An equation giving the composition of the solid forming during solidification as Cs = kC₀(1 − fs)^(k−1), assuming no diffusion in the solid and complete mixing in the liquid. It is used to estimate segregation in non-equilibrium solidification.
Japanese name: シャイルの式
Scheil equation appears in 2 practice questions across 1 topic, including Solidification and cast structures.
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.
Answer: C. About 7.8%
Writing the remaining liquid fraction as fL = 1 − fs, 33.2 = 4.0 × fL^(−0.83), so fL = (33.2/4.0)^(−1/0.83) = 8.3^(−1.20) ≈ 0.078, about 7.8%. In equilibrium solidification, 4.0 wt%Cu is below the α solubility limit of 5.65 wt%, so the alloy is all α at the end of solidification and no eutectic forms. 0% is the mistake of applying this equilibrium reasoning to real solidification, where diffusion in the solid is slow. About 12% is 4.0/33.2, which assumes that all the copper in the alloy collects in liquid of eutectic composition and ignores the copper left in the solid. Real castings likewise show eutectic containing Al₂Cu between the dendrites even at compositions within the solubility limit, so the solution treatment temperature is chosen below the melting point of this eutectic.
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.
Which is a correct assumption behind the Scheil equation Cs…Equilibrium partition coefficient Lever rule Chvorinov's rule Constitutional supercooling Dendrite Dendrite arm spacing Eutectic modification Macrosegregation