ISIJ International
Online ISSN : 1347-5460
Print ISSN : 0915-1559
ISSN-L : 0915-1559
Regular Article
Behavior of Non-metallic Inclusions in a Continuous Casting Tundish with Channel Type Induction Heating
Qiang WangFengsheng QiBaokuan LiFumitaka Tsukihashi
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2014 Volume 54 Issue 12 Pages 2796-2805

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Abstract

A generalized three dimensional mathematical model, which adopted the Euler-Lagrange approach, has been developed for the motion of inclusions in a continuous casting tundish with channel type induction heating. The inclusion trajectories were obtained by numerical solution of the motion equation including gravity, buoyancy, drag, lift, added mass, Brownian, electromagnetic pressure and thermophoretic forces. Besides, the collision and coalescence of inclusions and adhesion to the lining solid surfaces were also taken into account. The Brownian, Stokes and turbulent collision were clarified, separately. Then the effect of induction heating power on the inclusion behavior was demonstrated. A reasonable agreement between the experimental observation and numerical results was obtained. The results indicate that the electromagnetic pressure force significantly promotes the removal of inclusions, especially for the bigger inclusion. Although the thermophoretic force goes against the removal of inclusions, its influence is negligibly small. The removal ratio of inclusions in the tundish with induction heating increases from 67.45% to 96.43%, while the power varies from 800 kW to 1200 kW. The collision and coalescence should be included when model the inclusion motion, because it can promote the removal of inclusion. The turbulent and Brownian collision becomes more active with the increasing power, while the Stokes collision is just opposite.

1. Introduction

It is well known that non-metallic inclusions in the molten steel greatly influence the performance of steel products such as blisters, stress concentration, toughness degradation and poor surface finish.1,2,3,4,5) However, with increasing demands for high performance steel products year by year, removal of non-metallic inclusions from the matrix materials especially during the liquid state prior to casting into a final product has been pursued in the continuous casting process by means of various separation approaches such as tundish filters.6,7,8) In order to promote the removing of inclusions, the residence time of the molten steel should be prolonged. Many flow control devices therefore have been widely used in the tundish such as weirs, dams and turbulence inhibitors. Nevertheless, the temperature drop of the molten steel in the tundish becomes bigger with the increasing residence time which goes against the continuous casting process.9)

In recent years, tundish with induction heating has been identified as a good technology for the continuous casting process. The contradiction mentioned above can be solved by the technolog. A tundish with channel type induction heating patent was proposed by Ueda et al. in 1983.10) In the patent, tundish with a channeled induction-heating device at its sidewall was presented. Joule heating as well as electromagnetic force (EMF) would be generated in the channel. Thermal loss of the molten steel can be efficiently compensated by the Joule heating. Non-metallic inclusions are removed by the EMF which termed electromagnetic separation. Due to a lot of force terms acting on inclusions, the motion of inclusions is very complex in the tundish with induction heating, which including floatation, adhesion to the wall, rotation, collision and coalescence.11,12,13,14,15)

Much effort has been devoted to understand the motion of inclusions with electromagnetic separation.16,17,18,19,20,21) However, the temperature field was not taken into account in aforementioned works. The motion of inclusions is mainly controlled by the flow of the molten steel, and the heat transfer is critical to the flow which induces the natural convection.6,22) The inclusion removal efficiency was underestimated, because the adhesion to the lining solid surface as well as the collision and coalescence of inclusions was ignored.21) Takahashi and Taniguchi conducted an experiment and numerical simulation to study the separation of inclusions from liquid metal in an induction furnace.23) The temperature distribution and solidification time was also considered. A mathematical model including the turbulent collisions was developed by Miki and Thomas to predict the removal of inclusions in the tundish.24) The collision and coalescence of inclusions and adhesion to the lining solid surfaces were found in the mathematical model proposed by Zhang et al.25)

In the present work, a generalized three dimensional mathematical model, which adopted the Euler-Lagrange approach, has been developed for the motion of inclusions in a continuous casting tundish with channel type induction heating. Based on the flow and temperature fields of the molten steel, the motion of inclusions was investigated considering various acting forces such as electromagnetic pressure force and thermophoretic force. The collision and coalescence of inclusions and adhesion to the lining solid surfaces were also taken into account. Then the effect of induction heating power on the inclusion behavior was demonstrated by the model.

