Effect of an AC Magnetic-field on the Dead-zone Range of Inclusions in the Circular Channel of an Induction-heating Tundish

Abstract

In this paper, the radial electromagnetic force in the horizontal circular channel of an induction-heating tundish is derived. A dimensionless trajectory model of the inclusion is developed and numerically solved to acquire the trajectory of the moving inclusion. When the inclusion is in the lower half of the horizontal circular channel, the direction of the vertical component of the radial electromagnetic pinch force which acts on the inclusion is opposite to the buoyancy. Provided their magnitudes are the same, there is a balanced-position for the inclusions in a circular channel. Therefore, a dead-zone exists near the balanced-position, where the removal time of the inclusion with an AC magnetic-field is longer than without it. Then, the effect of the AC magnetic-field parameters on the range of the dead-zone is identified, which makes it possible to improve the removal efficiency of inclusions. The range of the dead-zone decreases with increasing magnetic-field intensity. When the dimensionless magnetic-field intensity is 56.3, the shielding parameter of 10–15.9 are optimal to decrease the range of the dead-zone.

1. Introduction

It is very significant to increase the cleanliness of liquid steel to meet the requirements of high-quality steel.^1,2) As the last refractory container of a metal-casting system, the tundish is the key to ensure continuous metal-casting.^3,4,5) To enable suitable flow conditions and avoid/reduce inclusions in the tundish, different systems, such as inhibitors, dams, weirs, and baffles are commonly used to control flow.^6,7,8,9) As these flow-control devices allow the liquid steel to reside longer in the tundish, the inclusion-removal efficiency can be increased. However, the liquid steel temperature decrease makes it difficult for the casting process to operate stably under a low superheat.^10,11,12)

A new tundish induction-heating technology has recently been developed,^{13,14,15,16,17)} which can compensate the molten steel’s temperature drop using an AC magnetic-field. The inclusions move towards the channel wall by the electromagnetic force acting on them, and they become trapped at the channel wall. Thus, this method has two functions: 1) heat loss compensation by Joule heating 2) inclusion removal by electromagnetic force.

Many studies were done to improve the understanding of the motion of inclusions when an AC magnetic-field was applied. Taniguchi et al.^18,19,20) used the trajectory method to solve the removal efficiency of nonmetallic inclusions in the molten steel with the AC magnetic-field. In their investigation, the molten steel was laminar flow in a circular channel, and the frequency of the AC magnetic-field was 60 Hz. Wang et al.^21,22) numerically simulated the motion of inclusions in a channel-type induction-heating tundish with an AC magnetic-field at 50 Hz. Unfortunately, these studies focused only on the commercial frequencies of 50 and 60 Hz. Maruyama and Zhang et al.^23,24,25,26) studied the effect of the AC magnetic-field parameters (including frequency and intensity) on the movement of the inclusions in the molten steel. However, in their research, the horizontal circular channel of a channel-type induction-heating channel was simplified to a 2-dimensional plate. The distributions of the electromagnetic forces, which act on the inclusion in a circular channel and a 2-D plate channel, are significantly different. For the 2-D plate channel, the AC magnetic-field is located on the upper and lower surfaces of the molten steel. This means that only a vertical electromagnetic force acts on the inclusion. In the circular channel, however, an AC coil surrounds it, and a radial electromagnetic force (acting on the inclusion) is produced - see Fig. 1. The radial electromagnetic force can be divided into vertical and horizontal electromagnetic forces acting on the inclusion. Thus, the trajectory of the inclusion in the circular channel with an AC magnetic-field is different from that of a 2-D plate.

Fig. 1.

Analytical model in this study.

In this paper, the radial electromagnetic force, which acts on the inclusion in a circular channel, was derived, and the trajectory of the inclusion in the circular channel was studied. In the lower half of the circular channel, the direction of the vertical component of the radial electromagnetic pinch force (which acts on the inclusion) is opposite to that of the buoyancy. Hence, a balanced-position exists for the inclusion, when the magnitudes of the vertical component of the radial electromagnetic pinch force and the buoyancy (which act on the inclusion) are identical. Thus, the dead-zone also exists near the balanced-position, the “dead-zone” is the region in the horizontal circular channel where the removal time of inclusions with the AC magnetic-field is longer than without it. This is because the net-time average force, which acts on the inclusion in the vertical direction, is almost zero near the balanced-position. To improve the removal efficiency of the inclusions from the molten steel, the range of the dead-zone needs to be decreased.

