The cooling of a body

In this section we will think about what happens in any experiment where heat is added to a body. The body will gain energy but it will also lose energy to the surroundings. In fact the bigger the temperature difference between an object and its surroundings the greater will be the rate of loss of heat energy. This becomes zero when the object reaches the same temperature as its surroundings. You can see an example of this is Figure 1.

The law governing the rate of loss of heat from a body to its surroundings was first proposed by Newton in 1701 and is therefore known as Newton's law of cooling. He proposed that:

This can be expressed mathematically as:


and if C is the thermal capacity of the body as:


This has been found to hold very well for forced convection, where the air velocity is >4 ms^-1 but not too well in still air, that is, for natural convection.

For natural convection the law has been shown experimentally to be a 5/4 power law.
That is:


The following set of results may be used to investigate the law of natural convection.

Excess temperature (oC)	20	30	40	50	60	70
Rate of loss of heat (Js-1)	0.212	0.350	0.501	0.660	0.830	1.01
lg(excess temperature)	1.301	1.477	1.602	1.699	1.778	1.845
lg(rate of loss of heat)	-0.674	-0.456	-0.300	-0.180	-0.081	0.004

If a graph is plotted of lg(excess temperature) against lg(rate of heat loss) the validity of the law can be checked.

Heat losses can be useful, in fact essential in some cases. Engines are cooled with water or air and integrated circuits that work at relatively high power are set in a piece of blackened aluminium.