# Numerical simulation of flow and solidification in continuous casting process with mold electromagnetic stirring

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## Abstract

The magnetic, heat transfer and flow phenomenon occurring in the continuous casting process under the mold electromagnetic stirring was further analyzed by solving the 3-D electromagnetic field mathematical model and flow solidification model with finite element method and finite volume method, respectively. The results indicate that the solidified shell thickness located in the effective stirring region fluctuates because of the unsteady scouring under the mold electromagnetic stirring. The maximum rotational velocity is a key parameter to the solidification of the billet when controlling the stirring intensity. When the rotational velocity reaches 0.32 m/s, the mush zone enlarges significantly and the solidification rate is further accelerated. The number of vortexes in the lower recirculation zone is not only two and depends on the stirring parameters. Besides, the secondary flow is closely associated with the solidification. Compared with the results of the model ignoring the influence of solidification on the flow of molten steel, the flow pattern within the lower recirculation region changes dramatically, and thus a coupling analysis of the flow, heat transfer, and solidification is essential when simulating the electromagnetic continuous casting process.

## Keywords

Continuous casting Electromagnetic stirring Solidification Numerical simulation Flow## 1 Introduction

The mold electromagnetic stirring (M-EMS) has been widely used in the continuous casting process to improve the surface quality, refine grain size, and reduce segregation [1]. The flow and heat transfer in the continuous casting process are considerably complex so that it is impossible to study these issues by field tests because of a lack of the effective detecting methods. Moreover, experimental researches are expensive, time-consuming, and difficult to simulate the solidification process [2, 3, 4]. Numerical simulation was widely used to study and optimize the process [5, 6, 7]. Natarajan and El-Kaddah [8] researched the flow in rotary stirring of steel during casting with the finite element method. It was shown that the electromagnetic force (EMF) at the corners of the billet drove the molten steel to rotate. Two approaches to obtain the EMF were compared by Javurek et al. [9]: One was the semi-empirical method, and the other was calculation with the MHD module built into FLUENT. The results demonstrated that the semi-empirical method was suspectable, while FLUENT MHD module produced an overprediction of the EMF. Rywotycki et al. [10] analyzed the heat transfer of the billet in the mold and the secondary cooling zone, and the finite element method was employed to solve the equations. Song et al. [11, 12] took the joule heat produced by the M-EMS into account, and the results indicated that the Joule heat had little effect on the heat transfer. Geng et al. [13] considered that equiaxial crystal ratio was more impressionable to current intensity than to current frequency, namely, compared with improving current frequency, it is more effective to promote the equiaxial crystal ratio by improving current intensity.

The works published previously have laid a strong foundation of the researches on the M-EMS; however, quite few mathematical models took the solidification process into account. In previous studies [14, 15, 16, 17], most of the analysis has been performed on the driven flow; nevertheless, the solidification of the molten steel has huge influence on the flow. In addition, the emphasis was always put on the rotational flow of the molten steel on the horizontal cross section [18, 19]. In fact, the recirculation zone called secondary flow caused by the stirring is essential, and this phenomenon should be explored in detail to address its metallurgical effects. Moreover, in order to save the simulation time, the computational domain was usually restricted in the mold, whereas the fact that M-EMS has a strong effect on the whole billet was proved [20].

In this paper, based on the commercial software of ANSYS APDL and CFX, the user-defined procedures were developed for the solidification process so that a mathematical model containing magnetic field, heat transfer, flow, and solidification simultaneously was established to gain a deep insight into the flow and solidification of the billet with the M-EMS. Moreover, the emphases of present work were posed on the relationships of the stirring current to the rotational velocity, flow pattern in vertical direction, and mush zone of the billet. Besides, a comparison of the flow pattern with M-EMS between the models considering solidification and without considering solidification was conducted to show how solidification deeply influences the flow of molten steel. In addition, the modeling domain consisted of a billet up to 9.3 m from the meniscus, which exposed the effect of stirring current on the surface temperature of billet, the length of liquid core, and the thickness of solidified shell.

