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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:

u

_{t}= 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:

u

_{t}= k(u

_{rr}+[itex]\frac{2}{r}[/itex]u

_{r})

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 = α

^{2}, 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:

α

_{n}= (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]A

_{0}+[itex]\sum[/itex][(A

_{n}cos(α

_{n}r)+B

_{n}sin(α

_{n}r))exp(-λ

_{n}kt)]

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?