Conservation of salt during freezing ($S_1=S_0/(1-f_i)$)

If a fraction $f_i$ of initial liquid mass freezes into salt-free ice, all original salt remains in a liquid mass fraction $1-f_i$. Writing salinity as salt concentration per mass gives $S_1(1-f_i)=S_0$, hence $\Delta S=S_0{f}_i/(1-f_i)$. For equal initial cooling and subsequent freezing heat extraction, $f_i=c_{p,w}\Delta T/L_f$. The liquid specific heat capacity, rather than the ice value, determines the initial heat removal. If ice retains salt, the mass balance instead reads $S_1(1-f_i)+S_i{f}_i=S_0$.