Why we don't need to overthink the critical radius of insulation when considering hot water pipes?

Ans: The critical radius of insulation is given by,

The usual values of heat transfer coefficients for natural convection are as low as 5 W/m^2.K and the thermal conductivity of common insulating material is about 0.05 W/m.K

Therefore,using the above formula the critical radius of insulation is nearly 1 cm.

The radius of hot water pipes is higher than 1 cm in most cases. Therefore, adding more and more insulation would decrease the heat transfer from pipes. (Which is what we want that is to have lesser and lesser heat loss from pipes)

We can almost always will be in green zone in case of hot water pipes where adding more insulation can mean reduction in heat transfer.

The blue zone corresponds to electric wires whose radii are very small(compared to critical radius) hence adding insulation increases the heat transfer in those cases.