In the previous article, we studied the heat flow through a cylindrical pipe without heat generation. Taking the discussion forward, we will now look at the concept of critical radius of insulation. 

Heat loss from in insulated cylindrical system to an external convective surrounding can be usually reduced by increasing the thickness of insulation. But this cannot be applied directly to small diameter systems, e.g. insulation of electrical wires, electrical resistors and other devices with current flow through them. Let us consider a wire with an insulation sleeve of conductivity 'k', which has an electrical resistivity 'Re' and carries a current 'I', as shown in the figure below. The heat generated in the wire is transferred to the outer surroundings via conduction throught the insulation and convection at the outer insulation surface.


The heat dissipated in the wire is lost to the ambient, and the formula used to give the value of heat transfer rate is,  , where the total resistance  is the sum of resistances for conduction through the insulation and external convectionm or


As per equation (2) it is clear that as the outer insulation radius 'r' increases, Rcond also increases whereas Rconv decreases due to the increase in the outer surface area. A considerable decrease in the latter means that there is an optimum value of 'r', or a critical radius rcr of insulation, for which the value of Rtotal is minimum and heat loss q is maximum. This value can be obtained by differentiating equation (2) with respect to 'r' and equating the derivative equal to zero. 



You can easily derive that the value of rcr will yield a positive value for the second derivative of equation (2), which will indicate minimum total resistance. 

The figure below depicts the variation of Rtotal given by equation (2) for an electrical resistor or any current carrying wire or conductor, and that the changes in the values of Rcond and Rconv with result in a minimum value of Rtotal is self evident. 

The condition described above is also evident in spherical system where, by undergoing the same mathematical analysis, the corresponding critical radius can be achieved as,

You should remember that the practical use of critical radius of insulation is limited to small diameter systems in very low convective coefficient surroundings. What this means is, that the radius of cylindrical system, which would need insulation for a "cooling" effect, should be less than .