Heat Equation: Cooking a Turkey

[Biljo6985]

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:
u_t = kΔu
Initial condition: u($\vec{x}$,0) = 35
Boundary condition: u($\vec{R}$,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:
u_t = k(u_rr+$\frac{2}{r}$u_r)
Then we let u(r,t) = X(r)Y(t)
Hence: $\frac{XY'}{k}$ = X''Y + (2/r)X'Y
This implies: $\frac{Y'}{kY}$ = (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+$\frac{1}{2}$)π
Hence our eigenvalues are:
α_n = (n+$\frac{1}{2}$)π/R

Then, solving 1:

T(t) = C exp(-kλt)

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

u(r,t) = $\frac{1}{2}$A₀+$\sum$[(A_ncos(α_nr)+B_nsin(α_nr))exp(-λ_nkt)]

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?

 

[edgepflow]

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.

 

[LawrenceC]

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.

 

[LCKurtz]

My wife uses a meat thermometer and the eyeball method.

 

[Norfonz]

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.