Melting Ice in a Glass

What is modelled

Ice cubes float in a glass of water in a warm room. Heat flows from the water and, much more weakly, from the air into the ice. First the cold cube warms up to 0 C; then every joule that arrives melts ice at 334 kJ per kilogram. The meltwater enters at 0 C and chills the drink. The cube is assumed to stay a cube whose side follows its mass, floating with about 92 % of its volume under the surface.

Equations

• Ice: m ci dTi/dt = Q while Ti < 0, then L dm/dt = -Q, with Q = hw Aw (Tw - Ts) + ha Aa (Ta - Ts) and surface temperature Ts = min(Ti, 0).
• Water: M cw dTw/dt = -hw Aw (Tw - Ts) + UA (Ta - Tw) - cw Tw dM/dt: the last term mixes in meltwater at 0 C. Energy is conserved: with no losses, Tf = (M cw T0 - m L) / ((M + m) cw), or all the water at 0 C with some ice left.
• Heat transfer at the cube, natural convection along a vertical wall: hw = 0.52 (k / a) Ra1/4, Ra = g β ΔT a³ / (ν α); the stirring slider multiplies h. The cube side is a = (m / ρi)1/3, so area and heat flow shrink as the ice melts.
• Archimedes: ρw g Vsub = ρi g V, so the submerged fraction is ρi / ρw = 0.919. The liquid level follows only the total mass (M + m) / ρw inside the glass profile, hence it does not change as the ice melts.

Simplifications

• The cube is treated as one lumped body at one temperature while warming (its Biot number is not small, so the warm-up time is only approximate), and the correlation constant 0.52 and the glass loss UA = 0.25 W/K are order-of-magnitude values, not measurements. Real cubes melt unevenly, round off, tilt and carry dissolved air.
• Water density and expansion are constants (the 4 C density maximum, which makes cold meltwater sink, is not modelled); salt and sugar in a real drink would lower the melting point and are ignored. Expect the melt times to be right to a factor of about two, and the energy balance and the level to be exact.

Numerical checks

Four checks run when the page loads (see Model check): the lossless energy balance against the closed form (both the all-melts and the some-ice-left case), buoyancy and the constant liquid level, step-size convergence of the Runge-Kutta integration, and the Blender glass dimensions against the profile used for the level.

• Start with the default cool drink and follow the cube: it first warms to 0 C, then melts while the water cools; the level in the glass never moves.
• Hot tea with three cubes: more ice, a faster cooling and a lower final temperature (third graph, against the lossless calorimetry line).
• Stir the drink (factor 3.5): the water-side heat transfer rises and the cube vanishes sooner; the batch button compares four stirring factors.