Heat loss through an insulated pipe

[yoghurt54]

Homework Statement

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

$$\vec{J} = \vec{h} \Delta T$$, where the magnitude h is assumed to be constant.

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

$$\frac{1}{hr} + \frac{1}{k} ln(\frac{r}{R})$$

Homework Equations

I guess that

$$\vec{J} = -\kappa \nabla T$$

is useful, as is the thermal diffusion equation:

$$\nabla^{2} T = - \frac{C}{\kappa} \frac{\partial T}{\partial t}$$

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:

$$T'(r') = T - constant \times ln(\frac{r'}{R})$$

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

$$\stackrel{.}{Q} = 2\pi r L J$$

but I have no idea where to proceed from here.

 

[Nate810]

I'm guessing that it's related to the thermal diffusion equation and setting it up in terms of the Fourier number but I'm not quite sure. Any help is appreciated!