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


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.



Average mass concentration of solute i (wt pct)


Average equilibrium mass concentration of solute i (wt pct)

Co, i

Mean concentration of solute i (wt pct)


Specific heat (J kg−1 K)


Drag co-efficient (–)


Diameter of inoculant particle (m)

D, i

Diffusion coefficient of solute i (m2 s−1)


Volume fraction (–)


Packing fraction (–)


Internal solid fraction (–)

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

Acceleration due to gravity (m s−2)


Averaged liquid enthalpy (J kg−1)


Averaged solid enthalpy (J kg−1)


Mixture enthalpy (J kg−1)


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


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

kp, i

Partition coefficient of solute i (–)


Permeability (m2)


Latent heat of fusion (J kg−1)


Characteristic length for permeability (m)

ml, i

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


Volumetric number density (m−3)


Grain density (m−3)


Liquid pressure (N m−2)


Perimeter of the ingot (m)


Water flow rate (m3 s−1)


Radius of the envelope (m)


Radius of the solid grain (m)


Reynolds number


Interfacial area density (m−1)


Schmidts number


Time (s)


Temperature (K)


Temperature of cooling water (K)


Temperature of boiling water (K)


Casting temperature (K)


Temperature of liquidus (K)


Melting temperature of pure Al (K)


Eutectic temperature (K)


Undercooling (K)


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)


Velocity of dendrite tip (ms−1)


Thermal expansion coefficient (K)

βC, i

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


Diffusion length of solute i (− m)


Dirac function


GIBBS–Thomson co-efficient (Km)


Growth rate (kg m−3 s−1)


Thermal conductivity (W m−1 K)


Liquid density (kg m−3)


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


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


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


Mixture density (kg m−3)


Liquid dynamic viscosity (Pa s)

Subscripts and Superscripts








Extragranular liquid


Intragranular liquid


Solid–liquid interface


Intra-extra granular liquid interface




Liquid buoyancy


Solid buoyancy



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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