Library / Heat Transfer / HTR-19
Heat Transfer
Critical insulation radius of a cylinder
Outer radius of insulation on a pipe or wire at which the heat loss is greatest.
- Ref
- HTR-19
- Rev
- v1
- Field
- Heat Transfer
- Result
- rcritical [mm]
For a long insulated cylinder (a pipe or wire) the total thermal resistance has two parts: conduction through the insulation, which rises with the outer radius, and convection from the outer surface, which falls as the surface area grows. Adding the two shows that the heat loss is greatest at an outer radius rcritical = kins/h. This is a radius, not a thickness. If the bare cylinder’s radius is below rcritical, adding insulation up to that radius increases the heat loss. Beyond it, more insulation reduces the loss, but not necessarily to below the bare-cylinder value.
Outer radius of insulation on a pipe or wire at which the heat loss is greatest. It is a radius, not a thickness. Below this radius, adding insulation increases the heat loss; above it, further insulation reduces it.
Calculator
- Critical diameter, 2k/h8 mm
- Convection coefficient that would give a critical radius of 25 mm1.6 W/(m² K)
- Heat-loss ratio for a bare cylinder of radius rc/2 insulated up to rc, relative to bare1.181
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Insulation with k = 0.04 W/(m K) in air with h = 10 W/(m² K) has a critical radius of 4 mm. A thin wire or small tube with an outer radius below 4 mm gains heat loss when it is insulated up to that radius: for a bare radius of 2 mm the loss rises by about 18 % at the critical radius.
- A pipe whose outer radius is already larger than 4 mm is above the critical radius, so any insulation reduces the loss. This is why insulation always works on large pipes but may make small cables or electric wires, which need to shed heat, lose more heat.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| kins Thermal conductivity of the insulation | 0.01 to 2 W/(m K) | 1 | 200 | 199 | +10% | +10 % |
| h External convection coefficient | 2 to 1,000 W/(m² K) | 0.04 | 20 | 19.96 | +10% | −9.09 % |
Sweep of External convection coefficient
| External convection coefficient (W/(m² K)) | Critical insulation radius (mm) |
|---|---|
| 2 | 20 |
| 126.8 | 0.3156 |
| 251.5 | 0.159 |
| 376.3 | 0.1063 |
| 501 | 0.07984 |
| 625.8 | 0.06392 |
| 750.5 | 0.0533 |
| 875.3 | 0.0457 |
| 1,000 | 0.04 |
Sweep one variable
Variables and how each one contributes
Critical insulation radius
What the equation gives you. Everything else feeds it.
Thermal conductivity of the insulation
The thermal conductivity of the insulation, in W/(m K) (about 0.03 to 0.05 for foam and mineral wool, 0.2 for rubber, 0.5 for some plastics). Raises the critical radius in direct proportion: a poorer insulator has a larger critical radius. Raising it by 0.01 (to 0.05) moves the result by +25 %, from 4 to 5 mm.
External convection coefficient
The convection coefficient on the outside surface, in W/(m² K) (about 5 to 25 for natural convection in air, higher for wind or flowing liquid). Lowers the critical radius in inverse proportion: stronger cooling of the outer surface moves the peak to a smaller radius. Raising it by 2 (to 12) moves the result by −16.7 %, from 4 to 3.333 mm.
Worked example with the values above
- Convert units: k = 0.04 W/(m K) and h = 10 W/(m² K).
- Divide: k/h = 0.04 / 10 = 0.004 m.
- rcritical = 0.004 m = 4 mm.
Assumptions and limits
- Steady radial conduction through a long cylindrical layer, with fixed inner-surface and ambient temperatures.
- Constant conductivity and convection coefficient, with radiation and end effects neglected.
Common mistakes
- Treating the result as a thickness: it is the outer radius of the insulation, so the thickness is rcritical minus the bare radius.
- Assuming insulation beyond the critical radius brings the loss below the bare value straight away: it only starts falling from the peak.
- Ignoring radiation, which adds to the effective h and so lowers the critical radius.
Statistics and uncertainty
The critical radius is a simple ratio, so its relative error is the quadrature sum of the errors in k and h. The convection coefficient h is often uncertain by 25 % or more, because it depends on wind, orientation and temperature difference, which makes rcritical uncertain by a similar amount; check the actual radius against both ends of that range.
Where it is used
- Deciding whether insulation helps on small pipes, tubing, electrical wire and cable.
- Choosing insulation thickness and material for steam, hot-water and refrigerant lines.
Maths you need
- Thermal conductivity: heat flow per metre per kelvin through a material
- Convection coefficient: heat flow per area per kelvin to a moving fluid