Graphical Statistics

Linearizing the Cooling Curves of the Insulation Data

In this activity, you will explore one of the uses of logarithms as a graphing tool. An experiment was conducted by a student from the Central Virginia Governor's School for Science and Technology to find out how heat flow through fiberglass batt insulation was affected when thin layers of high-efficiency insulation were added to the batts.

A great deal of energy is continually wasted due to inefficient home insulation such as cheap fiberglass batts. There are several insulation materials that are better insulators than fiberglass batts, but they are not widely used because they are expensive. Therefore, it was hypothesized that heat loss could be reduced by simply adding insulation to the fiberglass batts instead of totally replacing them.

In this experiment, three different combinations of insulation were tested. A small steel hot plate was heated to 150°F and surrounded on all sides by the type of insulation being tested. Then, temperature readings for the hot plate were taken every 10 minutes for a total of 80 minutes.

These readings were plotted with their respective times to graphically analyze the rate of cooling of the hot plate. This procedure was repeated 3 times for each insulation pack. The data you will plot in Excel represents the time and temperature data of one trial for the reference case (fiberglass batts only), and the time and temperature data of one trial for insulation pack #2 (in which a thin layer of Trymer 2000 was added to the fiberglass batts).

The data exists in three files; one is in Excel Data format and one is in Text format, one is in TI-83 Group format. If you want to plot the data using the TI-83, you will have to import the text data into the calculator from the computer after you have downloaded the file.

Using Excel or the TI-83, create a scatter plot of the temperatures with their respective times for each trial. You should plot time (the independent variable) on the x axis and temperature (the dependent variable) on the y axis. What do the graphs look like? Calculate the slope between t=0 minutes and t=10 minutes for each insulation pack. Now repeat the process between t=70 minutes and t=80 minutes. What do you notice about the two calculated slopes for each graph? A third column of data exists in the Excel file which contains the natural logarithms of the temperature values. Create new graphs with time on the x axis and the natural log on the y axis, for both trials. How did the graphs change? Is it now easier or more difficult to compare the slopes of the two graphs? Calculate the slopes of these new graphs as was done above or using a regression analysis. Why is it better to graph the logs of the temperature data rather than just the actual temperature?

For another activity dealing with cooling curves and logarithms, see the Cooling Activity.

Original work on this document was done by Central Virginia Governor's School students Jon Dodson and Jamie Williams (Class of '99).