Critical Thickness for Cylindric Insulation

Consider now the heat loss from an insulated pipe as a function of the insulation thickness. Note from the preceding example that the conductive resistivity of the pipe walls, compared with the conductive resistivity of the insulation, and the inner convective resistivity, compared with the outer convective resistivity, can be neglected. Consequently, the inner surface of the insulation assumes approximately the temperature of the inner fluid, 7.  

For notational convenience let the thermal conductivity and inner and outer radii of the insulation be k, n, r, respectively, and the outer heat transfer coefficient be h. Equation (2.29) may be then rearranged as 

The heat loss corresponding to the critical radius, inserting Eq. (2.33) into Eq. (2.32), is found to be 

Reconsider Ex. 2.2 for (1) fiberglass (k = 0.04 W/m-K), (2) plaster (gypsum) (k S 0.20 W/m-K). For each material we wish to determine the thickness of insulation for which the heat loss from the pipe is (a) maximum, (b) equal to that from the bare-pipe analysis. 

The minimum of total resistance is at a radius less than the outside radius of pipe So there is no physically realizable critical radius. The total resistance increases with increasing insulation and reduces the heat loss. 

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