2. Mathematical Model

In order to simplify the calculation, the inclusion is assumed to be an inert spherical solid with a constant density of 3960 kg/m3. Besides, the small inclusions are focused rather than all sizes in the present work, because the removal ratio of the small inclusion changes significantly in the tundish with induction heating.21) The diameters of inclusions are assumed to be 1 μm, 2 μm, 5 μm, 10 μm, 20 μm and 50 μm, respectively. The motion of inclusions cannot influence the arbitrary macroscopic flow pattern. Every inclusion is supposed to move independently of other until a collision happens. Upon contact, two inclusions coalesce each other instantaneously and form a new spherical inclusion that rises with a bigger velocity than its parents do. The inclusions are released uniformly at the tundish inlet with the same velocity of the molten steel. The inclusion is removed if its velocity is less than a threshold value. Otherwise, a non-perfect elastic collision occurs when the inclusions reach the top or lining solid surfaces.25,26,27,28)

2.1. Flow and Heat Transfer of Molten Steel

The transport model for the tundish with induction heating was proposed in our previous work.29) The main governing equations for the flow and heat transfer consist of the continuity equation, Navier-Stokes equation, turbulent kinetic energy and its dissipation rate equations of RNG k-ε turbulence model and energy conservation equation.

2.2. The Motion Equations of Inclusion

A pressure gradient is generated in the molten steel with the EMF. According to the law of action and reaction, the nonmetallic inclusion suspended in the molten steel suffers a pressure force as shown in Fig. 1.21,23) The thermophoretic phenomenon is first taken into consideration in the present work. The inclusion immersed in the molten steel which with a uneven temperature distribution experiences a thermophoretic force, and the force tends to move the inclusion against the temperature gradient as shown in Fig. 2.30,31) Furthermore, other hydrodynamic force terms are also included such as gravity, buoyancy, drag, lift, added mass and Brownian forces.32,33) Figure 3 presents the force balance of the inclusion and the widely used motion equation expressed as:   

ρ p π 6 d p 3 d v p dt = F p + F t + F g + F b + F d + F + F a + F brow (1)
Fig. 1.

Principle of the electromagnetic pressure force.

Fig. 2.

Principle of the thermophoretic force.

Fig. 3.

Schematic representation of the inclusion motion in the tundish with induction heating.

F p and F t represent the electromagnetic pressure and thermophoretic forces which are evaluated by:   

F p =- 3 2 σ - σ p 2 σ + σ p π d p 3 6 F (2)
where F p is the electromagnetic pressure force acting on the unit volume of the molten steel. In the case where the inclusion is non-conductive (σp = 0), the force is simplified as:21,23)   
F p =- 3 4 π d p 3 6 F (3)

The thermophoretic force is defined as:30,31)   

F t =- α β M T,   M= 1 6π μ d p (4)

Other forces in the Eq. (1) are expressed as:16,34)   

F g = ρ p π 6 d p 3 g F b = ρ π 6 d p 3 g F d = π 8 C D ρ d p 2 ( v p - v ) | v p - v | F =-1.615 μ d p 2 ( v p - v ) sgn( G ) ( ρ | G | μ ) ,      G= d v d n F α = ρ C A π 6 d p 3 ( d v dt - d v p dt ) F brow = ζ i π S o t ,       S o = C c μ k B T π 2 ρ p 2 d p 5 (5)

2.3. Random Walk Model

In order to consider the chaotic effect of the turbulent motion, the random walk model is applied to the calculation of the inclusion trajectory.24) A random velocity is added to the time-averaged velocity to obtain the inclusion transient velocity at each time step. Each random component of the inclusion transient velocity is proportional to the local turbulent kinetic energy of the molten steel:   

v p = v ¯ p + v ,       v = ζ i v ¯ 2 = ζ i 2 k 3 (6)
where ζi is a random number, normally distributed between –1 and 1 that changes at each time step.

2.4. Wall Effects

The presence of a solid surface in the vicinity of a moving inclusion will affect the drag force. In the present work, the drag force coefficient is approximated:16)   

C D = C D +( 24 / Re p ) ( W-1 ) (7)
where CD is the drag force coefficient in the absence of the solid wall, and W is given:   
W= 1-0.75857 λ 5 1-2.1050λ+2.0865 λ 3 -1.7068 λ 5 +0.72603 λ 6 (8)

The wall effects are estimated by using the tabulated correction factor for the drag coefficient on an inclusion moving perpendicular to a plane wall. The correction factor is a function of the ratio of the distance of the inclusion center from the wall to the inclusion radius and linear interpolation is employed to obtain the factor for the ratio.

