar to the form proposed by Savage.29) The convective heat transfer boundary condition is imposed on the secondary cooling zone of continuous caster, and the average heat transfer coefficient for wide and narrow face is 320 W/(m2·K) and 360 W/(m2·K), respectively.
q u 2 68 60. / L c ...................... (13)
Here, q is the heat flux density, W/m2, φ is constants for wide and narrow face which is 0.275 and 0.295,30) respec- tively, L is the distance from the meniscus, m.
(4) The top surface is treated as free-slipped boundary condition and considering the heat insulation of mold flux, adiabatic condition is applied to it.
3. Results and Discussion
3.1. Validation of Electromagnetic Field and Flow Field A simulation without molten steel (only the air within the
mold) was taken to validate the electromagnetic model. The calculated and measured magnetic flux density at 15 mm away from the mold outside radius of stirrer mid-plane are shown in Fig. 3 with operating condition of 600A/4.5 Hz. It can be seen that the numerical results show good agreement with measurement data, although there still exists some differences (possibly resulting from the simplified M-EMS geometry and the uncertainty of material properties, etc.),
the biggest magnetic flux density is about 0.091 T, which located near the mold left side.
Figure 4(a) shows the vector and contour distribution of magnetic flux density at stirrer mid-plane. The distribution of magnetic flux density is centrosymmetric, the magnetic flux density at the edges are larger than those in the inner part of cross-section, and the value of magnetic flux density ranges from 0.11 T to 0.01 T. Figure 4(b) presents the distribution of time-averaged electromagnetic force, as shown in this picture, the distribution of electromagnetic force is also centrosym- metric, which originated from the similar distribution of magnetic flux density. The tangential components of electro- magnetic force near the two parallel edges are the same, but their directions are opposite to each other, and thus, producing a horizontal recirculation. The distribution of time-averaged electromagnetic force at 15 mm away from the mold outside radius of stirrer mid-plane is shown in Fig. 4(c). It can be seen that electromagnetic force increases with the EMS current, as the EMS current increases from 400 A to 700 A, the maxi- mum electromagnetic force is 2 462 N/m3 and 7 540 N/m3. Additionally, the variation tendency of electromagnetic force is similar to that of electromagnetic field in Fig. 3.
In order to visualize the flow pattern and validate the mathematical model, a 1/5 scale physical model using mercury which coupled with electromagnetic field was established, the fluid velocity was measured by means of the Ultrasound Doppler Velocimetry (UDV), the model
Fig. 3. Distribution of magnetic flux density at stirrer mid plane (Z= 0.075 m). (Online version in color.)
Fig. 4. Vector and contour plots of (a) magnetic flux density and (b) time-average electromagnetic force at stirrer mid- plane, (c) distribution of time-averaged electromagnetic force along the wide face. (Online version in color.)
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dimensions and operating parameters are shown in Table 3. Figure 5 shows the flow field at a quarter longitudinal plane near the OR side of mold which obtained from the physical experiment and mathematical model under the cor- responding physical experiment model with 80A-EMS and casting speed of 1.2 m/min, respectively. It can be seen that the numerical results show good agreement with the experi- mental results. Due to the effect of electromagnetic force, the flow pattern in the mold changes significantly, the upper recirculating flow disappears, the jet flow in the left side of mold is suppressed, whereas it in the right side is intensified, and the fluid moves from the left side of mold to the right
side. Overall, by comparison the electromagnetic field and flow field between simulation and experimental results, the mathematical model is proven to be reliable.
3.2. Effect of Slide-gate Opening Rate on Bias Flow and Heat Transfer in the Mold
3.2.1. Effect of Slide-gate Opening Rate on Bias Flow Figure 6 shows the flow pattern within the SEN and mold
under 60% slide-gate opening rate when the casting speed is 1.2 m/min and EMS is off. As can be seen from Fig. 6(a), molten steel injected from the top of SEN, due to the block- ing effect of slide gate, molten steel will flow downstream through the opening region, this is the origin of bias flow, on the contrary of the opening region, the fluid formed an upper long recirculating flow under the slide gate. Figure 6(b) depicts the velocity iso-surface in the mold and the blue color expresses the velocity of 0.25 m/s, molten steel left the nozzle into the mold, then bias flow phenomenon has happened due to the off-center block effect of slide gate, the deviated direction is towards the Inside Radius (IR) side of mold, the main reason is that the opening direction of slide gate orients towards the Outside Radius (OR) side of mold. This bias flow may affect the distribution of flow field in OR side and IR side of mold.
Figure 7 demonstrates the flow behavior at steel/slag interface under different slide-gate opening rates when the casing speed is 1.2 m/min and EMS is off, and the steel/ slag interface is represented by using the steel phase vol- ume fraction αsteel = 0.5,31) the left-hand panels show the
Table 3. The mold dimensions and operating parameters used in physical experiment.
Parameters Value Unit Parameters Value Unit
Section size of mold 360×46 mm2 Slide-gate
opening rate, FL 100 %
from meniscus 10 mm
Outer diameter of SEN 24 mm Density 13 600 kg/m3
Inner diameter of SEN 16 mm Viscosity 0.00155 Pa·s
Submerged entry depth 34 mm EMS current 80 A
Inclination angle 15 ° EMS frequency 50 Hz
Fig. 6. Flow pattern in the SEN and mold under 60% slide-gate opening rate: (a) velocity vector within SEN, (b) veloc- ity iso-surface in the mold.
Fig. 5. Validation of fluid flow at a quarter longitudinal plane near OR with 1.2 m/min-80A and FL=100%: (a) experi- mental results, (b) numerical results.
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Fig. 7. Flow behavior at steel/slag interface under different slide-gate opening rate: (a) 100% opening rate, (b) 60% opening rate.
Fig. 8. Integral path for symmetric index.
velocity contour, whereas the right-hand panels show the velocity distribution 15 mm away from the mold wide face. As shown in Fig. 7(a), when the slide gate is fully opened (FL = 100%), the symmetry of flow field in the mold is great, the maximum velocity is 0.153 m/s, and the molten steel moves from narrow face toward the SEN. However, as shown in Fig. 7(b), when the slide-gate opening rate (FL) is 60%, the bias flow behavior has happened in the IR and OR side of the mold, and the velocity in the OR side is larger than that in the IR side, the maximum velocity is 0.11 m/s and 0.07 m/s, respectively. This illustrates that the slide-gate opening direction orients towards OR side of mold, it will result in the molten steel flows opposite to the OR side (shown in Fig. 6(b)) and causing the asymmetry of flow field. In addition, the velocity at steel/slag interface decreases as the slide-gate opening rate changes to 60%, it may not benefit to the melting of mold flux.
