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Numerical Investigation of Shell Formation in Thin Slab Casting of Funnel-Type Mold

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Abstract

The key issue for modeling thin slab casting (TSC) process is to consider the evolution of the solid shell including fully solidified strand and partially solidified dendritic mushy zone, which strongly interacts with the turbulent flow and in the meantime is subject to continuous deformation due to the funnel-type mold. Here an enthalpy-based mixture solidification model that considers turbulent flow [Prescott and Incropera, ASME HTD, 1994, vol. 280, pp. 59–69] is employed and further enhanced by including the motion of the solidifying and deforming solid shell. The motion of the solid phase is calculated with an incompressible rigid viscoplastic model on the basis of an assumed moving boundary velocity condition. In the first part, a 2D benchmark is simulated to mimic the solidification and motion of the solid shell. The importance of numerical treatment of the advection of latent heat in the deforming solid shell (mushy zone) is specially addressed, and some interesting phenomena of interaction between the turbulent flow and the growing mushy zone are presented. In the second part, an example of 3D TSC is presented to demonstrate the model suitability. Finally, techniques for the improvement of calculation accuracy and computation efficiency as well as experimental evaluations are also discussed.

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Abbreviations

c p :

Specific heat of liquid–solid mixture (J kg−1 K−1)

C 1ɛ ,C 2ɛ ,C μ :

Constants of the standard k–ɛ model (1)

E :

Young’s modulus (N m−2)

f ℓ,f s :

Volume fraction of liquid and solid phases (1)

\( f_{\text{s}}^{\text{integral}} \) :

Total solid phase in the calculation domain (1)

G :

Shear production of turbulence kinetic energy (kg m−1 s−3)

h :

Sensible enthalpy of liquid–solid mixture (J kg−1)

h ref :

Reference enthalpy at temperature T ref (J kg−1)

\( h_{\ell }^{\text{sensible}} \) :

Sensible enthalpy of liquid phase (J kg−1)

\( h_{\text{s}}^{\text{sensible}} \) :

Sensible enthalpy of solid phase (J kg−1)

\( h_{\ell }^{\text{total}} \) :

Total enthalpy of liquid phase (J kg−1)

\( h_{\text{s}}^{\text{total}} \) :

Total enthalpy of liquid phase (J kg−1)

H :

Total enthalpy of liquid–solid mixture (J kg−1)

H ℓ,H s :

Total enthalpy of liquid or solid phase (J kg−1)

HTC:

Heat transfer coefficient between mold and casting (W m−2 K−1)

k :

Turbulence kinetic energy per unit of mass (m2 s−2)

k p :

Partition coefficient of binary alloy (1)

K :

Permeability (m2)

L :

Latent heat (J kg−1)

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {n}_{\text{f}} \) :

Unit vector normal to the curved mold surface (1)

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {n}_{\text{z}} \) :

Unit vector normal in casting direction (1)

p :

Pressure (N m−2)

PDAS:

Primary dendrite arm space (m)

Prt,h :

Prandtl no. for energy equation (1)

Prt,k :

Prandtl no. for turbulence kinetic energy k (1)

Prt,ɛ :

Prandtl no. for turbulence dissipation rate ɛ (1)

S e :

Source term for energy equation (J m−3 s−1)

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {S}_{\text{mom}} \) :

Source term for momentum equation (kg m−2 s−2)

t :

Time (s)

T :

Temperature (K)

T ext :

External mold surface temperature

T f :

Melt point of pure solvent (K)

T inlet :

Inlet temperature (K)

T liquidus :

Liquidus temperature of alloy (K)

T ref :

Reference temperature for h ref (K)

T solidus :

Solidus temperature of alloy (K)

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {u} \)(u x,u y,u z):

Velocity of the liquid–solid mixture (m s−1)

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {u}_{\text{inlet}} \) :

Inlet velocity (m s−1)

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {u}_{\text{pull}} \) :

Casting velocity (m s−1)

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {u}_{\ell } \)(u xℓ ,u yℓ ,u zℓ ):

Liquid velocity (m s−1)

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {u}_{\text{s}} \)(u xs ,u ys ,u zs ):

Solid velocity (m s−1)

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {u}_{\text{s}}^{\text{surface}} \) :

Moving surface velocity (m s−1)

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\delta } \) :

Displacement vector (m)

Δx :

Mesh size (m)

ɛ :

Turbulence dissipation rate per unit of mass (m2 s−3)

λ :

1st Lamé parameter (N m−2)

λ mix :

Thermal conductivity of liquid–solid mixture (W m−1 K−1)

λ eff :

Effective thermal conductivity due to turbulence (W m−1 K−1)

λ t :

Turbulence thermal conductivity (W m−1 K−1)

ρ(=ρ ℓ = ρ s):

Density (kg m−3)

μ :

2nd Lamé parameter (N m−2)

μ eff :

Dynamic effective viscosity due to turbulence (kg m−1 s−1)

μ ℓ :

Dynamic liquid viscosity (kg m−1 s−1)

μ t :

Dynamic turbulence viscosity (kg m−1 s−1)

ν :

Poisson’s ratio (1)

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Acknowledgments

The financial support by RHI AG, the Austrian Federal Ministry of Economy, Family and Youth and the National Foundation for Research, Technology and Development is gratefully acknowledged. The authors acknowledge the fruitful discussions with Professor Brian G. Thomas, University of Illinois at Urbana-Champaign, USA, and Dr. Christian Chimani, Austrian Institute of Technology, Austria.

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Authors and Affiliations

  1. Christian-Doppler Lab for Adv. Process Simulation of Solidification & Melting, Department of Metallurgy, University of Leoben, Franz-Josef-Str. 18, 8700, Leoben, Austria

    A. Vakhrushev ( Senior Researcher) & M. Wu (Associate Professor)

  2. Department of Metallurgy, University of Leoben, Franz-Josef-Str. 18, 8700, Leoben, Austria

    M. Wu (Associate Professor) & A. Ludwig (University Professor)

  3. RHI AG, Technology Center, Magnesitstrasse 2, 8700, Leoben, Austria

    Y. Tang (Project Manager) & G. Hackl (Project Manager)

  4. RHI AG, Wienerbergstrasse 9, 1100, Vienna, Austria

    G. Nitzl (Product Manager)

Authors
  1. A. Vakhrushev
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  2. M. Wu
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  3. A. Ludwig
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  4. Y. Tang
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Corresponding author

Correspondence to M. Wu.

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Manuscript submitted July 3, 2013.

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Vakhrushev, A., Wu, M., Ludwig, A. et al. Numerical Investigation of Shell Formation in Thin Slab Casting of Funnel-Type Mold. Metall Mater Trans B 45, 1024–1037 (2014). https://doi.org/10.1007/s11663-014-0030-2

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