Cooling Curves: Dead bodies and fridges

Cooling Curves: Dead bodies and fridges

All the maths behind this fits for cooling bodies – whether objects placed in fridges or dead bodies cooling over time – and this idea is used in CSI investigations to work out the time of death of bodies.  I will do this investigation with a Microbit – which allows you to log data on temperature.  You can see the program needed here:  Environmental Data Logger. I changed the program to record the temperature every 1 second and then placed the Microbit in the fridge for about 15 minutes.  This data was then downloadable as an CSV file which could then be pasted into Desmos.  The graph of this data is shown below:

This fits what we would expect from a cooling curve, which is an exponential function.  So we can use Desmos’ regression to find a regression equation:

When this is plotted we can see that this is a very close fit (R squared 0.994). 

However we can also note that we don’t have the asymptote correctly given, we should be seeing this asymptote at y = 8 (or lower) rather than y = 8.60.

Derivation of the maths

Newton’s cooling curve for an object placed in a fridge with an ambient temperature of 8 degrees Celsius is given by the solution to the following differential equation:

This tells us that the rate of change in the temperature (T) of a body over time (t) is proportional to T-8 multiplied by a constant k.  We will use time in seconds and temperature in Celsius.  We can then separate the variables to give:

We can then integrate both sides to give:

We can then rename e^c as another constant an rearrange to get our desired equation:

We can then use some points on the curve to find the values of A and k.  If we take the initial temperature when t=0 as 32 then we have:

We can then choose another point to find k.  The points chosen can have a large impact on the k value – so it may be worth trying a few different points along the curve.  I chose when t = 200 to give:

When we plot this we can see that we get a close fit to our graph:

It’s nice to see how such a simple experiment can be done and then fit so neatly with the maths we’d expect. If we were doing this as a CSI investigation, we’d need to know the ambient temperature of the room (which would replace 8 in our equation).  We’d then take 2 measurements of the body temperature over a time period, and then use these to find the time when the body temperature was 37 degrees.