No Convective Mixing, Steady-State Solidification

If convective mixing can be avoided, after the first-to-freeze composition solidifies the rejected component will build up as a diffusion layer ahead of the moving solidification front, increasing the amount of the rejected component incorporated into the solid. Eventually, the composition in the melt will build up to C2 at the solidification interface and the solid will have composition Co and a steady-state situation is reached in which solute is being incorporated into the solid at the same rate as it is entering the advancing diffusion layer. 

Final composition profile of a directionally solidified ingot with complete convective mixing. The composition of the solid varies continually with distance and the solute that was rejected initially is later incorporated into the solid so that the overall average composition is the starting composition Co- 

Consider a reference system in which the solidification front remains stationary and xf is the distance into the melt. The melt flows in from the right-hand side at the solidification rate R. Balancing the solute flowing into the diffusion layer against the rate the solute is diffusing to the right. 

Under steady-state conditions, the composition of the solid being formed Cs is the same as the bulk melt, Cq. The boundary conditions are Cl(0) = Co/fc Cl(oo) = Cq. The solution can be written as 

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