It was found in Fig. 7 that symmetry is destroyed when slide-gate opening rate is less than 100%. So, in order to judge the symmetry under different process conditions, the symmetric index S has been introduced to quantitatively judge whether the flow pattern is symmetric or not. In this study, we choose the symmetry of transverse plane at the cen- ter of stirrer (Z=−0.075 m) to characterize the symmetry of flow field. The definition of symmetric index S is as followed:
S n
u u
u u ux y 2 2 ............................. (15)
where u is the 2D velocity of characterized point, m/s; subscript Mi and Ni denote the centrosymmetric points, n = 201; ux is the x-component velocity, m/s; and uy is the y-component velocity, m/s. The calculating data in Eq. (14) are descripted as the two black dash lines in Fig. 8, where both of them are 15 mm away from the mold wide face. S index indicates the symmetry of flow field, S is close to 1 when the flow field is close to symmetry, while S is close to 0 when the flow field is nearly symmetry. The symmet- ric index at Z = −0.075 m plane under different slide-gate opening rates when EMS is not applied are shown in Fig. 9.
Figure 9 demonstrates that when the slide-gate opening rate is 100%, the flow field in the mold is nearly symmetric, and it decreases with the decrease of slide-gate opening rate. Under a given casting speed, due to the blocking effect of slide gate, molten steel discharged from SEN is not sym- metric, a mass of molten steel moves opposite to the slide gate opening direction, and with the decrease of slide-gate opening rate, this phenomenon is even obvious. Therefore, the asymmetry of flow field in the mold aggravates. As the slide-gate opening rate decreases from 100% to 40%, the symmetric index decreases from 0.96 to 0.66, respectively.
3.2.2. Effect of Slide-Gate Opening Rate on Uniformity of Solidified Shell
Figure 10 shows the variation of solidified shell thick- ness at mold wide face under different slide-gate opening
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smaller than that in the OR side, especially at −0.3 m to −0.4 m away from the meniscus, this is the consequence of bias flow, molten steel poured from SEN deviated towards the IR side of mold, and resulting in the remelting phenom- enon of solidified shell, this may also lead to the nonunifor- mity of solidified shell. The maximal difference of solidified shell thickness between OR and IR side of mold is 3.98 mm when the slide-gate opening rate is 40%.
The aforementioned discussion illuminates that slide- gate opening rate may cause the asymmetric distribution of solidified shell, therefore, in order to quantitatively describe the uniformity of solidified shell thickness, we introduce variance of solidified shell thickness, δ 2, and the relevant definitions are depicted in Eqs. (16)–(17):
h h mi ave i j
j
m
i
j
m
i
2
1
...................... (17)
Here, i represents the Outside Radius Face (OR) and Inside Radius (IR) at X-Y mold plane, respectively. hi-j, hi-ave and mi are solidified shell thickness in mesh element j, mm, averaged solidified shell thickness, mm, and total mesh element number at the corresponding mold region, respectively, and the corresponding data for calculation are
Fig. 10. Variation of solidified shell thickness under different slide-gate opening rates: (a) 100% slide-gate opening rate, (b) 40% slide-gate opening rate. (Online version in color.)
Fig. 9. Symmetric index with different slide-gate opening rate at Z= − 0.075 m plane.
rates when the casting speed is 1.2 m/min and EMS is not applied, where the liquid fraction of 0.8 is defined as solidification front.
As shown in Fig. 10, solidified shell thickness increases along the casting direction. When slide-gate opening rate is 100%, solidified shell thickness in the OR and IR side of mold is nearly equal, the maximum of solidified shell thick- ness is about 12.96 mm. However, when slide-gate opening rate is 40%, the solidified shell thickness in the IR side is
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shown in Fig. 11. The smaller variance is, the more uniform solidified shell it will be. Figure 12 shows the variance of solidified shell in the mold under different slide gate opening rates when the casting speed is 1.2 m/min.
The flow field in the mold has great influence on the heat transfer and solidification in continuous casting, Fig. 12 demonstrated that the variance of solidified shell thickness increases along the casting direction, this is because that with the increase of distance from the meniscus, the influ- ence of jet flow induced by SEN enhances, high temperature liquid steel may remelt solidified shell, this can decrease the uniformity of solidified shell. Meanwhile, with the decrease of slide-gate opening rate, the variance of solidified shell thickness increases, especially at −0.3 m to −0.4 m distance from the meniscus, decreasing the slide-gate opening rate may aggravate the asymmetry of fluid field and the distri- bution of temperature, so the uniformity of solidified shell also decreases. As the slide-gate opening rate changes from 100% to 40%, the variance at −0.4 m from the meniscus of IR side change from 1.78 to 3.28.
Fig. 11. Data acquisition for Variance of solidified shell thickness. (Online version in color.)
In addition, the variance of solidified shell thickness in the IR side of mold is larger than that in the OR side of mold at −0.4 m distance from the meniscus when slide-gate opening rate is less than 100%. Because that the deviated direction of bias flow is towards the IR side of mold, the remelting phenomenon in the IR side of mold is much more serious than that in the OR side of mold.
3.3. Effect of EMS on Bias Flow and Heat Transfer with Slide Gate
3.3.1. Effect of EMS on Bias Flow with Slide Gate The previous discussions have shown that slide gate may
destroy the symmetry of flow field, inducing the asym- metry phenomenon in the mold, and thus, leading to some undesirable consequences, such as inactive free-surface flow, no-uniform solidified shell, et.al. Therefore, in order to alleviate or eliminate the adverse effects causing by slide gate, electromagnetic stirring has been used to optimize the asymmetric phenomenon in the mold.
Figure 13 shows the velocity vector on different cross
Fig. 12. Variance of solidified shell thickness under different slide-gate opening rates: (a) OR side of mold, (b) IR side of mold. (Online version in color.)
Fig. 13. Velocity vector on different cross section under different EMS currents: (a) without EMS current, (b) 600A- EMS current.
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sections under different EMS currents when the slide gate opening rate is 60% and casting speed is 1.2 m/min. It can be seen that when EMS is off (Fig. 13(a)), the jet flow poured from SEN deviated towards IR side of mold at Z = −0.3 m cross-section, while near the meniscus, the molten steel moved from the OR side to IR side, as a result, a bias flow phenomenon has happened in the mold. However, when EMS current is 600A (Fig. 13(b)), although the jet flow poured from SEN is similar to that without EMS, due to the effect of electromagnetic force, a horizontal recircu- lating flow is generated in the mold, and the fluid in the OR side will flow toward the IR side continuously, this can make the velocity distribution more uniform. Therefore, the asymmetry in the mold has been improved.
Figure 14 shows the symmetric index under different EMS currents as the casting speed and slide gate opening rate is 1.2 m/min and 60%, respectively, it indicates that EMS can greatly improve the symmetry of flow field in the mold, symmetric index increases with the increase of EMS current. When EMS is applied, a horizontal recirculating flow induced by horizonal electromagnetic force can make the velocity distribution more uniform, and the velocity difference in the OR side and IR side of the mold reduces, so the symmetry in the mold improves, however, it can not remove the asymmetry completely. As EMS current
increases from 0A to 700A, the symmetric index increases from 0.71 to 0.85, respectively.
