Numerical Simulation for Magnetohydrodynamic Flow and Solidification in an Ultra-wide Slab Continuous Caster with Electromagnetic Stirring Roller

Abstract

Electromagnetic stirring with segment roller in the secondary cooling zone is a very important metallurgical technology for the continuous casting of the ultra-wide slab. Thus, numerical simulation is applied to investigate magnetohydrodynamic flow and solidification in the continuous caster with strand electromagnetic stirring. Numerical results showed that, the predicted values agree well with the experimental data. If the electromagnetic stirring roller with the symmetric split structure forms the symmetric magnetic field, there are the symmetric electromagnetic force, the symmetric flow field and the symmetric solidified shell. If the single electromagnetic stirring roller with the symmetric split structure forms the symmetric electromagnetic force, the flow field is like a butterfly. If the two electromagnetic stirring rollers forms the symmetric electromagnetic force, the flow field is like two butterflies. The effect of strand electromagnetic stirring on the fluid flow in the mold can not be ignored in the case of SSR (Same direction for upper rollers, Same direction for lower roller, Reverse direction for relation between upper/lower rollers), and it can be ignored in the case of NAS (No upper rollers, Away direction for lower roller, Single roller for relation between upper/lower rollers) and CAR (Close direction for upper rollers, Away direction for lower roller, Reverse direction for relation between upper/lower rollers).

1. Introduction

In order to meet the demand of modern industry, the metallurgists try their best to improve the steel product quality. Electromagnetic stirring is one of the effective measures to improve slab quality and increase slab production.^1,2,3,4) According to the installation positions of electromagnetic stirrer, there are three types of electromagnetic stirrers: F-EMS (final electromagnetic stirrer), S-EMS (strand electromagnetic stirrer) and M-EMS (mold electromagnetic stirrer). In the past few decades, there were numerous documents about M-EMS,^{5,6,7,8,9,10,11,12)} and a few of papers about S-EMS.^13,14)

S-EMS is applied in the slab continuous caster because it can enlarge the equiaxed grain zone, and weaken solute segregation and porosity.^15,16) In the steelmaking mill, there are three types of S-EMS: box type, between-roller type and roller type. Among them, roller type S-EMS becomes more and more popular on the base of the following factors. (1) It is easier for the roller type S-EMS to be replaced and repaired because it has the same installation dimension as the actual supporting rollers. (2) It can generate strong electromagnetic force in the molten steel because of the short distance between the coil and the slab surface.

Usually, there is only one set of electromagnetic stirring device in an electromagnetic roller in the case of the traditional slab. Reference^13,14) gave the characteristics of the fluid flow and the solidification in this case, but they did not consider the interaction of the fluid flow between the mold and the secondary cooling zone. In the case of ultra-wide slab, there are two set of electromagnetic stirring devices in an electromagnetic roller shown in Fig. 1. In order to have a deep insight into the metallurgical phenomena of S-EMS for ultra-wide slab, it is necessary to investigate the effect of magnetic field on flow field and solidification in the region of mold and secondary cooling zone by numerical simulation. Thus, this paper is organized as follows. Section 2, 3 and 4 gives the mathematical model of magnetic field, flow field and solidification, and the related computational procedure. Section 5 validates the numerical results by experimental data, and gives the spatial distribution of magnetic field, electromagnetic field, flow field, and solidified shells.

Fig. 1.

Ultra-wide slab continuous caster. (Online version in color.)

2. Mathematical Model for Electromagnetic Field

2.1. Assumption

(1) The electromagnetic field in the case of S-EMS can be treated as the quasi-static magnetic field.

(2) Electromagnetic characteristics of iron core and molten steel are isotropic.

(3) The effect of fluid flow on the electromagnetic field can be neglected.

(4) The magnetic field is confined in the area three times of S-EMS geometry.

2.2. Governing Equations for Magnetic Field

Maxwell’s equations are the basic governing equation for electromagnetic field. In the case of electromagnetic stirring, the simplified Maxwell’s equations can be expressed as   

$$\nabla \cdot \mathit{B}=0$$(1)

  

$$\nabla \times \mathit{H}={\mathit{J}}_{\mathit{s}}$$(2)

  

$$\nabla \times E=-\frac{\partial \mathit{B}}{\partial t}$$(3)

With   

$$\mathit{B}=\mu \mathit{H}$$(4)

where B is magnetic flux density, H is the magnetic field strength, μ is the magnetic permeability, E is the electric field strength, J_s is the source current, σ is the electrical conductivity, and t is the time.