In this study, the range of the dead-zone was calculated, and it is related with the AC magnetic-field parameters (including frequency and intensity). In addition, the optimal operating-condition to remove the inclusion in the circular channel was clarified.

2. Mathematical Model

2.1. Conditions of the Analysis

A horizontal circular channel with a radius b is shown in Fig. 1. An AC magnetic-field (frequency is f and intensity is B₀) in z-direction surrounds the wall of the circular channel. Therefore, the induced current in the molten steel is the θ-direction, and the electromagnetic force is the r-direction.

2.2. Derivation of the Radial Electromagnetic Force Acting on the Molten Steel

The imposed AC magnetic-field is a time-harmonic electromagnetic field, which can be expressed using Eq. (1). According to the Maxwell equation,^27,28) Eq. (2) was determined. Thus, the magnetic field B_z in the circular channel was derived, Eq. (5), by solving Eq. (2) with the boundary conditions of Eqs. (3) and (4). Then, the current density distribution in the molten steel was obtained, Eq. (6), and the electromagnetic force (which acts on the molten steel) was solved – see Eq. (7).   

$$B_z\left(r,t\right)=B_z\left(r\right)e^{j2\pi ft}$$(1)

  

$${\nabla }^2{B}_z\left(r,t\right)=\mu \sigma \frac{\partial B_z\left(r,t\right)}{\partial t}\hspace{1em}0\le r\le b\hspace{1em}t>0$$(2)

  

$$\left|B_z\left(r,t\right)\right|<+\infty \mathrm{\hspace{0.17em} }(\text{finite value})\hspace{1em}r=0\hspace{1em}t>0$$(3)

  

$$Re\left(B_z\left(r,t\right)\right)=B_0\text{cos}\left(2\pi ft\right)\hspace{1em}r=b\hspace{1em}t>0$$(4)

  

$$\begin{array}{l}{B}_z\left(r,t\right)=\\ \frac{\pm B_0}{be{r}^2\sqrt{R_w}+be{i}^2\sqrt{R_w}}\sqrt{\left(be{r}^2\sqrt{R_w}+be{i}^2\sqrt{R_w}\right)\left(be{r}^2\left(\sqrt{R_w}\frac{r}{b}\right)+be{i}^2\left(\sqrt{R_w}\frac{r}{b}\right)\right)}{e}^{j\left(2\pi ft+{\varnothing }_1\right)}\end{array}$$(5)

  

$$\begin{array}{l}{J}_{\theta }\left(r,t\right)=-\frac{1}{\mu }\frac{\partial B_z}{\partial r}=\\ \frac{\pm B_0\sqrt{R_w}}{\mu b\left(be{r}^2\sqrt{R_w}+be{i}^2\sqrt{R_w}\right)}\sqrt{\left(be{r}^2\sqrt{R_w}+be{i}^2\sqrt{R_w}\right)\left(be{r}_1^2\left(\sqrt{R_w}\frac{r}{b}\right)+be{i}_1^2\left(\sqrt{R_w}\frac{r}{b}\right)\right)}{e}^{j\left(2\pi ft+{\varnothing }_2\right)}\end{array}$$(6)

  

$$\begin{array}{l}{F}_{mr}\left(r,t\right)=Re\left(B_z\left(r,t\right)\right)\times Re\left(J_{\theta }\left(r,t\right)\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=C_1\left[cos\left({\varnothing }_1-{\varnothing }_2\right)+cos\left(4\pi ft+{\varnothing }_1+{\varnothing }_2\right)\right]\end{array}$$(7)

  

$$\begin{array}{l}{\varnothing }_1=\\ \text{arctan}\left(\frac{ber\sqrt{R_w}\cdot bei\left(\sqrt{R_w}\frac{r}{b}\right)-bei\sqrt{R_w}\cdot ber\left(\sqrt{R_w}\frac{r}{b}\right)}{ber\sqrt{R_w}\cdot ber\left(\sqrt{R_w}\frac{r}{b}\right)+bei\sqrt{R_w}\cdot bei\left(\sqrt{R_w}\frac{r}{b}\right)}\right)\end{array}$$(8)

