ISIJ International
Online ISSN : 1347-5460
Print ISSN : 0915-1559
ISSN-L : 0915-1559
Casting and Solidification
Numerical Simulation of Collision-Coalescence and Removal of Inclusion in Tundish with Channel Type Induction Heating
Hong LeiBin YangQian BiYuanyou XiaoShifu ChenChangyou Ding
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2019 Volume 59 Issue 10 Pages 1811-1819

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Abstract

Numerical simulation is an effective tool to analyze the inclusion behavior in the tundish with channel induction heating. And the inclusion mass/population conservation model is applied to predict and describe inclusion physical field. Due to the channel induction heating, Archimedes slipping velocity and Archimedes collision are applied to describe the inclusion behavior in the tundish with channel induction heating. The predicted values agree with the experimental data for the inclusion model. Numerical results show that Joule heat and electromagnetic force can prompt the inclusion removal rate. Compared with Joule heat, electromagnetic force is a more important factor to affect the inclusion’s movement and Archimedes collision is also one of the important collision mechanisms for inclusion coalescence. The inclusion removal rate in the channels is up to one third of the inclusion removal rate in the tundish, and the inclusion removal rate in the tundish increases from 21.4% to 35.05% if channel induction heating is applied.

1. Introduction

The tundish is an important vessel for regulating the flow and temperature of molten steel between individual ladle and the continuous casting mold. Ideally, it is conducive to the removal of inclusions and the quality of steel products. Meanwhile, nonmetallic inclusions can cause various defects in steel products. And the property of metal materials are significantly affected by nonmetal inclusions.1) So it is essential to study the inclusion behavior in the molten steel. The inclusion removal is related to inclusion agglomeration and inclusion flotation,2,3) inclusion absorption by the top slag and the lining refractory,2,4) inclusion transport behavior, inclusion-bubble interaction,5,6) EM separation and Ostwald ripening growth.7,8,9,10)

Many previous papers have given many inclusion mathematical models in the tundish which can be divided into two branches. One branch is homogeneous model, and another branch is non-homogeneous model. The homogeneous model consist of particle collision dynamic model and fractal theory model,3,4,7,11,12) and the non-homogeneous model include Lagrange model and Euler model.13,14,15,16,17,18,19) Based on the concentration diffusion model, Shirabe proposed the inclusion population balance model20) to study the inclusion, and then such a model is well adopted by Sinha and Sahai,21) Wang et al.22) and Ling et al.23)

In the present work, the inclusion mass/population conservation model is developed to predict the spatial distribution of characteristic inclusion volume concentration, number density and radius in a tundish with channel type induction heating. There are two highlights in this paper, one is that the inclusion slipping velocity and Archimedes-Stokes collision is introduced to describe the inclusion behavior in the tundish with channel induction heating, another is the discussion about the effects of electromagnetic force and Joule heat on inclusion behaviors.

2. Calculation Model and Procedure

Figure 1 shows the structure and size of the tundish with channel type induction heating. The receiving chamber is connected to the discharging chamber by two channels. The iron core is installed on the back channel, and the coil is wound around the iron core.

Fig. 1.

Schematic of tundish with channel induction heating (all dimensions are in mm).

This work consists of three parts, as shown in Fig. 2. Part I is about the electromagnetic field,24) part II is about the flow field and temperature field,25) and part III is about the collision and coalescence among the inclusions in molten steel. The calculation procedure is as follows.

Fig. 2.

Computational procedure for transport in the tundish with channel induction heating.

(1) The software ANSYS is used to solve the Maxwell’s equation in order to obtain the physical fields (magnetic flux density, induction current, electromagnetic force and Joule heat).

(2) A Fortran program is applied to finish the interpolation process of electromagnetic force and Joule heat from ANSYS to CFX because the grid system for fluid field is different from that for electromagnetic field.

(3) The software CFX is used to solve the mass conservation equation, the momentum conservation with electromagnetic force, the energy conservation equation with Joule heat and the k–ε turbulence model in order to obtain the velocity field and the temperature field of molten steel.

(4) The software CFX is used to solve the inclusion mass/population conservation equations with Archimedes slip velocity and Archimedes collision in order to obtain the spatial distribution of the inclusion volumetric concentration, the inclusion number density and the inclusion characteristic radius.

