Cook a Turkey: Time Calculation Equation

[bluecheez]

According to this website, there is an equation obtained empirically to obtain the amount of time it takes to cook a Turkey:

t = W^(2/3)/1.5


Is there any theory to support such a relation? Yes, the internal structure of a turkey is complicated so I'm sure the proper way to do it is to numerically model a PDE of the heat equation.
But are there any approximates we can use in real life to get a good idea of how long it'll take to cook something?

 

[Chestermiller]

According to this website, there is an equation obtained empirically to obtain the amount of time it takes to cook a Turkey:

t = W^(2/3)/1.5


Is there any theory to support such a relation? Yes, the internal structure of a turkey is complicated so I'm sure the proper way to do it is to numerically model a PDE of the heat equation.
But are there any approximates we can use in real life to get a good idea of how long it'll take to cook something?

The W^2/3 seems to suggest that the heat transfer is dominated by conduction. Approximate the turkey by a sphere, and look up the solution to the transient heat conduction problem for a sphere, assuming that the temperature is T₀ at time zero and that the oven temperature is at T_∞. The solution to this heat conduction problem is in the literature, expressed as the average temperature change as a function of the Fourier number. Decide what temperature the center of the sphere has to reach for the turkey to be considered cooked. This will correspond to a specific value of the Fourier number. The Fourier number has an r² in the denominator, which is proportional to W^2/3.

 

[marcusl]

The diffusion of heat into many objects, including a sphere and a semi-infinite slab, results in the relationship t~x^2 to reach a given temperature where x is a linear distance. For a sphere, the appropriate distance measure is r, so t~r^2. Since weight is proportional to r^3, you have that t~W^(2/3). It is only the numerical factor of 1.5 that needs to be empirically determined.

 

[gmax137]

Interesting. For "normal" range of turkey weights, the Panofsky formula is nearly linear and is close to "90 minutes plus 11 minutes per pound." For turkeys from 8 to 24 pounds, the maximum deviation between the formula and this linear approximation is less than 8 minutes, which is negligible. It is important to get the bird close to room temperature before putting it in the oven, otherwise these "rules" won't work well.

 

[pickycat]

What units are the weight and the time? SI or Imperial?

 

[Vanadium 50]

w^2/3 is proportional to surface area. That's not crazy - I'd expect the true answer to be between 2/3 and 1. The 1.5 is just a constant from using conventional units. In Natural Turkey Units, this would be 1.

 

[vanhees71]

What units are the weight and the time? SI or Imperial?

The weight in SI are Newton in imperial units it's pound. I strongly recommend to stay within one system of units. There are tragic and expensive examples for technical failures only because two systems of units where at use. I think one Mars probe launched by the NASA only failed because one person thought in feet and the other in metres. I guess it was NASA's fault using SI units .

 

[pickycat]

The weight in SI are Newton in imperial units it's pound. I strongly recommend to stay within one system of units. There are tragic and expensive examples for technical failures only because two systems of units where at use. I think one Mars probe launched by the NASA only failed because one person thought in feet and the other in metres. I guess it was NASA's fault using SI units .

Ha, Ha! Very true! YET, I was asking about the time and the weight in the particular turkey cooking time function of the turkey's weight. In the meantime, I did search some more and I found that in this particular relationship the time has to be in hours, while the weight is in pounds. (Had I assumed SI units, like MOST physicists would use, I would have the same tragic outcome as NASA did!)

 

[vanhees71]

Well, use a thermometer, and you get your turkey right to the point!