3.3.2. Effect of EMS on Uniformity of Solidified Shell with Slide Gate
The variation of solidified shell thickness at mold wide face with and without EMS when slide-gate opening rate is 60%
Fig. 14. Symmetric index under different EMS currents at Z = − 0.075 m plane.
Fig. 15. Variation of solidified shell thickness under different EMS currents: (a) 0A, (b) 600A (c) schematic of bias flow and EMS driving flow. (Online version in color.)
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and casting speed is 1.2 m/min is shown in Fig. 15, where the liquid fraction of 0.8 is defined as solidification front.
When EMS is off (Fig. 15(a)), the solidified shell thick- ness at mold wide face is not equal, the solidified shell thick- ness in the IR side of mold is thinner especially in the region of IR side. When EMS is applied (Fig. 15(b)), a horizontal recirculating flow induced by electromagnetic force has gen- erated, shown in Fig. 15(c), this can make the distribution of solidified shell more uniform especially for the OR side of mold, while for the IR side of mold, horizonal recirculating flow can also suppress the direct impact of bias flow, there- fore, the remelting region caused by bias flow has decreases. Besides, the solidified shell thickness becomes thinner due to the superheat dissipation phenomenon caused by EMS, the maximal solidified shell thickness at the IR side of mold decreases from 14.57 mm to 13.13 mm as the EMS current increases from 0A to 600A.
Figure 16 depicts variance of solidified shell thickness under different EMS currents, it can be seen that EMS can improve the uniformity of solidified shell significantly. With the increase of EMS current, the variance decreases firstly and then increases, it means that there exists a critical EMS current to optimize the uniformity of solidified shell, while excessive EMS current may not benefit to uniform growth of solidified shell. The main reason is that when EMS current is smaller than critical EMS current, the electromagnetic force is relatively small that molten steel can not stir suf- ficiently, and this may lead to the uneven distribution of temperature and solidified shell. In contrast, as EMS current exceeds the critical value, due to the interaction of bias flow and diagonal jet flow induced by electromagnetic force, the remelting phenomenon is even serious, so the variance increases. When the EMS current changes from 0A to 700A, the variance at −0.4 m distance from the meniscus of IR side change from 3.35 to 2.59.
3.4. Effect of Casting Speed on Bias Flow and Heat Transfer with Slide Gate
3.4.1. Effect of Casting Speed on Bias Flow with Slide Gate
Figure 17 shows the velocity distribution and symmetric index under different casting speed at Z=−0.075 m plane when the slide-gate opening rate is 60% and EMS current
Fig. 16. Variance of solidified shell thickness under different EMS currents: (a) OR side of mold, (b) IR side of mold. (Online version in color.)
is 600A. As shown in Fig. 17(a), when the casting speed is 0.8 m/
min, due to the effect of electromagnetic force, a horizonal recirculating flow is generated, and the velocity distribution in the mold is nearly centrosymmetric which coincides with the distribution of electromagnetic force. However, when the casting speed is 1.4 m/min (Fig. 17(b)), although there still exists a recirculating flow, the velocity distribution in the OR and IR side of mold is not equal. Figure 17(c) shows the symmetric index under different casting speed, it can be seen that when EMS current is 600A, the sym- metric index under the casting speed of 0.8 m/min and 1.0 m/min is nearly equal, this also indicates that EMS can not remove the asymmetry of flow field completely. With the increase of casting speed, the symmetric index decreases, the main reason is that the intensity of jet flow poured from the SEN enhances with the casting speed increasing, this may deteriorate the asymmetry of flow field in the mold, and the optimization of EMS on asymmetric phenomenon weakens. As the casting speed increases from 0.8 m/min to 1.4 m/min, the symmetrical index decreases from 0.91 to 0.62, respectively.
3.4.2. Effect of Casting Speed on Uniformity of Solidified Shell with Slide Gate
The variation of solidified shell thickness at mold wide face under different casting speed is shown in Fig. 18 when the slide-gate opening rate 60% and EMS current is 600A, where the liquid fraction of 0.8 is defined as solidification front.
As illustrated in Fig. 18, the solidified shell thickness decreases with the increasing of casting speed, the maximal solidified shell thickness in the IR side of mold decreases 16.49 mm to 12.36 mm when the casting speed changes from 0.8 m/min to 1.4 m/min. For the OR side of mold, due to the washing effect of electromagnetic force, the solidified shell thickness is uniform even under different casting speed. However, for the IR side of mold, the remelt- ing region at the casting speed of 0.8 m/min is smaller than that of 1.4 m/min, the main reason is that the jet flow poured from the SEN is weak at the low casting speed (0.8 m/min), this may suppress the impact region of bias flow in the IR side of mold, furthermore, the diagonal jet flow
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induced by electromagnetic force can also aggravate the remelting phenomenon of solidified shell and lead to the inferior uniformity.
The variance of solidified shell thickness under different casting speeds is shown in Fig. 19, it can be seen that the variance of solidified shell thickness in the IR side of mold is greater than that in the OR side due to the obvious remelt- ing phenomenon induced by bias flow. With the increasing of casting speed, the variance of solidified shell thickness decreases firstly and then increases. The main reason is that under the low casting speed (0.8 m/min and 1.0 m/min), the present EMS current has exceeded the critical EMS current which is benefit to the uniform growth of solidified shell, whereas for the high casting speed (1.4 m/min), the intensity of upper recirculating flow increases, therefore, it may need
a larger EMS current to improve the uniformity of solidified shell thickness. When the casting speed changes from 0.8 m/ min to 1.2 m/min, the variance at −0.4 m from the meniscus of IR side changes from 2.63 to 1.72, respectively.
From all the results, in order to correct the bias flow caused by slide gate and improve the uniformity of solidi- fied shell, a suitable EMS current should be selected. For example, when the slide-gate opening rate is 60% and cast- ing speed is 1.2 m/min, the optimal EMS current is 600A in our study case. However, more detailed and general criterion requires further research considering additional factors, such as the slide-gate orientation, submerged entry depth and slab size, etc.
Fig. 17. Velocity distribution under different casting speeds: (a) 0.8 m/min, (b) 1.4 m/min and (c) symmetric index under different casting speed at Z = − 0.075 m plane.
Fig. 18. Variation of solidified shell thickness under different casting speeds: (a) 0.8/min, (b) 1.4 m/min.
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4. Conclusion
A three-dimensional mathematical model coupled with electromagnetic, flow, heat transfer and solidification has been developed to investigate the influence of EMS on bias flow under different slide-gate opening rates and casting speeds. The symmetric index and variance of solidified shell thickness have been defined to judge the symmetry of flow field and uniformity of solidified shell, respectively. The following conclusions can be drawn:
(1) When EMS is off, the bias flow phenomenon has happened in the mold, the deviated direction is opposite to the opening direction of slide gate. With the decrease of slide-gate opening rate, the velocity at steel/slag interface decreases.
