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# Scheil equation

In metallurgy, the Scheil-Gulliver equation (or Scheil equation) describes solute redistribution during solidification of an alloy. This approach approximates non-equilibrium solidification by assuming a local equilibrium of the advancing solidification front at the solid-liquid interface. This allows the use of equilibrium phase diagrams in solidifcation analysis.

Unlike equilibrium solidification, solute does not diffuse back into the solid and is rejected completely into the liquid. Complete mixing of solute in the liquid is also assumed as a result of convection and/or stirring.

## Derivation

The following figure shows the solute redistribution for non-equilibrium solidification where there is no diffusion of solute in the solid and complete mixing of solute in the liquid. $\ D_S = 0$ and $\ D_L = \infty$.

Equilibrium is assumed at the interface which allows the use of an equilibrium phase diagram.

The hatched areas in the figure represent the amount of solute in the solid and liquid. Considering that the total amount of solute in the system must be conserved, the areas are set equal as follows: $(C_L-C_S) \ df_S = (f_L) \ dC_L$.

Since the partitioning coefficient is $k = \frac{C_S}{C_L}$ (determined from the phase diagram)

and mass must be conserved $\ f_S + f_L = 1$

the mass balance may be rewritten as $C_L(1-k) \ df_S = (1-f_S) \ dC_L$.

Using the boundary condition $\ C_L = C_o$ at $\ f_S = 0$

the following integration may be performed: $\displaystyle\int^{f_S}_0 \frac{df_S}{1-f_S} = \frac{1}{1-k} \displaystyle\int^{C_L}_{C_o} \frac{dC_L}{C_L}$.

Integrating results in the Scheil-Gulliver equation for composition of the liquid during solidification: $\ C_L = C_o(1-f_S)^{k - 1}$

or for the composition of the solid: $\ C_S = kC_o(1-f_S)^{k - 1}$.

## References

• Gulliver, G.H., J. Inst. Met., 9:120, 1913.
• Kou, S., Welding Metallurgy, 2nd Edition, Wiley-Interscience, 2003.
• Porter, D. A., and Easterling, K. E., Phase Transformations in Metals and Alloys (2nd Edition), Chapman & Hall, 1992.
• Scheil, E., Z. Metallk., 34:70, 1942.