Effective Removal Zone of Inclusions in a Horizontal Channel under A.C. Magnetic Field Imposition

Abstract

A channel type horizontal induction heating tundish compensates for the heat loss of the molten steel due to Joule loss generated by an A.C. magnetic field. It also exhibits another function of inclusions removal because the A.C. magnetic field generates an electromagnetic pinch force. For the inclusions below the center of the horizontal channel, the direction of the electromagnetic pinch force and the buoyancy force acting on them are opposite. Thus, there is a possibility of the existence of the balanced position where the magnitudes of the electromagnetic pinch force and the buoyancy force are same. Around there the net time average force acting on the inclusions is almost zero, and there is a dead zone where the removal time of the inclusions under the imposed A.C. magnetic field is longer than that without it. In this study, non-dimensional models of the force balance and the inclusion trajectory were established and numerically solved to find out the relationship between the dead zone and the A.C. magnetic field parameters because the dead zone range should be reduced for effective removal of the inclusions. Consequently, the dead zone range decreased with the increase in the magnetic field intensity. Furthermore, the shielding parameter of 5–10 is one of the optimum conditions to reduce the dead zone range under the constant magnetic field condition because the dead zone range has the local and/or global minimum at this parameter.

1. Introduction

It is well known that nonmetallic inclusions such as alumina particles in steel lead to challenges in some processes such as rolling and wire drawing,^1,2,3) and they decrease fatigue strength of final products.^4,5,6) With increasing demands for high performance steel products year by year, removal of nonmetallic inclusions from molten steel has been systematically explored. In previous researches,^7,8,9,10) to enhance the removal of the inclusions, the inner configuration of the tundish was improved using the flow control devices such as turbulence inhibitors, dams, etc., in which the driving force of the inclusions removal was buoyancy force. Increasing velocity of inclusions is proportional to the square of their diameter^11,12) and the removal of micro-size inclusions requires long operating time, therefore, it hardly meets the cleanliness level requirement in many practical applications due to their low efficiency.^13,14) Thus, other effective method for the separation of inclusions are highly desirable.

Use of tundish with channel type induction heating has been identified as a good technology^{15,16,17,18,19)} to prevent temperature drop of the molten steel by imposing an A.C. magnetic field on the molten steel flowing in the channel. Furthermore, the inclusions move towards the channel wall by the electromagnetic force acting on them and are trapped on the channel wall. Thus, this method has two functions including the efficient compensation of the heat loss due to the Joule heating and the inclusion removal by the electromagnetic force.

Significant research efforts have been devoted for the comprehensive understanding of the variation of motion of the inclusions with the applied A.C. magnetic field. Taniguchi et al.^20,21,22) calculated the separation efficiency of nonmetallic inclusions from the molten steel flowing in a circular pipe by the trajectory method under laminar flow condition when the frequency of the A.C. magnetic field was 60 Hz. Wang et al.^23,24) numerically simulated the motion of inclusions in the tundish with channel type induction heating under the A.C. magnetic field condition with 50 Hz frequency. However, these studies treated only a commercial frequency of 50 or 60 Hz. Maruyama et al.^25,26) studied the effect of frequency and intensity of the A.C. magnetic field on the trajectories of the inclusions in the molten steel. They found the optimum frequency range for the removal of inclusions. However, their research focused only on the inclusions above the center of the channel, where the direction of the electromagnetic pinch force is always in the same direction with the buoyancy force acting on the inclusions, and the A.C. magnetic field is always beneficial to remove the inclusions.

In contrast, when the inclusion is present below the center of the channel, the effect of the A.C. magnetic field on the inclusion behavior is totally different. The direction of the buoyancy force acting on the inclusion in the channel is opposite to the gravitational direction (negative y-direction) as shown in Fig. 1. If the A.C. magnetic field with x-component is imposed on the molten steel flowing to the axis direction (x-direction) of the channel, the direction of the electromagnetic pinch force acting on the inclusion existing below the center becomes opposite to that of the buoyancy force. Thus, the balanced position where the magnitudes of the buoyancy force and the electromagnetic pinch force are same might exist. The net time average force acting on the inclusion is almost zero around there, and therefore there might be a region in the horizontal channel where the removal time of the inclusions using the A.C. magnetic field is longer than that without it. This region is called the “dead zone” in this study. Figure 2 demonstrates that at the upper boundary of the dead zone, the time of the inclusion floating up to the surface of the channel with the A.C. magnetic field is similar to that without the A.C. magnetic field. At the lower boundary of the dead zone, the times of the inclusion removal with and without the A.C. magnetic field are also same, though the moving directions of the inclusions are opposite. For effective removal of the inclusions from the molten steel, the dead zone should be reduced. However, the optimum operating condition for the inclusions removal in the circular channel has not been clarified till date.