In the molten steel, the induced electric current density can be determined by Ohm’s law.   

$${\mathit{J}}_I=\sigma \mathit{E}$$(5)

It is difficult to solve the above partial differential equations. In order to simplify these equations further, it necessary to introduce electric scalar potential φ and magnetic vector potential A which are defined as   

$$\mathit{E}=-\frac{\partial \mathit{A}}{\partial t}-\nabla \varphi$$(6)

  

$$\mathit{B}=\nabla \times \mathit{A}$$(7)

With   

$$\nabla \cdot \mathit{A}=0$$(8)

Consequently, we can obtain the following governing equations in different regions.

In the molten steel region:   

$$\nabla \times \left(\frac{1}{\mu }\nabla \times \mathit{A}\right)+\sigma \left(\frac{\partial \mathit{A}}{\partial t}+\nabla \varphi \right)=0$$(9)

  

$$\nabla \cdot \left(\sigma \frac{\partial \mathit{A}}{\partial t}-\sigma \nabla \varphi \right)=0$$(10)

In the other regions:   

$$\nabla \times \left(\frac{1}{\mu }\nabla \times \mathit{A}\right)={\mathit{J}}_s$$(11)

In this way, the induced current in the molten steel can be expressed as   

$${\mathit{J}}_i=\sigma \left(-\frac{\partial \mathit{A}}{\partial t}-\nabla \varphi \right)$$(12)

Further, the time averaged electromagnetic force caused by the induced current can stir the molten steel.   

$${\mathit{F}}_{em}=\frac{1}{2}Re({\mathit{J}}_i\times {\mathit{B}}^{*})$$(13)

where B^* is equal to the conjugate complex number of B, and Re is the real part of the complex quantity.

Joule heating power density Q caused by the induced current can heat the molten steel.   

$$Q={\mathit{J}}_i\cdot {\mathit{J}}_i/\sigma$$(14)

2.3. Boundary Conditions and Grid System

Figure 1(a) gives the schematic of roller type S-EMS. There are two pairs of rollers placed up and down, and the distance between the upper/lower rollers and the free surface are 6.2 m and 7.6 m, respectively. Each pair of rollers consists of two rollers placed on both side of the slab, and each roller includes two iron core with three coils and two half-coils, shown in Fig. 1(b).

Table 1 gives electromagnetic parameters of materials during the calculation. The electromagnetic roller consists of magnetic shielding ring, iron core and coil. The length of the magnetic shielding ring, coil and iron core are 3.3 m, 0.34 m and 1.49 m, respectively. And the inside and outside diameter of the coil are 100 mm and 110 mm. The opening angle of the magnetic shielding ring is 120°. The current frequency is 11 Hz, and the current is 200 A in the case of two pairs of electromagnetic stirring rollers and 280 A in the case of one pair of electromagnetic stirring rollers. The computational domain, which consists of air region, molten steel region, coil region and iron core region, is discretized by using about 590000 grid. And the magnetic flux-parallel condition is imposed at the boundary of the computational domain.

Table 1. Electromagnetic parameters of materials.v

Material	Relative magnetic permeability, –	Electrical conductivity, S/m
Molten steel	1.0	7.14×105
Iron core	1000	–
Coil	1.0	5.88×107
Air	1.0	–

3. Mathematical Model for Fluid Flow and Solidification

3.1. Assumption

(1) The flow of molten steel is at a stable state.

(2) The molten steel can be treated as the incompressible Newtonian fluid.

(3) In order to simplify the governing equations, the free surface is flat, and the top slag do not affect the fluid flow in the mold.

(4) The latent heat of δ-γ transformation is far less than the latent heat of fusion, so the latent heat of δ-γ transformation can be ignored.

3.2. Governing Equations

Fluid flow and solidification are the important metallurgical phenomenon in the continuous caster. The time-averaged governing equations for fluid flow and solidification in the Cartesian system can be formulated as follows:

Continuity equation   

$$\nabla \cdot (\rho \mathit{u})=0$$(15)

Here, ρ and u are the density and the velocity, respectively.

