Freeze Desalination Priced Honestly: Full Accounting of Washing, Brine Entrainment and Refrigeration for Natural and Refrigerated Freezing of Seawater

Freeze desalination is usually promoted on one number: the latent heat of fusion (334 kJ/kg) is about one seventh of the latent heat of vaporisation (2260 kJ/kg). This paper computes the full budget end-to-end for seawater at 35 g/L brought to a 0.5 g/L product target, charging brine entrainment, wash water, and refrigeration or ambient-cold heat transfer. At a central brine entrainment of 0.10 kg per kg of harvested ice with mean brine at 52.5 g/kg, unwashed ice melts at 5.25 g/L; ideal displacement washing consumes 0.235 kg of product per kg of ice, leaving a net yield of 0.765 and 382 L of net product per m3 of feed (two-stage freezing: 354 L per m3, freezing 900 kg per m3 of feed). Case A, natural freezing in shallow trays: at -10 C air in light wind (heat flux 72.7 W/m2) net production is 13.25 L/m2/day, so 75.5 m2 of tray gives 1 m3/day and 7.55 m2 gives 100 L/day within an assumed 100-day subarctic coastal window below -10 C, at a purchased-energy allowance of 0.2 kWh/m3. Case B, a refrigerated cycle at COP 2.5: 1307 kg of ice must be frozen per m3 of net product, a thermal load of 131.8 kWh/m3, giving 53.2 kWh/m3 including auxiliaries - 17.7 times the 3.0 kWh/m3 reverse-osmosis comparison figure; at COP 4.0 it is still 33.4 kWh/m3, and a generous heat-pump freeze-melt variant at COP 8.0 still requires 17.0 kWh/m3. Verdict: the latent-heat pitch survives only where the cold is free (natural winter freezing at household to small-community scale, or waste cold), and fails for refrigerated freezing of seawater against reverse osmosis. All results are simulated, not measured; computations are reproducible from the deposited compute.py and results.json.