Fig. 1.

Forces acting on the inclusion below the center of the channel.

Fig. 2.

Schematic diagram of “balanced position” and “dead zone”.

In this study, for optimum design of the horizontal channel to remove the inclusions, the balanced position and the dead zone range were numerically calculated as a function of the A.C. magnetic field parameters including intensity and frequency.

2. Mathematical Model

2.1. Configuration for Analysis

The horizontal circular channel was simplified as a two-dimensional plate model, wherein static molten steel was infinitely extended in horizontal direction and its vertical thickness was 2w, as shown in Fig. 3. This configuration is similar to that mentioned in the previous study.²⁵⁾ That is, the upper boundary is y = w, while the lower boundary is y = −w. The A.C. magnetic field with frequency f and intensity B₀ in x-direction is imposed on the molten steel from the top and bottom surfaces; therefore, the induced current flows in z-direction, and the electromagnetic force is in the y-axis direction.

Fig. 3.

Analytical system.

2.2. Forces Balance Acting on Inclusion

Based on the Maxwell’s equation,^27,28) Eq. (1) was acquired. The distribution of the magnetic field B_x in this analytical system is derived by using this equation with the boundary condition (2). Then the electromagnetic force acting on the molten steel is solved according to Eq. (3). In this equation, the electromagnetic force acting on the molten steel can be divided into two parts: the electromagnetic pinch force which is the first term in the right hand side and the electromagnetic oscillating force with the frequency of 2f which is the second term in the right hand side.   

$$\frac{\partial B_x}{\partial t}=\frac{1}{\sigma \mu }\frac{{\partial }^2{B}_x}{\partial y^2}$$(1)

  

$$B_x|{}_{y=\pm w}=B_0\text{cos(}2\pi ft\text{)}$$(2)

  

$$F_y=J_z\times B_x=C_1[\text{sinh(}2\varsigma \text{)}-sin(2\varsigma )]+C_1{C}_2\text{cos}\left(4\pi ft+a\right)$$(3)

  

$$C_1=\frac{-B_0^2\sqrt{R_w}}{2\mu w\left(cosh\sqrt{2R_w}+cos\sqrt{2R_w}\right)}$$(4)

  

$$\begin{array}{l}{C}_2=\\ {\displaystyle \frac{\sqrt{(cosh\left(4\varsigma \right)-cos\left(4\varsigma \right)\left[{\left(cosh\sqrt{2R_w}+cos\sqrt{2R_w}\right)}^2+sin{h}^2\left(2\varsigma \right)si{n}^2\left(2\varsigma \right)\right]}}{\sqrt{2}\left(cosh\sqrt{2R_w}+cos\sqrt{2R_w}\right)}}\end{array}$$(5)

  

$$\begin{array}{l}a={\displaystyle \frac{\pi }{4}}-\text{arccos}\\ \left[{\displaystyle \frac{4sinh\text{(}\varsigma \text{)}cosh\sqrt{{\displaystyle \frac{R_w}{2}}}\text{cos(}\varsigma \text{)}cos\sqrt{{\displaystyle \frac{R_w}{2}}}+cosh\text{(}\varsigma \text{)}sinh\sqrt{{\displaystyle \frac{R_w}{2}}}sin\text{(}\varsigma \text{)}sin\sqrt{{\displaystyle \frac{R_w}{2}}}}{2\sqrt{\left(cosh\text{(2}\varsigma \text{)}-cos\text{(2}\varsigma \text{)}\right)\left(cosh\sqrt{2R_w}+cos\sqrt{2R_w}\right)}}}\right]\end{array}$$(6)

  

$$\varsigma =\left(\frac{\text{y}}{w}\right)\sqrt{R_w/2}$$(7)

  

$$R_w=2\pi f\sigma \mu w^2=\frac{2w^2}{{\delta }^2}$$(8)

  

$$\delta =\sqrt{\frac{1}{\pi f\mu \sigma }}$$(9)

Where, C₁ and C₂ are coefficients indicated by Eqs. (4) and (5), respectively. a is initial phase angle of the electromagnetic oscillating force defined by Eq. (6), ς is inclusion position parameter, R_w is the shielding parameter which is an index of the ratio of the half thickness of the molten steel to the electromagnetic skin layer, δ is defined by using Eq. (9).

