Cook a Turkey: Time Calculation Equation

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Discussion Overview

The discussion revolves around the empirical equation used to calculate the cooking time for a turkey based on its weight, specifically the formula t = W(2/3)/1.5. Participants explore the theoretical underpinnings of this equation, including heat transfer principles and the implications of different cooking methods.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that the equation t = W(2/3)/1.5 is based on empirical observations, questioning whether there is theoretical support for this relationship.
  • Others suggest that the internal structure of a turkey is complex, and modeling the heat transfer using partial differential equations (PDEs) of the heat equation could provide a more accurate understanding.
  • A participant notes that the W(2/3) term indicates that heat transfer is likely dominated by conduction, and discusses approximating the turkey as a sphere to analyze heat conduction problems.
  • Another participant mentions that the diffusion of heat leads to a relationship where cooking time is proportional to the square of a linear distance, linking this to the weight of the turkey and the derived equation.
  • One participant introduces the Panofsky formula, which provides a nearly linear approximation for cooking times based on turkey weight, suggesting that for typical weights, the deviation from this approximation is minimal.
  • There is a discussion about the units of measurement for weight and time, with participants emphasizing the importance of consistency in unit systems to avoid errors, referencing historical examples of unit conversion failures.
  • Some participants express humor regarding the potential for confusion over units, while clarifying that in the context of the cooking time equation, weight should be in pounds and time in hours.
  • A later reply suggests using a thermometer to ensure the turkey is cooked properly, indicating a practical approach to the cooking process.

Areas of Agreement / Disagreement

Participants express a range of views on the theoretical basis for the cooking time equation, with some supporting its empirical nature while others advocate for a more rigorous theoretical framework. There is no consensus on the best approach to modeling the cooking process or the implications of the derived equation.

Contextual Notes

Participants highlight the complexity of heat transfer in cooking and the potential limitations of empirical formulas. The discussion also touches on the importance of unit consistency in scientific calculations, referencing specific historical failures related to unit conversion.

Who May Find This Useful

This discussion may be of interest to those studying heat transfer, cooking science, or anyone involved in culinary applications where precise cooking times are critical.

bluecheez
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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?
 
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bluecheez said:
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 W2/3 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 W2/3.
 
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.
 
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.
 
What units are the weight and the time? SI or Imperial?
 
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.
 
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pickycat said:
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 :biggrin:.
 
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vanhees71 said:
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 :biggrin:.
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!)
 
Well, use a thermometer, and you get your turkey right to the point!
 
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