Mpemba Effect: Hot vs Cold Water

What the Mpemba effect claims. Under some conditions a beaker of initially hot water is reported to freeze before an identical beaker of cooler water. It is measured in some experiments and absent in others, and the proposed explanations (evaporation, convection, dissolved gases, supercooling, frost on the container) are still argued about. The model here keeps the mechanisms that can be written as honest equations and lets you switch them on and off.

Heat budget of one beaker. Water and glass are treated as one lump at temperature T: (m cp + Cglass) dT/dt = - h A (T - Tair) - ṁevap Lv, with the surface coefficient h = 1.42 (ΔT/L)1/4 fair + 4 ε σ Tm3 (laminar natural convection on a vertical wall plus linearised radiation, ε = 0.9, L = 9 cm). Hot water loses heat faster simply because ΔT is bigger and because h grows as ΔT^1/4.

Evaporation. Mass leaves at ṁ = hm Atop (ρv,sat(T) - RH · ρv,ice(Tair)) with the Lewis analogy hm = hconv / (ρair cp,air Le2/3) and Magnus fits for the vapour pressure. Each kilogram carries away Lv, so a hot beaker loses a few percent of its mass and an extra tens of watts early on (graphs 2 and 3). It stops once a skin of ice forms.

Freezing. Water does not freeze at exactly 0 °C. It can supercool and then freezes at a random temperature (the “freezes at” sliders, typically -2 to -12 °C). At that moment the heat above 0 °C is released as ice: the fraction φ0 = (m cp + Cg) |Tn| / (m Lf) turns solid at once and the beaker jumps to 0 °C; then m Lf dφ/dt = h A |Tair| until it is a solid block. The page reports both first ice and completely solid.

What it shows. With identical water and identical supercooling the hot beaker usually loses: it has more heat to dump. Evaporation shortens its deficit (compare the evaporation switch) but in this model rarely erases it. The hot beaker wins when it happens to freeze closer to 0 °C than the cold beaker does, which is exactly the scatter experimenters see; graph 4 shows where the two cross as you move the starting temperature. That fits the reading that the effect appears only under special conditions and is often an artefact of uncontrolled supercooling.

Honest simplifications. Lumped (uniform-temperature) water, although internal convection is only roughly that fast (Biot number about 0.5); no glass-wall resistance, shelf contact, frost on the beaker, dissolved gases or salts, ice growing from the walls inward, or the 9 % expansion on freezing. The supercooling temperatures are inputs, not predictions. Times are for a 150 g beaker and are order-of-magnitude, not a prediction for any particular freezer. Water colour is false colour.