How Is Heat Transfer Calculated in a Cylindrical Tube with Insulation?

[joemama69]

Homework Statement

A Cylindrical Tube of Length 35m, inner radius = 2.5m
The tube is lines with 6cm thick insulation where k = 4X10^-5.
The inside temp=25 degrees celsius
Find the rate of heat delivered if the outside is -35 degrees celsius which keeps the inside temp at constant 25 degrees

Homework Equations

The Attempt at a Solution

Im having trouble finding the Area.

A = 2(pi)rL but when I plug in the radius do I use 2.5m or do I add the 6cm thk insulation to this

 

[rock.freak667]

I think you will need to add the 6cm to the 2.5m

 

[Mapes]

There are at least a couple ways of working the problem, but none involves adding 6cm to 2.5m. Since the pipe is expected to have a thermal conductivity far higher than the insulation (and because we don't know the pipe thickness), we need to assume the inside of the pipe is at approximately -35°C. The 6cm insulation is inside that, and it's this insulation that sustains the 60°C temperature gradient.

The most exact way to calculate the heat flux is to solve for the temperature distribution in the pipe by using cylindrical coordinates. But since the radius is much larger than the insulation thickness, we can get a close enough answer by assuming Cartesian coordinates and approximating the radius of the insulation with the average value: 2.47m.