Heat Equation: Cooking a Turkey

In summary, the conversation discusses a problem of determining the time it will take to cook a 15 pound turkey at 400 degrees to reach a center temperature of 180 degrees. The primary equation used is the heat equation and the problem is treated as solving the heat equation in spherical coordinates. The initial and boundary conditions are also provided. However, due to practical complications such as convection and radiation, a closed form solution is not possible. A scaling factor may be used to apply the solution to the new problem.
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Homework Statement


It's that time of the year. I'm trying to determine how long it will take to cook a 15 pound turkey at 400 degrees to reach a center temperature of 180 degrees, given that it takes 90 minutes to cook a 5 pound turkey to the desired center temperature. The roast is initially at 35 degrees (as was the 5 pound roast).

Homework Equations


The primary equation is the heat equation:
ut = kΔu
Initial condition: u([itex]\vec{x}[/itex],0) = 35
Boundary condition: u([itex]\vec{R}[/itex],t) = 400

The Attempt at a Solution


It seems to me that this can be treated as solving the heat equation in spherical coordinates, where the temperature varies radially.
So we get:
ut = k(urr+[itex]\frac{2}{r}[/itex]ur)
Then we let u(r,t) = X(r)Y(t)
Hence: [itex]\frac{XY'}{k}[/itex] = X''Y + (2/r)X'Y
This implies: [itex]\frac{Y'}{kY}[/itex] = (X''+(2/R)X')/X
We then take these equal to some constant, -λ say.
From here we get 2 ODE'S:

(1) Y' + kYλ = 0
(2) rX'' + 2X' + λrX = 0

We can let s(r) = r X(r)
s'(r) = X(r) + r X'(r)
s'' = X' + X' + r X'' = 2X' + rX''
Hence (2) becomes:

(3) s'' + λs = 0

This next part is where I start to get confused.
For the boundary conditions, I assume we can take X(R) = X(-R), where R is the radius of the sphere, since u(R,t) = u(-R,t) = 400 was provided.

Then, from (3), assuming λ = α2, α > 0
s(r) = A cos(αr) + B sin(αr)
Since s = rX, s(R) = R X(R) and s(-R) = -R X(-R) = -R X(R), so s(R) = -s(-R)
A cos(αR) + B sin(αr) = B sin(αR) - A cos(αR)
cos(αR) = 0
αR = (n+[itex]\frac{1}{2}[/itex])π
Hence our eigenvalues are:
αn = (n+[itex]\frac{1}{2}[/itex])π/R

Then, solving 1:

T(t) = C exp(-kλt)

We put our solutions together to get the full Fourier series:

u(r,t) = [itex]\frac{1}{2}[/itex]A0+[itex]\sum[/itex][(Ancos(αnr)+Bnsin(αnr))exp(-λnkt)]

Then I would plug in the initial condition to get the coefficients.

What I'm not sure of is how to scale this for the new mass. I guess I could assume that the radius is three times larger. I'm not even given the thermal conductivity, so I'm not sure if I am even approaching this right.

Does anyone have advice on how to proceed with this type of problem?
 
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  • #2
I like your attempt at a very analytical solution. But then we run into some practical problems of how to actually use this in calculating the time to cook the bird.

I would recommend a 1D transient conduction model. Most heat transfer texts have these. You can study the time behavior of the temperautre at the center and see how long to reach 180 F. I have set these up in MathCAD and they work good (at least good enough for most practical problems).

It usually takes me and my wife about 4 to 4.5 hours to cook a turkey this size - so this gives you a check of your model.
 
  • #3
When something is put in the oven there are two heat transfer boundary conditions mechanisms that are active. One is convection, the other is radiation with the walls of the oven. Further complicating the issue is the fact that the outer layer of the bird will experience some phase change where some of its water content evaporates. These complications eliminate a closed form solution.

Is it a fresh turkey or has it been frozen. And if frozen, is there still some frozen portion? Thus the initial condition is in itself a boundary value problem.

The problem is as complicated as you choose it to be.
 
  • #4
My wife uses a meat thermometer and the eyeball method. :approve:
 
  • #5
There is a slight flaw in your equation. You need to factor in the 1/r from the substitution you performed. Once you obtain your Fourier coefficients for the initial case, you should try a few tricks to apply it to your new problem. Hint: use a scaling factor.
 

1. What is the heat equation and how does it apply to cooking a turkey?

The heat equation is a mathematical formula used to describe how heat is transferred within a material. In cooking, this equation can be used to determine the amount of time and temperature needed to cook a turkey based on its size and the type of cooking method being used.

2. How does heat transfer affect the cooking process of a turkey?

Heat transfer is essential in cooking a turkey because it allows for the transfer of heat from the oven or cooking source to the turkey, resulting in the desired level of doneness. The rate of heat transfer also affects the cooking time and can impact the texture and taste of the turkey.

3. What factors can influence the heat equation when cooking a turkey?

Some factors that can influence the heat equation when cooking a turkey include the size and weight of the turkey, the type of cooking method being used (roasting, grilling, etc.), the temperature of the cooking source, and the type of oven or equipment being used.

4. How can I use the heat equation to ensure my turkey is cooked safely and thoroughly?

By using the heat equation, you can calculate the necessary cooking time and temperature for your turkey based on its size and the cooking method being used. This will ensure that the turkey is cooked to a safe internal temperature and is thoroughly cooked throughout.

5. Are there any tips for using the heat equation to cook a turkey?

When using the heat equation to cook a turkey, it is important to make sure that the temperature of the cooking source is consistent throughout the cooking process. It is also helpful to use a meat thermometer to check the internal temperature of the turkey to ensure it has reached the recommended temperature for safe consumption.

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