3. Mathematical Model

Our previous papers has described the numerical simulation for the electromagnetic field, the flow field and the temperature field in detail. It need no repetition in here.24,25) In the mathematical model, Joule heating (σJ2) acts as the source term in the energy equation, it increases the temperature of molten steel in the tundish, and related the thermal buoyancy (β(T0 − T) ρfg) change the fluid flow. The electromagnetic force ( J × B ) acts the source term in the momentum conservation equation, it stirs the molten steel in the tundish.

This paper pays more attention to the mathematical model of inclusion collision and coalescence.

3.1. Assumption

The inclusion behavior in the molten steel is very complicated, so some reasonable assumptions are applied to simplify the mathematical model for inclusion transport in a tundish.

(1) There is no chemical reaction in the molten steel.

(2) The inclusion is a spherical particle.

(3) Each inclusion moves independently until two inclusions collide.

(4) Brownian collision among inclusions can be neglected.26)

(5) The existence of inclusions does not affect macroscopic flow of molten steel because the volume fraction of inclusion particles is too small.

(6) The inclusion can be treated as the continuum phase.

(7) The fractional inclusion number density decreases exponentially as the inclusion size increases, f(r) = Ae−Br, and the parameters A and B are the function of the time and the coordinates.

3.2. Governing Equations

The inclusion transport process is a steady state on the premise of the steady fluid flow in the tundish. Consequently, the inclusion collision–coalescence model can be expressed by the inclusion mass conservation equation and the inclusion population conservation equation.   

( ρ f u C C)=( ρ f D eff,C C) (1)
  
( ρ f u N N)=( ρ f D eff,N N)+ S N (2)

With Deff = D0 + vt/Sct

3.2.1 Inclusion Characteristic Radius

According to the mass conservation equation:   

4 3 π r *3 N= 0 4 3 π r 3 f(r)dr (3)

The inclusion characteristic radius can be given by r * = 3C 4πN 3 .

3.2.2. Inclusions Velocity

The inclusion convection velocity u is the sum of fluid velocity u f and inclusion slipping velocity u p . Thus, the inclusion convection velocities u C and u N in the inclusion mass/population conservation equation have the similar form.   

u C = u f + u p,C (4)
  
u N = u f + u p,N (5)

After an electromagnetic field is imposed on the molten steel, the important forces acting on the inclusion particle are Archimedes force, the gravity, the buoyancy and the viscous drag. In the stationary molten steel, if the inclusion is in the force equilibrium state, the inclusion trajectory equation can be simplified as   

4π r 3 3 ρ l g - 4π r 3 3 ρ p g + π r 2 2 C D ρ p | u P | u P - 3 4 4π r 3 3 J × B =0 (6)

Here, r is the inclusion radius.

In this way, the terminal floating velocity of a single inclusion can be expressed as follows:   

u p = 2 9 r 2 g Δρ μ l - 1 6 r 2 μ J × B (7)

Thus, the inclusion slipping velocities can be expressed as the sum of Stokes floating velocity and Archimedes slipping velocity.   

u p,C = 0 u p f(r)dr = 40 9 6 2 3 Δρ r *2 μ l g - 10 3 6 2 3 r *2 μ l J × B (8)
  
u p,N = 0 u p f(r) 4 3 π r 3 dr = 4 9 6 2 3 Δρ r *2 μ l g - 1 3 6 2 3 r *2 μ l J × B (9)

Archimedes slipping velocity appears in the inclusion population conservation equation and inclusion mass conservation equation, but Archimedes collision only appears in the inclusion population conservation equation.

In the tundish with channel induction heating, the inclusion collisions have four mechanisms: Brownian collision, Stokes collision, Archimedes collision and turbulent collision. Brownian collision is two orders smaller than the other collisions if the inclusion size is greater than 1 micro,26) so the collision source term SN can be expressed as:   

S N = S turb + S Arch + S Stokes (10)
where Sturb, SArch and SStokes come from turbulent collision and Archimedes collision and Stokes collision, respectively.   
S turb = 1 2 0 0 1.3α π ρ f ε μ l ( r i + r j ) 3 N i N j d r i d r j =2.6α π ρ f ε μ l N *2 r *3 (11)

Similar to Stokes collision, Archimedes-Stokes collision has the following form.   

