Question

Critical thickness of insulation for spheres is given by which of the following relationship ?

• A. k/h
• B. 2k/h
• C. h/2k
• D. 4k/h

Hint:

Remember the critical radius of insulation formulas:
- For a Cylinder: \( r_c = \frac{k}{h} \)
- For a Sphere: \( r_c = \frac{2k}{h} \)

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Adding insulation to a pipe or sphere increases the conduction resistance but decreases the external convection resistance due to the increased surface area.
The critical radius of insulation (\( r_c \)) is the outer radius at which the total heat transfer rate is maximized.
Any insulation added beyond this critical radius will successfully reduce heat transfer.

Step 2: Detailed Explanation:
Let us analyze the mathematical derivation for a sphere:
The total thermal resistance of an insulated sphere is:
\[ R_{\text{total}} = R_{\text{conduction}} + R_{\text{convection}} \]
\[ R_{\text{total}} = \frac{r_c - r_i}{4 \pi k r_i r_c} + \frac{1}{4 \pi r_c^2 h} \]
To find the minimum resistance (maximum heat transfer), we differentiate \( R_{\text{total}} \) with respect to \( r_c \) and set it to zero:
\[ \frac{d(R_{\text{total}})}{dr_c} = 0 \]
Solving this differentiation yields:
\[ r_c = \frac{2k}{h} \]
where:
\( k \) is the thermal conductivity of the insulation material,
\( h \) is the convective heat transfer coefficient of the surrounding air.
Therefore, the critical radius (often referred to as critical thickness in standard questions) is given by \( \frac{2k}{h} \).

Step 3: Final Answer:
The critical thickness (radius) of insulation for a sphere is \( \frac{2k}{h} \).