2.3. Trajectory of Inclusion

The non-metallic inclusion is assumed to be spherical and non-conductive in this study. Thus, the electromagnetic force acting on the inclusions is −3/4 times the electromagnetic force acting on the molten steel,²⁹⁾ which is also divided into two parts: the electromagnetic pinch force acting on the inclusions, F_p, and the electromagnetic oscillating force acting on the inclusions, F_o.

Furthermore, the vertical motion of the inclusion was analyzed in which the electromagnetic pinch force, the electromagnetic oscillating force, the viscous drag force, and the buoyancy force acting on the inclusion were considered. The Basset force was not considered.²⁵⁾

Momentum balance for the inclusion is indicated by Eq. (10).   

$${\displaystyle \frac{du}{dt}}={\displaystyle \frac{1}{\left({\rho }_s+{\displaystyle \frac{{\rho }_f}{2}}\right)}}\left[-{\displaystyle \frac{3C_D{\rho }_f}{4D}}{u}^2+F_p+F_o+g({\rho }_f-{\rho }_s)\right]$$(10)

  

$$C_D=\frac{24}{R{e}_s}=\frac{24\eta }{{\rho }_{f}Du} \mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }R{e}_s<1$$(11)

In this study, the inclusion diameter is 100 μm and the inclusion velocity is around 0.003 m.s⁻¹. Thus, the assumption that the particle Reynolds number Re_s is small enough was adopted in the calculation.

Moreover, in order to generalize the optimized design of the channel, non-dimensional numbers were further introduced. The first one is non-dimensional magnetic field intensity ${\check{B}}_0$ which is a ratio of the electromagnetic pinch force to the buoyancy force. The second one is non-dimensional position $\check{y}$ normalized by the half thickness of the molten steel. Moreover, the third one is non-dimensional time $\check{t}$ normalized by the inclusion removal time t_s when the initial inclusion position is channel center under the no-magnetic field condition. These are defined as Eqs. (12), (13), (14), respectively.   

$${\check{B}}_0=\frac{B_0^2}{\mu wg({\rho }_f-{\rho }_s)}$$(12)

  

$$\check{y}=\frac{y}{w}$$(13)

  

$$\check{t}=\frac{D^{2}g({\rho }_f-{\rho }_s)}{18\eta w}t=\frac{t}{t_s}$$(14)

  

$$t_s=\frac{18\eta w}{D^{2}g({\rho }_f-{\rho }_s)}$$(15)

The non-dimensional form of Eq. (10) under steady state condition is   

$$\begin{array}{l}{\displaystyle \frac{d\check{y}}{d\check{t}}}=-{\displaystyle \frac{3}{4}}{C}_1\left[sinh\left(\sqrt{2R_w}\check{y}\right)-sin\left(\sqrt{2R_w}\check{y}\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{0.5em}-{\displaystyle \frac{3}{4}}{C}_1{C}_{2}cos\left(4\pi f\times {\displaystyle \frac{18\eta w}{D^{2}g({\rho }_f-{\rho }_s)}}\times \check{t}+a\right)+1\end{array}$$(16)

The fourth-order Runge–kutta method was used to calculate this equation, and in order to improve calculation accuracy, a very small time step of $\frac{1}{20f{t}_s}$ was used.

The electromagnetic pinch force and the buoyancy force are balanced at the balanced position; therefore, non-dimensional form of the balanced position is expressed as Eq. (17).   

$$-\frac{3}{4}{C}_1\left[\text{sinh}\left(\sqrt{2R_w}{\check{y}}^{*}\right)-sin\left(\sqrt{2R_w}{\check{y}}^{*}\right)\right]=1$$(17)

Physical properties used in the calculations are presented in Table 1. Table 2 summarizes the calculation conditions. The parameters in the calculations are non-dimensional magnetic field intensity and the shielding parameter. For convenience, these two non-dimensional parameters are related with the actual channel size and the imposed A.C. magnetic field intensity and frequency.

