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Ursula Kattner
Bill Boettinger
Dilip Banerjee
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Back Diffusion SolidificationUnlike the lever rule or Scheil
approach, solidification calculation considering solid or back
diffusion can not be carried out without assumptions regarding the
microgeometry of the liquid plus solid region and the time evolution of
the fraction solid. Details of the back diffusion calculation are given
by Boettinger et al. (1998).
For a well mixed liquid, a system of simultaneous equations is solved for a given time increment, dt, and enthalpy change, dH. The solution gives the change in temperature dT consistent with the solidification model, as well as the changes in liquid concentrations, dCLi, solid phase fractions, dfS, and average solid concentrations, d<CSi>,
for each of the solutes and solid phases since the previous time
step. Instead of solving diffusion equations for the solid,
an approximation following the approach of Wang and Beckermann is
used. This approach takes into account the secondary
dendrite arm spacing, lambda2, and the interdiffusion coefficients, DSi
(off diagonal diffusion coefficients are neglected). A system
of simultaneous equations is constructed to guarantee that heat
balance and mass balance are maintained and that concentrations of the
liquid phase remain on the primary phase liquidus using the liquidus
slopes, mSi.
Back diffusion is only considered while the primary phase is being
formed. The computational procedure is simplified by assuming a fixed cooling rate, i.e., the heat from the phase change is not considered.
Back diffusion in the primary phase is assumed during
secondary solidification while lever rule behavior is assumed for the
secondary phase. The basis for the latter assumption is that the
particle size of the secondary phase is fairly small and, therefore,
diffusion will be sufficient to approach thermodynamic equilibrium for
the secondary phase.
Examples for the lever rule and Scheil paths of Ni-Al-Ta and Sn-Bi-Pb alloys
are shown.
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