Question

The critical radius of insulation is

• A. \( \frac{\text{The thermal conductivity of an insulating material}}{\text{Heat transfer coefficient at the outer surface of insulation}} \)
• B. \( \frac{\text{The thermal conductivity of metal to be insulated}}{\text{Heat transfer coefficient at the outer surface of insulation}} \)
• C. \( \frac{\text{The thermal conductivity of an insulating material}}{\text{Heat transfer coefficient at the inside surface of insulation}} \)
• D. \( \frac{\text{The thermal conductivity of metal to be insulated}}{\text{Heat transfer coefficient at the inside surface of insulation}} \)

Hint:

The critical radius of insulation for a cylinder is \( r_c = \frac{k}{h} \), where \( k \) is the insulation’s thermal conductivity, and \( h \) is the outer convective heat transfer coefficient.

Answer 1

The Correct Option is A

Step 1: Understand the concept of critical radius of insulation.
The critical radius of insulation applies to cylindrical or spherical objects (e.g., pipes or wires) where insulation is added to reduce heat loss. The critical radius is the outer radius of insulation at which the heat loss is maximized. Adding insulation beyond this radius reduces heat loss, while adding insulation below this radius increases heat loss due to the increased surface area for convection.
Step 2: Derive the critical radius for a cylindrical pipe.
Consider a pipe with inner radius \( r_1 \), outer radius of insulation \( r \), thermal conductivity of the insulation \( k \), and convective heat transfer coefficient at the outer surface \( h \). The heat transfer rate \( Q \) through the cylindrical insulation (steady-state, radial conduction) is determined by combining the conduction resistance through the insulation and the convection resistance at the outer surface:
Conduction resistance: \( R_{\text{cond}} = \frac{\ln(r/r_1)}{2\pi L k} \), where \( L \) is the length of the pipe.
Convection resistance: \( R_{\text{conv}} = \frac{1}{h A} = \frac{1}{h (2\pi r L)} \).
The total resistance is: \[ R_{\text{total}} = \frac{\ln(r/r_1)}{2\pi L k} + \frac{1}{h (2\pi r L)}. \] Heat transfer rate: \[ Q = \frac{\Delta T}{R_{\text{total}}} = \frac{\Delta T}{\frac{\ln(r/r_1)}{2\pi L k} + \frac{1}{h (2\pi r L)}}. \] To find the critical radius, maximize \( Q \) with respect to \( r \). This is equivalent to minimizing the total resistance. Take the denominator of \( Q \), multiply through by \( 2\pi L \): \[ \text{Denominator} = \frac{\ln(r/r_1)}{k} + \frac{1}{h r}. \] Differentiate with respect to \( r \) and set to zero: \[ \frac{d}{dr} \left( \frac{\ln(r/r_1)}{k} + \frac{1}{h r} \right) = \frac{1}{k} \cdot \frac{1}{r} - \frac{1}{h r^2} = 0, \] \[ \frac{1}{k r} = \frac{1}{h r^2}, \] \[ r = \frac{k}{h}. \] This \( r \) is the critical radius \( r_c \), where \( k \) is the thermal conductivity of the insulating material, and \( h \) is the convective heat transfer coefficient at the outer surface.
Step 3: Evaluate the options.
(1) \( \frac{\text{The thermal conductivity of an insulating material}}{\text{Heat transfer coefficient at the outer surface of insulation}} \): Correct, as \( r_c = \frac{k}{h} \), where \( k \) is the thermal conductivity of the insulation, and \( h \) is the outer convective coefficient. Correct.
(2) \( \frac{\text{The thermal conductivity of metal to be insulated}}{\text{Heat transfer coefficient at the outer surface of insulation}} \): Incorrect, as the thermal conductivity of the metal is not relevant for the critical radius; it’s the insulation’s conductivity that matters. Incorrect.
(3) \( \frac{\text{The thermal conductivity of an insulating material}}{\text{Heat transfer coefficient at the inside surface of insulation}} \): Incorrect, as the critical radius depends on the outer surface heat transfer coefficient, not the inside. Incorrect.
(4) \( \frac{\text{The thermal conductivity of metal to be insulated}}{\text{Heat transfer coefficient at the inside surface of insulation}} \): Incorrect, as both the metal’s conductivity and the inside coefficient are irrelevant. Incorrect.
Step 4: Select the correct answer.
The critical radius of insulation is \( \frac{\text{The thermal conductivity of an insulating material}}{\text{Heat transfer coefficient at the outer surface of insulation}} \), matching option (1).