 Finite-inventory directional salt rejection

For pure solid rejecting all solute, pulling speed $V>0$, solutal diffusivity $D$ and finite solute inventory per unit area $N$, a stationary planar front has $C(z)=C_I{e}^{-Vz/D}$ in its liquid half-space. The advection-diffusion equation gives this profile, and ${\int }_0^{\infty }Cdz=N$ fixes $C_I=VN/D$. A liquidus $T_I=-m{C}_I$ fixes its position in an imposed temperature gradient. The pure-solid branch requires $C_I<C_E$, where $C_E$ is the eutectic composition.