Momentum conservation equation   

$$\nabla \cdot (\rho \mathit{u}\mathit{u})=-\nabla \mathrm{p}+\nabla \cdot (\rho {\nu }_{\mathrm{e}\mathrm{f}\mathrm{f}}\nabla \mathit{u})+\rho \mathit{g}+{\mathit{F}}_{\mathit{e}\mathit{m}}+{\mathit{S}}_{\mathit{u}}+{\mathit{S}}_{\mathit{T}}$$(16)

where g, p and F_em are the gravitational acceleration, the pressure and the electromagnetic force, respectively. S_u represents the velocity source term from the interaction between the solidified shell and the molten steel, and S_T represents the thermal buoyancy. The effective viscosity ν_eff is determined by the low Reynolds k-ε turbulence model.^17,18,19) Here, k is turbulent kinetic energy, ε is the dissipation rate of turbulent kinetic energy.

Energy conservation equation   

$$\nabla \cdot (\rho {\mathrm{C}}_{\mathrm{p}}\mathit{u}T)=\nabla \cdot ({\mathrm{K}}_{\mathrm{e}\mathrm{f}\mathrm{f}}\nabla T)-\nabla \cdot [\rho f_{\mathrm{l}}{\mathit{u}}_{\mathrm{l}}\mathrm{\Delta }{\mathrm{H}}_{\mathrm{f}}]$$(17)

With   

$$f_{\mathrm{l}}=\{\begin{array}{ll}0\hfill & \mathrm{i}\mathrm{f}\mathrm{\hspace{0.17em} } T<{\mathrm{T}}_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}\mathrm{u}\mathrm{s}}\hfill \\ {\displaystyle \frac{T-{\mathrm{T}}_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}\mathrm{u}\mathrm{s}}}{{\mathrm{T}}_{\mathrm{l}\mathrm{i}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{d}\mathrm{u}\mathrm{s}}-{\mathrm{T}}_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}\mathrm{u}\mathrm{s}}}}\hfill & \mathrm{i}\mathrm{f}\mathrm{\hspace{0.17em} } {\mathrm{T}}_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}\mathrm{u}\mathrm{s}}<T<{\mathrm{T}}_{\mathrm{l}\mathrm{i}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{d}\mathrm{u}\mathrm{s}}\hfill \\ 1\hfill & \mathrm{i}\mathrm{f}\mathrm{\hspace{0.17em} } T>{\mathrm{T}}_{\mathrm{l}\mathrm{i}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{d}\mathrm{u}\mathrm{s}}\hfill \end{array}$$(18)

Here, C_p and K_eff are the specific heat and the effective thermal conductivity, respectively. T is the temperature, f_l and u_l is the liquid fraction and velocity of molten steel, ΔH_f is the latent heat of fusion. T_solidus and T_liquidus are solidus temperature and liquidus temperature, respectively.

3.3. Boundary Conditions and Grid System

Table 2 gives geometric and physical parameters during the calculation. The computational domain is covered by about 290000 grids. The boundary conditions consist of two part: flow field and temperature.

Table 2. Geometric parameters and physical parameters during numerical simulation.

Item	Value
Slab size	150 mm×3160 mm×10560 mm
Nozzle angle	15°
Submerged depth	100 mm
Casting speed	1.2 m/min
Density of molten steel	7020 kg/m3
Viscosity of molten steel	0.0062 Pa·s
Thermal conductivity of molten steel	35 W/(m·K)
Specific heat	680 J/(kg·K)
Latent heat of fusion	270000 J/kg
Melt superheat	20 K
Liquidus temperature	1787 K
Solidus temperature	1729 K

3.3.1. Flow Field

At the inlet, the velocity at the nozzle port can be obtained according to the mass balance law, other variables are set on the base of the inlet velocity.^20,21,22) At the outlet of continuous caster, the normal gradients of all variables are equal to zero. The normal velocity is zero and the normal gradients of other variables are equal to zero near the free surface, and the solidified shell moves down at the casting speed.

3.3.2. Temperature Field

At the inlet, the superheat of molten steel is 20 K. At the free surface and the outlet, the normal gradients of the temperature is zero. At the mold wall, the heat flux comes from an empirical formula.^23,24) At the secondary cooling zone, the heat loss comes from the convective heat transfer. And the convective heat transfer coefficient at the wide/narrow face are 175 W/m²·K and 125 W/m²·K.

