Subtopic: Steady state heat transfer in conduction, convection and radiation
Question Statement:
A thin-walled steam pipe of outer radius $r_o = 40\text{ mm}$ is covered with insulation having thermal conductivity $k = 0.08\text{ W/(m}\cdot\text{K)}$. The surrounding air has a convective heat transfer coefficient $h_o = 4.0\text{ W/(m}^2\cdot\text{K)}$. The critical radius of insulation is $r_c = k / h_o$. Adding this insulation to the pipe will:
Options:
(A) Always decrease heat loss because the outer radius of the pipe exceeds the critical radius of insulation
(B) Initially increase heat loss until the outer radius reaches $80\text{ mm}$
(C) Have zero effect on heat loss
(D) Cause thermal runaway because $k < h_o$
Answer Key:
A
Step-by-Step Solution & Explanation
The critical radius of insulation for a cylinder is:
$r_c = \frac{k}{h_o} = \frac{0.08\text{ W/(m}\cdot\text{K)}}{4.0\text{ W/(m}^2\cdot\text{K)}} = 0.020\text{ m} = 20\text{ mm}$
Since the outer radius of the bare pipe ($r_o = 40\text{ mm}$) is already greater than $r_c$, adding insulation only adds conduction resistance while convective resistance changes minimally, so total thermal resistance increases and heat loss strictly decreases.