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Application of an Equiaxed Grain Growth and Transport Model to Study Macrosegregation in a DC Casting Experiment

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Abstract

A simplified three-phase, multiscale macrosegregation model which describes the growth kinetics of equiaxed grains and the coupling between microstructure morphology and the macroscopic transport has been proposed previously. In this paper, the model is validated by comparing the numerical model predictions to the experimental data from DC casting of an AA7050 alloy billet. The morphology of the equiaxed grains has an important influence on the macrosegregation, and we show that the model predictions are accurate when the grain morphology is described correctly.

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Abbreviations

〈C i,〉:

Average mass concentration of solute i (wt pct)

〈C *, i 〉:

Average equilibrium mass concentration of solute i (wt pct)

C o, i :

Mean concentration of solute i (wt pct)

c p :

Specific heat (J kg−1 K)

C D :

Drag co-efficient (–)

d :

Diameter of inoculant particle (m)

D , i :

Diffusion coefficient of solute i (m2 s−1)

g :

Volume fraction (–)

g pack :

Packing fraction (–)

g intern :

Internal solid fraction (–)

\( \vec{\varvec{g}} \) :

Acceleration due to gravity (m s−2)

〈h l〉l :

Averaged liquid enthalpy (J kg−1)

〈h s〉s :

Averaged solid enthalpy (J kg−1)

h m :

Mixture enthalpy (J kg−1)

h primary :

Primary cooling heat-transfer coefficient (W m−2 K−1)

h secondary :

Secondary cooling heat-transfer coefficient (Wm−2 K−1)

k p, i :

Partition coefficient of solute i (–)

K :

Permeability (m2)

L f :

Latent heat of fusion (J kg−1)

l kc :

Characteristic length for permeability (m)

m l, i :

Liquidus slope of solute i, K (wt pct−1)

N inuc :

Volumetric number density (m−3)

N g :

Grain density (m−3)

p l :

Liquid pressure (N m−2)

P :

Perimeter of the ingot (m)

Q water :

Water flow rate (m3 s−1)

R env :

Radius of the envelope (m)

R s,eq :

Radius of the solid grain (m)

Re :

Reynolds number

S v :

Interfacial area density (m−1)

Sc :

Schmidts number

t :

Time (s)

T :

Temperature (K)

T water :

Temperature of cooling water (K)

T sat :

Temperature of boiling water (K)

T cast :

Casting temperature (K)

T liq :

Temperature of liquidus (K)

T m :

Melting temperature of pure Al (K)

T eut :

Eutectic temperature (K)

ΔT :

Undercooling (K)

ΔT c :

Critical undercooling for nucleation (K)

\(\langle \vec{v}_{\text{l}}\rangle^{\text{l}} \) :

Intrinsic average velocity of liquid phase (ms−1)

\( \langle\vec{v}_{\text{s}}\rangle^{\text{s}} \) :

Intrinsic average velocity of solid phase (ms−1)

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

Casting velocity (ms−1)

V tip :

Velocity of dendrite tip (ms−1)

β T :

Thermal expansion coefficient (K)

β C, i :

Solutal expansion coefficient of solute i, (pct w−1)

δ i :

Diffusion length of solute i (− m)

δ(t):

Dirac function

ΓGT :

GIBBS–Thomson co-efficient (Km)

Γ:

Growth rate (kg m−3 s−1)

κ :

Thermal conductivity (W m−1 K)

ρ l :

Liquid density (kg m−3)

ρ s :

Solid density used to account for shrinkage (kg m−3)

ρ l,b :

Liquid buoyancy density used to account for Bousinessq approximation (kg m−3)

ρ s,b :

Solid buoyancy density used to account for grain motion (kg m−3)

ρ m :

Mixture density (kg m−3)

μ l :

Liquid dynamic viscosity (Pa s)

l:

Liquid

s:

Solid

env:

Envelope

e:

Extragranular liquid

d:

Intragranular liquid

s–d:

Solid–liquid interface

e–d:

Intra-extra granular liquid interface

* :

equilibrium

l,b:

Liquid buoyancy

s,b :

Solid buoyancy

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Acknowledgments

This study was conducted within the framework of PRIMAL project, of which Hydro Aluminium ASA, Alcoa Norway ANS, Aleris Rolled Products Germany GmbH, Institute of Energy Technology (IFE), NTNU, and SINTEF are the partners. This project is supported by the Research Council of Norway. A.P and M.M acknowledge the support of NOTUR High Performance Computing program. H.C and M.Z. acknowledge the support by the French State through the program “Investment in the future” run by the National Research Agency (ANR) and referenced by ANR-11 LABX-0008-01 (LabEx DAMAS).

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Authors and Affiliations

  1. Department of Materials Technology, NTNU, 7491, Trondheim, Norway

    Akash Pakanati, Knut Omdal Tveito & Mohammed M’Hamdi

  2. SINTEF Materials and Chemistry, 0314, Oslo, Norway

    Mohammed M’Hamdi

  3. Université de Lorraine, CNRS, Institut Jean Lamour – IJL, 54000, Nancy, France

    Hervé Combeau & Miha Založnik

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  1. Akash Pakanati
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  2. Knut Omdal Tveito
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  3. Mohammed M’Hamdi
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  4. Hervé Combeau
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  5. Miha Založnik
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Correspondence to Mohammed M’Hamdi.

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Manuscript submitted July 9, 2018.

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Pakanati, A., Tveito, K.O., M’Hamdi, M. et al. Application of an Equiaxed Grain Growth and Transport Model to Study Macrosegregation in a DC Casting Experiment. Metall Mater Trans A 50, 1773–1786 (2019). https://doi.org/10.1007/s11661-019-05133-z

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