Question

The critical radius \(r\) of insulation on a pipe is given by

• A. \(r=\frac{2k}{h}\)
• B. \(r=\frac{k}{h}\)
• C. \(r=\frac{k}{2h}\)
• D. \(r=\frac{h}{k}\)

Hint:

For cylindrical insulation, critical radius is \(r_c=\frac{k}{h}\). For spherical insulation, it is \(r_c=\frac{2k}{h}\).

Answer 1

The Correct Option is B

Critical radius of insulation is the radius at which heat loss from an insulated cylinder becomes maximum. For a cylindrical pipe, the critical radius of insulation is: \[ r_c=\frac{k}{h} \] where, \[ k=\text{thermal conductivity of insulation} \] and \[ h=\text{outside heat transfer coefficient} \] When insulation is added to a pipe, two effects occur. First, conduction resistance increases. Second, outside surface area for convection also increases. For small radii, increase in surface area may dominate, so heat loss may increase. At critical radius, heat loss becomes maximum. For a cylinder: \[ r_c=\frac{k}{h} \] For a sphere, the critical radius is: \[ r_c=\frac{2k}{h} \] Since the question asks for insulation on a pipe, the formula is: \[ r=\frac{k}{h} \]