Solidification Scheil equation

The treatment above is the traditional derivation of the Scheil equation. However, it is not possible to derive this equation, using the same mathematical method, if the partition coefficient, k, is dependent on temperature and/or composition. The Scheil equation is applicable only to dendritic solidification and cannot, therefore, be applied to eutectic-type alloys such Al-Si-based casting alloys, or even for alloys which may be mainly dendritic in nature but contain some final eutectic product. Further, it cannot be used to predict the formation of intermetallics during solidification. 

The procedure described above is simple to model in a computer programme and has a number of significant advantages (1) The Scheil equation is only applicable to binary alloys and is not easily derived with multiple k values, which would be necessary for a multi-component alloy. A calculation as described above can be applied to an alloy with any number of elements. (2) The partition coefficients need not be constant, which is a prerequisite of the Scheil equation . (3) The Scheil equation carmot take into account other phases which may form during such a solidification process. This is handled straightforwardly by the above calculation route. 

II.3.3.2 Modifying the Scheil solidification model The Scheil equation can be modified to allow some back diffusion into the solid during a small isothermal increment of solidification. The composition of the liquid is then modified to a new... 

During a plane front solidification process, partitioning of species between the solid and liquid phases can take place, which results in a nonuniform compositional profile at the end of the solidification process. The severity of the compositional nonuniformity depends on the degree of solute partitioning between the solid and the liquid as well as the rate of solidification relative to the rate of solid-state diffusion. If solid-state diffusion is fast, then the composition gradient in the solid can be quickly erased, leading to a more uniform composition profile. However, in most cases, solid-state diffusion is slow compared to solidification, and thus these nonuniform composition gradients tend to be frozen in. The Scheil equation provides a way to model the nonuniform concentration profile that arises in such a plane front solidification process ... 

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