Cooling Models.

Elementary cooling can be modeled by the function
where Ta is the ambient temperature, To is the
temperature at time zero and k is a constant. If
the ambient temperature and the temperature at time zero are known,
it is a very easy process to calculate k and write the
equation. What if we have a set of cooling data and the
person who created the data forgot to record the ambient temperature?
Then we must use more of a trial and error method to find the
cooling function.

The data set shown below is the cooling data from a cooling
experiment done with water. The water was heated to boiling
and then allowed to cool. The temperature of the water was
recorded every minute for 30 minutes. No ambient temperature
was given. We will find the cooling function and, in the
process, estimate the ambient temperature.

Let's make an initial guess at a cooling function. Assume
an ambient temperature of 75 degrees F and calculate the corresponding
value of k at t = 30 minutes. (see below)

Let's use this value for k and write an initial guess
at the cooling function.

Using this guess, let's create another column in the spreadsheet
and compare the values from the function with the empirical values
in the spreadsheet.

I also added a column for residuals; that is, the differences
between our function estimates and the empirical data. Let's
look at the last few rows of the spreadsheet and see what the
sum of the residuals is. We will use this value as a goodness
of fit test as we make adjustments to the parameters in the function

.

As our function gets to be a better model for the cooling curve,
sum of the residuals should decrease. We can also graph
this data to get a visual idea of how well our function fits the
data.

Not a very good fit, is it? What is wrong? First,
from the shape of the blue diamonds representing the empirical
data it would seem that the ambient temperature is somewhat greater
than 75 degrees. We can say this because the graph is much
less steep between 25 and 30 minutes, indicating that an asymptote
was being approached. Let's try an ambient temperature of
100 degrees F and see what parameters we get and what the effect
is on the residuals. With an assumed ambient of 100 degrees,
k = -0.04391. Using this value for k we
replace the function and look at the result on the residuals and
the graph.

Much better. The sum of the residuals has decreased to
-91.176 and the graph of the function looks much closer to the
empirical data. However, the size of the sum of the residuals
indicates that we can still do better. Lets try 110 degrees
F as an estimate of the ambient temperature and see what happens.
At 110 degrees k = -0.054308. Using these values
in the function we will again look at the residuals and the graph.

Another dramatic decrease in the sum of the residuals to -34.28,
and the graph appears to be almost perfect. Using the sum
of the residuals as a control we can continue incremental increases
in the estimate of the ambient temperature. At some point,
the sum of the residuals will no longer decrease. Then
we must back up and use smaller increments. Using this method,
I found that the probable ambient temperature for this experiment
was 114.2 degrees F. This seems a little high, but if the
container for the water was left on something like a stovetop,
this is not unreasonable. Using this value for the ambient
temperature, k = -0.06076, and we get the following:

The sum of the residuals is now less than 5 one hundredths
and the graphs of the function and the empirical data seem to
lay one on top of the other. This manual technique is similar
to the more rigorous statistical techniques in that the objective
is to minimize the sum of the residuals. In the absence
of sophisticated statistical software, such as SPSS or MiniTab,
this technique can be useful. There are four more spreadsheets
available for different types of data where I have used this technique
to do curve fitting. If you wish to view the spreadsheets,
click on one of the selections below.