  

$$\begin{array}{l}{\varnothing }_2=\\ arctan\left[\frac{\left(ber\sqrt{R_w}-bei\sqrt{R_w}\right)be{i}_1\left(\sqrt{R_w}\frac{r}{b}\right)-\left(ber\sqrt{R_w}+bei\sqrt{R_w}\right)be{r}_1\left(\sqrt{R_w}\frac{r}{b}\right)}{\left(ber\sqrt{R_w}-bei\sqrt{R_w}\right)be{r}_1\left(\sqrt{R_w}\frac{r}{b}\right)+\left(ber\sqrt{R_w}+bei\sqrt{R_w}\right)be{i}_1\left(\sqrt{R_w}\frac{r}{b}\right)}\right]\end{array}$$(9)

  

$$\begin{array}{l}{C}_1=\\ \frac{\sqrt{R_w}{B}_0^2}{2\mu b\left(be{r}^2\sqrt{R_w}+be{i}^2\sqrt{R_w}\right)}\sqrt{\left(be{r}^2\left(\sqrt{R_w}\frac{r}{b}\right)+be{i}^2\left(\sqrt{R_w}\frac{r}{b}\right)\right)\left(be{r}_1^2\left(\sqrt{R_w}\frac{r}{b}\right)+be{i}_1^2\left(\sqrt{R_w}\frac{r}{b}\right)\right)}\end{array}$$(10)

  

$$R_w=2\pi f\sigma \mu b^2$$(11)

2.3. Trajectory Model for Inclusions in a Circular Channel

This study assumes that a non-metallic inclusion is non-conducting and spherical. The electromagnetic force acting on the inclusions is –3/4 times the electromagnetic force acting on the molten steel.²⁹⁾ Thus, the electromagnetic force acting on the inclusion is composed of two parts: 1) The radial electromagnetic pinch force acting on the inclusion, F_pr. 2) The radial electromagnetic oscillating force acting on the inclusion, F_or:   

$$F_{pr}\left(r,t\right)=-\frac{3}{4}{C}_1\left[cos\left({\varnothing }_1-{\varnothing }_2\right)\right]$$(12)

  

$$F_{or}\left(r,t\right)=-\frac{3}{4}{C}_{1}cos\left(4\pi ft+{\varnothing }_1+{\varnothing }_2\right)$$(13)

The Cartesian coordinate system was used to determine the trajectory of the inclusion, which is shown in Fig. 2. The trajectory of the inclusion, when the initial position was in the lower half of the circular channel, was studied. As the trajectory of the inclusion was symmetric along the y axis, the initial position was set in the fourth quadrant or on the -y-axis. In addition, the momentum balance for the inclusion in the fourth quadrant was indicated by Eqs. (14) and (15), where the electromagnetic pinch force, the electromagnetic oscillating force, the viscous drag force, and the buoyancy acting on the inclusion were taken into account.   

$$\frac{d{u}_x}{dt}=\frac{1}{\left({\rho }_s+\frac{{\rho }_f}{2}\right)}\left[-\frac{3C_{Dx}{\rho }_f}{4D}{u}_x^2+F_{pr}cos\left(\theta \right)+F_{or}cos\left(\theta \right)\right]$$(14)

  

$$\begin{array}{l}\frac{d{u}_y}{dt}=\\ \frac{1}{\left({\rho }_s+\frac{{\rho }_f}{2}\right)}\left[-\frac{3C_{Dy}{\rho }_f}{4D}{u}_y^2+F_{pr}sin\left(\theta \right)+F_{or}sin\left(\theta \right)+g\left({\rho }_f-{\rho }_s\right)\right]\end{array}$$(15)

  

$$\theta =\text{arctan}\left(\frac{y}{x}\right)\hspace{1em}\frac{3\pi }{2}\le \theta \le 2\pi$$(16)

  

$$C_{Dx}=\frac{24}{R{e}_{sx}}=\frac{24\eta }{{\rho }_{f}D{u}_x} \mathrm{\hspace{0.17em} } \text{when}\mathrm{\hspace{0.17em} } R{e}_{sx}<1$$(17)

  

$$C_{Dy}=\frac{24}{R{e}_{sy}}=\frac{24\eta }{{\rho }_{f}D{u}_y} \mathrm{\hspace{0.17em} } \text{when}\mathrm{\hspace{0.17em} } R{e}_{sy}<1$$(18)

Fig. 2.

