How long does it take to melt some butter? You always wanted to know, but were afraid to ask. Well, the physics involved
is pretty basic - the Heat Equation first derived by Joseph Fourier in 1822. However, direct application of the heat equation is complicated by the phase change from solid to liquid.
Here is a very simple numerical simulation based on the heat equation, but with
a provision for the phase change. The butter is in stick form (1/2 of a stick - 32 X 32 X 60 mm)
and is placed vertically on a constant temperature hot plate (stove).
The butter melts very slowly in real-time (poor heat conduction only 0.2 W/m-K due to the fat content - that's how fat critters survive the winter), so you can
accelerate the action by increasing the Simulation Time Scale. As an added bonus, there are several other melting
materials to try.
Nomenclature: Tplate is the temperature of the hot plate, Tamb is the room air temperature, Tinit is the initial
temperature of the butter (about -18 °C out of the freezer, 3 °C out of the refrigerator). Cooling/heating due to the room air is controlled
by the Convective Heat Coefficient h.
Details of the model can be found by clicking HERE.
An experimental 2-D version is found HERE.