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Equiaxed dendritic solidification with convection: Part I. Multiscale/multiphase modeling

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Abstract

Equiaxed dendritic solidification in the presence of melt convection and solid-phase transport is investigated in a series of three articles. In part I, a multiphase model is developed to predict com-position and structure evolution in an alloy solidifying with an equiaxed morphology. The model accounts for the transport phenomena occurring on the macroscopic (system) scale, as well as the grain nucleation and growth mechanisms taking place over various microscopic length scales. The present model generalizes a previous multiscale/multiphase model by including liquid melt convec-tion and solid-phase transport. The macroscopic transport equations for the solid and the interdendritic and extradendritic liquid phases are derived using the volume averaging technique and closed by supplementary relations to describe the interfacial transfer terms. In part II, a numerical application of the model to equiaxed dendritic solidification of an Al-Cu alloy in a rectangular cavity is dem-onstrated. Limited experimental validation of the model using a NH4C1-H2O transparent model alloy is provided in part III.

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Abbreviations

A :

interfacial surface area, m2

A s :

area of the solid/interdendritic liquid interface, m2

A e :

area of the dendrite envelope, m2

b:

body force vector, N/m3

C :

concentration of a chemical species, wt pct

c :

specific heat, J/kg K

C ε :

settling ratio

C p :

shape factor function

d s :

mean characteristic length or diameter of the solid phase, m

d e :

mean characteristic diameter of the dendrite envelope, m

D :

mass diffusion coefficient, m2/s

h :

enthalpy, J/kg

Iv:

Ivantsov function

j :

species diffusion flux, kg/m2 s

J :

interfacial species transfer rate per unit volume

k :

thermal conductivity, W/m K

l :

species diffusion length, m

m l :

liquidus line slope, K/wt pct

Ms d :

solid/liquid interfacial drag, N/m s

n :

equiaxed nuclei density (m-3), or an index in Eqs. [38] and [41]

n:

outwardly directed unit normal vector

PЕε :

multiphase Pelcet number, εlvl - vsde/Dl

PЕt :

solutal Peclet number at the dendrite tip, VtRt/2Dl

PЕ∞ :

ambient Pelcet number for dendrite tips, vl

q :

heat flux, W/m2

Q :

interfacial heat-transfer rate, W/m3

R t :

tip radius

S :

interfacial area concentration, A/V0 (m-1)

Sc:

Schmidt number,vID

t :

time, s

T :

temperature, K

v:

velocity vector, m/s

V k :

volume of phase k, m3

V 0 :

averaging volume, m3

V t :

dendrite tip velocity, m/s

w :

interface velocity, m/s

β :

dimensionless parameter, Eq. [38]

γ :

momentum dispersion coefficient, Eq. [18]

г:

interfacial phase change rate (kg/m3 s) or Gibbs-Thomson coefficient (m K)

г*:

macroscopic transport property

δh:

latent heat of phase change, J/kg

ε:

volume fraction

εsi :

internal solid fraction, εs/(εs + εd )

k :

partition coefficient, wt pct/wt pct

kε v :

flow partition coefficient

λ2 :

secondary dendrite arm spacing, m

Ø :

sphericity

ρ :

density, kg/m3

Μ :

viscosity, Pa s

Σ* :

stability constant

Τ:

shear stress

Ф:

a general transfer

ψ:

a field property

Ώ:

solutal supersaturation, Eq. [30]

d :

interdendritic liquid

e :

dendrite envelope

E :

eutectic

f :

total liquid phase (d +l)

g :

grain

j :

phasej

k :

phasek

kj:

pertinent to phasek on thek- j interface

l :

extradendritic liquid

Id :

extradendritic-interdendritic liquid interface

m :

melting point of pure metals

n :

normal direction

N :

nucleation

0:

initial state

s :

solid

sd :

solid-interdendritic liquid interface

t :

dendrite tip or tangential

c :

critical

d :

due to diffusion

j :

due to species gradients

t :

macroscopic dispersion

F:

due to interface movement

-:

interfacial area-averaged

*:

effective

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Hawaii at Manoa, 96822, Honolulu, HI

    C. Y. Wang (Assistant Professor)

  2. Department of Mechanical Engineering, The University of Iowa, 52242, Iowa City, IA

    C. Beckermann (Associate Professor)

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Wang, C.Y., Beckermann, C. Equiaxed dendritic solidification with convection: Part I. Multiscale/multiphase modeling. Metall Mater Trans A 27, 2754–2764 (1996). https://doi.org/10.1007/BF02652369

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