A First Look at Temperature Change -- Newton's Model of Cooling

This material is an excellent introduction to mathematical modeling. We look at a real problem -- understanding the impact of wind on cooling -- and we use that problem to illustrate some interesting mathematics and, more importantly, the practice of building and interpreting mathematical models.

Ideally, this material would come up on a cold day in winter with television weather people excited about a wind chill factor of 10 or more degrees below zero. Ask your students the difference between a calm day with a thermometer reading of 20 degrees Fahrenheit and a windy day with the same thermometer reading but a wind chill factor of 10 degrees below zero. You might also pose some other questions about weather -- At what time during the day is the sun strongest? -- What is the hottest time of the day? -- At what time of the year is the sun strongest? -- What is the hottest time of the year? -- If you live near the ocean or a large body of water, at what time of the day or year is the water hottest? -- At what time of the day or year is the air warmest?

We begin by setting up an experiment in the front of the room, recording the temperature of a hot cup of water as it cools during the class period.

While it is cooling we ask students to make some educated guesses about what the data will look like. As one possibility we draw a graph like the one below on the blackboard

and discuss what this graph says about the cooling water -- for example,

The initial temperature is about 190 degrees.
The water cools to room temperature in about 30 minutes.
It takes the same length of time to cool from 190 degrees to 150 degrees as it does to cool from 150 degrees to 110 degrees.
The water temperature appears to drop steadily to room temperature and then stop dropping.

Then we ask students to draw graphs showing what they think actual data will look like. We ask them to draw two graphs -- one showing the water cooling in calm air and another showing water cooling in windy air. We will eventually look at data from two cups of cooling water, one under a fan and one protected from the fan. You can use data supplied with the CAS files below or, better yet, you can use data that you collect in class.

Click here for more about data collection

The table and graph below show some simple data we collected with just a single cup of water cooling in calm air. This is real data, so it is not as clean as one might wish.

missing graphic

The same data is ready for use in the Maple worksheet and Mathematica notebook obtained by clicking on the icons below. Click on the icon for your CAS system and then arrange the CAS window that comes up so that you can easily move back-and-forth between this browser window and your new CAS window.

Using this data or, better yet, data collected by your students as motivation we discuss mathematical modeling. Our models are based on Newton's Law of Cooling which we prefer to call Newton's Model of Cooling to emphasize that it is not enacted by Congress or handed down on stone tablets but is just one possible model and that it may or may not be a good model.

Newton's Model of Cooling says that if we make a series of temperature measurements of a cooling object at equally spaced times then the temperature changes according to the equation

where A is the ambient or room temperature and k is a constant that depends on the object that is cooling. If the object is small and poorly insulated then k will be relatively large but if the object is more massive and well insulated then k will be relatively small.

If we know the ambient temperature A then we can determine the value of the constant k using any two data points by noticing that

If we don't know the ambient temperature then we can determine the values of both constants, A and k using any three equally spaced data points as shown below.

Once we have determined the constants A and k using these three data points we have a model given by

where S(i) is the temperature predicted by the model at the time of the i-th data point.

Now students can use their CAS window to compare the data with various models based on Newton's Model of Cooling. The picture below shows part of a Maple session.

Notice that the model (the red curve) matches the data (the black curve) exactly at times 20, 40, and 60 -- the three points used to determine the model.

Your students should experiment with data, ideally data that they have collected themselves, using a CAS system and curve fitting using various data points to see how well Newton's Model of Cooling describes a cooling object. They can try several variations on the basic theme.

Using a second temperature probe keep track of the ambient temperature as well as the temperature of the cooling water. The graph below shows one experiment we did. Notice that the ambient temperature varied quite a bit, apparently because the furnace cycled on and off several times during data collection.
Using an ice water bath -- that is, a mixture of ice and water -- try to keep the ambient temperature constant.

