Critical thickness of insulation

Definition-

Critical thickness is the maximum radius of insulation up to which heat transfer increases and after which when we add more insulation on it, will decrease the rate of heat transfer.

Physical significance-

• When r(critical) less than R 

Conduction  - slightly decreases
Convection. - rapidly increases
The result- overall effect being a decrease in total resistance to an increase in heat transfer rate.

• When r(critical) greater than R

Conduction- rapidly decreases 
Convection- slightly increases
The result- overall resistance increased so heat transfer will be decreased

Application-
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 rcr) would increase the total resistance and therefore decrease the heat loss. This behavior would be desirable for insulation of pipes, where insulation is added to reduce heat loss to the surroundings.