(2) When EMS is off, the symmetry of flow field and the uniformity of solidified shell thickness decreases with the decrease of slide-gate opening rate.
(3) EMS can improve the symmetry of flow field sig- nificantly, but it can not be eliminated completely.
(4) EMS can improve the uniformity of solidified shell in the mold, but the excessive EMS current may not benefit to the uniform growth of solidified shell.
(5) As the casting speed increases, for a given slide-gate opening rate and EMS current, the symmetry of flow field decreases, while the uniformity of solidified shell increases firstly and then decrease.
(6) There exists an optimal EMS current to balance the symmetry of flow field and uniformity of solidified shell. For example, under the operating condition of 60% slide- gate opening rate and 1.2 m/min casting speed, the optimal EMS current is 600A.
Acknowledgement This project financially supported by National Science
Foundation of China (NO. U1860107 and 52074181) and the Science and Technology Commission of Shanghai Municipality (No. 19DZ2270200).
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* Corresponding author: E-mail: [email protected]
© 2021 The Iron and Steel Institute of Japan. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs license (https://creativecommons.org/licenses/by-nc-nd/4.0/).
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1. Introduction
During the continuous casting, molten steel enters the casting mold through a Submerged Entry Nozzle (SEN), the flow rate of the molten steel is commonly controlled by either slide gate or stopper rod. The off-center slide gate may generate an asymmetric or bias flow in the mold, which may induce asymmetrical level fluctuations and possibly to enhance the risks of superficial defects, such as hook formation, longitudinal cracks.1,2) Therefore, understanding quantitatively the molten steel flow affected by the slide gate and optimizing the asymmetric flow in the mold are of great importance to improve steel quality.
Extensive and elaborate past work were conducted on the different aspects of slide gate such as slide-gate open- ing rate,3–7) slide gate orientation,3) slide gate design8,9) and the influence of these parameters on the mold flow structure.10,11) Bai et al.3,4) simulated two phase flow under different gate opening fractions and orientations, the results were verified using experimental data and found that the off-
Numerical Simulation of In-mold Electromagnetic Stirring on Slide Gate Caused Bias Flow and Solidification in Slab Continuous Casting
Haibiao LU,1) Bin LI,1) Jiaxun LI,1) Yunbo ZHONG,1) Zhongming REN1) and Zuosheng LEI1,2)*
1) State Key Laboratory of Advanced Special Steel & Shanghai Key Laboratory of Advanced Ferrometallurgy, Shanghai Univer- sity, Shanghai, 200444 China. 2) Sino-European School of Technology, Shanghai University, Shanghai, 200444 China.
(Received on December 9, 2020; accepted on February 26, 2021)
A mathematical model coupled with electromagnetic field, flow, heat transfer and solidification has been developed to simulate the bias flow caused by slide gate under different slide-gate opening rates, EMS currents and casting speeds. Through comparing the magnetic flux density and flow field with measured results, the reliability of mathematical model is proved. The symmetric index, variance of solidified shell thickness have been introduced to judge the symmetry of flow field and uniformity of solidified shell, respectively. The results show that bias flow phenomenon has happened when the slide-gate opening rate is less than 100%, the deviated direction is opposite to the opening direction of slide gate. As the slide- gate opening rate decreases, the symmetry of flow field and uniformity of solidified shell decrease. Increasing the EMS current and decreasing the casting speed, the symmetry of flow field caused by slide gate increases, but it can not eliminate completely, while the uniformity of solidified shell increases firstly and then decreases. There exists an optimal EMS to balance the symmetry of flow field and uniformity of solidified shell.
KEY WORDS: continuous casting; slide gate; bias flow; electromagnetic stirring; uniformity of solidified shell.
center blocking effect of the slide gate generated asymmetric flow and the 0-deg gate orientations the worst bias flow between the left and right ports. Gursoy et al.6) developed mathematical model for different slide gate and stopper rod opening rates in a slab continuous casting, the flow structure and meniscus velocity were discussed. Cedillo et al.7) used the water model and mathematical model to investigate the effects of slide gate opening on the flow field in the mold and demonstrated that the slide gate opening will cause the bias flow in the mold. Kononov et al.8) designed new slide gate systems for ingot and shaped casting. However, although great advances have been made in optimizing the flow pattern in the mold and submerged entry nozzle with slide gate, there still exist distinct bias flow in the mold.
ElectroMagnetic (EM) systems have been validated as an attractive method to control relevant phenomena related to fluid flow in continuous casting. Recently many research- ers12–17) have previously reported that electromagnetic brake (EMBr) can improve the asymmetric flow either in SEN or mold. Garcia-Hernandez12) et al. studied the effects of mold curvature, slide gate and EMBr fields on steel flow in a slab mold with a mathematical model, the results showed
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that the symmetry of flow field increases with the increase of magnetic flux density. Jin et al.13) constructed a math- ematical model to study the effect of electromagnetic brake on the bias flow in the mold together with slide gate, the results show that EMBr can reduce the extent of asymmetric flow caused by slide gate. Additionally, Electromagnetic stirring (EMS), as another electromagnetic controller, has shown the ability to change the flow pattern18–21) and opti- mize the adverse mold flow such as the bias flow caused by SEN clogging.22,23) Li et al.22,23) reported that M-EMS can greatly improve the asymmetry of flow field in the slab mold with SEN clogging, and the stirring intensity should be controlled below a safe level to avoid slag entrapment. However, relatively little study has reported on the influence of EMS on flow field in the mold under slide gate system. Geng et al.24) investigated the effect of rotating magnetic field on the flow field in the SEN and the billet mold under different slide-gate opening rates, and showed that the EMS current can improve the bias flow in the mold, but the elec- tromagnetic stirrer was equipped around the SEN but not the mold, furthermore, most of research did not consider the solidified shell in their work, in addition, some other process parameters, such as the casting speed and SEN depth, are also lack of consideration.
The current work presents a mathematical model to inves- tigate the influence of EMS on bias flow, heat transfer and solidification behavior in an actual slab mold under different slide-gate opening rates and casting speeds. By comparisons with the magnetic flux density data and flow field, the reli- ability of our mathematical model is proved. The symmetric index and variance of solidified shell have been introduced to judge the symmetry of flow field and uniformity of solidi- fied shell thickness, respectively.
2. Mathematical Modeling
In this mathematical model, the Maxwell’s equation was solved to calculate the electromagnetic field and electromag- netic force, and the latter was acted as the momentum source term into the Navier-Stokes equation. The electromagnetic field, fluid flow, heat transfer and solidification have been coupled to investigate the phenomenon occurring in con- tinuous casting under different slide-gate opening rates and casting speeds.
2.1. Assumption In order to simplify the numerical simulation, the present
work includes the following assumptions: (1) The influence of flow field on the electromagnetic field
is ignored due to the small magnetic Reynolds number,25) and the electromagnetic field is assumed to be quasi-static.
