**what is critical radius of insulation**We all know that by adding more insulation to a wall we can increase its thermal resistance (by reducing the heat transfer) and adding insulation always increases the thermal resistance of the wall without affecting the convection resistance.

Adding insulation to a cylindrical piece or a spherical shell, however, is a different matter. The additional insulation increases the conduction resistance of the insulation layer but decreases the convection resistance of the surface because of the increase in the outer surface area for convection. The heat transfer from the pipe may increase or decrease, depending on which effect dominates.

Consider a cylindrical pipe of outer radius r1 whose outer surface temperature T1 is maintained constant . The pipe is now insulated with a material whose thermal conductivity is k and outer radius is r2. Heat is lost from the pipe to the surrounding medium at temperature tܣ, with a convection heat transfer coefficient h . The rate of heat transfer from the insulated pipe to the surrounding air can be expressed as

critical radius of insulation heat transfer rate reaches maximum is determined differentiating the above equation with respect to r and equate it to zero

which will provide Rcr. = k/h,