Table 1. Physical properties of materials used in the calculation.

Steel density ρf/(kg/m3)	Inclusion density ρs/(kg/m3)	Steel viscosity η/(Pa·s)	Steel conductivity σ/(S/m)	Magnetic permeability μ/(H/m)
6958	3880	5.28 × 10−3	7.2 × 105	4π × 10−7

Table 2. Shielding parameter and non-dimensional magnetic field intensity used in this analysis and the corresponding values in actual process.

			Rw
			0.159	1.59	5	10	15.9	159
${\check{B}}_0$	7.9	B0(T)	0.3	0.3	0.3	0.3	0.3	0.3
		f(Hz)	0.3	3	10	20	31	311
		w(m)	0.3	0.3	0.3	0.3	0.3	0.3
	14.1	B0(T)	0.2	0.2	0.2	0.2	0.2	0.2
		f(Hz)	5	50	156	313	500	5000
		w(m)	0.075	0.075	0.075	0.075	0.075	0.075
	31.7	B0(T)	0.3	0.3	0.3	0.3	0.3	0.3
		f(Hz)	5	50	156	313	500	5000
		w(m)	0.075	0.075	0.075	0.075	0.075	0.075
	56.3	B0(T)	0.4	0.4	0.4	0.4	0.4	0.4
		f(Hz)	5	50	156	313	500	5000
		w(m)	0.075	0.075	0.075	0.075	0.075	0.075

3. Results and Discussion

3.1. Effect of the A.C. Magnetic Field on Balanced Position

Figure 4 shows the distribution of the electromagnetic pinch force normalized by the buoyancy force acting on the inclusion under the constant non-dimensional magnetic field intensity of ${\check{B}}_0$ = 14.1. Noteworthy, the magnitude of the electromagnetic pinch force increases from the center to the lower surface of the channel under all calculation conditions, and the frequency of the A.C. magnetic field significantly affects the distribution of the electromagnetic pinch force. Obviously, the A.C. magnetic field affects the balanced position of the inclusion in the horizontal channel. When the shielding parameter is 0.159, the electromagnetic pinch force is always less than the buoyancy force, and there is no balanced position. In other words, influence of the electromagnetic pinch force on the inclusion motion might be neglected. When the shielding parameter is 159, the electromagnetic pinch force gets concentrated on the lower surface vicinity region in the channel because the electromagnetic skin layer is very small. Thus, the balanced position of −0.764487 is close to the lower surface of the channel, and the effect of the electromagnetic pinch force on the inclusion motion might be limited only in the surface vicinity region. When the shielding parameter is 1.59 or 15.9, the balanced positions are −0.607406 and −0.473914, respectively. Owing to the large electromagnetic skin layer, the region where the electromagnetic pinch force affects the inclusion motion is the largest under these calculation conditions.

Fig. 4.

Distribution of electromagnetic pinch force normalized by buoyancy force (${\check{B}}_0$ = 14.1).

Figure 5 presents the effect of the shielding parameter and the non-dimensional magnetic field intensity on the balanced position of the inclusion. If the shielding parameter is small enough, there is no balanced position in the channel because the electromagnetic pinch force acting on the inclusion is less than the buoyancy force even when the non-dimensional magnetic field intensity is 56.3. The greater the non-dimensional magnetic field intensity, the closer the balanced position to the center of the channel under the constant shielding parameter condition. The balanced position-shielding parameter curves exhibit the existence of a peak, indicating that the balanced position is the closest to the center of the channel under a certain shielding parameter. This shielding parameter is around 5 or 10 under this calculation condition.

Fig. 5.

Effect of A.C. magnetic field on balanced position of inclusions.

