## 1. Introduction

The technology of TSCE-ESR adopts the mode of two electrodes connected in series, which can reduce the inductive reactance of short net (transformer-water cooled copper plate-water cooled cable-electrode), improve the power factor, reduce the active power consumption of the short net, thus greatly reducing the power consumption. It is widely used in the preparation of tool steel, die steel, military steel, and other special alloy materials [

1]. Compared with the process of OE-ESR, the current flow direction of the TSCE-ESR process has changed. On the one hand, it will affect the heating area of the slag. On the other hand, it will change the direction of the electromagnetic force and the flow field of slag pool, which will have an important impact on the temperature field and the structure of the molten metal pool, and ultimately affect the quality of the electroslag remelting steel ingot. Therefore, the systematic study on the heating and heat-transfer mechanisms of TSCE-ESR process is of great significance to the development of the process. In recent years, some scholars have studied the remelting process of TSCE-ESR. For example, Wang [

2] established a mathematical model of coupled harmonic electromagnetic equations, obtained the distribution of current density and electromagnetic force in the process of TSCE-ESR, and analyzed the influence of frequency on current density distribution. Li [

3] established a three-dimensional finite element model of TSCE-ESR process, and studied the influence of the ingot height on the molten metal pool. However, there are still few systematic studies on the multi-physics field, droplet formation, and dripping behavior and molten metal pool structure in the process of TSCE-ESR, which limits the technological development and the improvement of product quality.

Because of the high temperature and invisibility, it is difficult to observe a series of physical and chemical phenomena during the electroslag remelting process. In recent years, many scholars have studied the electroslag remelting process by means of numerical simulation. Kelkar [

4] established an electroslag remelting model taking into account electromagnetic, flow, and heat transfer. He calculated the Joule heat, flow field, and metal pool profile. Weber [

5] established a two-dimensional transient mathematical model of the electroslag remelting process, and analyzed the influence of filling ratio on the flow field distribution. Li [

6] studied the distribution of electromagnetic field, Joule heat, and temperature field during the electroslag remelting process under the influence of skin effect. The above studies have been able to realize the coupling of multiple physical fields to the mathematical model of electroslag remelting. However, the electroslag remelting process involves the two phases of slag and liquid metal, which is a typical multiphase flow problem. Obviously, the influence of droplet formation and droplet dropping on the electroslag remelting process is still ignored in the above studies. With the development of multiphase flow models, researchers have tried to couple multiphase flow models to directly simulate the dripping process of droplets. Ruckert [

7] used VOF (Volume of Fluid) module to track the slag/metal (slag and metal pool) interface in order to study the droplet formation and dripping behavior. Li [

8] simulated the flow between slag and metal in the slag pool based on the VOF algorithm, and obtained the formation and dripping process of metal droplets. However, these studies did not take into account the effects of buoyancy and electromagnetic forces on metal droplets. Giesselmann [

9] considered the effects of buoyancy and electromagnetic force on metal droplets, but he assumed that the effects of electromagnetic field, Joule heat and Lorentz force were independent of each other. However, the two actually affected each other, so this simulation method would also lead to inaccurate simulation results. Liu [

10] established a mathematical model of droplet behavior of the electroslag remelting process based on magnetohydrodynamic. The electromagnetic force and joule heat in the process of droplet formation and dropping were studied in detail. However, the shape of electrode tip was specified in the study and a constant mass flow was applied, while the effect of slag temperature on the electrode melting process was ignored, so the droplet formation process could not be reflected. In conclusion, it is difficult to find a comprehensive mathematical model to describe the multi-physical field, droplet effect, and metal pool structure in the electroslag remelting process.

By constructing a simultaneous sequential coupling numerical model of three models, which means a multi-physical steady-state model, a transient model of electrode melting and droplet dropping, and a multiphase transient model of metal pool, it can be more accurate to simulate the multi-physical field, droplet dropping behavior, and molten pool structure in the process of TSCE-ESR. In summary, it is necessary to comprehensively simulate the electroslag remelting process considering the droplet effect. By comparing OE-ESR process with TSCE-ESR process, the influence caused by the change of current direction in the process of TSCE-ESR was systematically studied.

In this paper, the multi-physical steady-state model, the transient model of electrode melting and droplet dropping, and the multiphase transient model of molten metal pool are established in the process of OE-ESR and TSCE-ESR. The electromagnetic field is calculated by the user-defined function (UDF), and the Joule heat and electromagnetic force are coupled to the equation by the source term to calculate the flow field and temperature field of OE-ESR and TSCE-ESR at steady state. The steady-state calculation results are used as the initial conditions for the transient model of electrode melting and droplet dropping and multiphase transient model of metal pool. Then the formation and dripping behavior of the droplets were tracked by VOF method. By using the remelting rate calculated from the transient model of electrode melting and droplet dropping as the inlet boundary condition of multi-phase transient model of molten metal pool, the shape of the molten metal pool considering the droplet effect is calculated. Based on the above methods, this paper makes a comprehensive comparison and study on the process of OE-ESR and TSCE-ESR, and provides a theoretical basis for the development of the process of TSCE-ESR and the improvement of product quality.

