Consider a very thin pipe with negligible conduction resitance with a hot fluid inside and cold ambient temperature. Also, consider an insulation material with known thermal conductivity. Our aim is to insulate the pipe with this insulation material with no option to choose another insulation material.

A little thing to remember is the relation to calculate the crtical radius of insulation

Consider the plot drawn below which shows the thermal resistance of the system versus the outer radius of insulation. The outer radius of the pipe (which is the inner radius of the insulation) is at point (1), and the critical radius of insulation -calculated using the previous formula- for the insulation material and the ambient fluid is at point (2).

At point (1) the thickness of insulation is zero (because radius of insulation equals the outer radius of the pipe) and so the thermal resistance of the system

Note that by adding insulation (increasing radius of insulation) starting from point (1) to point (2), the thermal resistance of the system decreases that it becomes

By adding extra insulation starting from point (2) to point (3), the thermal resistance of the sytem increases, but it is

Adding more insulation starting from point (3) is going to increase the thermal resistance of the system that it will be

So, Based on the previous example, it is economically-recommended to select an insulator material that has a critical radius of insulation smaller than the pipe radius so that insulation will take place whatever the insulation thickness.

A little thing to remember is the relation to calculate the crtical radius of insulation

**In the following paragraphs, bare pipe means a pipe without insulation.***Note:*Consider the plot drawn below which shows the thermal resistance of the system versus the outer radius of insulation. The outer radius of the pipe (which is the inner radius of the insulation) is at point (1), and the critical radius of insulation -calculated using the previous formula- for the insulation material and the ambient fluid is at point (2).

At point (1) the thickness of insulation is zero (because radius of insulation equals the outer radius of the pipe) and so the thermal resistance of the system

__is only__convection.Note that by adding insulation (increasing radius of insulation) starting from point (1) to point (2), the thermal resistance of the system decreases that it becomes

__lower than__the resistance of the bare pipe. That means the insulation in this case improves heat transfer rather that insulation.By adding extra insulation starting from point (2) to point (3), the thermal resistance of the sytem increases, but it is

__still lower than__the thermal resistance of the bare pipe.Adding more insulation starting from point (3) is going to increase the thermal resistance of the system that it will be

__higher than__the thermal resistance of the bare pipe. So, insulation will actually take place starting from point (3) and therefore, the insulation material between points (1) and (3) is just a wasted material.So, Based on the previous example, it is economically-recommended to select an insulator material that has a critical radius of insulation smaller than the pipe radius so that insulation will take place whatever the insulation thickness.