Details

The density of ice at 0 °C is 0.917 ; the density of water is approximated as 1.000 over the temperature range 0–60°C. Archimedes' principle states that a floating object displaces its own weight of water. Thus 91.7% of the cube lies beneath the surface. For the cube to lower its center of gravity, it reorients itself such that one of its three-fold axes is vertical. The remaining 8.3% forms a triangular pyramid of altitude and volume , with projecting surface area . Thus, . (This is a nice exercise in solid geometry.)

As a function of immersion time, the volume of the ice cube can be approximated by the differential equation , where the constant is a lumped parameter depending on the ice/water interface heat transfer rate (e.g., in joules per unit area per unit time), the heat of fusion of ice, as well as the heat capacities of ice and water. Taking , the temperature of the water in the beaker is in °C, and is approximated to remain constant during the melting process. For a cube in the water, and . Using the measurements obtained from our simple experiment, the instantaneous width of the ice cube (initially ) is approximated by , with in cm and in minutes.

It is assumed for simplicity that the ice maintains its cubic shape as it shrinks. Actually it deforms more irregularly and sometimes shows a beautiful fractal structure before it disappears.

Snapshot 1: beginning configuration

Snapshot 2: drop ice cube into water; note reorientation to lower center of gravity

Snapshot 3: ice cube begins to melt

Snapshot 4: melting completed

Snapshot 5: faster melting at higher water temperature

Snapshot 6: water at 0 °C; no net melting but dynamic equilibrium as ice and water exchange molecules

Reference

[1] Wikipedia. "Ice." (Aug 5, 2013) en.wikipedia.org/wiki/Ice