# Thermal conductivity resistance for a squre pipe

I would like to derive an expression for the thermal conduction resistance through a square pipe.

For example, consider a cylindrical pipe, the conduction resistance can easily be obtained from Fourier's law in polar coordinates:

$$q=-kA\frac{dT}{dr}$$ where we integrate from $r_{i}$ to $r_{o}$ and $T_{i}$ to $T_{o}$. After integration, we end up getting the expression: $$q=2\pi Lk\frac{T_i-T_o}{ln(\frac{r_o}{r_i})}$$ Where we take $\frac{ln(\frac{r_o}{r_i})}{2\pi Lk}$ as the conductivity resistance through a cylindrical pipe wall of thickness $L$.

So what about a pipe with a square cross section with wall thickness $L$? How can this be done using integration?

• have you googled? what did you find? – aaaaa says reinstate Monica Feb 22 '17 at 2:23
• nothing. I think what I am going to do is consider a single (trapezoidal) face, and where the area that heat passes through increases as you go further towards the outer surface. Integrate like that, then multiply by 4 (for each edge). Either that or find an "effective" radius of the square and use the above formula – Mike James Johnson Feb 22 '17 at 2:28