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A General Coupled Mathematical Model of Electromagnetic Phenomena, Two-Phase Flow, and Heat Transfer in Electroslag Remelting Process Including Conducting in the Mold

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Abstract

A transient three-dimensional finite-volume mathematical model has been developed to investigate the coupled physical fields in the electroslag remelting (ESR) process. Through equations solved by the electrical potential method, the electric current, electromagnetic force (EMF), and Joule heating fields are demonstrated. The mold is assumed to be conductive rather than insulated. The volume of fluid approach is implemented for the two-phase flow. Moreover, the EMF and Joule heating, which are the source terms of the momentum and energy sources, are recalculated at each iteration as a function of the phase distribution. The solidification is modeled by an enthalpy-porosity formulation, in which the mushy zone is treated as a porous medium with porosity equal to the liquid fraction. An innovative marking method of the metal pool profile is proposed in the experiment. The effect of the applied current on the ESR process is understood by the model. Good agreement is obtained between the experiment and calculation. The electric current flows to the mold lateral wall especially in the slag layer. A large amount of Joule heating around the metal droplet varies as it falls. The hottest region appears under the outer radius of the electrode tip, close to the slag/metal interface instead of the electrode tip. The metal pool becomes deeper with more power. The maximal temperature increases from 1951 K to 2015 K (1678 °C to 1742 °C), and the maximum metal pool depth increases from 34.0 to 59.5 mm with the applied current ranging from 1000 to 2000 A.

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Abbreviations

\( \vec{A} \) :

Magnetic potential vector [(V s)/m]

A mush :

Mushy zone constant

\( \vec{B} \) :

Magnetic flux density (T)

c p :

Heat capacity [J/(kg K)]

\( C_{1\varepsilon } \) :

1.42

\( C_{2\varepsilon } \) :

1.68

\( C_{3\varepsilon } \) :

0.09

\( \vec{D} \) :

Electric flux density (C/m2)

\( \vec{E} \) :

Electric field intensity (N/C)

f H :

Electric current frequency (Hz)

f ℓ :

Liquid fraction

\( \vec{F}_{\text{b}} \) :

Buoyancy force (N/m3)

\( \vec{F}_{\text{e}} \) :

Electromagnetic force (N/m3)

\( \vec{F}_{\text{ed}} \) :

Effective electromagnetic force (N/m3)

\( \vec{F}_{\text{p}} \) :

Pressure drop (m/s)

\( \vec{g} \) :

Gravitational acceleration (m2/s)

G k :

Generation of turbulence kinetic energy due to the mean velocity gradients (Pa s)

G b :

Generation of turbulence kinetic energy due to buoyancy (Pa s)

h :

Sensible enthalpy (J/kg)

\( \vec{H} \) :

Magnetic field intensity (A/m)

H :

Enthalpy (J/kg)

I rms :

Root mean square current (A)

\( \vec{J} \) :

Current density (A/m2)

k :

Turbulent kinetic energy (m2/s2)

k eff :

Effective thermal conductivity [W/(m K)]

L :

Latent heat (J/kg)

p :

Pressure (Pa)

Q Joule :

Joule heating (W)

R :

Radius (m)

R max :

Radius of the mold (m)

R ɛ :

Additional term of dissipation rate of turbulent kinetic energy (m2/s3)

S k :

Source term of solidification in the turbulent kinetic energy equation (m2/s3)

S ɛ :

Source term of solidification in the dissipation rate of turbulent kinetic energy equation (m2/s3)

t :

Time (s)

T :

Temperature (K)

T ref :

Reference temperature (K)

T s :

Solidus temperature (K)

T ℓ :

Liquidus temperature (K)

\( \vec{u} \) :

Velocity (m/s)

\( \vec{u}_{\text{cast}} \) :

Casting velocity (m/s)

V drop :

Voltage drop of the lateral wall (V)

Y M :

Contribution of the fluctuating dilatation to the overall dissipation rate (m2/s3)

α :

Volume fraction of fluid

α k :

Inverse effective Prandtl number for turbulent kinetic energy

α ɛ :

Inverse effective Prandtl number for dissipation rate of turbulent kinetic energy

β :

Thermal expansion coefficient (1/K)

δ :

Electromagnetic skin thickness (m)

ɛ :

Dissipation rate of turbulent kinetic energy (m2/s3)

η :

Correction coefficient

μ :

Magnetic permeability (F/m)

μ 0m :

Vacuum permittivity of metal (F/m)

μ eff :

Effective viscosity (Pa s)

ρ :

Density (kg/m3)

ρ em :

Electric resistance of metal (Ω m)

σ :

Electrical conductivity [1/(Ω m)]

σ m :

Electrical conductivity of the metal [1/(Ω m)]

σ s :

Electrical conductivity of the slag [1/(Ω m)]

ϕ :

Electric potential (V)

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Acknowledgments

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

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Authors and Affiliations

  1. School of Materials and Metallurgy, Northeastern University, Wenhua Road 3-11, Heping District, Shenyang, 110819, People’s Republic of China

    Qiang Wang (PhD Candidate) & Baokuan Li (Professor)

  2. Key Laboratory for Ferrous Metallurgy and Resources Utilization, Ministry of Education, Wuhan University of Science and Technology, Wuhan, 430081, People’s Republic of China

    Zhu He (Associate Professor)

  3. Department of Advanced Materials Science, Graduate School of Frontier Sciences, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba, 277-8561, Japan

    Fumitaka Tsukihashi (Professor)

Authors
  1. Qiang Wang
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Correspondence to Baokuan Li.

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Manuscript submitted November 25, 2013.

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Wang, Q., He, Z., Li, B. et al. A General Coupled Mathematical Model of Electromagnetic Phenomena, Two-Phase Flow, and Heat Transfer in Electroslag Remelting Process Including Conducting in the Mold. Metall Mater Trans B 45, 2425–2441 (2014). https://doi.org/10.1007/s11663-014-0158-0

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  • DOI: https://doi.org/10.1007/s11663-014-0158-0

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