Critical radius of insulation

There are instances when the addition of insulation to the outside surfaces of cylindrical or spherical walls (geometries which have non-constant cross-sectional areas) does not reduce the heat loss.

 In cylindrical and spherical coordinates, the addition of insulation also increases the outer surface, which decreases the convection resistance at the outer surface. Moreover, in some cases, a decrease in the convection resistance due to the increase in surface area can be more important than an increase in conduction resistance due to thicker insulation. As a result, the total resistance may actually decrease resulting in increased heat flow.

The thickness up to which heat flow increases and after which heat flow decreases is termed as critical thickness. In the case of cylinders and spheres, it is called critical radius. The critical radius of insulation depends on the thermal conductivity of the insulation k and the external convection heat transfer coefficient h.

                             

 For the spherical shell, the critical radius of thickness is 2*k/h.

Where k is thermal conductivity & h is heat transfer coefficient.

 

 This(r₁ < r_cr )is desirable for cooling of electrical wire since the addition of electrical insulation would aid in transferring heat dissipated in the wire to the surroundings. On the other hand, any further addition of material (beyond r_cr) would increase the total resistance and therefore decrease the heat loss. This behavior would be desirable for insulation of pipes(i.e Steam Pipes), where insulation is added to reduce heat loss to the surroundings.