Question

The optimum thickness of insulation of steam pipe line is

• A. \(k / h_o\)
• B. \(h_o / k\)
• C. \(k + h_o\)
• D. \(k - h_o\)

Hint:

Critical Radius of Insulation:
Cylinder (Pipe) = $\mathbf{r_c = \frac{k}{h_o}}$.
Sphere = $r_c = \frac{2k}{h_o}$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Critical radius of insulation for cylindrical pipes: the critical radius ($r_c$) at which heat transfer reaches a maximum (below which adding insulation increases heat loss and above which insulation reduces heat loss) is given by $r_c = k / h_o$, where $k$ is thermal conductivity and $h_o$ is the external convective heat transfer coefficient.
Key Formula or Approach:
\[ \mathbf{Critical \text{ } Radius \text{ } of \text{ } Insulation \text{ } (r_c)} = \mathbf{\frac{k}{h_o}} \quad [k = \text{Insulation Conductivity}, \; h_o = \text{Outside Film Coefficient}] \]

Step 2: Detailed Explanation:
In thermal insulation engineering for steam pipelines and chilled glycol lines:
- Adding radial insulation around a cylinder produces two competing heat transfer effects:
1. Conduction Resistance: Increases logarithmically ($R_{\text{cond}} = \frac{\ln(r_2/r_1)}{2\pi k L}$).
2. Convection Surface Area: Increases linearly ($R_{\text{conv}} = \frac{1}{2\pi r_2 L h_o}$), decreasing convective resistance.
- Differentiating total thermal resistance with respect to outer radius $r_2$ and setting $\frac{dR_{\text{total}}}{dr_2} = 0$:
\[ \mathbf{r_c = \frac{k}{h_o}} \]
- For a steam pipe where outer radius $r_1 > r_c$, any insulation added beyond $r_c = k/h_o$ decreases heat loss, establishing $k / h_o$ (A) as the fundamental critical/optimum design parameter.

Step 3: Final Answer:
Hence, the optimum thickness/critical radius parameter is \(k / h_o\), matching option (A).