4. Computational Procedure and Convergence Criterion

The commercial software ANSYS²⁵⁾ is applied to solve the governing equations about electric scalar potential φ and magnetic vector potential A to obtain the magnetic field and the induced current, and the convergence criterion is that the root mean square should be less than 10⁻⁵. Further, the commercial software FLUENT²⁵⁾ is applied to solve the flow field and the continuity equation, the momentum conservation equation and the energy conservation equation to obtain the flow field and the solidified shell, and the convergence criterion is that the residual value should be less than 10⁻⁵. It should be noted that a FORTRAN source codes is developed to interpolate electromagnetic force between two grid systems since the grid system in ANSYS is different from that in FLUENT.

Table 3 gives three experimental scheme on S-EMS. Among these schemes, the spatial distribution of electromagnetic force is symmetric in the cases of the NAS (No upper rollers, Away direction for lower roller, Single roller for relation between upper/lower rollers) scheme and the CAR (Close direction for upper rollers, Away direction for lower roller, Reverse direction for relation between upper/lower rollers) scheme, and the spatial distribution of electromagnetic force is asymmetric in the case of the SSR (Same direction for upper rollers, Same direction for lower roller, Reverse direction for relation between upper/lower rollers) scheme.

Table 3. Electromagnetic stirring schemes.

Experimental scheme	Upper rollers	Lower rollers	Relation between upper/lower rollers	Direction of electromagnetic force
NAS	None	Away	Single roller	none
				←→
SSR	Same direction	Same direction	Reverse direction	←←
				→→
CAR	Close	Away	Reverse direction	→←
				←→

5. Result and Discussion

5.1. Model Validation

In order to check the validity of the numerical results, we used a digital Gaussmeter CH-1800 to measure the magnetic field in the slab continuous caster. Figure 2 shows that the predicted magnetic field conformed well with the experimental data. When the current is 200 A and the frequency is 11 Hz, the predicted maximum magnetic flux density is 117.25 mT, and the measured maximum magnetic flux density is 125 mT. In other words, the relative error is only 6.2%.

Fig. 2.

Verification of magnetic field.

If electromagnetic stirring is stronger, white bands appear in the slab cross-section. The position of white band can give the solidified shell thickness. Figure 3 shows that, when the distances from the free surface are 6.2 m and 7.6 m, the solidified shell thicknesses are 36 mm and 46 mm, respectively. And the related numerical results are 38.5 mm and 46.7 mm. In other words, the relative errors are 6.9% and 1.5%, respectively.

Fig. 3.

Thickness distribution of solidified shells.

5.2. Electromagnetic Field

Figure 4 shows that the magnetic field has the following character. (1) This is a symmetric magnetic field because the electromagnetic stirring roller has the symmetric split structure. (2) The magnetic field has ‘M’ shape because the phase difference of adjacent coil current is 90° and there is three coil and two half-coil at each side of the electromagnetic stirring roller shown in Fig. 1(b). (3) There are six peak magnetic induction intensities, and the maximum value is 265 mT.

Fig. 4.

Magnetic field in the central horizontal cross section in the case of NAS.

Figure 5 shows the electromagnetic force is in the opposite direction in the region between two pairs of electromagnetic stirring rollers. The electromagnetic force increase first and then decrease with the increase of the distance from the nozzle centerline. The maximum electromagnetic force is 87 kN/m³.

Fig. 5.

Electromagnetic force field in the central longitudinal section in the case of NAS.

5.3. Fluid Flow and Solidification

Figure 6(a) shows that the upper recirculation zone and the lower recirculation zone appear in the mold, and there is fully developed pipeline flow under the mold in the case of no S-EMS. And Figs. 6(b)–6(d).

Fig. 6.

Flow field in the central longitudinal section of continuous caster.

Figure 6 give the multi-vortex structures caused by S-EMS have some similar features in the case of different S-EMS schemes. (1) There are two pairs of vortex on both sides of the electromagnetic roller because the electromagnetic stirring roller with the symmetric split structure forms the symmetric electromagnetic force. (2) The vortex near the upper electromagnetic stirring roller is bigger than that near the lower electromagnetic stirring roller because the thickness of solidified shell near the upper electromagnetic stirring roller is greater than that near the lower electromagnetic stirring roller. (3) The effect of S-EMS on the fluid flow in the mold can not be ignored in the case of SSR, and it can be ignored in the case of NAS and CAR because the region of asymmetric flow in the case of SSR is greater than the region of symmetric flow in the case of NAS and CAR.