Calculation model for the inclusion trajectory using the Cartesian coordinate system.

In this study, the inclusion diameter is 100 μm, and the inclusion velocity in x and y direction is around 0.0025 m/s. Thus, the assumption that the particle Reynolds number Re_sx and Re_sy are small enough was adopted in the calculation.

Moreover, to summarize the optimized design of the channel, dimensionless numbers such as the dimensionless magnetic-field intensity ${\stackrel{⌣}{B}}_0$, the dimensionless position $\stackrel{⌣}{x},\stackrel{⌣}{y}$, and dimensionless time $\stackrel{⌣}{t}$ are also introduced, similar to a previous study.²⁶⁾ The dimensionless forms of Eqs. (14) and (15) under steady-state conditions are:   

$$\begin{array}{l}\frac{d\stackrel{⌣}{x}}{d\stackrel{⌣}{t}}=-\frac{3}{4}{C}_{1}cos\left({\varnothing }_1-{\varnothing }_2\right)cos\left(\theta \right)\\ \hspace{1em}\hspace{1em}-\frac{3}{4}{C}_{1}cos\left(4\pi f\frac{18\eta b}{D^{2}g\left({\rho }_f-{\rho }_s\right)}\stackrel{⌣}{t}+{\varnothing }_1+{\varnothing }_2\right)cos\left(\theta \right)\end{array}$$(19)

  

$$\begin{array}{l}\frac{d\stackrel{⌣}{y}}{d\stackrel{⌣}{t}}=-\frac{3}{4}{C}_{1}cos\left({\varnothing }_1-{\varnothing }_2\right)sin\left(\theta \right)\\ \hspace{1em}\hspace{1em}-\frac{3}{4}{C}_{1}cos\left(4\pi f\frac{18\eta b}{D^{2}g\left({\rho }_f-{\rho }_s\right)}\stackrel{⌣}{t}+{\varnothing }_1+{\varnothing }_2\right)sin\left(\theta \right)+1\end{array}$$(20)

  

$${\stackrel{⌣}{B}}_0=\frac{B_0^2}{\mu bg\left({\rho }_f-{\rho }_s\right)}$$(21)

  

$$\stackrel{⌣}{x}=\frac{x}{b}$$(22)

  

$$\stackrel{⌣}{y}=\frac{y}{b}$$(23)

  

$$\stackrel{⌣}{t}=\frac{D^{2}g\left({\rho }_f-{\rho }_s\right)}{18\eta b}t$$(24)

The fourth-order Runge-Kutta method was adopted to solve these equations with double precision.³⁰⁾ Table 1 shows the physical parameters of the materials for the calculation. Table 2 shows the AC magnetic-field conditions for the calculation. The shielding parameter and the dimensionless magnetic-field intensity are related with the circular channel radius and the AC magnetic-field frequency and intensity.

Table 1. Physical parameters for the calculation.

Density of steel ${\rho }_f/\left(\frac{kg}{m^3}\right)$	Density of inclusion ${\rho }_s/\left(\frac{kg}{m^3}\right)$	Viscosity of steel η/(Pa·s)	Conductivity of steel $\sigma /\left(\frac{S}{m}\right)$	Permeability of steel $\mu /\left(\frac{H}{m}\right)$
6958	3880	5.28×10−3	7.2×105	4π×10−7

Table 2. Shielding parameter and dimensionless magnetic-field intensity in this analysis, and corresponding conditions.

			Rw
			0.159	1.59	5	10	15.9	159
${\stackrel{⌣}{B}}_0$	7.9	B0 (T)	0.2	0.2	0.2	0.2	0.2	0.2
		f (Hz)	1.5	15	50	100	150	1500
		b (m)	0.135	0.135	0.135	0.135	0.135	0.135
	14.1	B0 (T)	0.2	0.2	0.2	0.2	0.2	0.2
		f (Hz)	5	50	156	313	500	5000
		b (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
		b (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
		b (m)	0.075	0.075	0.075	0.075	0.075	0.075

At the balanced-position, the magnitude of the buoyancy equals the y-component electro-magnetic pinch force. The balanced-position was solved by the dimensionless equation, Eq. (25).   

