Subtopic: Steady state heat transfer in conduction, convection and radiation
Question Statement:
A cylindrical stainless steel pipe of outer radius $r_1 = 0.025\text{ m}$ carries chilled refrigerant. The pipe is covered with mineral wool insulation of thermal conductivity $k = 0.040\text{ W/(m}\cdot\text{K)}$. The external ambient air has a convective heat transfer coefficient of $h = 8.0\text{ W/(m}^2\cdot\text{K)}$. The critical radius of insulation for the cylinder ($r_c = \frac{k}{h}$) in $\text{mm}$ is ________ (answer in integer).
Answer Key:
5
Step-by-Step Solution & Explanation
Multi-Approach Solution:
**Method 1: Critical Radius Formula**
For a cylinder:
$r_c = \frac{k}{h} = \frac{0.040\text{ W/(m}\cdot\text{K)}}{8.0\text{ W/(m}^2\cdot\text{K)}} = 0.0050\text{ m} = 5.0\text{ mm}$
**Method 2: Derivative of Total Thermal Resistance**
Thermal resistance per unit length:
$R' = \frac{\ln(r/r_1)}{2 \pi k} + \frac{1}{2 \pi r h}$
Setting $\frac{dR'}{dr} = 0 \implies \frac{1}{2 \pi k r} - \frac{1}{2 \pi h r^2} = 0 \implies r_c = \frac{k}{h} = 5\text{ mm}$.