Warming, Cooling, and Urban Ozone Pollution

Part 1: Background: Newton's Law of Cooling

In this module we study situations that may be modeled by differential equations of the form

dy/dt = - k (y - b).

In particular, the quantity y has an equilibrium value at y = b, and this equilibrium is stable because dy/dt is negative when y > b and positive when y < b. In words, the differential equation says that y approaches b at a rate proportional to how far it is from b. Given data that appear to fit this model, our objectives will be

• to determine if the data really do fit the model, and, if so,
• to determine values of the parameters k and b.

We start with a warming and cooling experiment. The following figure shows data gathered by students with a temperature probe (DigiDial PDT 300). In the warming phase (red dots), the probe -- initially at ambient room temperature -- was held in a clenched fist until the temperature leveled off. Then it was allowed to cool in the ambient air (blue dots). At least the cooling phase of this experiment should satisfy Newton's Law of Cooling: The rate at which an object cools is proportional to the difference between its temperature and the ambient temperature. In mathematical symbols, that's the differential equation

dT/dt = - k (T - a),

where a is the ambient temperature. The shape of the warming curve suggests the possibility that it too rises at a rate proportional to the difference from the ambient temperature -- in this case, the temperature of the clenched fist.

The cooling curve is a little easier to study, because we actually know the ambient temperature of the room: That's the y-coordinate of the first red dot, which happens to be 72.3oF. The last red dot is at 95.5oF -- which must be close to the fist temperature -- but we don't know if the actual temperature was slightly higher. Thus, we will study the cooling curve first.

We will consider two ways to examine whether our data are actually modeled by Newton's Law of Cooling.

First, we can make a change of variable, y = T - a -- which is the same thing as setting 0 on the vertical scale at the ambient temperature. Then dy/dt = dT/dt, and the differential equation becomes

dy/dt = -ky.

That's the defining equation for decaying exponential functions (recall the Radioactive Decay module), and we can determine whether the data fit such a model by looking at a semilog graph. If we see a straight line, we know the model fits -- and the slope of the line determines the decay constant.

Second, we can estimate values of dT/dt from the data, for example, by using symmetric difference quotients:

(Ti+1 - Ti-1) / (ti+1 - ti-1).

We can plot these slope estimates against Ti to see if we get a straight line -- after all, Newton's Law of Cooling says the slope should be a linear function of T. Furthermore, as we will see, the slope and y-intercept of the line will give us both parameters in the model -- we don't need to know the ambient temperature.

1. Explain why the solution of the initial value problem
dT/dt = - k(T - a), T(0) = T0
is
T = a + (T0 - a) e-kt.
2. Explain why (Ti+1 - Ti-1) / (ti+1 - ti-1) is a good estimate of dT/dt at t = ti.

Note: If you have access to a data gathering device (e.g., CBL or PSL with temperature probe), you should do the experiment yourself and substitute your own data for the sample data given here.

Send comments to the authors <modules at math.duke.edu>