(2) The influence of Joule heat generated by currents is ignored in simulation of heat transfer and solidification due to its low frequency.
(3) The liquid steel and the liquid slag behave as incom- pressible Newtonian fluids, and each phase is considered as a homogeneous phase.
(4) The effects of mold oscillation and mold curvature are not taken into account.
(5) The mushy zone is modeled as a porous medium, where the flow follows Darcy’s law.
(6) The sintering layers and the raw material layers of the mold flux are not considered.
2.2. Governing Equation 2.2.1. Electromagnetic Model
Electromagnetic field was determined by solving Maxwell’s equations:
where
H is magnetic field strength, A/m. The time averaged electromagnetic force can be calcu-
lated by:
B* is the complex conjugate of
B; and Re denotes the real part of the complex quantity.
2.2.2. Fluid Flow Model Continuity equation:
u 0 ................................ (6)
Momentum equation (N-S):
t uu p u S Fm ... (7)
Where u is the fluid velocity, m/s; p is the pressure, Pa; ρ is the fluid density, kg/m3; and μeff is the effective viscos- ity, Pa·s, namely:
eff t pc 0 0 2
................. (8)
The standard κ-ε turbulence model is used to determine the effective viscosity, μeff.
Energy equation:
T
................... (10)
Where λ is the thermal conductivity of the fluid, W/ (m·K); cp is the specific heat capacity of the fluid, J/(kg·K), Prt is the turbulence Prandtl number, 0.85, H is enthalpy, J/kg; href is reference enthalpy relative to temperature, Tref, J/kg; fl is liquid fraction of liquid steel, %; and Ls is latent heat of solidification, J/kg.
The enthalpy-porous model is used to simulate the solidi- fication of steel in a slab continuous casting mold, the liq- uid-solid mushy zone is treated as a porous zone. The sink
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term (Sm) is added to the turbulence equations to account for reduced porosity in the mushy region, as follows:
S
f
2
3 .
............. (11)
Where Amush is a mushy zone constant which was set as 108 in current study;26) and uc is casting speed, m/s.
t u 0 ...................... (12)
Where t is time, s; u is the viscosity of the molten steel, Pa·s; and αsteel is the volume fraction of the molten steel in the mold, where αsteel = 0 indicates liquid slag, 0 < αsteel < 1 indicates steel slag interface, and αsteel = 1 means molten steel.
2.3. Geometry and Boundary Condition 2.3.1. Geometry Model
A three-dimensional mathematical model coupled with electromagnetic stirring has been established in continuous casting with slide gate. The flow rate of the molten steel through the SEN into the mold is controlled by a slide gate that moves between the geometric center and the Outside Radius (OR) of the mold, as shown in Fig. 1. The linear fraction of opening distance is defined as slide-gate opening rate (FL).4) Figure 2(a) shows the geometry for electromag- netic simulation, two electromagnetic stirrers are installed in both side of the mold wide face, the center of stirrer is
located at 75 mm below the meniscus. Figure 2(b) indicates the geometry for fluid simulation with slide gate system, in order to simulate the behavior in steel/slag interface, a liq- uid slag with 15 mm-thickness is involved in computation domain. The model mesh type was a hexahedral structure mesh, the grid was encrypted in special regions and grid- independent verification was performed. The number of grids are about 1.6 million, and the effective height of the mold is 800 mm, considering the influence of reflow on the flow field of molten steel in the upper mold, we extended the mold model to 3 000 mm. The process conditions and physical properties of numerical simulation are given in Tables 1 and 2.
Fig. 1. Schematic of slide gate configuration. (Online version in color.)
Fig. 2. Schematics of the calculation model. (a) model used for electromagnetic simulation, (b) model used for fluid simu- lation. (Online version in color.)
Table 1. Process parameters used in mathematical simulation.
Parameters Value Unit Parameters Value Unit
Section size of mold 1 800× 230 mm2 Casting speed 0.8, 1.0, 1.2, 1.4 m/min
Size of SEN port 80×60 mm2 Slide-gate opening rate, FL 40, 50, 60, 70, 100 %
Outer diameter of SEN 120 mm Coil number of each stirrer 36 –
Inner diameter of SEN 80 mm Turn number of each coil 20 –
Inclination angle 15 ° Stirrer center from meniscus 75 mm
Submerged entry depth 170 mm EMS frequency 4.5 Hz
Total nozzle length 905 mm EMS current 0, 400, 500, 600, 700 A
Table 2. Physical parameters used in mathematical simulation.
Molten steel parameters Value Unit Liquid slag
parameters Value Unit
Density 7 000 kg/m3 Density 2 600 kg/m3
Specific heat 720 J/(kg·K) Specific heat 1 200 J/(kg·K)
Thermal conductivity 31 W(m·K) Thermal
conductivity 3 W(m·K)
Latent heat 275 000 J/kg Solidus temperature 1 307 K
Solidus temperature 1 724 K Liquidus
temperature 1 486 K
tension 1.2 N/m
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2.3.2. Boundary Conditions for Electromagnetic Simula- tion
The whole geometry model was taken to be surrounded by an air cuboid (2.95 m × 1.2 m × 3.6 m) in which most of the magnetic flux lines are closed. Boundary conditions are applied on the external surface of this cuboid with mag- netically flux parallel boundary.27)
2.3.3. Boundary Conditions for Fluid Field and Tempera- ture Field Simulation
(1) The inlet velocity of SEN was calculated based on the mass conservation, and turbulent kinetic energy and the energy dissipation rate are estimated by the semiempirical equations,28) the casting temperature is set as 1 815 K.
(2) The outlet boundary in the mold bottom is a fully developed outflow condition.
(3) The mold wall is treated with the nonslip boundary condition and the heat flux on wide and narrow face of the mold is a function of distance down the mold, as shown in Eq. (13), which is similar to the form proposed by Savage.29) The convective heat transfer boundary condition is imposed on the secondary cooling zone of continuous caster, and the average heat transfer coefficient for wide and narrow face is 320 W/(m2·K) and 360 W/(m2·K), respectively.
q u 2 68 60. / L c ...................... (13)
Here, q is the heat flux density, W/m2, φ is constants for wide and narrow face which is 0.275 and 0.295,30) respec- tively, L is the distance from the meniscus, m.
(4) The top surface is treated as free-slipped boundary condition and considering the heat insulation of mold flux, adiabatic condition is applied to it.
3. Results and Discussion
3.1. Validation of Electromagnetic Field and Flow Field A simulation without molten steel (only the air within the
mold) was taken to validate the electromagnetic model. The calculated and measured magnetic flux density at 15 mm away from the mold outside radius of stirrer mid-plane are shown in Fig. 3 with operating condition of 600A/4.5 Hz. It can be seen that the numerical results show good agreement with measurement data, although there still exists some differences (possibly resulting from the simplified M-EMS geometry and the uncertainty of material properties, etc.),
the biggest magnetic flux density is about 0.091 T, which located near the mold left side.