2.5. Collision and Coalescence of Inclusions

Figure 4 shows the collision models of inclusions including the Brownian collision, Stokes collision and turbulent collision. The number of collisions per unit volume and unit time between two inclusions with size ri and rj is defined:   

N ij =β( r i , r j ) n( r i ) n( r j ) (9)
where β(ri, rj) is the collision rate constant with a dimension of volume/time, also called the “collision volume” which represent the probabilities of collision and coalescence for inclusions.25,35)
Fig. 4.

Schematic representation of the collision and coalescence. (a) Brownian collision. (b) Stokes collision. (c) Turbulent collision.

2.5.1. Brownian Collision

Inclusions will contact, collide and coagulate each other due to the Brown movement. The collision rate constant of the Brown collision is given:36)   

β 1 ( r i , r j ) = 2 k T 3 μ ( 1 r i + 1 r j ) ( r i + r j ) (10)

2.5.2. Stokes Collision

Bigger inclusion floats with a larger rising velocity in the molten steel. The collision and coalescence would appear when the bigger inclusion catches up with and contacts with the smaller one. The collision rate constant of the Stokes collision is derived:37)   

β 2 ( r i , r j ) = 2gΔρ 9 μ | r i 2 - r j 2 |π ( r i + r j ) 2 (11)

2.5.3. Turbulent Collision

Inclusions would collide each other under the effect of the molten steel turbulent motion. The collision rate constant of the turbulent collision is:38)   

β 3 ( r i , r j ) =1.3 ( r i + r j ) 3 ρ ε / μ (12)

The following factors make a great effect on the collision rate constant: inclusion size, turbulent kinetic energy dissipation rate, viscosity of molten steel and density difference between molten steel and inclusion.

3. Results and Discussion

3.1. Validation of the Mathematical Model

The experiment performed by Takahashi and Taniguchi is used to validate the mathematical model.23) The apparatus is composed of a 15 turn induction coil with 94 mm diameter, a silica crucible with 40 mm inside diameter. The applied current is 50A, 100A and 150A, respectively. A certain amount of aluminum is melt in the crucible. When the temperature reaches 1073 K, a piece of aluminum alloy containing SiC particles is put into the melt and agitated about 30 s. The power is turned off after a prefixed time, and the melt is cooled rapidly by the spray-cooling unit. In our calculation, around 20000 inclusions with a diameter of 20.5 μm are released at the center of the crucible.

Figure 5 shows the simulated EMF and velocity fields with the applied current of 50A at the vertical section of the half crucible. A pinch effect is applied on the molten aluminum by the EMF. Due to the skin effect, the EMF on the periphery of the molten aluminum is bigger than that at the centre. The flow at the centre is dominated by the buoyancy force, while by the EMF at the outer side. The ratio of the number of inclusions adhered to the wall to the total inclusions is defined as the removal efficiency. They demonstrate the removal efficiency obtained from the experiment. The inclusions moving to the wall are counted in our simulation. Figure 6 shows the comparison of inclusions removal efficiency between the experiment and simulation. It is clear that the calculated results are in good agreement with measured tendencies. The discrepancy can be attributed to the ignoring of the solidification and the uncertainties in model parameters. The mathematical model therefore is validated quantitatively.

Fig. 5.

Simulated results with the applied current of 50A at the vertical section of the half crucible. (a) EMF field. (b) Velocity field.

Fig. 6.

Comparison of the removal efficiency between the experiment and simulation.

3.2. Motion of Inclusions in the Channel

Figure 7 shows the geometrical model of the tundish with induction heating. The calculation domain includes the molten steel in the receiving chamber, channels and discharging chamber. More details such as properties of the molten steel can be found in our previous work.29) About 1500 inclusions are released uniformly at the inlet of the tundish with the same velocity of the molten steel at each time step. Figures 8(a) and 8(b) demonstrate the typical streamlines and temperature field of the molten steel and the inclusions trajectories in the tundish with a 800 kW heating power. The inclusions trajectories are similar to the streamlines of the molten steel, because inclusions mainly follow the movement of the molten steel. The motion of inclusions is most complex in the channel. Due to the turbulent motion, a jagged path instead of smooth path is obtained. Figure 9 displays the vertical projection of a typical inclusion trajectory in the channel. The inclusion enters from the center of the channel inlet. Because of the electromagnetic pressure force, the inclusion owns a rotating velocity. Thus, it moves to the wall when travels through the channel.