$$-\frac{3}{4}{C}_{1}cos\left({\varnothing }_1-{\varnothing }_2\right)sin\left(\theta \right)=1$$(25)

3. Results and Discussion

3.1. Distribution of Radial Electromagnetic Pinch Force and the Balanced-position in the Circular Channel

Figure 3 shows the distribution of the electromagnetic pinch force (normalized with the buoyancy acting on the inclusions in the circular channel and the 2-D plate model,²⁶⁾ respectively), when the dimensionless magnetic-field intensity is 31.7. According to Fig. 3(a), the closer the location is to the wall of the circular channel, the stronger is the radial electromagnetic pinch force for all calculated conditions. In addition, the effect of the AC magnetic-field frequency on the distribution of the radial electromagnetic pinch force is similar to the calculated results for the 2-D plate model. However, the electromagnetic pinch force, which acts on the inclusions in the 2-D plate channel, is stronger than for the circular model for all considered conditions.

Fig. 3.

Distribution of electromagnetic pinch force normalized with the buoyancy (${\stackrel{⌣}{B}}_0$ = 31.7).

For the inclusions on the −y axis, the direction of the radial electromagnetic pinch force is opposite to that of the buoyancy. Figure 3(a) also shows the balanced-position of the inclusions on the −y axis. When the shielding parameter is 0.159, there is no balanced-position on the −y axis. When the shielding parameter is 1.59, the balanced-position is −0.828410, which is closer to the wall of the circular channel than for the 2-D plate. This is because the radial electromagnetic pinch force is relatively small. When the shielding parameter is 159, the balanced-position is −0.725575, which is also closer to the wall of the circular channel because the radial electromagnetic pinch force concentrates near the wall at high frequency. When the shielding parameter is 15.9 or 10, the balanced-positions are almost the same and they are closer to the center of the circular channel.

The buoyancy, which acts on the inclusion, is always pointing vertically upwards, i.e., in the +y direction. On the other hand, the direction of the radial electromagnetic force is not always the −y direction when the inclusion is not on the −y axis. Thus, the y-direction component of the radial electromagnetic pinch force (acting on the inclusion) was calculated to determine the balanced-position for the inclusion in the fourth quadrant. Figure 4 shows the balanced-position of the inclusion for all calculation conditions shown in Table 2. It can be seen that the balanced-position changes not only with the AC magnetic-field parameters but also with the x position in the circular channel. When the distance from the inclusion to the center of the circular channel is constant, the radial electromagnetic pinch force remains unchanged. However, as the x-coordinate value of the inclusion increases, the y-component of the radial electromagnetic pinch force acting on the inclusion decreases. Therefore, the y-component of the electromagnetic pinch force and the buoyancy can be balanced if the inclusion is closer to the wall of the circular channel. In other words, when the dimensionless magnetic-field intensity and shielding parameter are fixed, the balanced position is closer to the wall of the circular channel with the x coordinate value increasing.

Fig. 4.

Balanced-position of inclusions in the lower half of the circular channel for different AC magnetic-field conditions.

3.2. Trajectories of the Inclusions in the Lower Half of the Circular Channel

The trajectories of the inclusion, assuming the dimensionless magnetic-field intensity and the shielding parameter of the magnetic-field were 31.7 and 15.9, were calculated when the initial positions were close to the balanced-position (0.0, −0.394412) and on the y axis. The inclusion then moves along the y axis in these cases. Figure 5 shows the results of the trajectory with and without the AC magnetic-field. When the initial position is (0.0, −0.390000), the inclusion floats toward the upper surface of the circular channel, and the removal time for the inclusion with the AC magnetic-field is shorter than that without the AC magnetic-field. When the initial position is (0.0, −0.394410), which is closer to the balanced-position, the inclusion also floats toward the upper surface of the horizontal channel. However, the removal time for the inclusion with the AC magnetic-field is longer than without the AC magnetic-field. This initial position is located within the dead-zone. When the inclusion is set at (0.0, −0.394415) as initial position, the inclusion sinks to the lower surface of the circular channel in the case with the AC magnetic-field, while it floats without the AC magnetic-field. This initial position is also located in the dead-zone. Furthermore, the inclusion also sinks to the lower surface of the circular channel (in the case with the AC magnetic-field) if the initial position is (0.0, −0.394450). However, in this case, the initial position is not located within the dead-zone. The result is similar to the one reported in a previous study.²⁶⁾

Fig. 5.