Figure 4(a) shows the vector and contour distribution of magnetic flux density at stirrer mid-plane. The distribution of magnetic flux density is centrosymmetric, the magnetic flux density at the edges are larger than those in the inner part of cross-section, and the value of magnetic flux density ranges from 0.11 T to 0.01 T. Figure 4(b) presents the distribution of time-averaged electromagnetic force, as shown in this picture, the distribution of electromagnetic force is also centrosym- metric, which originated from the similar distribution of magnetic flux density. The tangential components of electro- magnetic force near the two parallel edges are the same, but their directions are opposite to each other, and thus, producing a horizontal recirculation. The distribution of time-averaged electromagnetic force at 15 mm away from the mold outside radius of stirrer mid-plane is shown in Fig. 4(c). It can be seen that electromagnetic force increases with the EMS current, as the EMS current increases from 400 A to 700 A, the maxi- mum electromagnetic force is 2 462 N/m3 and 7 540 N/m3. Additionally, the variation tendency of electromagnetic force is similar to that of electromagnetic field in Fig. 3.
In order to visualize the flow pattern and validate the mathematical model, a 1/5 scale physical model using mercury which coupled with electromagnetic field was established, the fluid velocity was measured by means of the Ultrasound Doppler Velocimetry (UDV), the model
Fig. 3. Distribution of magnetic flux density at stirrer mid plane (Z= 0.075 m). (Online version in color.)
Fig. 4. Vector and contour plots of (a) magnetic flux density and (b) time-average electromagnetic force at stirrer mid- plane, (c) distribution of time-averaged electromagnetic force along the wide face. (Online version in color.)
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dimensions and operating parameters are shown in Table 3. Figure 5 shows the flow field at a quarter longitudinal plane near the OR side of mold which obtained from the physical experiment and mathematical model under the cor- responding physical experiment model with 80A-EMS and casting speed of 1.2 m/min, respectively. It can be seen that the numerical results show good agreement with the experi- mental results. Due to the effect of electromagnetic force, the flow pattern in the mold changes significantly, the upper recirculating flow disappears, the jet flow in the left side of mold is suppressed, whereas it in the right side is intensified, and the fluid moves from the left side of mold to the right
side. Overall, by comparison the electromagnetic field and flow field between simulation and experimental results, the mathematical model is proven to be reliable.
3.2. Effect of Slide-gate Opening Rate on Bias Flow and Heat Transfer in the Mold
3.2.1. Effect of Slide-gate Opening Rate on Bias Flow Figure 6 shows the flow pattern within the SEN and mold
under 60% slide-gate opening rate when the casting speed is 1.2 m/min and EMS is off. As can be seen from Fig. 6(a), molten steel injected from the top of SEN, due to the block- ing effect of slide gate, molten steel will flow downstream through the opening region, this is the origin of bias flow, on the contrary of the opening region, the fluid formed an upper long recirculating flow under the slide gate. Figure 6(b) depicts the velocity iso-surface in the mold and the blue color expresses the velocity of 0.25 m/s, molten steel left the nozzle into the mold, then bias flow phenomenon has happened due to the off-center block effect of slide gate, the deviated direction is towards the Inside Radius (IR) side of mold, the main reason is that the opening direction of slide gate orients towards the Outside Radius (OR) side of mold. This bias flow may affect the distribution of flow field in OR side and IR side of mold.
Figure 7 demonstrates the flow behavior at steel/slag interface under different slide-gate opening rates when the casing speed is 1.2 m/min and EMS is off, and the steel/ slag interface is represented by using the steel phase vol- ume fraction αsteel = 0.5,31) the left-hand panels show the
Table 3. The mold dimensions and operating parameters used in physical experiment.
Parameters Value Unit Parameters Value Unit
Section size of mold 360×46 mm2 Slide-gate
opening rate, FL 100 %
from meniscus 10 mm
Outer diameter of SEN 24 mm Density 13 600 kg/m3
Inner diameter of SEN 16 mm Viscosity 0.00155 Pa·s
Submerged entry depth 34 mm EMS current 80 A
Inclination angle 15 ° EMS frequency 50 Hz
Fig. 6. Flow pattern in the SEN and mold under 60% slide-gate opening rate: (a) velocity vector within SEN, (b) veloc- ity iso-surface in the mold.
Fig. 5. Validation of fluid flow at a quarter longitudinal plane near OR with 1.2 m/min-80A and FL=100%: (a) experi- mental results, (b) numerical results.
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Fig. 7. Flow behavior at steel/slag interface under different slide-gate opening rate: (a) 100% opening rate, (b) 60% opening rate.
Fig. 8. Integral path for symmetric index.
velocity contour, whereas the right-hand panels show the velocity distribution 15 mm away from the mold wide face. As shown in Fig. 7(a), when the slide gate is fully opened (FL = 100%), the symmetry of flow field in the mold is great, the maximum velocity is 0.153 m/s, and the molten steel moves from narrow face toward the SEN. However, as shown in Fig. 7(b), when the slide-gate opening rate (FL) is 60%, the bias flow behavior has happened in the IR and OR side of the mold, and the velocity in the OR side is larger than that in the IR side, the maximum velocity is 0.11 m/s and 0.07 m/s, respectively. This illustrates that the slide-gate opening direction orients towards OR side of mold, it will result in the molten steel flows opposite to the OR side (shown in Fig. 6(b)) and causing the asymmetry of flow field. In addition, the velocity at steel/slag interface decreases as the slide-gate opening rate changes to 60%, it may not benefit to the melting of mold flux.
It was found in Fig. 7 that symmetry is destroyed when slide-gate opening rate is less than 100%. So, in order to judge the symmetry under different process conditions, the symmetric index S has been introduced to quantitatively judge whether the flow pattern is symmetric or not. In this study, we choose the symmetry of transverse plane at the cen- ter of stirrer (Z=−0.075 m) to characterize the symmetry of flow field. The definition of symmetric index S is as followed:
S n
u u
u u ux y 2 2 ............................. (15)
where u is the 2D velocity of characterized point, m/s; subscript Mi and Ni denote the centrosymmetric points, n = 201; ux is the x-component velocity, m/s; and uy is the y-component velocity, m/s. The calculating data in Eq. (14) are descripted as the two black dash lines in Fig. 8, where both of them are 15 mm away from the mold wide face. S index indicates the symmetry of flow field, S is close to 1 when the flow field is close to symmetry, while S is close to 0 when the flow field is nearly symmetry. The symmet- ric index at Z = −0.075 m plane under different slide-gate opening rates when EMS is not applied are shown in Fig. 9.
