Predicting
the Temperature of Cooled Water
(or not...)
By: Lauren
Wright and Andy Tyminski
In this experiment, we took a pot of
boiled water and measured its initial temperature (time t = 0)
and then recorded temperature readings for the next 30 minutes.
You would think that this would be an easy task for two graduate
students...
After measuring the temperatures, we
entered them on a spreadsheet and ATTEMPTED to use Newton's Law
of Cooling to model the data and predict the temperature of the
water after 45 minutes, 60 minutes, or 300 minutes.
First, we will show you the data that
we collected with a room temperature of 74 degrees Fahrenheit.
Minute 
Temperature 
Minute 
Temperature 
Minute 
Temperature 
0 
212 




1 
204 
11 
155 
21 
136 
2 
192 
12 
153 
22 
135 
3 
186 
13 
151 
23 
134 
4 
180 
14 
149 
24 
132 
5 
175 
15 
147 
25 
131 
6 
171 
16 
145 
26 
130 
7 
168 
17 
143 
27 
129 
8 
164 
18 
142 
28 
128 
9 
162 
19 
139 
29 
127 
10 
159 
20 
139 
30 
125 
The formula for Newton's Law of Cooling
is:
with t = time in minutes, =
temperature of the air, = initial temperature,
and k = experimental constant.
First, we set out finding what our
k would be. We did this by calculating the value of k at five
different data points, namely T(6), T(11), T(19), T(26), and T(29).
Then, we took the average of those five k values to get an estimate
of our experimental constant.
The k that we used in Newton's formula
is k = 0.042901.
Here is a table of the values that
were returned.
Minute 
Temperature 
Minute 
Temperature 
Minute 
Temperature 
0 
212 




1 
206.204859462488 
11 
160.085646064217 
21 
130.05496264225 
2 
200.653078735481 
12 
156.470585069981 
22 
127.701003320969 
3 
195.334438222578 
13 
153.007334122942 
23 
125.445895630749 
4 
190.239147486782 
14 
149.68951814659 
24 
123.285488418732 
5 
185.357827228474 
15 
146.511029777924 
25 
121.215804854627 
6 
180.681492020207 
16 
143.46601812515 
26 
119.233035110248 
7 
176.201533766522 
17 
140.548877997494 
27 
117.333529346465 
8 
171.909705858361 
18 
137.754239587282 
28 
115.51379099466 
9 
167.798107992885 
19 
135.076958585322 
29 
113.77047032032 
10 
163.859171630773 
20 
132.512106711368 
30 
112.10035825692 
As you can see, these values differ
greatly from what we recorded. Here is a graph of the two sets
of data.
So, the next step in our experiment
was to see exactly how bad the error was.
We took the square of the difference
for each time, summed the squares, and divided by the number of
data points. This gave us an average error of approximately 60.3
degrees. For illustration, take a look at the graph of the residuals...
So, this experiment was a complete
disaster using Newton's Law of Cooling. We could attribute this
to many factors  a faulty thermometer or human error are the
most likely causes (I mean, ANDY was in charge of reading
the thermometer. So, anything is possible.)
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