Question

Given below are two statements:
Statement (I): The critical radius of insulation is the thickness of insulation at which the rate of heat transfer through the body is minimum.
Statement (II): Rate of heat transfer increases with an increase in the thickness of insulation up to the critical radius of insulation.
In light of the above statements, choose the most appropriate answer from the options given below.

• A. Both Statement (I) and Statement (II) are true.
• B. Both Statement (I) and Statement (II) are false.
• C. Statement (I) is true but Statement (II) is false.
• D. Statement (I) is false but Statement (II) is true.

Hint:

Keep this summary in mind:
- Critical radius (\(r_c\)) \(\rightarrow\) Minimum thermal resistance \(\rightarrow\) Maximum heat loss.
- If \(r_{\text{outer}} < r_c\) \(\rightarrow\) Adding insulation increases heat loss.
- If \(r_{\text{outer}} > r_c\) \(\rightarrow\) Adding insulation decreases heat loss.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
For cylindrical or spherical geometries, adding insulation has two competing effects: it increases the conductive thermal resistance (reducing heat transfer) but also increases the outer surface area, which decreases the convective thermal resistance (increasing heat transfer). The critical radius of insulation is the size where these two opposing effects balance.

Step 2: Detailed Explanation:
Let us analyze each statement:

Statement (I): The total thermal resistance (\(R_{\text{total}}\)) of an insulated cylindrical pipe is the sum of conductive resistance and convective resistance: \[ R_{\text{total}} = \frac{\ln(r_{\text{out}}/r_i)}{2\pi k L} + \frac{1}{2\pi r_{\text{out}} h L} \] Differentiating \(R_{\text{total}}\) with respect to \(r_{\text{out}}\) and setting it to zero yields the critical radius of insulation: \[ r_c = \frac{k}{h} \] At this specific radius, the total thermal resistance is at aminimum.
Because the rate of heat transfer is inversely proportional to thermal resistance (\(Q = \Delta T / R_{\text{total}}\)), a minimum thermal resistance results in amaximum rate of heat transfer.
Therefore, Statement (I) isfalse because it states that the rate of heat transfer is minimum.

Statement (II): If the outer radius of the insulation is less than the critical radius (\(r_{\text{out}} < r_c\)), adding more insulation reduces the total thermal resistance. This is because the reduction in convective resistance is greater than the increase in conductive resistance.
Consequently, the rate of heat transfer increases with additional insulation up to the critical radius. Thus, Statement (II) istrue.

Step 3: Final Answer:
Statement (I) is false and Statement (II) is true, which corresponds to Option (D).