Heat loss through an insulated pipe

by yoghurt54
Tags: cylinder, cylindrical, heat equation, thermal conductivity
yoghurt54 is offline
Jan11-11, 11:58 AM
P: 19
1. The problem statement, all variables and given/known data

A pipe of radius R is maintained at temperature T. It is covered in insulation and the insulated pipe has radius r. Assume all surfaces lose heat through Newton's law of cooling

[tex] \vec{J} = \vec{h} \Delta T[/tex], where the magnitude h is assumed to be constant.

Show that the heat loss per length of pipe is inversely proportional to

[tex] \frac{1}{hr} + \frac{1}{k} ln(\frac{r}{R}) [/tex]

2. Relevant equations

I guess that

[tex]\vec{J} = -\kappa \nabla T[/tex]

is useful, as is the thermal diffusion equation:

[tex] \nabla^{2} T = - \frac{C}{\kappa} \frac{\partial T}{\partial t} [/tex]

3. The attempt at a solution

I'm guessing that this is the steady state, and that because there's no azimuthal or translational variance in temperature, then we can find T(r') to be:

[tex]T'(r') = T - constant \times ln(\frac{r'}{R})[/tex]

If we define the length of the pipe to be L and the rate of heat loss to be

[tex]\stackrel{.}{Q} = 2\pi r L J [/tex]

but I have no idea where to proceed from here.
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