Subtopic: Steady state heat transfer in conduction, convection and radiation
Question Statement:
Critical thickness of insulation (r_cr) for a pipe having thermal conductivity (k) and convective heat transfer coefficient (h_o) is __________.
Options:
(A) 2k/h_o
(B) k/h_o
(C) h_o/k
(D) 2h_o/k
Official Answer Key:
B
Step-by-Step Solution & Explanation
For a cylinder (pipe), adding insulation has two opposing effects: it increases the conduction resistance, but it also increases the outer surface area and hence reduces the convection resistance. Total resistance per unit length:
R = ln(r/r₁)/(2πk) + 1/(2πr h_o)
Setting dR/dr = 0:
1/(2πkr) − 1/(2πr²h_o) = 0
⇒ r = k/h_o
This is the critical radius: below it, adding insulation increases heat loss; above it, insulation begins to reduce heat loss.
Important distinction: for a sphere the corresponding result is 2k/h_o — option (A) is planted for candidates who confuse the two geometries.
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