- #1
MisterX
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Homework Statement
Problem 1.60. A frying pan is quickly heated on the stovetop to 200 C. It has an iron handle that is 20 cm long. Estimate how much time should pass before the end of the handle is too hot to grab with your bare hand. (Hint: The cross-sectional area of the handle doesn't matter. The density of iron is about 7.9 g/cm3 and its specific heat is 0.45 J/g-C).
For iron [itex]k_t = 80 \frac{W}{m\cdot K}[/itex]
Homework Equations
[itex] \frac{Q}{\Delta t} = -k_t A \frac{dT}{dx}[/itex]
The Attempt at a Solution
So I might consider a little section at the end of the handle with length d which is receiving heat.
[itex]m = \rho A d[/itex]
[itex]T_{end} = \frac{Q_{end}}{c \cdot m} = \frac{Q_{end}}{c \rho A d}[/itex]
[itex]c \rho A d T_{end} = Q_{end}[/itex][itex] \frac{ c \rho A d \Delta T_{end}}{\Delta t} = -k_t A \frac{dT}{dx}[/itex]
The area cancels
[itex] \frac{ c \rho d \Delta T_{end}}{\Delta t} = -k_t \frac{dT}{dx}[/itex]
But we still don't know what is [itex]\frac{dT}{dx}[/itex], which presumably depends upon time. There is also that d still there.
Note that we are asked to derive the heat equation in a later problem, so I'm assuming I'm not supposed to use heat equation for this problem, but perhaps I am wrong. (I have already derived the heat equation from the Fourier Law of Heat Conduction).
I supposed I could assume d = dx = 20 cm, and dT = T - 200, with the initial condition for T being at room temperature and solve that differential equation. Is that what I'm supposed to do?
This problem is 1.60 from Schroeder Thermal Physics. It's not coursework or homework, as I am doing this independently, but I like you to treat it as if it were.