## 2 Mathematical model

### 2.1 Model assumptions

- 1.
The displacement current is neglected.

- 2.
The flow and solidified shell of the steel have little effect on the magnetic field.

- 3.
The meniscus is assumed flat, adiabatic, and the slag layer is not modeled.

- 4.
The density, viscosity, specific heat, and thermal conductivity of steel are constant with temperature.

- 5.
The influence of the mold oscillation on the billet is not considered.

- 6.
The solidification shrinkage is irrespective.

### 2.2 Electromagnetic model

*is the electric field strength;*

**E***is the magnetic flux density;*

**B***is the current density;*

**J***μ*is the magnetic permeability;

*ρ*

_{m}is the charge density; and

*t*is the time.

*is the time-average Lorentz force acting on the molten steel;*

**F***is the velocity of molten steel;*

**u***ρ*

_{e}is the total charge density; and

*q*is the charge carried by every infinitesimal element.

*ρ*

_{m}plays no significant part, and the electric force,

*q*

*, is negligible, compared with the Lorentz force. Besides, the influence of flow velocity on the magnetic field can be ignored. Thus, the Maxwell’s equations can be simplified as:*

**E***is the magnetic field intensity.*

**H***σ*is the electric conductivity.

**F**_{mag}is expressed as follows:

### 2.3 Flow and solidification model

*k*–

*ω*model, which is more suitable in low Reynolds number flow than

*k*–

*ε*model [23, 24], where

*k*is the turbulent kinetic energy;

*ω*is the specific dissipation rate; and

*ε*is the turbulent dissipation rate. The equations are expressed as follows:

*P*is pressure;

*ρ*is the density of molten steel;

*μ*

_{l}and

*μ*

_{t}are molecular viscosity and turbulent viscosity, respectively;

*G*

_{k}and

*G*

_{ω}represent the generation of turbulence kinetic energy and specific dissipation rate, respectively;

*σ*

_{k}and

*σ*

_{ω}are the Prandtl numbers corresponding to the turbulent kinetic energy and the turbulent energy dissipation rate, respectively; and

*S*

_{m},

*S*

_{k},

*S*

_{ω}are source items.

*S*

_{m}and

*S*

_{ϕ}(

*ϕ*=

*k*,

*ω*) into the Navier–Stokes equation and the governing equations of turbulence model. According to the Darcy’s law, it can be expressed as [25, 26, 27]:

*ξ*is a constant less than 0.0001;

*λ*

_{2}is the secondary dendrite arm spacing;

**u**_{c}is the casting velocity; and

*f*

_{l}is the liquid fraction defined as:

*T*is the temperature of molten steel;

*T*

_{s}is the solidus temperature; and

*T*

_{l}is the liquidus temperature.

*H*is the total enthalpy;

*h*

_{s}is the enthalpy of solid phase;

*h*

_{l}is the enthalpy of liquid phase; and

*λ*

_{eff}is the effective thermal conductivity.

*Pr*and

*Pr*

_{t}are the Prandtl number and turbulent Prandtl number, respectively;

*h*

_{ref}is the reference enthalpy;

*T*

_{ref}is reference temperature;

*T*

_{tim}is the timely temperature of the molten steel;

*c*

_{p}is the specific heat of molten steel; and

*L*is latent heat.

### 2.4 Boundary conditions

*v*

_{in}is the velocity of molten steel at inlet;

*d*

_{0}is the diameter of inlet; and subscript in indicates the nozzle inlet.

The normal gradients of all variables are set equal to zero at the meniscus and outlet. Moreover, the velocity perpendicular to meniscus is also equal to zero. The wall of the billet is treated as moving wall, whose velocity is specified as casting velocity.

*q*

_{m}is the heat flux of the mold;

*l*is the distance of molten steel from meniscus;

*β*is an empirical coefficient;

*L*

_{m}is the effective length of the mold; \(\bar{q}\) is the mean heat flux of the mold;

*ρ*

_{w}is the density of cooling water;

*C*

_{w}is the specific heat of cooling water;

*W*

_{L}is the flow rate of cooling water located in mold; ∆

*T*is the temperature difference of cooling water between inlet and outlet of the mold cooling system; and

*S*

_{c}is the effective contact surface between mold wall and molten steel.