The boundary whether the balanced position exists or not under the shielding parameter - the non-dimensional magnetic field intensity map was calculated by using Eq. (17). The calculated results are plotted in solid dots in Fig. 6. When the shielding parameter is less than 1.0, the non-dimensional magnetic field intensity is inversely proportional to the square of the shielding parameter. However, the non-dimensional magnetic field intensity is inversely proportional to the square-root of the shielding parameter if the shielding parameter is far greater than 1.0. In contrast, when the shielding parameter is near unity or in the range between 1.0 and 5.0, derivation of a simple relationship between the shielding parameter and the non-dimensional magnetic field intensity is difficult from Eq. (17). Thus, it was derived from the numerically calculated values. These are expressed as Eq. (18).   

$$\{\begin{array}{l}{\check{B}}_0=5.66R_w^{-2}\\ {\check{B}}_0=5.9286R_w^{-1.083}\\ {\check{B}}_0=2.667R_w^{-0.5}\end{array}\begin{array}{l}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{R}_w \le 1.0\\ \hspace{1em}\hspace{1em}1.0< R_w\le 5.0\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{R}_w>5.0\end{array}$$(18)

Fig. 6.

Boundary whether the balanced position exists or not in the shielding parameter - the non-dimensional magnetic field intensity map.

These equations are useful for evaluating the critical magnetic field intensity between the two regions. For example, when the shielding parameter is 0.159, the balanced position in the channel exists if the non-dimensional magnetic field intensity is greater than 223.9.

3.2. Trajectories of the Inclusions below the Center of the Horizontal Channel

When the non-dimensional magnetic field intensity and the shielding parameter of the A.C. magnetic field are 14.1 and 1.59, respectively, the non-dimensional balanced position is −0.607406 as shown in Fig. 4. Then the inclusions trajectories with and without A.C. magnetic field under the four different initial positions which were close to this balanced position were calculated and the results are presented in Fig. 7. When the initial position is −0.570000, the inclusion floats at the upper surface of the horizontal channel, and the removal time of the inclusion under the imposed A.C. magnetic field is shorter than that without the A.C. magnetic field as shown in Fig. 7(a). In other words, this initial position does not belong to the dead zone. When the initial position of the inclusion is set at −0.607405, which is slightly above the balanced position, the inclusion floats at the upper surface of the horizontal channel; however the removal time of the inclusion with the A.C. magnetic field is longer than that without the A.C. magnetic field as shown in Fig. 7(b). This initial position is in the dead zone. When the inclusion is set at −0.607690 as the initial position which is slightly lower than the balanced position, the inclusion sinks to the lower surface of the horizontal channel in the case with the A.C. magnetic field, while it floats without the A.C. magnetic field as shown in Fig. 7(c). In the cases of Figs. 7(b) and 7(c), the trajectories of the inclusion is completely different though the initial positions in these cases are very close. This is because the initial position in the case of the Fig. 7(b) is slightly above the balanced position while that in the case of the Fig. 7(c) is slightly lower than the balanced position. The inclusion also sinks to the lower surface of the horizontal channel in the case with the A.C. magnetic field if the initial position is at −0.610000 as shown in Fig. 7(d). Nonetheless, this initial position is not in the dead zone. The inclusions trajectories with the A.C. magnetic field in the cases shown in Figs. 7(b) and 7(c) are quite different, though the difference of the initial positions in these cases is only 0.000285. That is, the inclusion removal time depends significantly on the initial position of the inclusion.

Fig. 7.

Inclusions trajectories when ${\check{B}}_0$ = 14.1, R_w = 1.59.

3.3. Effect of the A.C. Magnetic Field on the Dead Zone Range

The critical non-dimensional magnetic field intensity for the existence of balanced position was calculated to be 223.9 with the shielding parameter of 0.159 by using Eq. (18). This is larger than the maximum non-dimensional magnetic field intensity of 56.3 under the calculation condition presented in Table 2. Thus, the dead zone range was acquired by calculating the inclusions trajectories under the A.C. magnetic field conditions of R_w = 1.59 – 159 and ${\check{B}}_0$ = 7.9 − 56.3. Figure 8 demonstrates the effect of the shielding parameter and the non-dimensional magnetic field intensity on the dead zone range. The greater the non-dimensional magnetic field intensity, the smaller the dead zone range under the constant shielding parameter condition. The ratio of the dead zone range in the case ${\check{B}}_0$ = 7.9 to that ${\check{B}}_0$ = 56.3 is greater than 200 when the shielding parameter is 5 or 10. On the other hand, when the shielding parameter is as large as 159, the effect of the A.C. magnetic field intensity on the dead zone range is relatively small. The dead zone range has the local minimum around the shielding parameters of 5 or 10 for every non-dimensional magnetic field intensity. This is also the global minimum in the case when the non-dimensional magnetic field intensity is 31.7 or 56.3. However, the global minimum is observed when the shielding parameter is 159 in the case with the non-dimensional magnetic field intensity of 7.9 or 14.1. Thus, the optimum frequency or the shielding parameter to reduce the dead zone range depends on the magnetic field intensity. However, when the shielding parameter is greater than or equal to 5 and the non-dimensional magnetic field intensity is greater than or equal to 7.9, the dead zone range is less than or equal to 0.01. Thus, the industrial operation in this range might be suitable for the inclusion removal.

