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Mathematical modeling of microsegregation in binary metallic alloys

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Abstract

A mathematical model was developed to calculate microsegregation in binary metallic alloys. This model utilized the mathematical techniques of the method of lines combined with invariant imbedding (MOL/II) to solve the problem of combined heat and mass transfer during and after solidification. Model predictions were compared to experimental measurements in the Al-Cu system and to other microsegregation models. The MOL/II model predicted nonequilibrium second-phase contents within ±3 pct at low and intermediate cooling rates, when dendrite-arm coarsening was included in the model. It also was able to reproduce concentration profiles reasonably well. The analytical models commonly used (equilibrium cooling, Scheil equation, Brody/ Flemings model, Clyne/Kurz model, Solari/Biloni model, and Basaran equation) in micro-segregation calculations were shown to be considerably less accurate than the numerical models (MOL/II and the Ogilvy/Kirkwood model).

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Abbreviations

A :

constant used in coarsening equations

C e :

solute concentration in liquid at eutectic temperature (mole pct)

Cg227p:

heat capacity (J/kg K) Co initial solute concentration (mole pct)

C :

concentration of solute (mole pct)

D :

chemical diffusivity of solute (m2/s)

ΔH f :

enthalpy of fusion (J/kg)

h :

location of interface (m)

k :

equilibrium partition coefficient (mole pct solute in solid/mole pct solute in liquid)

m :

slope of liquidus curve (K/mole pct)

n :

exponent from Eq. [1]

q :

mass flux, defined by Eq. [13] (mole pct × m/s)

R :

Riccati parameter, defined by Eq. [18] (m2 K/W)

S :

Riccati parameter, defined by Eq. [19] (m/s)

T :

temperature (K) 70 melting point of pure solvent (K)

T ∗ :

temperature at interface (K)

t :

time (s)

t s :

solidification time (s) v heat flux, defined by Eq. [12] (W/m2)

w :

Riccati parameter, defined by Eq. [18] (K)

x :

position (m)

z :

Riccati parameter, defined by Eq. [19] (mole pct x m/s) α

ρCpΔt:

(W/m3 K)

β :

pre-exponential term in Eq. [1] (m)

Γ:

Gibbs-Thomson coefficient (K m)

ε:

cooling rate (K/s)

κ:

thermal conductivity (W/m K)

λ :

dendrite-arm spacing (m)

λ 0 :

initial secondary dendrite-arm spacing (m)

λ 1 :

primary dendrite-arm spacing (m)

λ 2 :

secondary dendrite-arm spacing (m)

ρ:

mass density (kg/m3)

Σ:

parameter used in determination of interface position (mole pct × m/s) Subscripts/Superscripts

S :

solid

L :

liquid

n :

current time-step

i :

current mesh point int at interface

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Authors and Affiliations

  1. Centre for Numerical Modelling and Process Analysis, Thames Polytechnic, Wellington Street, SEI 8 6PF, Woolwich, London, UK

    Thomas P. Battle (Postdoctoral Research Fellow)

  2. Department of Materials Science and Engineering, The University of Michigan, 48109-2136, Ann Arbor, MI

    Robert D. Pehlke (Professor)

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  1. Thomas P. Battle
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  2. Robert D. Pehlke
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Formerly Graduate Research Assistant, The University of Michigan

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Battle, T.P., Pehlke, R.D. Mathematical modeling of microsegregation in binary metallic alloys. Metall Trans B 21, 357–375 (1990). https://doi.org/10.1007/BF02664204

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