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Solidification and cast structures

Which is a correct assumption behind the Scheil equation Cs = kC₀(1 − fs)^(k−1) (fs is the fraction solid), which gives the composition of the solid forming during solidification?

  1. ADiffusion is fast enough in both the solid and the liquid that both have uniform compositions at each temperature
  2. BDiffusion in the solid is negligible, the liquid is completely mixed and uniform, and equilibrium partitioning holds at the interface
  3. CSolute in the liquid is carried only by diffusion with no mixing, so a solute-rich layer forms ahead of the interface
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Answer: B. Diffusion in the solid is negligible, the liquid is completely mixed and uniform, and equilibrium partitioning holds at the interface

The Scheil equation is derived by assuming that solute does not diffuse at all in the solid once it has formed, that the liquid is completely mixed and uniform through convection and the like, and that at the solid–liquid interface solute always partitions in the ratio of the equilibrium partition coefficient k. As a result, with k < 1 the remaining liquid keeps becoming richer as solidification proceeds, and the last liquid eventually reaches the eutectic composition. Uniform solid and liquid is the assumption of equilibrium solidification, in which the lever rule holds. When the liquid is not mixed and solute moves only by diffusion, a concentration layer builds up ahead of the interface until a steady state is reached in which solid of composition C₀ forms, described by a different expression. Real solidification lies between equilibrium and the Scheil equation, to the extent that there is some diffusion in the solid.

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More Solidification and cast structures questions

  1. An Al-4.0 wt%Cu alloy is solidified. Assume the Scheil equation holds (equilibrium partition coefficient k = 0.17, constant), so the concentration of the remaining liquid rises as CL = C₀(1 − fs)^(k−1). Which is closest to the fraction of liquid remaining when the liquid reaches the eutectic composition of 33.2 wt%Cu (this becomes non-equilibrium eutectic)? On the equilibrium phase diagram, this composition lies within the maximum solubility of the α phase (5.65 wt%).
  2. Which is the correct pairing of additions to a hypoeutectic Al-Si casting alloy and their main effects?
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  4. Take the densities of pure aluminum at its melting point (660 °C) as 2.375 g/cm³ for the liquid and 2.55 g/cm³ for the solid. Which is closest to the decrease in volume relative to the liquid volume (solidification shrinkage) when the liquid turns entirely into solid at the melting point?
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