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Numerical Approaches for microscopic modelling of solute redistribution during solidification of binary alloys

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Abstract

In the present study, two numerical approaches for single-domain modelling of microsegregation during solidification of binary alloys are presented. In the first approach, the concentration jump at the moving solid/liquid interface is formulated using a volumetric term and a Boolean function. The governing solute redistribution equation, valid for the whole domain comprising the solid and liquid regions, is derived in terms of the liquid phase composition. The effects of microstructure coarsening on microsegregation has been described and included in the model. In the second approach, the continuum mixture theory is utilized to derive a single domain solute redistribution equation in terms of the mixture composition. The solidification front motion and dendrite arm coarsening effects are accommodated by considering the representative elementary volume to consist of solid, interdendritic, and extradendritic liquid phases. Numerical solutions have been obtained using a control-volume based finite-difference method with a fixed grid. Good agreement has been observed between the predictions of the present fixed-domain models and the exact analytical and experimental results.

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Abbreviations

A :

Constant (m3/s)

C m :

Total solute quantity in the domain

C o :

Initial alloy concentration

D α :

Mass diffusion coefficient (m2/s)

f :

Mass fraction

f α :

Solute concentration

f k :

Mass fraction of phase k

Fo :

Fourier number (D αs t f/X 2f )

g :

Volume fraction

i f :

Phase change node

J k :

Surface diffusion flux (kg/ms)

k p :

Partition coefficient

M :

Number of space steps in x direction

N :

Number of space steps in y direction

\(\vec n\) :

Outward normal to the s/l interface

S k :

Volumetric source term

t :

Time (s)

T :

Temperature (K)

t f :

Final solidification time (s)

u, v :

Velocities in x and y directions (m/s)

V d :

Coarsening velocity (m/s)

V f :

Solidification front velocity (m/s)

X F :

1-D domain size (m)

X s :

Position of the interface (m)

τ:

Dimensionless time = t/t f

Δt :

t time step (s)

Δx, Δy :

Control volume size (m)

\(\bar{\rho}_{{{\rm k}}}\) :

Partial density of phase k (kg/m3)

λ2 :

Secondary dendrite arm spacing (m)

λ1 :

Primary dendrite arm spacing (m)

-:

Volume averaged value

α:

Solute

o :

Previous time step

d:

Dendritic liquid

eut:

Eutectic

f:

Final

i:

Interface

k:

Phase

l:

Liquid (extradendritic liquid)

liq:

Liquidus

m:

Mixture

p:

Central node point

s:

Solid

s, n, w, e:

South, north, west, east faces of control volume

S, N,W, E:

South, north, west, east nodes

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Authors and Affiliations

  1. Department of Mech. Eng. Technology, Benha High Institute of Technology, Benha, 13 512, Egypt

    M. A. Rady

  2. TREFLE-UMR CNRS 8508 Site ENSCPB, Université Bordeaux I, 16 Avenue Pey Berland, 33607, Pessac Cedex, France

    M. A. Rady & E. Arquis

Authors
  1. M. A. Rady
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  2. E. Arquis
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Correspondence to M. A. Rady.

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Rady, M.A., Arquis, E. Numerical Approaches for microscopic modelling of solute redistribution during solidification of binary alloys. Heat Mass Transfer 42, 347–358 (2006). https://doi.org/10.1007/s00231-005-0022-5

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