## Author Contributions

Conceptualization, W.L.; methodology, W.T. and W.L.; formal analysis, W.T. and W.L.; investigation, W.T.; resources, X.Z. and H.L.; writing—original draft preparation, W.T.; writing—review and editing, W.T. and W.L.; supervision, Z.J. and D.L. All authors have read and agreed to the published version of the manuscript.

## Funding

The authors gratefully express their appreciation to Natural Science Foundation of China (No.51974153, No. U1960203), and the Joint Fund of State key Laboratory of Marine Engineering and University of Science and Technology Liaoning (SKLMEA-USTLN-201901, SKLMEA-USTL-201707), and the China Scholarship Council (201908210457).

## Conflicts of Interest

The authors declare that there are no conflicts of interest.

## Abbreviations

E | Electric Field (V·m^{−1}) |

H | Magnetic field intensity (A·m^{−1}) |

J | Current density (A·m^{−2}) |

B | Magnetic flux density (T) |

t | Time (s) |

ρ | Density of fluid (kg·m^{−3}) |

v | Velocity (m·s^{−1}) |

P | Pressure (Pa) |

μ_{eff} | Effective viscosity of the fluid (Pa·s) |

F_{loc} | Electromagnetic force (N·m^{−3}) |

μ_{0} | Vaccum permeabilitbility (H·m^{−1}) |

Q_{j} | Joule heat per unit volume (W·m^{−3}) |

σ | Electroconductibility (S·m^{−1}) |

u_{t} | Turbulent velocity (m·s^{−1}) |

U | Velocity of droplet (m·s^{−1}) |

r_{d} | Radius of droplet (m) |

q_{d} | Density of droplet (kg·m^{−3}) |

C_{D} | Resistance coefficient |

λ | Conductivity of liquid (S·m^{−1}) |

C_{p} | The thermal capacity of slag (J·kg^{−1}·K^{−1}) |

T_{dp} | Temperature of droplet (K) |

LST | Local solidification time (s) |

V_{r} | Local solidification rate (mm·s^{−1}) |

R_{c} | Local cooling rate (K·s^{−1}) |

d_{e} | The Diameter of Droplet (m) |

σ_{d} | Surface tension (N·m^{−1}) |

ρ_{m} | Density of metal (kg·m^{−3}) |

ρ_{s} | Density of slag (kg·m^{−3}) |

H_{S} | Coefficient of position (1) |

τ | Droplet residence time in the slag pool (s) |

V_{t} | Terminal velocity (m·s^{−1}) |

W_{e} | Weber number (ρ_{m}·V_{t}^{2}·d·r·σ_{d}^{−1}) |

P_{d} | Physical property group |

R_{e} | Reynolds number (ρ_{m}·V_{t}^{2}·d·μ^{−1}) |

m_{e} | Melt rate (kg·s^{−1}) |

T_{0} | Reference temperature (K) |

R | Radius of the electrode (m) |

q_{se} | Heat flux from slag to electrode (W·m^{−2}) |

T_{L} | Liquidus temperature (K) |

T_{me} | Melting point of electrode (K) |

I_{0} | Current (A) |

Q_{c} | Heat of convection heat transfer (W·m^{−2}·K^{−1}) |

A | Heat exchange area (m^{2}) |

h | Heat transfer coefficient (W·m^{−2}·K^{−1}) |

C_{P,d} | Heat capacity of the droplet (J·kg^{−1}·K^{−1}) |

X | Solid-liquid two-phase zone width (mm) |

G | Liquid temperature gradient (K·mm^{−1}) |

d | Secondary dendrite spacing (mm) |

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**Figure 1.**
Schematic diagram of OE-ESR (one-electrode electroslag remelting) and TSCE-ESR (electroslag remelting with two series-connected electrodes).

**Figure 2.**
Computational domains. (**a**) OE-ESR. (**b**) TSCE-ESR. 1-electrode, 2-slag pool, 3-ingot.

**Figure 3.**
Schematic diagram of boundary conditions.

**Figure 4.**
Numerical simulation flow charts. (**a**) Multi-physical steady-state model. (**b**) Transient model of electrode melting and droplet dropping. (**c**) Multi-phase transient model of molten metal pool.

**Figure 5.**
Current density distribution in slag pool. (**a**) OE-ESR. (**b**) TSCE-ESR. The black line represents the current flow line.

