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"&quot;Explanation:\n\nConcept of the critical thickness of insulation:\n\n\n\tIn a plane wall thicker the insulation, lower the heat transfer rate. This is due to the fact the outer surface have always the same area.\n\tBut in cylindrical and spherical coordinates, the addition of insulation also increases the outer surface area which decreases the convection resistance at the outer surface.\n\tAt the same time conduction resistance increases due to increase in insulation thickness. So, to optimize the balance of conductive and convective resistance, a critical radius is decided. At this critical radius, the equivalent conduction and convection resistance will be minimum and heat transfer will be maximum.\n\tMoreover, in some cases, a decrease in the convection resistance due to the increase in surface area can be more significant than an increase in conduction resistance due to thicker insulation.\n\n\nAs a result, the total resistance may actually decrease resulting in increased heat flow.\n\n\n\nThickness up to which heat flow increases and beyond which heat flow decreases is termed as critical thickness. For cylinders and spheres, it is the critical radius (rc ).\n\nThat can be derived from the critical radius of insulation depends on the thermal conductivity of the insulation k and the external convection heat transfer coefficient h.\n\nFor cylinder\n\n({{bf{r}}_{bf{c}}} = ;frac{k}{h})\n\nFor sphere\n\n({{bf{r}}_{bf{c}}} = ;frac{{2k}}{h})&quot;"
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