Freezing-to-cooling density efficiency

Solution

Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 77 / 2 / Solution

Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 80 / 4 / Solution

{title2=$S_1=S_0/(1-f_i)$}

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 $S_1(1-f_i)+S_i f_i=S_0$.

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 Incoming links: Conservation of salt during freezing