*q*

_{S}is the heat flux of secondary cooling zone;

*W*

_{2}is the spray water flux;

*t*

_{w}is the temperature of cooling water;

*α*is a coefficient related to guide roller;

*T*

_{sur}and

*T*

_{w}are the surface temperature of billet and environment temperature, respectively;

*q*

_{a}is the heat flux of air cooling zone;

*ε*

_{r}is the radiation heat transfer coefficient; and

*σ*

_{B}is the Boltzmann constant.

### 2.5 Numerical methods and geometric model

^{−4}was obtained and the sum of mass residual in all nodes was less than 0.01% of the initial total mass. All calculation was done in steady state to save computation time. The geometric model of the billet and M-EMS is represented in Fig. 1. The main material properties and operating parameters are listed in Table 1.

Properties of molten steel and process parameters used in calculation

Parameter | Value |
---|---|

Conductivity/(S m | 7.14 × 10 |

Relative permeability | 1 |

Density/(kg m | 7100 |

Viscosity/(Pa s) | 6.7 × 10 |

Specific heat/(J kg | 643.8 |

Thermal conductivity/(W m | 30 |

Latent heat/(J kg | 2.556 × 10 |

Solidus temperature/K | 1751.15 |

Liquidus temperature/K | 1663.15 |

Casting temperature/K | 1767.15 |

Ambient temperature/K | 300 |

Casting velocity/(m min | 1.75 |

Cross-sectional dimensions/(m × m) | 0.16 × 0.16 |

Radius of caster/m | 10 |

Length of mold/m | 1 |

Length of billet/m | 9.3 |

## 3 Results and discussion

### 3.1 Model validation

*l*

_{1}and

*l*

_{2}is 69.76 mm and 63.30 mm, respectively. The comparison of the calculated and measured solidified shell thickness is shown in Fig. 4. It can be seen that the calculated and measured values are in good agreement. Besides, this model is also validated by comparing with the experiments conducted by Deng and He [28].

### 3.2 Distribution of Lorentz force

### 3.3 Flow and solidification around the stirrer

### 3.4 Solidification of whole billet

## 4 Conclusions

- 1.
When the M-EMS is applied, the flow pattern of the molten steel on the cross section is rotary, and the velocity of the billet increases firstly and then decreases from the center to the surface. Moreover, the rotational velocity goes up with the increase in the current intensity, and the maximum velocity reaches 0.32 m/s when the current intensity reaches 640 A.

- 2.
The flow pattern of the molten steel on the transverse and longitudinal cross sections is changed by the M-EMS. The number of the vortexes within the lower swirl zone is related to the current intensity. The service area of the stirrer can be obtained accurately by investigating the maximum tangential speed of the molten steel along the casting direction of the billet.

- 3.
The thickness and the shape of the solidified shell fluctuate within the effective area of the stirrer with the M-EMS, and the fluctuation is restricted to the lower half of the mold and the foot roller section. Moreover, because of the improvement in the heat transfer condition, the mush zone expands, which will promote the increase in the equiaxed crystal ratio, accelerate the solidification process, and shorten the length of liquid core. Compared with the stirring current intensity of 320 A and 452 A, when the current intensity reaches 640 A, namely, when the maximum rotational velocity reaches 0.32 m/s, the liquid region shrinks, the mush zone enlarges, and the solidification process is obviously different from the others. It is essential to obtain an appropriate current intensity to improve the quality of the billet by using the M-EMS.

- 4.
The flow, heat transfer, and solidification of the molten steel are closely correlated; therefore, the solidification must be considered when the numerical simulation about the continuous casting process is conducted.

## Notes

### Acknowledgements

This work was supported by National Natural Science Foundation of China (51474065, 51574083), the Doctoral Scientific Research Foundation of Liaoning Province of China (20141008), and the Program of Introducing Talents of Discipline to Universities of China (B07015).

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