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Freckle criteria for the upward directional solidification of alloys

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Abstract

Macrosegregation defect freckles result from the convective flow due to a density gradient in the mushy zone. Several freckle criteria were proposed in the literature for the evaluation of the freckle tendency of alloys and processes. These criteria were examined using experimental data of the upward directional solidification of Pb-Sn, Pb-Sb, and Pb-Sn-Sb systems. It is found that the freckle criterion given by Flemings and co-workers gives the best description of the freckle tendency. A Rayleigh number is derived from Flemings’ criterion. The critical value of the Rayleigh number is found to be Ra cr ≈ 1.

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Abbreviations

T :

temperature, °C

T 0 :

alloy melting temperature, °C

C :

composition of a solute element, wt pct

C 0 :

averaged composition of a solute element in an alloy, wt pct

ρ :

liquid density, kg/cm3

ρ 0 :

liquid density at the melting point of an alloy, kg/cm3

Π:

permeability in mushy zone, m2

f L :

liquid fraction in mushy zone

▽T, G :

thermal gradient in mushy zone

R :

growth rate of crystal

Ra:

Rayleigh number

Ra T :

thermal Rayleigh number

Ra s :

solutal Rayleigh number

g :

gravity, 9.81 m/s2

v :

viscosity of liquid

ε :

cooling rate, °C/s; ε=G · R

V :

flow velocity of interdendritic liquid

d1:

primary dendrite arm spacing

D :

solutal diffusivity

κ :

thermal diffusivity

α :

thermal expansion coefficient of density, \(\left. {\alpha = \frac{1}{{\rho _0 }}\left( {\frac{{\partial \rho }}{{\partial T}}} \right)} \right|_{C_0 ,T_0 } \)

β :

solutal expansion coefficient of density, \(\left. {\beta = \frac{1}{{\rho _0 }}\left( {\frac{{\partial \rho }}{{\partial C}}} \right)} \right|_{C_0 ,T_0 } \)

m :

slope of the liquidus line for binary alloys, m=(dT/dC)

β*:

composite thermal and solutal expansion coefficient of density, β*=β − α/m

ℓ:

characteristic length in Rayleigh number

z :

depth into the mushy zone, measured from the mush-bulk liquid interface

Δρ :

density change, Δρ=(ρ-ρ0/ρ0)=β*(C-C0)

dρ/dz :

density gradient in mushy zone, dp/dz=-α(dT/dz)+β(dC/dz)

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Authors and Affiliations

  1. the Department of Mechanical and Aerospace Engineering, West Virginia University, 26506, Morgantown, WV

    Wanhong Yang (Research Assistant Professor) & Keh-Minn Chang (Professor)

  2. Department of Mechanical and Aerospace Engineering, West Virginia University, USA

    Wei Chen (Graduate Research Assistant, Materials Engineer)

  3. GE Power Systems, 12308, Schenectady, NY

    Wei Chen (Graduate Research Assistant, Materials Engineer)

  4. Advanced, Research & Development, USA

    Sarwan Mannan (Metallurgist)

  5. Special Projects, the Special Metals Corporation, 25705, Huntington, WV

    John DeBarbadillo (Manager)

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Yang, W., Chang, KM., Chen, W. et al. Freckle criteria for the upward directional solidification of alloys. Metall Mater Trans A 32, 397–406 (2001). https://doi.org/10.1007/s11661-001-0271-y

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