Fig. 7.

Geometrical model of the tundish with induction heating.

Fig. 8.

(a) Typical streamlines and temperature field of the molten steel in the tundish with a 800 kW heating power. (b) Typical inclusions trajectories in the tundish with induction heating of 800 kW.

Fig. 9.

Vertical projection of a typical inclusion trajectory in the channel.

In order to investigate the effect of the induction heating on various inclusions, seven diameters inclusions are released respectively at the inlet of the channel. The amount and size of inclusions trapped by the channel wall are exported. The removal ratio is the ratio of the quantity of inclusions trapped by the wall and the total released inclusions. Figure 10 illustrates the removal ratio of each diameter inclusion in the channel under different powers. It is obvious that the removal ratio is significantly improved by the induction heating for all inclusions. Moreover, the removal ratio of all inclusions increases with more power. According to Eq. (3), the electromagnetic pressure force is proportional to the diameter of the inclusion. Therefore, the removal ratio increasing rate of the bigger inclusion is larger than that of the smaller inclusion. For instance, the removal ratio difference of the 50 μm inclusions between no heating and a 800 kW heating is 39.83%, while the difference of the 1 μm inclusions is only 15.69%.

Fig. 10.

Removal ratio of each diameter inclusion in the channel.

The evolution of the inclusion size is also revealed. The coalescence occurs if the inclusion becomes bigger. Figure 11 represents the size distribution of inclusions trapped by the channel wall with the induction heating of 800 kW. The proportion of 1–2 μm trapped inclusions reaches up to 47.63% when the initial diameter is 1 μm (Fig. 11(a)), which indicate that most inclusions suffer the collision and coalescence before trapped. This phenomenon is also observed when the initial diameter is 2 μm and 5 μm. However, for the inclusions with a bigger initial diameter such as 10 μm, 20 μm, 30 μm and 50 μm, the probability of collision and coalescence reduces significantly. When the initial diameter is 10 μm, the proportion of 10 μm trapped inclusions is 52.69% and the proportion of 10–20 μm trapped inclusions is 41.32% (Fig. 11(d)). The proportion of 50 μm trapped inclusions reaches up to 73.68%, while the proportion of other trapped inclusions is only 24.67% when the initial diameter is 50 μm (Fig. 11(g)). The collision and coalescence is an important removal way for smaller inclusions.

Fig. 11.

Size distribution of trapped inclusions in the channel with the induction heating of 800 kW.

The effect of thermophoretic force on the motion of inclusions is clarified. Figure 12 shows the removal ratio of each diameter inclusion in the channel without and with thermophoretic force. Because of the small temperature difference, there is slight difference between the removal ratios without and with thermophoretic force when the heating power is 800 kW. Nevertheless, the effect of the thermophoretic force becomes more significant with the increasing power, especially for smaller inclusions. With a 1000 kW heating power, only 1 μm, 2 μm and 5 μm inclusions own a small difference of the removal ratio. The difference of the removal ratio is observed in all sizes inclusions when the power increases to 1200 kW. The difference of the smaller inclusion is larger than that of the bigger inclusion. However, the removal ratio difference is very small (less than 3%) even with a 1200 kW power. Therefore, the effect of thermophoretic force on the removal ratio can be ignored.

Fig. 12.

Removal ratio of each diameter inclusion in the channel without and with thermophoretic force with the different heating powers. (a) 800 kW. (b) 1000 kW. (c) 1200 kW.

3.3. Collision Rate Constant of Inclusions in the Channel

A lot of inclusions would collide and coagulate each other into a bigger one in the channel. This section presents the collision rate constants in the channel with different heating powers. Seven diameters inclusions are released simultaneously instead of respectively at the channel inlet. The removal ratio and each collision rate constant in the channel are calculated.

Figure 13 displays the removal ratio in the channel without and with the collision and coalescence. The effect of collision and coalescence is embodied in the improving of the removal ratio. The removal ratio with collision and coalescence is much bigger than that without collision and coalescence. For example, the removal ratio without collision and coalescence is about 65%, while around 94% with collision and coalescence when the power is 1200 kW. The removal ratio increases by approximately 45%. It is necessary to include the collision and coalescence in the simulation of the inclusions behavior. Bigger inclusions in the channel would float up and then adhere to the channel wall, and other would flow into the discharging chamber.