Figure 6 shows that there are some different features in the case of different S-EMS schemes. (1) the upper vortex is near the solidified shell and the lower vortex is close to the nozzle centerline because the direction of fluid flow is from the top to the bottom and then the lower vortex is affected by the upper vortex. (2) The flow fields are like a butterfly in the case of NAS and two butterflies in the case of CAR. (3) There are irregular vortex in the case of SSR because the one-way electromagnetic force destroys the symmetry of the flow field.

Figure 7 gives the thickness distribution of solidified shells growing along the centers of the slab broad face and along the center of the slab narrow face, and shows the close relationship between the thickness distribution of solidified shells and flow field. Figures 6(a) 6(b) 6(d) and 7(a) 7(b) 7(d) indicate that the symmetric flow field leads to the symmetric thickness distribution of solidified shells. Figures 6(c) and 7(c) indicate that the asymmetric flow field leads to the asymmetric thickness distribution of solidified shells. Such an interesting phenomenon comes from the following reasons. (1) The symmetric/asymmetric flow field can form the same/different convective heat transfer at the solidification front. (2) The same/different convective heat transfer leads to the same/different solidified shells at the narrow faces or the broad faces of the slab.

Fig. 7.

Thickness distribution of solidified shells.

Figure 7(a) shows that the solidified shell is about 88 mm at the narrow face and 47 mm at the wide face if the distance from the free face is 8 m because the cooling intensity at the narrow face is greater than that at the wide face. Figure 7(b) shows that, in the case of NAS, the solidified shell is about 84 mm at the narrow face and 49 mm at the wide face if the distance from the free face is 8 m because there is stronger flow near the center of the narrow face and weaker flow near the center of the wide face. Figure 7(c) shows that, in the case of SSR, the solidified shell at the narrow-left face is 11 mm greater than that at the narrow-right face if the distance from the free face is 8 m because of the asymmetric flow caused by the one-way stirring from each electromagnetic roller. Figure 7(c) shows that, in the case of CAR, the solidified shell is about 89 mm at the narrow face and 49 mm at the wide face when the distance from the free face is 8 m because of the uniform stirring.

Figure 7 also shows that the steel shell profile on wide face get thicker with very high increase rate after 9 m below meniscus because the convective heat transfer at the wide face is greater than that at the narrow face.

6. Conclusions

(1) If the electromagnetic stirring roller with the symmetric split structure forms the symmetric magnetic field, there are the symmetric electromagnetic force, the symmetric flow field and the symmetric solidified shell.

(2) If a pair of electromagnetic stirring rollers with the symmetric split structure forms the symmetric electromagnetic force, the flow field is like a butterfly because of the symmetric physical field.

(3) If the two pairs of electromagnetic stirring rollers forms the symmetric electromagnetic force, the flow field is like two butterflies because of the symmetric physical field.

(4) The effect of S-EMS on the fluid flow in the mold can not be ignored in the case of SSR because of the asymmetric flow field, and it can be ignored in the case of NAS and CAR because of the symmetric flow field.

Nomenclatures

B: Magnetic flux density (T)

H: Magnetic field strength (A/m)

μ: Magnetic permeability (H/m)

E: Electric field strength (V/m)

J_s: Source current (A/m²)

σ: Electrical conductivity (S/m)

t: time (s)

φ: Electric scalar potential (V)

A: Magnetic vector potential (T·m)

ρ: Fluid density (kg·m⁻³)

u: Fluid velocity (m/s)

g: gravitational acceleration (m·s⁻²)

p: Pressure (Pa)

F_em: Electromagnetic force (N/m³)

S_u: Velocity source term from the interaction between the solidified shell and the molten steel (N/m³)

S_T: Thermal buoyancy (N/m³)

ν_eff: Effective viscosity (m²/s)

k: Turbulent kinetic energy (m²/s²)

ε: Dissipation rate of turbulent kinetic energy (m²/s³)

C_p: Specific heat (J/kg⁻¹·K⁻¹)

K_eff: Effective thermal conductivity (W/m⁻¹·K⁻¹)

T: Temperature (K)

f_l: Liquid fraction (–)

u_l: Velocity of molten steel (m/s)

ΔH_f: Latent heat of fusion (J/kg)

T_solidus: Solidus temperature (K)

T_liquidus: Liquidus temperature (K)

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