Figure 9 demonstrates that when the slide-gate opening rate is 100%, the flow field in the mold is nearly symmetric, and it decreases with the decrease of slide-gate opening rate. Under a given casting speed, due to the blocking effect of slide gate, molten steel discharged from SEN is not sym- metric, a mass of molten steel moves opposite to the slide gate opening direction, and with the decrease of slide-gate opening rate, this phenomenon is even obvious. Therefore, the asymmetry of flow field in the mold aggravates. As the slide-gate opening rate decreases from 100% to 40%, the symmetric index decreases from 0.96 to 0.66, respectively.
3.2.2. Effect of Slide-Gate Opening Rate on Uniformity of Solidified Shell
Figure 10 shows the variation of solidified shell thick- ness at mold wide face under different slide-gate opening
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smaller than that in the OR side, especially at −0.3 m to −0.4 m away from the meniscus, this is the consequence of bias flow, molten steel poured from SEN deviated towards the IR side of mold, and resulting in the remelting phenom- enon of solidified shell, this may also lead to the nonunifor- mity of solidified shell. The maximal difference of solidified shell thickness between OR and IR side of mold is 3.98 mm when the slide-gate opening rate is 40%.
The aforementioned discussion illuminates that slide- gate opening rate may cause the asymmetric distribution of solidified shell, therefore, in order to quantitatively describe the uniformity of solidified shell thickness, we introduce variance of solidified shell thickness, δ 2, and the relevant definitions are depicted in Eqs. (16)–(17):
h h mi ave i j
j
m
i
j
m
i
2
1
...................... (17)
Here, i represents the Outside Radius Face (OR) and Inside Radius (IR) at X-Y mold plane, respectively. hi-j, hi-ave and mi are solidified shell thickness in mesh element j, mm, averaged solidified shell thickness, mm, and total mesh element number at the corresponding mold region, respectively, and the corresponding data for calculation are
Fig. 10. Variation of solidified shell thickness under different slide-gate opening rates: (a) 100% slide-gate opening rate, (b) 40% slide-gate opening rate. (Online version in color.)
Fig. 9. Symmetric index with different slide-gate opening rate at Z= − 0.075 m plane.
rates when the casting speed is 1.2 m/min and EMS is not applied, where the liquid fraction of 0.8 is defined as solidification front.
As shown in Fig. 10, solidified shell thickness increases along the casting direction. When slide-gate opening rate is 100%, solidified shell thickness in the OR and IR side of mold is nearly equal, the maximum of solidified shell thick- ness is about 12.96 mm. However, when slide-gate opening rate is 40%, the solidified shell thickness in the IR side is
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shown in Fig. 11. The smaller variance is, the more uniform solidified shell it will be. Figure 12 shows the variance of solidified shell in the mold under different slide gate opening rates when the casting speed is 1.2 m/min.
The flow field in the mold has great influence on the heat transfer and solidification in continuous casting, Fig. 12 demonstrated that the variance of solidified shell thickness increases along the casting direction, this is because that with the increase of distance from the meniscus, the influ- ence of jet flow induced by SEN enhances, high temperature liquid steel may remelt solidified shell, this can decrease the uniformity of solidified shell. Meanwhile, with the decrease of slide-gate opening rate, the variance of solidified shell thickness increases, especially at −0.3 m to −0.4 m distance from the meniscus, decreasing the slide-gate opening rate may aggravate the asymmetry of fluid field and the distri- bution of temperature, so the uniformity of solidified shell also decreases. As the slide-gate opening rate changes from 100% to 40%, the variance at −0.4 m from the meniscus of IR side change from 1.78 to 3.28.
Fig. 11. Data acquisition for Variance of solidified shell thickness. (Online version in color.)
In addition, the variance of solidified shell thickness in the IR side of mold is larger than that in the OR side of mold at −0.4 m distance from the meniscus when slide-gate opening rate is less than 100%. Because that the deviated direction of bias flow is towards the IR side of mold, the remelting phenomenon in the IR side of mold is much more serious than that in the OR side of mold.
3.3. Effect of EMS on Bias Flow and Heat Transfer with Slide Gate
3.3.1. Effect of EMS on Bias Flow with Slide Gate The previous discussions have shown that slide gate may
destroy the symmetry of flow field, inducing the asym- metry phenomenon in the mold, and thus, leading to some undesirable consequences, such as inactive free-surface flow, no-uniform solidified shell, et.al. Therefore, in order to alleviate or eliminate the adverse effects causing by slide gate, electromagnetic stirring has been used to optimize the asymmetric phenomenon in the mold.
Figure 13 shows the velocity vector on different cross
Fig. 12. Variance of solidified shell thickness under different slide-gate opening rates: (a) OR side of mold, (b) IR side of mold. (Online version in color.)
Fig. 13. Velocity vector on different cross section under different EMS currents: (a) without EMS current, (b) 600A- EMS current.
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sections under different EMS currents when the slide gate opening rate is 60% and casting speed is 1.2 m/min. It can be seen that when EMS is off (Fig. 13(a)), the jet flow poured from SEN deviated towards IR side of mold at Z = −0.3 m cross-section, while near the meniscus, the molten steel moved from the OR side to IR side, as a result, a bias flow phenomenon has happened in the mold. However, when EMS current is 600A (Fig. 13(b)), although the jet flow poured from SEN is similar to that without EMS, due to the effect of electromagnetic force, a horizontal recircu- lating flow is generated in the mold, and the fluid in the OR side will flow toward the IR side continuously, this can make the velocity distribution more uniform. Therefore, the asymmetry in the mold has been improved.
Figure 14 shows the symmetric index under different EMS currents as the casting speed and slide gate opening rate is 1.2 m/min and 60%, respectively, it indicates that EMS can greatly improve the symmetry of flow field in the mold, symmetric index increases with the increase of EMS current. When EMS is applied, a horizontal recirculating flow induced by horizonal electromagnetic force can make the velocity distribution more uniform, and the velocity difference in the OR side and IR side of the mold reduces, so the symmetry in the mold improves, however, it can not remove the asymmetry completely. As EMS current
increases from 0A to 700A, the symmetric index increases from 0.71 to 0.85, respectively.
3.3.2. Effect of EMS on Uniformity of Solidified Shell with Slide Gate
The variation of solidified shell thickness at mold wide face with and without EMS when slide-gate opening rate is 60%
Fig. 14. Symmetric index under different EMS currents at Z = − 0.075 m plane.
Fig. 15. Variation of solidified shell thickness under different EMS currents: (a) 0A, (b) 600A (c) schematic of bias flow and EMS driving flow. (Online version in color.)
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and casting speed is 1.2 m/min is shown in Fig. 15, where the liquid fraction of 0.8 is defined as solidification front.