Fig. 13.

Removal ratio of inclusions in the channel without and with the collision and coalescence.

Figure 14 shows the isoline of collision rate constants in the channel which are expressed in the logarithmic form (log β(ri, rj)). The collision rate constant of the Brown collision is very small, less than 10–15 m3/s (Fig. 14(a)). The collision rate constant of the Stokes collision (Fig. 14(b)) is larger than that of the Brown collision. Moreover, it is zero of the Stokes collision rate constant for the two inclusions with the same sizes, because they own the same rising velocities so cannot contact each other. Because of the intense flow, the turbulent collision rate constant is 104 times larger than the Brown and Stokes collision rate constants (Fig. 14(c)). Therefore, the turbulent collision is the dominant way of collision and coalescence in the channel.

Fig. 14.

Isoline of the collision rate constants in the channel which are expressed in the logarithmic form (log β(ri, rj)) as a function of inclusion size. (a) Brownian collision. (b) Stokes Collision. (c) Turbulent collision.

According to the definition (Eqs. (10), (11), (12)), the Brown, Stokes and turbulent collision rate constants are mainly affected by the molten steel temperature, the density difference between the inclusion and molten steel and the turbulent kinetic energy dissipation rate, respectively. So these variables are used to express the evolution of the Brown, Stokes and turbulent collision rate constants with different heating powers. Figure 15 shows the variation of the mean temperature of the molten steel in the channel, the density difference between the inclusion and molten steel and the mean value of the turbulent kinetic energy dissipation rate of molten steel (log ε) with different powers. It is obvious that the mean temperature and turbulent kinetic energy dissipation rate increase, while the density difference becomes smaller with more power. Therefore, the Brown and turbulent collision are promoted, and the Stokes collision is suppressed by the increasing power.

Fig. 15.

Variation of the mean temperature of molten steel in the channel, the density difference between inclusion and molten steel in the channel (Δρ) and the mean value of molten steel turbulent kinetic energy dissipation rate which is expressed in the logarithmic form (log ε) in the channel with different powers.

3.4. Collision and Coalescence of Inclusions in the Tundish

The collision and coalescence of inclusions in the tundish with induction heating is presented in this section. Figure 16 illustrates the removal ratio in the tundish without and with the collision and coalescence. The tundish is composed of the receiving chamber, channel and discharging chamber. Most inclusions are removed in the discharging chamber of tundish without induction heating. Due to more space, the molten steel moves slower in the discharging chamber.29) Inclusions therefore get more opportunities to float up. Once with induction heating, most inclusions are removed in the channel because of the electromagnetic pressure force. Furthermore, the removal ratio is improved significantly with the increasing power. The removal ratio increases from 67.45% to 96.43%, while the power ranges from 800 kW to 1200 kW. The proportion of inclusions removed in the channel is bigger than that of inclusions removed in the receiving or discharging chambers due to the induction heating.

Fig. 16.

Removal ratio of inclusions in the tundish without and with the collision and coalescence.

Due to the strong turbulent flow, the collision and coalescence promote the proportion of inclusions removed in the channel and receiving chamber. Figure 17 displays the isoline of the collision rate constants of the three models which are expressed in the logarithmic form in the tundish with a 800 kW induction heating. The Brown and Stokes collision rate constants in the three parts of the tundish own little difference, so the mean values of the two quantities are showed. The turbulent collision rate constant changes a lot which is divided into the value of the receiving chamber, channel and discharging chamber, respectively. It is obvious that the turbulent collision is the dominant way and the collision rate constant of the turbulent collision is the maximal. Furthermore, the turbulent collision rate constant in the channel is about 10 times larger than that in the receiving chamber and approximate 1000 times in the discharging chamber. It can be referred that the flow in the channel experiences violent turbulence which can promote the coalescence. Figure 18 shows the distribution of the turbulent kinetic energy dissipation rate which is expressed in the logarithmic form under different powers. It is responsible for the magnitude of the turbulent collision rate constant. The turbulent kinetic energy dissipation rate in the channel is always the maximum whether or not with the induction heating. The effect of induction heating power on the three models in the tundish is presented in Fig. 19. The variation of the three collision rate constants in the tundish is similar to that in the channel mentioned in Fig. 15. The turbulent collision rate constant in the channel is bigger than that in the receiving or discharging chambers. More opportunities are provided for inclusions to collide and coalesce when go through the channel.

