Scheil equation

In Fig. 22.1a one-dimensional solidification is depicted a liquid binary alloy initially of uniform composition cq is placed in a bar-shaped crucible of length L. The bar is progressively cooled from one end, so it solidifies from one end to the other 

Kinetics of Materials. By Robert W. Balluffi, Samuel M. Allen, and W. Craig Carter. 543 Copyright 2005 John Wiley Sons, Inc. 

As solidification continues and solute is continuously rejected into the remaining liquid, the concentration in the bulk liquid increases slowly and the quasi-steady-state solute distribution in the boundary layer evolves. This, in turn, produces 

Because the solute diffusivity in the solid is far smaller than in the liquid, any diffusion in the solid will be neglected. In most cases of interest, the transient period required to produce a quasi-steady-state solute distribution at the interface is relatively small.1 At a relatively short time after the establishment of the quasi-steady-state concentration spike, the flux relative to an origin at the interface moving at velocity v is 

Expressions for cLS and cSL can now be found by using the Stefan continuity condition at the interface 

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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... 

Equation (10.14) is called the Scheil equation. Figure 10.8 shows its predictions of how the composition of the ingot changes with fs for r o = 5% and k = 1/5. It should be noted that Equation (10.14) is valid as long as no diffusion occurs in the solid and there is perfect mixing in the liquid. It applies for even dendritic growth. 

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 ... 

Segregation is always included in crystal growth of solid solution systems from the melt The Bridgman and (in parts) the Czochalski method are termed normal freezing methods, i.e. the whole melt volume will be transferred into the solid state. In this well-arranged case the distribution function is described by the so-called Pfaim/Scheil equation. A complete mixing of the melt at each time is assumed to exist for the derivation of this equation... 

Table 9.2 Set of Parameters Used in the Extended Scheil Equation.

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