Trajectories of inclusions when initial positions are located on the -y axis and around the balanced-position (${\stackrel{⌣}{x}}_{ini}$ = 0.0, ${\stackrel{⌣}{B}}_0$ = 31.7, Rw = 15.9).

When the initial position of the inclusion is not on the −y axis, it moves not only in vertical direction but also in the horizontal direction. Under this condition, the trajectories of the inclusions with the four initial positions near the balanced-position of (0.02, −0.394073) were calculated – see Fig. 6. The distribution of the electromagnetic pinch force acting on these inclusions is shown in Fig. 7. According to Figs. 6 and 7, for these four cases, when the inclusion is near the balanced position, the magnitudes of the y-component of radial electromagnetic pinch force and buoyancy are almost equal. Thus, the net-time average-force (which acts on the inclusion) in y direction is almost zero, and the electromagnetic oscillating force represents the main force acting on the inclusion. The inclusion moves in a zigzag manner along y direction and the movement distance of the inclusion in y direction is almost zero, meanwhile in x direction, the x-component of the radial electromagnetic pinch force pushes the inclusion in the positive x direction. Furthermore, the closer the inclusion is to the balanced-position, the longer the electromagnetic oscillating force affects the trajectory of the inclusion and the farther the inclusion moves along the x direction during the initial stage. When the inclusion is far from the balanced position, the effect of the electromagnetic oscillating force on the inclusion trajectory is small because the amplitude of the electromagnetic oscillating force is relatively small, and the trajectory of the inclusion is controlled by the x-component and y-component of the radial electromagnetic pinch force.

Fig. 6.

Trajectories of an inclusion, when the initial position is not on the y axis and near the balanced-position (${\stackrel{⌣}{x}}_{ini}$ = 0.02, ${\stackrel{⌣}{B}}_0$ = 31.7, Rw = 15.9).

Fig. 7.

Electromagnetic pinch force (which acts on the inclusion), when its initial position is not on the y axis and near the balanced-position (${\stackrel{⌣}{x}}_{ini}$ = 0.02, ${\stackrel{⌣}{B}}_0$ = 31.7, Rw = 15.9).

When ${\stackrel{⌣}{y}}_{ini}$ is −0.392000, the inclusion moves in the x direction. With a short movement, the y-direction component of the radial electromagnetic pinch force (which acts on the inclusion) decreases albeit its absolute value increases. Thus, the inclusion moves along the y direction due to the buoyancy.

In the case ${\stackrel{⌣}{y}}_{ini}$ = −0.393300, the inclusion moves along the x direction for a long time because the initial position is closer to the balanced-position compared to y_ini = −0.392000. Then, the inclusion also moves along the y direction driven by the buoyancy with decreasing y-component of the radial electromagnetic pinch force.

In the case ${\stackrel{⌣}{y}}_{ini}$ = −0.393430, the inclusion first moves a larger distance along the x direction during the initial stage. Then, the inclusion moves upward (due to the buoyancy) because the y-component of the radial electromagnetic pinch force decreases. At the same time, the inclusion also moves along the x direction because the larger distance movement in the x direction of the inclusion increases the x-component of the radial electromagnetic pinch force acting on the inclusion. Furthermore, when the inclusion is closer to the wall of the circular channel, the radial electromagnetic pinch force (which acts on the inclusion) increases substantially, and the y-component of the radial electromagnetic pinch force starts to increase. The latter rebalances with the buoyancy, and even exceeds the buoyancy, which causes the inclusion to move slightly downward.

When ${\stackrel{⌣}{y}}_{ini}$ is −0.393500, the inclusion moves a greater distance along the x direction because the initial position is the closest to the balanced-position for these four conditions. As the inclusion moves toward the wall of the circular channel, the radial electromagnetic pinch force increases substantially – see Fig. 3(a). In addition, its y-component also increases and exceeds the buoyancy. As a result, the inclusion moves downward quickly.