S Arch-Stokes = 1 2    0        0     | 2π 9 g Δρ μ - π 6 J × B μ l || r i - r j | ( r i + r j ) 3 N i N j d r i d r j = 10π 9 6 3 | 3 4 J × B - g Δρ | μ l N 2 r *4 (12)

3.3. Initial and Boundary Conditions

At the inlet, the inclusion velocity is the same as the local fluid velocity, and the inclusion volume concentration and the inclusion number density are fixed at 203 ppm and 4.89 × 1012/m3 respectively. The other boundary conditions about electromagnetic field, fluid field and temperature field can be found in our previous paper.24,25)

Diffusion and convection are two mechanisms for inclusion’s transport in the molten steel. Thus, the inclusions flux at the tundish boundaries (top slag, bottom wall, incline wall and channel wall) can be expressed by the convection flux and the diffusion flux.   

F= F c + F d (13)

The convection flux in the inclusion mass conservation equation and the inclusion population conservation equation can be expressed as follows:   

F C c = u p,c C (14)
  
F N c = u p,N N (15)

The diffusion flux in the inclusion mass conservation equation and the inclusion population conservation equation are listed as follows:   

F C d = 16π 3 C 2 6 3 N r *4 n (16)
  
F N d = C 2 6 3 N r * n (17)
with   
C 2 = 0.01 τ 0 ρυ (18)

Table 1 gives four types of boundary conditions: inlet, outlet, free surface and wall in the tundish with channel induction heating. Figure 3 shows that there are two classic walls where the inclusions are adsorbed. One is the tundish wall, another is the channel wall.

Table 1. Boundary conditions for inclusion transport.
ParameterCN
InletC0N0
Outle C n =0 N n =0
Tundish wallJc,wallJN,wall
Tundish channelJc,wallJN,wall
Free surfaceJc,wallJN,wall
Fig. 3.

The schematic diagram of wall adsorption.

Whether it is the tundish wall or the channel wall, the inclusion flux at wall can be expressed:   

J C,wall =[max(0, F c d + F c c ) n ] (19)
  
J N,wall =[max(0, F N d + F N c ) n ] (20)
where n is normal vector.

3.4. Grid System and Convergence Criteria

The tundish domain is discretized by using a non-uniform grid system which encloses the fluid domain. The number of grid is about 300000. For the inclusion field, the convergence criteria is that the value of the root mean square normalized residual for variables was less than 10−5.

4. Results and Discussion

In order to describe the spatial distribution of inclusion in the tundish clearly, it is necessary to define some characteristic cross sections, as shown in Fig. 4. Section A-A and section B-B pass through the center axis of the channel, and sections C-C (inlet of channel), D-D and E-E (outlet of channel) are the central cross section of the channel.

Fig. 4.

Characteristic sections in the tundish. (Online version in color.)

In Navier-Stokes equation, there are two source terms: thermal buoyancy and electromagnetic force. But in the inclusion model, the effects of Joule heat on the inclusion movement (Brownian motion) and on the inclusion collision (Brownian collision) can be ignored because Brownian motion and Brownian collision are too weak, and the effects of the electromagnetic force on the inclusion behavior can be described as Archimedes velocity and Archimedes collision.

In order to have a deep insight into the effects of electromagnetic force and Joule heat, Table 2 gives the four cases in the numerical simulation. In Case 2, Joule heat appears in the energy conservation equations. In Case 3, electromagnetic force appears in Navier-Stokes equation, and Archimedes slipping velocity and Archimedes collision should be included in the inclusion mass/population conservation equations.

Table 2. Four cases in the numerical simulation.
No.Joule heatElectromagnetic forceNotes
1NoNoNo channel induction heating
2YesNoOnly Heat effect
3NoYesOnly Electromagnetic Stirring
4YesYesChannel induction heating

4.1. Model Validation

The model is validated by a tundish with a capacity of 35 t molten steel, and its size and the related parameters are shown in Table 3.

Table 3. Main dimensions and parameters of model validation tundish and properties of molten steel.
ParametersValue
Tundish capacity35 t
Number of strand2 strands
Inlet velocity0.81 (m/s)
Inlet temperature1763 (K)
Density of molten steel7020 (kg/m3)
Heat capacity750 (J/kg·K)
Thermal conductivity41 (W/m·K)
Thermal expansion coefficient1.0 × 10−4 (K−1)
Heat flux at the top surface15 (kW/m2)
Heat flux at the bottom wall1.4 (kW/m2)
Heat flux at the long walls3.2 (kW/m2)
Heat flux at the short walls3.8 (kW/m2)
Viscosity of molten steel0.0067 (Pa·s)
Diameter of the outlet50 (mm)
Distance between the two outlets4200 (mm)
Depth of the molten steel925 (mm)

The experiment sample is taken from a 200-series stainless steel in the industrial plant. The sample and slag are taken from the 300 mm below the slag layer of the pouring zone and above the outlet in the tundish. The inclusions are detected by SEM-EDS system. The scanning area is 183 mm2, the inclusion diameter is from 1.00 μm to 108.6 μm.23) In the numerical verification process, the initial inclusion concentration and number density are 271.5 ppm and 3.33 × 1013/m3.

