The Turkey Equation

2009 November 25
by Daniel Lakeland

There's a problem that occurs every year for millions of Americans that can be solved with the proper application of Mathematical Modeling: how should you cook a Turkey, and how long will it take?

I have an opinion on how to pose this model. It's an optimization. We want to maximize the quality of the bird. The quality decreases with "overcooking" and strongly decreases with "under cooking" (due to safety concerns). The issue in a turkey is that there is no single "temperature" there are different temperatures at different points within the bird. We need a method so that by the time the bird is served, the minimum temperature within the entire bird is sufficiently high to have eliminated the risk of bacterial infection, but we also need that the maximum temperature has been controlled well enough so that the outer portions of the turkey are not dry, tough and ruined.

I propose two families of cooking methods:

  1. The variable temperature oven method. Start the turkey at refrigerator temperature, put it into a hot oven, and adjust the oven temperature through time so that the equilibrium temperature at infinite time is 165F the perfect serving temperature. By including a time preference into the model we can modify this temperature vs time profile to produce optimal results at finite time. The resulting function T_{\rm oven}(t,t_f,T_0,m_{\rm bird})  tells you how to adjust the oven temperature in time from time t=0 to time t=t_f when the bird is taken out and served.
  2. The more simple alternative is the "let it rest" method. In this method, the oven temperature is held constant, but after some period of time the bird is removed from the oven and insulated, being allowed to equilibriate to the ideal constant temperature 165F. This is a much easier to understand model. The input to the model is m_{\rm bird} and t_f the desired maximum cooking time, the output is an oven temperature, an oven time, and a resting time.

To generate these models we should take into account the physical phenomenon of heat transfer, but also we will need to calibrate our models to real data. The first step is to get a classroom full of students to each go home and cook a Thanksgiving turkey according to a single randomized temperature between say 300 and 400F, record the turkey weight, initial temperature, oven actual temperature, oven temperature as shown on the oven panel, and the internal temperature at 3 different depths within the bird at several different times during cooking. With this data, we can build an appropriate deterministic heat transfer model with uncertain coefficients, and then calibrate the coefficients to data. Once we have the coefficients we can model the cooking process either through simulation or analytical results, produce a relationship that would finally answer the question we all have: "How should I cook a turkey and how long will it take?"

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