Question

Time to Cook a Turkey Model a turkey as a sphere of radius a. Let T(r, t) be the temperature of the turkey at distance r from the center and time t. Assume T is the (uniform) temperature of the turkey when it is put in an oven at temperature T1. The heat equation says ?²T/?r² + 2/r ?T/?r = 1/k ?T/?t, 0 < r < a, t > 0 (1) where k is the thermal diffusivity. Fourier Series Solution Show T(r, t) = T1 + ?_{n=1}^? An (sin(n?r/a) / r) e^-(n²?²k/a²)t satisfies (1) and the boundary condition T(a, t) = T1, t > 0. Here the An are constants. Initial Conditions Assume the turkey is at the constant temperature T0 when it is put in the oven, so T(r, 0) = T0, 0 ? r ? a. Use the Euler formula from Fourier series to find An, n = 1, 2, 3.... The Time to Cook The turkey is cooked when the central temperature reaches the desired value T2. Write down this equation for t as explicitly as possible (you will need to take a limit as r ? 0). Numerical Values Find the time to cook a turkey of radius 12 cm and temperature 40 °F if it is put into a 350 °F oven and is done when the central temperature reaches 185 °F. The thermal diffusivity of turkey meat is roughly 1.3 × 10?? m²/s.

          Time to Cook a Turkey

Model a turkey as a sphere of radius a. Let T(r, t) be the temperature of the turkey at distance r from the center and time t. Assume T is the (uniform) temperature of the turkey when it is put in an oven at temperature T1. The heat equation says
?²T/?r² + 2/r ?T/?r = 1/k ?T/?t, 0 < r < a, t > 0 (1)
where k is the thermal diffusivity.

Fourier Series Solution

Show T(r, t) = T1 + ?_{n=1}^? An (sin(n?r/a) / r) e^-(n²?²k/a²)t satisfies (1) and the boundary condition T(a, t) = T1, t > 0. Here the An are constants.

Initial Conditions

Assume the turkey is at the constant temperature T0 when it is put in the oven, so T(r, 0) = T0, 0 ? r ? a. Use the Euler formula from Fourier series to find An, n = 1, 2, 3....

The Time to Cook

The turkey is cooked when the central temperature reaches the desired value T2. Write down this equation for t as explicitly as possible (you will need to take a limit as r ? 0).

Numerical Values

Find the time to cook a turkey of radius 12 cm and temperature 40 °F if it is put into a 350 °F oven and is done when the central temperature reaches 185 °F. The thermal diffusivity of turkey meat is roughly 1.3 × 10?? m²/s.

Time to Cook a Turkey

Model a turkey as a sphere of radius a. Let T(r, t) be the temperature of the turkey at distance r from the center and time t. Assume T is the (uniform) temperature of the turkey when it is put in an oven at temperature T1. The heat equation says
?²T/?r² + 2/r ?T/?r = 1/k ?T/?t, 0 < r < a, t > 0 (1)
where k is the thermal diffusivity.

Fourier Series Solution

Show T(r, t) = T1 + ?n=1^? An (sin(n?r/a) / r) e^-(n²?²k/a²)t satisfies (1) and the boundary condition T(a, t) = T1, t > 0. Here the An are constants.

Initial Conditions

Assume the turkey is at the constant temperature T0 when it is put in the oven, so T(r, 0) = T0, 0 ? r ? a. Use the Euler formula from Fourier series to find An, n = 1, 2, 3....

The Time to Cook

The turkey is cooked when the central temperature reaches the desired value T2. Write down this equation for t as explicitly as possible (you will need to take a limit as r ? 0).

Numerical Values

Find the time to cook a turkey of radius 12 cm and temperature 40 °F if it is put into a 350 °F oven and is done when the central temperature reaches 185 °F. The thermal diffusivity of turkey meat is roughly 1.3 × 10?? m²/s.

Added by Kimberly A.

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