Presented By

Dana TeCroney

 

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.

 

Polynomial Model

 

Power Model

 

Logarithmic Model