Introduction: Statistical Analysis of Turkey Cooking
The following is my analysis of cooking a turkey. I will be measuring temperature and time for a 20 pound turkey cooked in a Rival brand roaster.
Step 1: Instruments
TEMPRITE Digital Cooking Thermometer with Alarm
Resolution: .1 degrees
Temperature range of -40 to 450 degrees F
6 inch metal probe with silicone coated wire
"Slow" response time and low temperature range suggests this uses a thermistor
Rival Top Browning Roaster Oven
Possibly broken temperature knob. It feels as if no internal component is moving.
Step 2: Calibration
Whenever you take measurements, your instruments should be calibrated (if applicable). The probe of the TEMPRITE thermometer was submerged in ice water. Phase change (freezing/melting point) for water occurs at 32 degrees F. Therefore, the ice water should be approximately 32 degrees F.
5 Measurements were taken. Crushed ice was poured into a cup and then filled with water. Between each measurement, the water was poured off, the ice was topped off and the water refilled.
Mean: 32.4 degrees F
Standard Deviation: 0.777
r squared: .13
For the purposes of the analysis, it is assumed that the instrument is accurate. Deviation from 32 degrees F can be attributed to minerals and other "imperfections" in the water in addition to human error etc. Horrible r squared value can be attributed to very low sample size (an ideal r square value is 1 or very very close to 1).
Normally, I would have also tested on boiling water. Unfortunately, my stove and oven at home is out of commission (and on thanksgiving too :( ). For some reason it won't hold oven temperature and will be replaced the Saturday after thanksgiving...
Below is a graph with the calibration data - if the r squared value was close to 1, the pink regression line would be very close to horizontal (zero slope).
Step 3: Background and Methodology
Cooking time is dependent on cooking temperature, food starting temperature, food volume, food shape (spherical), food surface area, and food density.
Lets assume spherical turkey and that all turkeys have the same (or very similar) density. The volume of a given turkey will be (4/3)*pi*r3 where r is dependent on food density. So for a given r, we can find the surface area as 4*pi*r2 (FYI: Volume of a sphere is the Integral of Surface Area).
The following was calculated assuming a 10lb turkey has a radius of 10 units.
So a 10lb (and a radius of 10) turkey has a volume of : 1333 * pi
and a surface area of: 400 * pi
and a radius: 10
Now let us double our mass and go for 20lb. Remember that the density is the same (and our only mathematical constant).
The following is based of the 10lb turkey calculations (keep in mind, these are quick estimate calculations).
Volume: 2666 * pi
Surface Area: 634 * pi
What we find is that if we double the mass, the volume is increased by a factor of 2. However, the surface area is only increased 1.59 times. So we have twice the amount of mass to heat, but only a 1.59 times the surface area to move said heat. On the bright side, our heat does not have to travel twice the distance :)
So it can clearly be seen that cooking time estimates is some sort of voodoo best described by thermodynamisits (? :P) OR by those with meat probes.
Before collecting data, the instruments were calibrated (if applicable) and then cleaned. The probe was placed in the thigh in what is usually considered the "normal" location to get an accurate temperature reading for dark meat (which takes longest to cook).
The data was then recorded at regular intervals. When the correct "doneness" was reached, a final reading was taken before removing and cleaning the probe. Using the collected data, a basic analysis was completed although time constraints due to final examination schedule made this a bit hasty.
Step 4: Measurements
Measurements began at 47.1 degrees F with an ambient temperature of 71.2 degrees. For about 15 minutes the turkey was rubbed and seasoned buttered and punctured. The pan was filled with potatoes veggies and neck.
We reached 160 and to my horror we passed 200 degrees F. Something was horribly wrong here. We opened up the roaster and discovered that the probe had worked its way out a little (exposed to the much warmer surrounding air). The probe was pushed back in, but the temperature stayed around 210.
Yes, something fowl was afoot!
The deadline is fast approaching, I have some Gauss-Legendre quadratures to attend and the analysis is not yet complete. So hastily, I put some things together so everyone can witness the overcooked (and dry) turkey data. That being said, it wasn't bad - it was quite good and fit well on the plate alongside the 2 pounds of fresh whipped potato.
Perhaps one day I will return and give you a proper cubic line estimate and an uncertainty analysis (that's if I'm really bored :P).
Step 5: Afterthought
I had meant to do this project with a thermocouple wire I made a couple weeks ago. I was going to build a bridge, make my own calibration curve and take measurements with a multimeter. I would then do some simple math on said calibration curve and get data.
As fate would have it, the wire insulation is not food safe. Bugger. But here's a picture of the spot welded thermocouple :)