The predicted magnetic field, fluid field and temperature field have been validated in the previous paper.24,25) Certainly, the inclusion size distribution should also be validated. Figure 5 shows that predicted value of inclusion number density agrees well with the experimental data when the inclusion diameter is less than 15.5 micro at the ladle shroud and 21.21 micro at the tundish exit. The difference between the numerical result and the experimental result comes from the following reasons. (1) In the mathematical model, the inclusion number density and the inclusion radius is assumed to satisfy the exponential function.27) Figure 5 shows that such an assumption is not inaccurate if the inclusion size is greater than 15.5 micro at the ladle shroud. (2) The experimental data comes from 2D inclusion analysis in the experiment. In some cases, a cluster-inclusion maybe is regarded as several smaller inclusions. But in the mathematical model, the inclusion is the three dimensional spherical particle. (3) The experimental data for inclusion maybe includes the exogenous inclusion which comes from refractory erosion or slag entrapment. But the endogenous inclusion is not considered by the current mathematical model.

Fig. 5.

Measured and predicted inclusion number density in tundish.

4.2. The Characteristic Inclusion Volume Concentration

Figure 6 shows that the spatial distribution of the inclusion volume concentration without channel induction heating has the following features. (1) The inclusion concentration decreases gradually from receiving chamber to discharging chamber because inclusions can be removed by the tundish wall and the slag layer. (2) The inclusion concentration has a great change in receiving chamber. (3) In the channel, the inclusion concentration near the center is greater than that near the channel wall because some inclusions are removed by the channel wall. But the change of the inclusion concentration is not obvious because the fluid flow in the channel is a developed pipe flow and the channel wall is not big.

Fig. 6.

Characteristic inclusion volume concentration in the tundish without channel induction heating.

Figure 7 shows that the inclusion volume concentration distribution with channel induction heating has some striking features: (1) The inclusion concentration drops rapidly from receiving chamber to discharging chamber. (2) Channel is an important place to remove the inclusions because the pinch force forces the inclusion to move to the channel wall.28)

Fig. 7.

Inclusion volume concentration in the tundish with channel induction heating.

4.3. The Characteristic Inclusion Radius

Figure 8 shows that the spatial distribution of the characteristic inclusion radius in the tundish without channel induction heating has some remarkable features: (1) The inclusion radius gradually increases from receiving chamber to the discharging chamber duo to Stokes collision and turbulent collision. (2) In the receiving chamber, the inclusion in the lower part is larger than that in the upper part, because the jet flow from the channel divides the receiving chamber into the lower recirculation zone and the upper recirculation zone. The upper recirculation zone is larger than the lower recirculation zone, and the inclusions in the upper recirculation zone have more chances to be removed by the slag layer than that in the lower recirculation zone.

Fig. 8.

Characteristic inclusion radius in the tundish without channel induction heating.

Figure 9 gives the spatial distribution of inclusion radius distribution in the tundish with channel induction heating, some interesting phenomena are as followed: (1) The change of inclusion size in the receiving chamber with channel induction heating is smaller than that without channel heater because there is a larger upper recirculation in the receiving chamber with channel induction heating.25) (2) The inclusion size in the channel with channel induction heating is greater than that without channel induction heating because of the strong Archimedes collision among the inclusions and the strong turbulent collision caused by the strong spiral flow in the channel.25)

Fig. 9.

Characteristic inclusion radius in the tundish with channel induction heating.

4.4. The Inclusion Number Density

Figures 10 and 11 give the spatial distribution of the inclusion number density in the tundish with/without induction heating, they have the following features. (1) The inclusion number density gradually decreases from receiving chamber to discharging chamber due to the coalescence among the inclusions. (2) The inclusion number density in the channel with induction heating is less that without induction heating because of the strong spiral flow in the channel and the strong Archimedes collision.26)

Fig. 10.

Characteristic inclusion number density in the tundish without channel induction heating.

Fig. 11.

Characteristic inclusion number density in the tundish with channel induction heating.