**Figure 6.**
Current density value at z = −0.31 m.

**Figure 7.**
Joule heat distribution in slag pool. (**a**) OE-ESR. (**b**) TSCE-ESR.

**Figure 8.**
Vector diagram of electromagnetic force. (**a**) OE-ESR. (**b**) TSCE-ESR.

**Figure 9.**
Velocity field and temperature field distribution in slag pool. (**a**) Velocity field of OE-ESR. (**b**) Temperature field of OE-ESR. (**c**) Velocity field of TSCE-ESR. (**d**) Temperature field of TSCE-ESR.

**Figure 10.**
Surface map of temperature distribution in slag pool. (**a**) OE-ESR. (**b**) TSCE-ESR.

**Figure 11.**
Formation and dripping process of droplet of OE-ESR.

**Figure 12.**
Formation and dripping process of droplet of TSCE-ESR.

**Figure 13.**
Distribution of velocity field during droplet dripping process of OE-ESR.

**Figure 14.**
Distribution of the temperature field during the droplet dripping process of OE-ESR.

**Figure 15.**
Distribution of velocity field during droplet dripping process of TSCE-ESR.

**Figure 16.**
Distribution of the temperature field during the droplet dripping process of TSCE-ESR.

**Figure 17.**
Overall heat balance. (**a**) OE-ESR. (**b**) TSCE-ESR.

**Figure 18.**
Profile of molten metal pool without droplet effect. (**a**) OE-ESR. (**b**) TSCE-ESR.

**Figure 19.**
Profile of molten metal pool with droplet effect. (**a**) OE-ESR. (**b**) TSCE-ESR.

**Figure 20.**
The experiment of TSCE-ESR process. (**a**) Photo of the TSCE-ESR plant experiment. (**b**) Molten metal pool structure in the process of TSCE-ESR.

**Figure 21.**
Comparison of molten metal pool structure between the experimental value and the numerical value.

**Figure 22.**
Comparison of the width of mushy zone between the experimental value and the numerical value.

**Figure 23.**
Relationship of local solidification time and cooling rate in ESR process.

**Figure 24.**
Local solidification time (LST) at different processes.

**Table 1.**
Physical properties and geometric parameters for the simulation.

Parameter | Value |
---|

**Physical Properties of Slag** |

Density, kg·m^{−3} | 2850 |

Specific capacity, J·kg^{−1}·K^{−1} | 1404 |

Thermal conductivity, W·m^{−1}·K^{−1} | 10.45 |

Viscosity, kg·m^{−1}·s^{−1} | 0.01 |

Emissivity | 0.6 |

Expansion coefficient, K^{−1} | 0.0001 |

**Physical Properties of Steel** |

Density, kg·m^{−3} | 7200 |

Specific capacity, J·kg^{−1}·K^{−1} | 502 |

Thermal conductivity, W·m^{−1}·K^{−1} | 31.9 |

Steel solidus temperature, K | 1723 |

Steel liquidus temperature, K | 1693 |

Latent heat of solidification, J·kg^{−1} | 247,000 |

**Process Parameters** |

Electrode immersion depth, m | 0.02 |

Mold diameter, m | 0.14 |

Electrode length, m | 0.31 |

Slag height, m | 0.07 |

Current, A | 4000 |

**Table 2.**
Average temperature in the slag pool area and the radial temperature gradient at the slag/electrode interface.

Process | Average Temperature in Slag Pool (K) | Radial Temperature Gradient at Slag/Electrode Interface (K) |
---|

OE-ESR | 1996.7 | 338.17 |

TSCE-ESR | 1838.4 | 201.78 |

**Table 3.**
The calculated and theoretically predicted diameter of droplets.

OE-ESR | TSCE-ESR |
---|

No. | Calculated Diameter | Average | Theoretical Diameter | No. | Calculated Diameter | Average | Theoretical Diameter |

1 | 14.92 | 15.06 | 15.1 | 1 | 12.99 | 13.11 | 13.47 |

2 | 15.17 | 2 | 13.27 |

3 | 15.02 | 3 | 13.16 |

4 | 14.88 | 4 | 12.94 |

5 | 15.31 | 5 | 13.19 |

**Table 4.**
Average velocity in steady and transient states in slag pool.

Items | OE-ESR | TSCE-ESR |
---|

Average velocity of slag pool at steady state (m/s) | 0.052 | 0.051 |

Maximum average velocity of slag pool at transient state (m/s) | 0.062 | 0.093 |

The rate of growth (%) | 19.2 | 82.3 |

**Table 5.**
Droplet average diameter and remelting rate.

Items | Average Droplet Diameter (mm) | Remelting Rate (kg·s^{−1}) |
---|

OE-ESR | 15.06 | 0.0151 |

TSCE-ESR | 13.11 | 0.0321 |

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