Heat loss through an insulated pipeby yoghurt54 Tags: cylinder, cylindrical, heat equation, thermal conductivity 

#1
Jan1111, 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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