We know that by adding more insulation to a wall always decreases heat transfer. The thicker the insulation, the lower the heat transfer rate. This is expected, since the heat transfer area *A * is constant, 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 *r*_{1} whose outer surface temperature *T*_{1} is maintained constant (Figure. 2.15). The pipe is now insulated with a material whose thermal conductivity is *k * and outer radius is *r*_{2}. Heat is lost from the pipe to the surrounding medium at temperature, with a convection heat transfer coefficient *h *. The rate of heat transfer from the insulated pipe to the surrounding air can be expressed as (Figure. 2.16)

Note that the critical radius of insulation depends on the thermal conductivity of the insulation *k * and the external convection heat transfer coefficient *h *. The rate of heat transfer from the cylinder increases with the addition of insulation for * **r*_{2}< r_{cr}, reaches a maximum when *r*_{2}=* r*_{cr}, and starts to decrease for *r*_{2}> * r*_{cr}. Thus, insulating the pipe may actually increase the rate of heat transfer from the pipe instead of decreasing it when *r*_{2}*< r*_{cr} .

Figure 2.16 Variation Of Heat Transfer Rate With Radius

The important question to answer at this point is to whether we need to be concerned about the critical radius of insulation when insulating hot water pipes or even hot water tanks. Should we always check and make sure that the outer radius of insulation exceeds the critical radius before we install any insulation? Probably not, as explained below.

The value of the critical radius *r*_{cr} will be the largest when *k * is large and *h * is small. Noting that the lowest value of *h * encountered in practice is about 5 W/m^{2}K for the case of natural convection of gases, and that the thermal conductivity of common insulating materials is 0.05 W/m^{2}K, the largest value of the critical radius we are likely to encounter is

This value would be even smaller when the radiation effects are considered. The critical radius would be much less in forced convection, often less than 1 mm, because of much larger *h * values associated with forced convection. Therefore, we can insulate hot water or steam pipes freely without worrying about the possibility of increasing the heat transfer by insulating the pipes.

The radius of electric wires may be smaller than the critical radius. Therefore, the plastic electrical insulation may actually enhance the heat transfer from electric wires and thus keep their steady operating temperatures at lower and thus safer levels.

The discussions above can be repeated for a sphere, and it can be shown in a similar manner that the critical radius of insulation for a spherical shell is