Subtopic: Steady state heat transfer in conduction, convection and radiation
Question Statement:
Regarding heat transfer insulation on cylindrical pipes and spherical vessels in food processing plants, which of the following statements are TRUE?
Options:
(A) The critical radius of insulation for a circular pipe is $r_c = k/h$, where $k$ is thermal conductivity and $h$ is outer convective coefficient
(B) The critical radius of insulation for a sphere is $r_c = 2k/h$
(C) Adding insulation to a bare cylinder whose outer radius $r_o < r_c$ increases the heat loss rate until $r = r_c$
(D) For steam pipes with outer radius $r_o > r_c$, adding any insulation will decrease heat loss
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
1. In radial systems, adding insulation increases conduction resistance ($R_{cond} = \frac{\ln(r/r_i)}{2\pi k L}$) while decreasing outer convection resistance ($R_{conv} = \frac{1}{2\pi r h L}$).
2. For a cylinder, differentiating total thermal resistance yields minimum resistance at critical radius $r_c = k/h$. (Statement A is True)
3. For a sphere, minimum resistance occurs at $r_c = 2k/h$. (Statement B is True)
4. If $r_o < r_c$, adding insulation decreases total resistance, thereby INCREASING heat loss until $r = r_c$. (Statement C is True)
5. If $r_o > r_c$, adding insulation always increases total thermal resistance, strictly decreasing heat loss. (Statement D is True)