User:Baldoovino/sandbox

ince the partition coefficient (related to solute distribution) 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:


 * $$\ C_L = C_o(f_L)^{k - 1}$$

or for the composition of the solid:


 * $$\ C_S = kC_o(1-f_S)^{k - 1}$$.

\[ {\frac {\partial }{\partial \mathit{fs}}}\,\mathrm{T}(\mathit{fs} ) \]