Explain the concept of critical radius of insulation.

Critical radius of insulation is the outer radius at which adding more insulation actually increases heat transfer instead of decreasing it. For a cylinder, r_cr = k/h, where k is insulation thermal conductivity and h is external convective coefficient. Below critical radius, adding insulation increases surface area for convection more than it adds thermal resistance, thus increasing heat loss. This concept is crucial for small-diameter pipes and electrical wires. For typical insulation (k~0.05 W/m-K) in air (h~10 W/m²-K), r_cr ≈ 5mm.