Fig. 8.

Effect of A.C. magnetic field on dead zone range of inclusions.

To find out the reason for the existence of the local minimum in Fig. 8, gradient of the electromagnetic pinch force acting on the inclusions at the balanced position was computed, and the corresponding results are presented in Fig. 9. The gradient of the electromagnetic pinch force has the local maximum around the shielding parameter of 5 or 10 except when the non-dimensional magnetic field intensity is 7.9. When the non-dimensional magnetic field intensity is 7.9, the gradient does not exhibit the local minimum value; however, it is almost constant around the shielding parameter of 5 or 10. If the gradient of the electromagnetic pinch force is large, a slight change in the position of the inclusion causes a large change in the electromagnetic pinch force acting on the inclusion. Then the balance between the electromagnetic pinch force and the buoyancy force becomes loose, and the inclusions move rapidly toward the channel wall. This might be the reason why the dead zone range has the local minimum around the shielding parameter of 5 or 10. When the shielding parameter is 159, the gradient of electromagnetic pinch force is the largest, however, the dead zone range is not the smallest. This is attributed to the fact that, in this case, the balanced position of the inclusion is closer to the bottom of the horizontal channel as shown in Fig. 5. That is, the effective region of the pinch force for a floating up inclusion to the upper surface of the channel is the smallest in this case. Thus, the range of the dead zone is not only determined by the gradient of the electromagnetic pinch force at the balanced position but also related to the balanced position.

Fig. 9.

Gradient of electromagnetic pinch force at balanced position under different A.C magnetic field conditions.

4. Conclusions

For the optimum design of the horizontal channel to remove the inclusions, the balanced position where the magnitudes of the buoyancy force and the electromagnetic pinch force are same and the dead zone range where the removal time of the inclusions under the A.C. magnetic field is longer than that without it were numerically calculated as a function of the A.C. magnetic field parameters of intensity and frequency.

The main conclusions drawn from this study are as follows.

(1) Relationship between the critical magnetic field intensity and the critical shielding parameter whether the balanced position exists or not is obtained.

(2) The dead zone range decreases with the increase in the magnetic field intensity.

(3) The shielding parameter of 5–10 is one of the optimum conditions to reduce the dead zone range under constant magnetic field condition because the dead zone range has the local and/or global minimum around there.

List of symbols:

a: initial phase angle (rad)

B: magnetic field intensity (T)

${\check{B}}_0$: non-dimensional magnetic field intensity (–)

C₁: constant defined by Eq. (4) (–)

C₂: constant defined by Eq. (5) (–)

C_D: drag coefficient (–)

D: inclusion diameter (m)

f: magnetic field frequency (Hz)

F: electromagnetic force (N/m³)

F_p: electromagnetic pinch force (N/m³)

F_o: electromagnetic oscillating force (N/m³)

J: current density (A/m²)

Re_s: particle Reynolds number (–)

R_w: shielding parameter (–)

t: inclusion removal time (s)

$\check{t}$: non-dimensional removal time of the inclusions (–)

u: inclusion velocity in the molten steel (m/s)

w: vertical thickness of the horizontal channel (m)

y: inclusion position in y direction (m)

$\check{y}$: non-dimensional position of the inclusions (–)

${\check{y}}^{*}$: non-dimensional balanced position of the inclusions (–)

δ: electromagnetic skin layer (m)

η: steel viscosity (Pa·s)

σ: steel conductivity (S/m)

μ: magnetic permeability (H/m)

ρ_f: steel density (kg/m³)

ρ_s: inclusion density (kg/m³)

ς: inclusion position parameter (–)

Subscripts:

x: x direction

y: y direction

z: z direction

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