3.3. Effect of the AC Magnetic-field Parameters on the Range of the Dead-zone

Figure 8 shows the removal time for the inclusion if the initial position is close to the balanced-position assuming that the dimensionless magnetic-field intensity is 31.7 and the shielding parameter is 15.9. When the inclusion is located on the −y axis, i.e., ${\stackrel{⌣}{x}}_{ini}$ = 0.0, the removal time for the inclusions with an AC magnetic-field is longer than without AC magnetic-field provided the initial position in the y direction is between −0.393320 and −0.394417. Thus, the range of the dead-zone is 0.001097. When the initial x position is ${\stackrel{⌣}{x}}_{ini}$ = 0.02, the removal time longer range in y direction by imposing the AC magnetic-field is between −0.392600 and −0.393432. In other words, the range of the dead-zone is 0.000832. It can be seen that the dead-zone range changes with the x-coordinate. As the x coordinate value increases, the balanced-position moves closer to the wall of the circular channel and the electromagnetic oscillating force is relatively larger. Thus, it is easier for the inclusion to escape from the dead-zone, and the range of the dead-zone decreases. When the range of the dead-zone is less than 0.000001, the dead-zone is considered “disappeared”, in this paper. Thus, when the dimensionless magnetic-field intensity is 31.7 and the shielding parameter is 15.9, the dead-zone disappears under the condition that the x coordinate value is equal to or larger than 0.0307.

Fig. 8.

Removal time for the inclusion, when its initial position is near the balanced-position (${\stackrel{⌣}{B}}_0$ = 31.7, Rw = 15.9).

According to Fig. 4, the balanced-position exists except for the combination (${\stackrel{⌣}{B}}_0$, R_w) = (7.9, 0.159), (7.9, 1.59), (14.1, 0.159), (14.1, 1.59), (31.7, 0.159), (56.3, 0.159) for the 24 conditions shown in Table 2. The dead-zone is shown in Fig. 9, except for the conditions without the balanced-position. The dead-zone shape resembles half a crescent moon and is elongated in the x-direction. Furthermore, the absolute value of the electromagnetic pinch force at the balanced-position increases with increasing distance from the y-axis. This is because the y-component of the electromagnetic pinch force is balanced by the buoyancy. Thus, the electromagnetic oscillating force, which is the driving force for the inclusion escaping from the dead-zone, also increases. Furthermore, the balanced-position is an arc shape shown in Fig. 4. This is the reason, why the dead-zone shape resembles half a crescent moon.

Fig. 9.

Dead-zone range for different AC magnetic-field conditions. (Online version in color.)

The length in x-direction and the width in y-direction of the dead zone were then evaluated and are shown in Fig. 10. Both the dead-zone length in x-direction and the dead-zone width in y-direction decrease with increasing dimensionless magnetic-field intensity.

Fig. 10.

Effect of the AC magnetic-field on the range of the dead-zone of inclusions.

Physical interpretation of the distribution of the dead-zone width in y direction shown in Fig. 10(a) is essentially similar with the two-dimensional model,²⁶⁾ because the electromagnetic forces acting on the inclusion on the y axis are only in the y direction. The large gradient of the electromagnetic pinch force enables an easy escape of the inclusion from the dead-zone because the balance between the buoyancy and the electromagnetic pinch force can easily be disturbed. According to Fig. 11, the gradient of the electromagnetic pinch force in the cylindrical model increases monotonically for an increasing shielding parameter. In other words, a large shielding parameter reduces the dead-zone width in y direction. In addition, the ratio of the effective distance of the electromagnetic pinch force to the total moving distance of the inclusion is also an important factor. When this ratio is high, the inclusion-removal-time change is relatively large by small change of the initial position of the inclusion. Thus, the dead-zone width is reduced. This ratio reaches a maximum for Rw = 10 or 15.9 because the balanced-position is the closest to the channel center - see Fig. 4. The dead-zone width in y direction is determined by these factors.

Fig. 11.

Gradient of the electromagnetic pinch force at the balanced-position on the y-axis for different AC magnetic-field conditions.

