Discussion
Critical thickness of insulation (rcr) for a pipe having thermal conductivity (k) and convective heat transfer coefficient (ho) is __________.
Answer: B
The critical radius of insulation for a pipe is \(k/h_{o}\).
For insulation of outer radius r on a pipe, the total resistance per unit length is conduction through the insulation plus convection from its outer surface:
\[ R' = \frac{\ln(r/r_{i})}{2\pi k} + \frac{1}{2\pi r h_{o}} \]
Setting \(dR'/dr = 0\) gives \(\frac{1}{kr} = \frac{1}{h_{o}r^{2}}\), so \(r_{cr} = k/h_{o}\). Below this radius, adding insulation actually increases heat loss, because the larger surface gains more convection than the extra conduction resistance costs.
\(2k/h_{o}\) is the critical radius for a sphere. \(h_{o}/k\) and \(2h_{o}/k\) invert the ratio and do not even have the units of length.
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