= Final salt budget of a sealed mush column
{title2=$C_f(h_0-h_f)=\int_0^{h_0}(1-\phi_0)C\,dz$}
Assume a column is sealed to salt at both ends, diffusion occurs only in liquid, and its imposed temperature increases strictly upward. A finite-width equilibrium <mushy layer> cannot be stationary: its solute flux is nonzero unless its liquid fraction vanishes. The final state is pure ice below a height $h_f$ and uniform-composition brine above it. Their interface obeys $C_f=-(T_E+Gh_f)/m$. The displayed <mass conservation> equation fixes $h_f$ from the initial salt content. For a <diffusive steadily pulled equilibrium mush>, let $H=-T_E/G$, $c=mC_0/G$, $d=D/V$ and $h_0=H-c-d$. The initial salt content is $N=C_0[h_0+d\log(1+h_0/c)]$. Thus the liquid-cap depth $\ell_f=h_0-h_f$ obeys $\ell_f(\ell_f+c+d)=c[h_0+d\log(1+h_0/c)]$, which has one positive root. This conclusion assumes no subsequent exchange with an underlying eutectic reservoir.
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