An Analogy for Cooling

  There is an experiment that one could have done in ancient times that results in an equation similar to Newton's Law of Cooling.

Consider a vessel filled with water with a small opening at the bottom like an ancient clepsydra or water clock. The rate at which water drips out would be proportional to the pressure of the water at the bottom minus the ambient pressure. As the level of the water decreases the flow rate would decrease. One could use a more accurate water clock in which the level of the water is kept fixed with a steady drip rate for comparison. The amount of water collected is a measure of the elapsed time.

So one might think of a cooling object as a vessel containing a quantity of heat with a porous skin through which the heat escapes. In the derivation below P is the pressure at the bottom of the vessel, Q is the volume of the quantity of water, C=A/(ρg) is the "capacitance" of the vessel, R is the "resistance" to flow of the opening, ρ is the density of water and g is the acceleration due to gravity.

For equal steps in time the ratio of the two pressure differences will result in a geometric series.

Supplemental (3/15): A more contemporary analogy for Newton would be a leaky vessel charged with air above atmospheric pressure for which the internal pressure slowly decreases. The inner pressure on the vessel's walls would be proportional to the change in momentum of the particles striking it in a given time. With the outer pressure less the number of particles striking the outer surface would also be less so there would be a net flux out. Newton thought in terms of the corpuscular theory. If light consisted of corpuscles one might say the same about heat particles. The air pump dates from 1649 and Boyle made use of one.

Reference

Milham - Time & Timekeepers, clepsydra

Boyle's law