Newton's Law of Cooling

Newton's Law of Cooling

The exact, corrected radiation law of Section 10.14, $P_{\text{net}} = e\sigma A(T^4 - T_0^4)$, is

completely general but mathematically awkward to use in many everyday situations, since it involves the

fourth power of two absolute temperatures. Newton's law of cooling provides a much simpler

approximate description that works well whenever a body's own temperature is not too far above the

temperature of its surroundings.

Statement of the law

Newton's law of cooling states that the rate at which a moderately hot body loses heat -- and hence the

rate at which its own temperature falls with time -- is directly proportional to the difference

between the body's instantaneous temperature $T$ and the (assumed constant) temperature of its

surroundings, $T_s$:

the negative sign simply indicating that temperature is decreasing with time as the body cools.

Newton's law as a small-temperature-difference approximation

Newton's law can be obtained directly from the more exact Stefan-Boltzmann law with Boltzmann's

correction, $P \propto (T^4 - T_s^4)$, in the special case that $T$ is only a little greater than $T_s$.

Writing $T = T_s + \Delta T$ with $\Delta T$ small compared with $T_s$, the difference of fourth powers

can be shown (by a standard binomial-expansion argument, kept to first order in the small quantity

$\Delta T/T_s$) to reduce, approximately, to a term directly proportional to $\Delta T = (T - T_s)$ --

exactly the form Newton's law states. This is why Newton's law of cooling is described as an

approximation, valid only when the body being studied is not enormously hotter than its surroundings;

for very large temperature differences, the full Stefan-Boltzmann treatment with Boltzmann's correction

must be used instead.

The average-temperature working form

For solving numerical problems, Newton's law of cooling is almost always applied in a convenient

practical form. Suppose a body's temperature falls from $\theta_1$ to $\theta_2$ over a (short) time

interval $t$, while the surrounding temperature is held fixed at $\theta_s$ throughout. Replacing the

instantaneous rate $-dT/dt$ by the average rate of fall $\dfrac{\theta_1 - \theta_2}{t}$, and replacing

the instantaneous excess temperature $(T - T_s)$ by the excess of the average body temperature over the

interval, $\dfrac{\theta_1 + \theta_2}{2} - \theta_s$, gives:

where $k$ is a proportionality constant that depends on the body's surface area, its emissive power, and

the nature of the surrounding medium, but which stays the same for that same body cooling under the same

surrounding conditions. In a typical problem, $k$ is first found from one measured cooling interval

(known $\theta_1$, $\theta_2$, $t$, and $\theta_s$), and the same value of $k$ is then used to predict how

long the body will take to cool through a second, different temperature interval under identical …