When EMS is off (Fig. 15(a)), the solidified shell thick- ness at mold wide face is not equal, the solidified shell thick- ness in the IR side of mold is thinner especially in the region of IR side. When EMS is applied (Fig. 15(b)), a horizontal recirculating flow induced by electromagnetic force has gen- erated, shown in Fig. 15(c), this can make the distribution of solidified shell more uniform especially for the OR side of mold, while for the IR side of mold, horizonal recirculating flow can also suppress the direct impact of bias flow, there- fore, the remelting region caused by bias flow has decreases. Besides, the solidified shell thickness becomes thinner due to the superheat dissipation phenomenon caused by EMS, the maximal solidified shell thickness at the IR side of mold decreases from 14.57 mm to 13.13 mm as the EMS current increases from 0A to 600A.
Figure 16 depicts variance of solidified shell thickness under different EMS currents, it can be seen that EMS can improve the uniformity of solidified shell significantly. With the increase of EMS current, the variance decreases firstly and then increases, it means that there exists a critical EMS current to optimize the uniformity of solidified shell, while excessive EMS current may not benefit to uniform growth of solidified shell. The main reason is that when EMS current is smaller than critical EMS current, the electromagnetic force is relatively small that molten steel can not stir suf- ficiently, and this may lead to the uneven distribution of temperature and solidified shell. In contrast, as EMS current exceeds the critical value, due to the interaction of bias flow and diagonal jet flow induced by electromagnetic force, the remelting phenomenon is even serious, so the variance increases. When the EMS current changes from 0A to 700A, the variance at −0.4 m distance from the meniscus of IR side change from 3.35 to 2.59.
3.4. Effect of Casting Speed on Bias Flow and Heat Transfer with Slide Gate
3.4.1. Effect of Casting Speed on Bias Flow with Slide Gate
Figure 17 shows the velocity distribution and symmetric index under different casting speed at Z=−0.075 m plane when the slide-gate opening rate is 60% and EMS current
Fig. 16. Variance of solidified shell thickness under different EMS currents: (a) OR side of mold, (b) IR side of mold. (Online version in color.)
is 600A. As shown in Fig. 17(a), when the casting speed is 0.8 m/
min, due to the effect of electromagnetic force, a horizonal recirculating flow is generated, and the velocity distribution in the mold is nearly centrosymmetric which coincides with the distribution of electromagnetic force. However, when the casting speed is 1.4 m/min (Fig. 17(b)), although there still exists a recirculating flow, the velocity distribution in the OR and IR side of mold is not equal. Figure 17(c) shows the symmetric index under different casting speed, it can be seen that when EMS current is 600A, the sym- metric index under the casting speed of 0.8 m/min and 1.0 m/min is nearly equal, this also indicates that EMS can not remove the asymmetry of flow field completely. With the increase of casting speed, the symmetric index decreases, the main reason is that the intensity of jet flow poured from the SEN enhances with the casting speed increasing, this may deteriorate the asymmetry of flow field in the mold, and the optimization of EMS on asymmetric phenomenon weakens. As the casting speed increases from 0.8 m/min to 1.4 m/min, the symmetrical index decreases from 0.91 to 0.62, respectively.
3.4.2. Effect of Casting Speed on Uniformity of Solidified Shell with Slide Gate
The variation of solidified shell thickness at mold wide face under different casting speed is shown in Fig. 18 when the slide-gate opening rate 60% and EMS current is 600A, where the liquid fraction of 0.8 is defined as solidification front.
As illustrated in Fig. 18, the solidified shell thickness decreases with the increasing of casting speed, the maximal solidified shell thickness in the IR side of mold decreases 16.49 mm to 12.36 mm when the casting speed changes from 0.8 m/min to 1.4 m/min. For the OR side of mold, due to the washing effect of electromagnetic force, the solidified shell thickness is uniform even under different casting speed. However, for the IR side of mold, the remelt- ing region at the casting speed of 0.8 m/min is smaller than that of 1.4 m/min, the main reason is that the jet flow poured from the SEN is weak at the low casting speed (0.8 m/min), this may suppress the impact region of bias flow in the IR side of mold, furthermore, the diagonal jet flow
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induced by electromagnetic force can also aggravate the remelting phenomenon of solidified shell and lead to the inferior uniformity.
The variance of solidified shell thickness under different casting speeds is shown in Fig. 19, it can be seen that the variance of solidified shell thickness in the IR side of mold is greater than that in the OR side due to the obvious remelt- ing phenomenon induced by bias flow. With the increasing of casting speed, the variance of solidified shell thickness decreases firstly and then increases. The main reason is that under the low casting speed (0.8 m/min and 1.0 m/min), the present EMS current has exceeded the critical EMS current which is benefit to the uniform growth of solidified shell, whereas for the high casting speed (1.4 m/min), the intensity of upper recirculating flow increases, therefore, it may need
a larger EMS current to improve the uniformity of solidified shell thickness. When the casting speed changes from 0.8 m/ min to 1.2 m/min, the variance at −0.4 m from the meniscus of IR side changes from 2.63 to 1.72, respectively.
From all the results, in order to correct the bias flow caused by slide gate and improve the uniformity of solidi- fied shell, a suitable EMS current should be selected. For example, when the slide-gate opening rate is 60% and cast- ing speed is 1.2 m/min, the optimal EMS current is 600A in our study case. However, more detailed and general criterion requires further research considering additional factors, such as the slide-gate orientation, submerged entry depth and slab size, etc.
Fig. 17. Velocity distribution under different casting speeds: (a) 0.8 m/min, (b) 1.4 m/min and (c) symmetric index under different casting speed at Z = − 0.075 m plane.
Fig. 18. Variation of solidified shell thickness under different casting speeds: (a) 0.8/min, (b) 1.4 m/min.
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4. Conclusion
A three-dimensional mathematical model coupled with electromagnetic, flow, heat transfer and solidification has been developed to investigate the influence of EMS on bias flow under different slide-gate opening rates and casting speeds. The symmetric index and variance of solidified shell thickness have been defined to judge the symmetry of flow field and uniformity of solidified shell, respectively. The following conclusions can be drawn:
(1) When EMS is off, the bias flow phenomenon has happened in the mold, the deviated direction is opposite to the opening direction of slide gate. With the decrease of slide-gate opening rate, the velocity at steel/slag interface decreases.
(2) When EMS is off, the symmetry of flow field and the uniformity of solidified shell thickness decreases with the decrease of slide-gate opening rate.
(3) EMS can improve the symmetry of flow field sig- nificantly, but it can not be eliminated completely.
(4) EMS can improve the uniformity of solidified shell in the mold, but the excessive EMS current may not benefit to the uniform growth of solidified shell.
(5) As the casting speed increases, for a given slide-gate opening rate and EMS current, the symmetry of flow field decreases, while the uniformity of solidified shell increases firstly and then decrease.
(6) There exists an optimal EMS current to balance the symmetry of flow field and uniformity of solidified shell. For example, under the operating condition of 60% slide- gate opening rate and 1.2 m/min casting speed, the optimal EMS current is 600A.
Acknowledgement This project financially supported by National Science
Foundation of China (NO. U1860107 and 52074181) and the Science and Technology Commission of Shanghai Municipality (No. 19DZ2270200).
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