The dead-zone length in x-direction for Rw = 10 represents the local minimum given that ${\stackrel{⌣}{B}}_0$ = 31.7 or 56.3. It represents the local minimum for Rw = 15.9 if ${\stackrel{⌣}{B}}_0$ = 7.9 or 14.1 - see Fig. 10(b). Both the gradient of the electromagnetic pinch force as well as the ratio of the electromagnetic pinch force’s effective distance to the total moving distance of the inclusion affect the dead-zone length in the x-direction. The intensity change of the electromagnetic force at the balanced-position is another important factor. To evaluate the electromagnetic-force increase, the distance from the channel-center to the balanced-position, r_b, is introduced. It increases as the balanced-position moves far from the y-axis - see Fig. 4. The differential of r_b with respect to x, dr_b/dx, was calculated as a function of x for different AC magnetic-field conditions - see Fig. 12. The differential reaches a maximum when Rw = 10 for every dimensionless magnetic-field intensity. In other words, the inclusion can escape more easily from the dead-zone when Rw = 10. Because of these factors, the x-direction dead-zone length for Rw = 10 represents a local minimum if ${\stackrel{⌣}{B}}_0$ = 31.7 or 56.3. It is a local minimum for Rw = 15.9 if ${\stackrel{⌣}{B}}_0$ = 7.9 or 14.1.

Fig. 12.

Rate of increase of the distance between balanced-position and channel-center along the x-direction for different AC magnetic-field conditions.

4. Conclusions

To optimize the circular channel to remove inclusions, the balanced-position (the magnitude of the buoyancy equals the y-component electromagnetic pinch force) and the range of the dead-zone (the removal period of inclusions with the AC magnetic-field is longer than without it) were calculated by numerical method. Both depend on the AC magnetic-field parameters (intensity and frequency).The following are the conclusions of this research:

(1) When the dimensionless magnetic-field intensity is 7.9 or 14.1, there is a balanced-position, unless the shielding parameter is 0.159 or 1.59. When dimensionless magnetic field is 31.7 or 56.3, there is a balanced-position unless the shielding parameter is 0.159.

(2) When the magnetic-field intensity increases, the range of the dead-zone decreases.

(3) For the dead-zone in y direction, the larger the shielding parameter is, the smaller is the dead-zone range unless ${\stackrel{⌣}{B}}_0$ = 56.3. When the dimensionless magnetic-field intensity is 56.3, the dead-zone range in y direction is minimal if the shielding parameter is 10.

(4) For the dead-zone in x direction, the dead-zone range is smallest when the shielding parameter is 10 for the dimensionless magnetic-field intensities 31.7 or 56.3.

Overall, when the magnetic field is high, the shielding parameters 10 and 15.9 represent the optimum condition to reduce the range of the dead-zone and to improve the removal efficiency of inclusions within the circular channel.

List of symbols:

b: radius of circular channel (m)

B₀: magnetic field intensity (T)

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

ber, bei, ber₁, and bei₁: Kelvin functions

C_D: drag coefficient (–)

C₁: constant defined by Eq. (4) (–)

D: inclusion diameter (m)

F_m: electromagnetic force acting on the molten steel (N/m³)

F_p: electromagnetic pinch force acting on the inclusion (N/m³)

F_o: electromagnetic oscillating force acting on the inclusion (N/m³)

j: imaginary unit

J: current density distribution in the molten steel (A/m²)

r: radial position of the inclusions in the circular channel (m)

R_w: shielding parameter (–)

Re_s: particle Reynolds number (–)

t: inclusion removal time (s)

$\stackrel{⌣}{t}$: non-dimensional removal time of the inclusion (–)

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

$\stackrel{⌣}{x},\stackrel{⌣}{y}$: non-dimensional x direction and y direction position of the inclusions (–)

${\stackrel{⌣}{x}}_{ini}$, ${\stackrel{⌣}{y}}_{ini}$: non-dimensional x direction and y direction initial position of the inclusions (–)

η: molten steel viscosity (Pa·s)

ρ_f: molten steel density (kg/m³)

ρ_s: inclusion density (kg/m³)

μ: molten steel magnetic permeability (H/m)

σ: molten steel electrical conductivity (S/m)

${\varnothing }_1$: constant defined by Eq. (8) (–)

${\varnothing }_2$: constant defined by Eq. (9) (–)

θ: circumferential angle of inclusion in the circular channel (rad)

Subscripts:

r: radial direction in Cylindrical coordinate system

x: x direction in Cartesian coordinate system

y: y direction in Cartesian coordinate system

z: axial direction in Cylindrical and Cartesian coordinate system

θ: circumferential direction in Cylindrical coordinate system

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