Heat Cool and Insulate

gintable:

It is best not to think about R-value for cylindrical situations. R-value only has meaning for prismatic insulation where the area available for heat transfer on both sides of the insulation is common. By definition, the R value (R") is R" = R*A, where A is the area available for heat transfer. If it is a uniform material, R" = L/k, the insulation length of thickness/the thermal conductivity.

If you wanted to define an equivalent R" value for cylindrical situations, you’d have a different version for both the inner surface and the outer surface.

Instead, you need the actual thermal conductivity of the material (k), the inner and outer radii of the insulation (ri and ro), and the length of the system (L).

To find the thermal resistance of a cylindrical insulation or pipe, use
R = ln(ro/ri)/(2*Pi*k*L)

So, observing the trends, they are mostly obvious. The larger the ratio of radii, the more thermal resistance (learn what a natural logarithm is so you can understand this trend). The larger the conductivity or total length, the less thermal resistance offered.

The phenomena you are asking about is the critical insulation radius. This comes in to play when convection on the outer surface of the system is involved (due to air flow usually).

Combine the insulation wall resistance (R1) in series with the convection resistance (R2) to find the total thermal resistance:

Rnet = R1 + R2

R1 = ln(ro/ri)/(2*Pi*k*L)

And if you know the convection coefficient h,
R2 = 1/(h*A), where A is the surface area exposed to convection (assuming uniform value of h).

Re-construct:
Rnet = ln(ro/ri)/(2*Pi*k*L) + 1/(h*A)

And recall the area of a cylindrical surface:
A = 2*Pi*ro*L

Thus:
Rnet = ln(ro/ri)/(2*Pi*k*L) + 1/(2*Pi*ro*L*h)

Assuming that ri, k, L, and h are independent of ro (which makes sense), we can find the critical value of ro, using the optimization methods of Calculus. Whether it is a minimum or maximum, can also be determined with an inspection method of Calculus. I will not show the details, but the results are below.

As it turns out, the critical value of ro is:
ro_crit = k/h

And it turns out that ro is a minimum value of the total resistance.

A typical conductivity of an insulation is on the order of k=0.05 W/m-C.
A typical heat transfer coefficient for natural convection in air is h=15 W/m^2-C

Plugging in the numbers yields ro_crit = 3 millimeters. 3 mm is less than the outer radius of a typical pipe (without insulation). So for normal pipe insulation sizes, adding more insulating material is still an insulation advantage.

Critical insulation radius is more of interest for wire insulation than for pipe insulation. With wires, we only add the insulation, such that it insulates electrically. Since there is no such thing as electrical convection, only the wire insulation itself serves the purpose of electrically insulating.

Thermally, we want the wire insulation to conduct away as much residual heat as possible, and thus it is a good design idea to make the outer radius of wire insulation, the critical outer radius value. Wires are typically thinner than pipes, so these critical radius values are much more practical.