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

  • Akash Pakanati
  • Knut Omdal Tveito
  • Mohammed M’HamdiEmail author
  • Hervé Combeau
  • Miha Založnik
Article

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.

Nomenclature

Ci,

Average mass concentration of solute i (wt pct)

Ci*,

Average equilibrium mass concentration of solute i (wt pct)

Co, i

Mean concentration of solute i (wt pct)

cp

Specific heat (J kg−1 K)

CD

Drag co-efficient (–)

d

Diameter of inoculant particle (m)

D, i

Diffusion coefficient of solute i (m2 s−1)

g

Volume fraction (–)

gpack

Packing fraction (–)

gintern

Internal solid fraction (–)

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

Acceleration due to gravity (m s−2)

hll

Averaged liquid enthalpy (J kg−1)

hss

Averaged solid enthalpy (J kg−1)

hm

Mixture enthalpy (J kg−1)

hprimary

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

hsecondary

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

kp, i

Partition coefficient of solute i (–)

K

Permeability (m2)

Lf

Latent heat of fusion (J kg−1)

lkc

Characteristic length for permeability (m)

ml, i

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

Nnuci

Volumetric number density (m−3)

Ng

Grain density (m−3)

pl

Liquid pressure (N m−2)

P

Perimeter of the ingot (m)

Qwater

Water flow rate (m3 s−1)

Renv

Radius of the envelope (m)

Rs,eq

Radius of the solid grain (m)

Re

Reynolds number

Sv

Interfacial area density (m−1)

Sc

Schmidts number

t

Time (s)

T

Temperature (K)

Twater

Temperature of cooling water (K)

Tsat

Temperature of boiling water (K)

Tcast

Casting temperature (K)

Tliq

Temperature of liquidus (K)

Tm

Melting temperature of pure Al (K)

Teut

Eutectic temperature (K)

ΔT

Undercooling (K)

ΔTc

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)

Vtip

Velocity of dendrite tip (ms−1)

βT

Thermal expansion coefficient (K)

βC, i

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

δi

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)

ρl

Liquid density (kg m−3)

ρs

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

ρl,b

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

ρs,b

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

ρm

Mixture density (kg m−3)

μl

Liquid dynamic viscosity (Pa s)

Subscripts and Superscripts

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

Notes

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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Copyright information

© The Minerals, Metals & Materials Society and ASM International 2019

Authors and Affiliations

  • Akash Pakanati
    • 1
  • Knut Omdal Tveito
    • 1
  • Mohammed M’Hamdi
    • 1
    • 2
    Email author
  • Hervé Combeau
    • 3
  • Miha Založnik
    • 3
  1. 1.Department of Materials TechnologyNTNUTrondheimNorway
  2. 2.SINTEF Materials and ChemistryOsloNorway
  3. 3.Université de Lorraine, CNRS, Institut Jean Lamour – IJLNancyFrance

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