# Heat loss through an insulated pipe

by yoghurt54
Tags: cylinder, cylindrical, heat equation, thermal conductivity
 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 $$\vec{J} = \vec{h} \Delta T$$, where the magnitude h is assumed to be constant. Show that the heat loss per length of pipe is inversely proportional to $$\frac{1}{hr} + \frac{1}{k} ln(\frac{r}{R})$$ 2. Relevant equations I guess that $$\vec{J} = -\kappa \nabla T$$ is useful, as is the thermal diffusion equation: $$\nabla^{2} T = - \frac{C}{\kappa} \frac{\partial T}{\partial t}$$ 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: $$T'(r') = T - constant \times ln(\frac{r'}{R})$$ 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.

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