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# The Semester’s Over, Let’s Drink Some Beer

An Application of Newton’s Law of Cooling

Spreadsheets provide a utility in mathematics that allow us to easily organize and adapt large amounts of data.  One feature is that spreadsheet programs such as Excel or ClarisWorks make available is that of quickly producing charts and graphs from the matrix of data.  In this investigation, I will illustrate some of the features and limitations of Excel.  While it provides very nice scatter plots and even curves that fit the data, it fails to have give an accurate regression for the provided data.  So where did the data come from?  Beer

One of the hobbies I have accrued over the years is making my own beer.  In the process of making beer, the wort (mixture before yeast is added) is boiled for one hour, then cooled to 70oF before the yeast is added and fermentation takes place.  The following data was collected on December 7, 2006 when I was making a “Comin’ Home for Christmas Ale.”

 Time (Minutes) Temp. (degrees Fahrenheit) 0 183.2 1 169.6 2 151.1 3 138.91 4 129.43 5 121.58 6 115.4 7 109.97 8 105.85 9 102.2 10 98.82 11 96.01 12 93.414 13 91.357 14 89.474 15 87.633 16 86.126 17 84.414 18 83.237 19 81.73 20 80.726 21 79.721 22 78.718 23 77.9 24 77.082 25 76.414 26 75.744 27 75.071 28 74.557 29 73.871 30 73.359 31 72.868 32 72.377 33 72.05 34 71.723 35 71.386 36 71.043 37 70.7 38 70.357 39 70.168 40 69.843 41 69.674 42 69.507 43 69.172 44 69.005 45 68.837 46 68.67 47 68.502 48 68.335 49 68.167 50 68.167 51 68 52 67.829 53 67.657 54 67.486 55 67.486

Excel provides a nice scatter plot of the data shown below.

Another option Excel offers is to draw in a line versus data points.  Based on the graph below, one might think that and accurate regression could be created, but this is not the case…

When choosing a regression model it’s often helpful to consider the phenomenon you are observing.   Will a liquid cooling fit a linear, quadratic, exponential, or some other model?  This is a commonly know application of Newton’s Law of Cooling which is involves an exponential model, so let’s start there.  Notice, the regression

Notice, the regression doesn’t give such a good model.  This is not a surprising result however, if you consider where the data points lie.  Are the majority of the points in the curved part of the graph (left side), or where the graph flattens out?  As you can see above, they are in the flatter part of the graph, which causes the regression model to be flatter that we wish.  One way we could attempt to remedy this situation is to make a piecewise function where we approximate the left side using one function and the other part of the graph using a different function.  Below, I used the first 20 minutes and found an exponential regression, the regression still fails to provide an accurate model.

Would Newton’s law do any better in this case?  Newton’s Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature (i.e. the temperature of its surroundings).  It is represented by the equation:

T(t) = Ta +(To + Ta)e-kt

T(t) – Temperature at time t

Ta – Ambient Temperature

To – Original Temperature

k – cooling constant

t – time

Applying Newton’s Law of Cooling to our data, we obtain the equation

T(t) = 67.53 + 115.67e0.0020847076t

Using a TI-83 Plus Silver Edition, the following graph was produced.  The x and y axes represent the same things as above.  As you can see, this is a much better approximation.

As a further exploration, I tried some different models to see which one actually Excel actually fit the data best with.  The results are below.  As you can see the power model and the logarithmic model give much better approximations.

Power Model