You will discover that Newton's Model of Cooling does an excellent job of describing a cooling cup of water if the ambient temperature is steady unless the temperature of the cooling cup of water is too hot. When it is close to boiling then another mode of cooling evaporative cooling becomes important. Temperature is a measure of the random motion of molecules -- essentially their average speed. Individual molecules may be moving quite a bit faster than this average and some of them may be moving fast enough to escape from the cup of water. This is called evaporation and, although it occurs at any temperature, its effects are strongest at high temperatures. As the fastest molecules escape from the cup of water the average speed of the molecules remaining in the cup of water drops -- that is, the temperature drops. This effect is accentuated when a breeze is blowing across the surface of the water carrying the escaping molecules away.

The picture below shows one experimental set up with two temperature probes recording the temperature of the cooling cup of water in calm air and the temperature of the cooling cup of water in windy air simultaneously.

The graph below shows the data we collected in this experiment. Notice that not only does the water in windy air cool faster but that its temperature actually drops lower than the water in calm air.

This concludes our first in class look at cooling. Notice that we have not answered all the questions we posed at the beginning. Point out to your students that we will return to these questions later in the year. They may want to do some explorations on their own using the ideas we have developed so far. Here are some possibilities.

Modify the Maple worksheet, Mathematica notebook or TI-92 program fitdata to take into account varying ambient temperature. This is a particularly nice idea because it fits in well with later work.
Some students may point out other possible explanations for more rapid cooling in the wind. For example, in still air the air around the cooling cup of water may warm up and act as insulation (one student advanced the idea that a pillow of warmer air formed just above the surface of the water in calm air but not in windy air) slowing down further cooling. In windy air, a continual supply of fresh cool air keeps the temperature right around the cooling cup of water cool.
This is a great opportunity to talk about the interplay between theory and experiment. The two possible theoretical explanations for for the observed difference in calm air and windy air cooling suggest experiments that might be able to distinguish between the two suggested mechanisms.
- Compare cooling for wet and dry objects. For dry objects evaporative cooling is no longer operative and, thus, the windy and calm cooling should be the same but the insulating pillow mechanism would still be operative and, thus, the windy and calm cooling would still be different.
- For the same reason one might look at the behavior of a different, more volatile, liquid.
- Instead of starting with very hot water start with very cold water. Now the pillow mechanism predicts that in calm air the cold water warms up more slowly than in windy air but the evaporative cooling mechanism implies that in calm air the cold water warms up more quickly than in calm air because evapoartive cooling keeps the windy water cooler.

Of course, the best explorations are those that students think up entirely on their own. With real data you and they will get some surprizing and interesting results. Real data is often dirty.

The following two data sets are particularly interesting. The first was collected in class and is unusually clean. This class met from 7:30 - 8:20 and during the course of the period the sun came up over the mountains and warmed the room -- hence, the steadily increasing ambient temperature. We used very shallow bowls of water and were able to observe clearly the fact that in windy air the temperature of the water drops below the ambient temperature. Notice that it is still falling even after it is below the ambient temperature. With only a 50 minute class period and some time lost to set-up it can be difficult to observe "long term" behavior.

Missing graph

The next data set was collected in my living room and is far from clean. Notice the oscillation of the ambient temperature as the furnace cycles on-and-off. This data is extraordinarily rich. Here we are looking at what happens as very cold water warms up and we see clearly the effects of both the pillow mechanism and evaporative cooling. The cold water under windy air does warm up faster at first because of the absence of the insulating pillow but then it "stabilizes" below the ambient temperature because of evaporative cooling. Notice that the oscillations of the windy air water mimic and lag behind the oscillations of the ambient temperature. The experiment didn't go on long enough to observe the "steady state" behavior of the calm air water.

Missing graph

Copyright c 1995 by PWS Publishing Company, a division of International Thomson Publishing Inc. Comments to Frank Wattenberg, Department of Mathematics, Carroll College, Helena, MT 59625.