Fig. 17.

Isoline of the collision rate constants of the three models which are expressed in logarithmic form (log β(ri, rj)) in the tundish with a 800 kW heating power as a function of inclusion size. (a) Brownian collision. (b) Stokes Collision. (c) Turbulent collision in the channel. (d) Turbulent collision in the receiving chamber. (e) Turbulent collision in the discharging chamber.

Fig. 18.

Distribution of the turbulent kinetic energy dissipation rate which is expressed in the logarithmic form under different powers. (a) Without induction heating. (b) 800 kW induction heating power. (c) 1000 kW induction heating power. (d) 1200 kW induction heating power.

Fig. 19.

Variation of the mean temperature of molten steel in the tundish, the density difference between inclusion and molten steel in the tundish (Δρ) and the mean values of the molten steel turbulent kinetic energy dissipation rate which are expressed in the logarithmic form (log ε) in the channel, receiving chamber and discharging chamber with different powers.

4. Conclusions

(1) A generalized three dimensional mathematical model, which adopted the Euler-Lagrange approach, has been developed for the motion of inclusions in a continuous casting tundish with channel type induction heating. On the basis of the molten steel flow and temperature fields, inclusion trajectories were obtained by numerically solving the motion equation including gravity, buoyancy, drag, lift, added mass, Brownian, electromagnetic pressure and thermophoretic forces. Besides, the collision and coalescence of inclusions and adhesion to the lining solid surfaces were taken into consideration. A reasonable agreement was obtained between the experiment observation and numerical results. The effect of induction heating power on the inclusion behavior was demonstrated by the model.

(2) The electromagnetic pressure force induced by the induction heating can significantly promote the removal of inclusions, especially for the bigger inclusion.

(3) The thermophoretic force is very small (less than 3%) even with a 1200 kW heating power. It can be ignored, though it reduces the removal ratio.

(4) The removal ratio of inclusions in the tundish increases from 67.45% to 96.43%, while the power varies from 800 kW to 1200 kW.

(5) The removal ratio of inclusions in the channel with collision and coalescence is much bigger than that without collision and coalescence. It is necessary to include the collision and coalescence in the modeling of inclusions motion.

(6) The turbulent collision is the strongest in the three collisions. The turbulent collision rate constant in the channel is 10 times larger than that in the receiving chamber and 103 times in the discharging chamber.

(7) The turbulent and Brownian collision becomes more active with the increasing heating power, while the Stokes collision is just opposite.

Acknowledgements

The authors’ gratitude goes to National Natural Science Foundation of China [NO. 51210007].

Symbols

CD drag force coefficient

CD drag force coefficient in the absence of the solid wall

CA added mass force coefficient

CC Brownian force coefficient

dp inclusion diameter

F p  electromagnetic pressure force

F t  thermophoretic force

F g  gravity

F b  buoyancy force

F d  drag force

F  lift force

F a  added mass force

F brow  Brownian force

F  electromagnetic force

g gravitational acceleration

G wall-normal gradient of the streamwise molten steel velocity

kB Boltzmann constant

k  molten steel turbulent kinetic energy

n(r) number density of the inclusion with radius r

n  unit vector perpendicular to the wall

Nij number of collision per unit volume and per time

r inclusion radius

Rep inclusion Reynolds number

t time

T  molten steel temperature

v p  inclusion transient velocity

v ¯ p  inclusion mean velocity

v′ inclusion random velocity

v  molten steel velocity

α  molten steel thermal diffusion

β  molten steel thermal expansion coefficient

β(ri, rj) collision rate constant of ri and rj inclusions

Δρ density different between the molten steel and inclusion

ε  molten steel turbulent kinetic energy dissipation rate

λ ratio of the diameter of the inclusion to the channel

μ  molten steel dynamic viscosity

ζi random number, normally distributed between –1 and 1 that changes at each time-step

ρp inclusion density

ρ  molten steel density

σ  molten steel electrical conductivity

σp inclusion electrical conductivity

sgn(G) signum function

Subscript

i, j inclusion sequence number

 parameters of molten steel

p parameters of inclusion

References
 
© 2014 by The Iron and Steel Institute of Japan
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