4.5. The Characteristic Inclusion Distribution for Four Cases

Figure 12 shows that the inclusion removal ratio in the case of no channel induction heating is similar to that in the case of Joule heat and no electromagnetic force. Two reasons lead to this interesting phenomenon. (1) The natural heat convection caused by the temperature difference has little effect on the fluid flow in the tundish. (2) Fluid temperature is the key factor to affect Brownian collision among inclusions, but Brownian collision rate is far less than Stokes collision rate and turbulent collision rate if the inclusion size is greater than 1 micro.26)

Fig. 12.

Inclusion removal rate for four cases. (Online version in color.)

Figure 12 also shows that the inclusion removal ratio in the case of no Joule heat and electromagnetic force is similar to that in the case of channel induction heating. Such an interesting phenomenon come from several reasons. (1) Electromagnetic force is the key factor to affect the fluid flow in the tundish. (2) Archimedes collision is one of the important inclusion collision mechanisms. (3) Channel is one of the important places to remove inclusions in the case of channel induction heating because the inclusion removal ratio in the channel is up to one third of the inclusion removal ratio in the tundish.

Figure 13 gives the following facts at the tundish exit. (1) The inclusion characteristic radius falls from 4.36 micro (without channel induction heating) to 4.02 micro (only with Joule heat) and to 3.77 micro (only with electromagnetic force), and the inclusion concentration falls from 159.6 ppm (without channel induction heating) to 158.6 ppm (only with Joule heat) and to 132.1 ppm (only with electromagnetic force). (2) At the tundish exit, the inclusion characteristic radius in the case of only electromagnetic force equals to that in the case of channel induction heating, and the inclusion concentration in the case of only electromagnetic force is close to that in the case of channel induction heating. These facts can give the following information. (1) The inclusion behavior in the tundish without channel induction heating is similar to that in the tundish with Joule heat (no electromagnetic force). (2) The inclusion behavior in the tundish with channel induction heating is similar to that in the tundish with electromagnetic force (no Joule heat). (3) Both electromagnetic force and Joule heat can promote the inclusion removal rate. And the effect of electromagnetic force on the inclusion behavior is stronger than the effect of Joule heat.

Fig. 13.

Inclusion radius and concentration. (Online version in color.)

5. Conclusions

(1) In the inclusion mass and population conservation model, a three-dimensional Archimedes velocity is applied to describe the inclusion movement and Archimedes collision is applied to describe the inclusion coalescence in the tundish with channel induction heating.

(2) In the tundish with channel induction heating, Joule and electromagnetic force can prompt the inclusion removal rate. And electromagnetic force is more important for inclusion behavior than Joule heat.

(3) In the tundish with channel induction heating, channel is an important place for inclusion removal.

(4) At the tundish exit, inclusion size and inclusion concentration in the case of channel induction heating is less than that in the case of no channel induction heating.

(5) The inclusion characteristic radius falls from 4.36 micro to 3.77 micro (with electromagnetic force), and inclusion concentration falls from159.6 ppm (without channel induction heating) to 158.6 ppm (only with Joule heat) and to 131.8 ppm (channel induction heating).

(6) The inclusion removal rate increases from 21.40% (without channel induction heating) to 35.05% (channel induction heating).

Acknowledgement

The work is supported by National Natural Science Foundation of China and Shanghai Baosteel (No. U1460108) and National Natural Science Foundation of China (No. 51574074).

Nomenclature

N: Inclusion number density (1/m3)

C: Inclusion volume concentration

ρf: Density of fluid (Kg/m3)

u C ,    u N : Inclusion slipping velocity (m/s)

Deff: Effective diffusivity of molten steel (m2/s)

A: Constant

B: Constant

r*: Inclusion characteristic radius (μm)

r: Inclusion radius (μm)

f: Fractional inclusion number density (1/m4)

μ: Dynamic viscosity [Kg/(m·s)]

υ: Kinematic viscosity (m2/s)

F: Electromagnetic force (N/m3)

CD: Drag coefficient of particle (m2/s)

ρp: Density of inclusion (Kg/m3)

J: Current density (A/m2)

B: Magnetic flux density (T)

u l : Velocity of molten steel (m/s)

α: Coagulation coefficient

ε: Turbulence dissipation rate (m2/s)

F C c : Transport flux of inclusion volume concentration (m/s)

F N c : Transport flux of inclusion number density [1/(m2·s)]

F C d : Diffusion flux of inclusion volume concentration (m/s)

F N d : Diffusion flux of inclusion number density [1/(m2·s)]

τ0: Wall friction [kg/(s2·m)]

n : Normal vector

g: Gravitational acceleration (m/s2)

References
 
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