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Simulation of Crystal Sedimentation and Viscoplastic Behavior of Sedimented Equiaxed Mushy Zones

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Abstract

During solidification of castings, equiaxed crystals that is formed sink downwards, sediment and form a packed bed. The behavior of separated moving crystals can be described by a submerged object approach, whereas the viscoplastic behavior of a semi-solid slurry follows a volume-averaged viscoplastic constitutive equation. In this work, a two-phase Eulerian–Eulerian volume-averaging approach is used to combine both flow regimes. The transition happens at a certain solid volume fraction, the so-called coherency limit. Starting with a uniform distribution of crystals at rest, sedimentation and packing of crystals are described. In addition, the material density of the crystal is assumed to increase on cooling and thus the domain shrinks which is also accounted for in this report. It is demonstrated how sensitive the model is, on the considered crystal diameters and on the assumed value for the coherency limits.

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Abbreviations

α, β:

Rheological function taken from [9]

c p,ℓ , c p,s :

Specific heat (J/kg/K)

C ε :

Settling ratio (–)

d :

Average crystal diameter (m)

g ℓ , g s :

Volume fraction (–)

\( g_s^{\text{pack}} \) :

Packing limit for hard spheres (–)

\( g_s^{\text{cohe}} \) :

Coherency limit (–)

h ℓ , h s :

Volume averaged enthalpy (J/kg)

H*:

Volume heat transfer coefficient (W/m3/K)

H ℓs :

Enthalpy exchange (J/m3/s)

I :

Identity tensor (–)

k ℓ , k s :

Heat conductivity (J/m2/K)

K :

Permeability (m2)

K v :

Viscoplastic consistency (kg sm−2/m)

K ℓs :

Drag coefficient (kg/m3/s)

m :

Strain rate sensitivity coefficient (–)

M ℓs :

Mass transfer (kg/m3/s)

p ℓ , p s :

Pressure (N/m2)

T ℓ , T s :

Volume average temperature (K)

v ℓ , v s :

Flow velocity (m/s)

\({\dot{ \varvec{ \varepsilon }}_\ell },\;{\dot{ \varvec{ \varepsilon }}_s} \) :

Strain rate tensor (–)

\( \dot \varepsilon_{eq}^s \) :

Equivalent strain rate (–)

μ ℓ , μ s :

Viscosity (kg/m/s)

ρ ℓ , ρ s :

Density (kg/m3)

σ ℓ , σ s :

Stress tensor (N/m2)

τ ℓ , τ s :

Deviatoric stress tensor (N/m2)

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Acknowledgments

This work was financially supported by the FWF Austrian Science Fund (P22614-N22), FFG Bridge Early Stage (No. 3893791), and the Austrian Federal Ministry of Economy, Family and Youth and the National Foundation for Research, Technology and Development within the framework of the Christian Doppler Laboratory for Advanced Process Simulation of Solidification and Melting.

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Authors and Affiliations

  1. Chair for Modeling and Simulation of Metallurgical Processes, Montanuniversitaet Leoben, Leoben, Austria

    Andreas Ludwig, Menghuai Wu, Tobias Holzmann & Abdellah Kharicha

  2. Christian-Doppler Laboratory for Advanced Process Simulation of Solidification & Melting, Montanuniversitaet Leoben, Leoben, Austria

    Alexander Vakhrushev & Menghuai Wu

Authors
  1. Andreas Ludwig
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  2. Alexander Vakhrushev
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  4. Tobias Holzmann
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  5. Abdellah Kharicha
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Correspondence to Andreas Ludwig.

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Ludwig, A., Vakhrushev, A., Wu, M. et al. Simulation of Crystal Sedimentation and Viscoplastic Behavior of Sedimented Equiaxed Mushy Zones. Trans Indian Inst Met 68, 1087–1094 (2015). https://doi.org/10.1007/s12666-015-0651-4

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