Conduction: Walls, Pipes and Insulation
Fourier’s law, composite walls and U-values, cylinders, insulation thickness, the critical radius and contact resistance.
Fourier’s law
In steady conduction the heat flux is proportional to the temperature gradient and flows down it:
q″ = −k dT/dx plane wall: Q̇ = k A (T₁ − T₂) ÷ L
The conductivity k is a material property that varies a little with temperature (and a lot with moisture: wet insulation can conduct ten times as much as dry). The values are in Heat, Temperature and the Three Modes.
A composite wall
A wall made of several layers is a chain of resistances in series: each layer has R″ = L ÷ k (m²·K/W per square meter), and the total is their sum. The same heat flux passes through every layer, so each layer’s temperature drop is q″ R″.
Worked example: heat loss through a cavity wall
A wall of 15 mm plaster, 200 mm brick, 100 mm mineral wool and a 100 mm outer brick leaf, with air at 20 °C inside (film coefficient 8 W/m²·K) and −5 °C outside (25 W/m²·K in wind).
| Layer | Thickness | k (W/m·K) | R″ = L ÷ k (m²K/W) | Drop at q″ (K) |
|---|---|---|---|---|
| Inside air film | — | — | 0.125 | 1.0 |
| Plaster | 15 mm | 0.500 | 0.030 | 0.2 |
| Brick | 200 mm | 0.720 | 0.278 | 2.2 |
| Mineral wool | 100 mm | 0.040 | 2.500 | 20.1 |
| Outer brick leaf | 100 mm | 0.720 | 0.139 | 1.1 |
| Outside air film | — | — | 0.040 | 0.3 |
| Total | 3.112 | 25.0 |
- Total resistance R″ = 3.112 m²·K/W, so the overall coefficient is U = 1 ÷ R″ = 0.321 W/m²·K.
- Heat flux q″ = U (Ti − To) = 0.321 × 25 = 8.0 W/m²; a 12 m² wall loses 96 W.
- The 100 mm of mineral wool is 2.50 of the 3.11 total: 80 percent of the resistance. Replace it with the same thickness of brick and the loss rises to 33 W/m², 4.1 times as much.
- The inside wall surface sits at 19.0 °C, only 1.0 K below the room: warm enough to avoid condensation (see Insulation, Heat Loss and Troubleshooting).
Films and the overall U-value
The air films on each side add resistance, and the overall heat transfer coefficient U combines everything: 1 ÷ U = 1 ÷ hi + Σ L ÷ k + 1 ÷ ho. Typical surface coefficients are about 8 W/m²·K inside (still air, including radiation) and 25 W/m²·K outside (wind). Building codes set maximum U-values: roughly 0.15 to 0.3 W/m²·K for new walls and roofs in cold climates, 0.8 to 1.2 for good windows.
Where there is a thermal bridge (a steel beam, a concrete slab edge, mortar between insulation boards) heat flows round the insulation, and the real loss is higher than the one-dimensional U-value says. Software or a thermal-bridge catalog is needed for the detail.
Cylinders: insulating a pipe
Conduction through a cylinder of inner radius r₁ and outer radius r₂ has a resistance per meter of R′ = ln(r₂ ÷ r₁) ÷ (2 π k). Because the area grows with radius, the first layers of insulation do the most good.
Worked example: insulating the 80 °C pipe
- The bare 60.3 mm pipe of the previous page loses about 164 W/m (natural convection plus radiation, h ≈ 14.4 W/m²·K).
- Cover it with mineral wool (k = 0.040) and assume a combined outer coefficient of 10 W/m²·K. For 40 mm thickness: R′ = ln(70.15 ÷ 30.15) ÷ (2π × 0.04) + 1 ÷ (10 × 2π × 0.0702) = 3.36 + 0.23 = 3.59 m·K/W, so q′ = 60 ÷ 3.59 = 16.7 W/m, a 90 percent saving.
- The surface of the insulation is only 23.8 °C: safe to touch. Over a year (8 000 h) a 100 m run saves about 117 557 kWh of heat.
| Insulation thickness | Heat loss | Saving against bare | Surface temperature |
|---|---|---|---|
| 20 mm | 25.6 W/m | 84 % | 28.1 °C |
| 40 mm | 16.7 W/m | 90 % | 23.8 °C |
| 60 mm | 13.2 W/m | 92 % | 22.3 °C |
| 100 mm | 10.1 W/m | 94 % | 21.2 °C |
The saving rises quickly for the first 20 to 40 mm and then flattens: doubling from 40 to 80 mm only saves a further few percent. The economic thickness (see Insulation, Heat Loss and Troubleshooting) balances the cost of the insulation against the value of the heat.
The critical radius
There is a surprise for very thin conductors. Adding insulation raises the conduction resistance but also enlarges the outer surface, which lowers the convection resistance. For a small wire the second effect can win, and a thin layer of insulation increases the loss until the outer radius passes the critical radius rc = k ÷ h.
With k = 0.04 and h = 10 the critical radius of mineral wool is only 4 mm, far below any pipe, so insulating pipes always helps. The effect matters for fine wires, small cables, heating elements and thin tubing, where the insulation is chosen to shed heat (it is why a thin enamel coating on a wire does not make it hotter and why power cables are rated with their insulation).
Contact resistance
Where two solid surfaces touch, the real contact is only at the high points, so there is a thermal resistance at the interface, R″c (m²·K/W). It is typically 0.0001 to 0.0005 m²·K/W for pressed metal joints and falls with pressure, smoother surfaces, and a filler such as thermal grease or a gap pad. For a heat flux of 100 kW/m² (a power module), a joint with R″c = 2.0 × 10⁻⁴ gives a temperature step of 20 K: a significant part of the budget (see Cooling Electronics and Thermal Design).
Related
- Previous: Heat, Temperature and the Three Modes. Next: Convection and Heat Transfer Coefficients.
- Material Properties and the Material Properties Table.
- Heat Sink and Thermal Resistance Calculator.
- Heat Transfer: the whole guide.
FAQ
Why is the insulation’s share of resistance so big?
Resistance is thickness divided by conductivity. Mineral wool has a conductivity about 18 times lower than brick, so 100 mm of it is equal to about 1.8 m of brick. Metals have almost no resistance at normal thicknesses.
What happens when insulation gets wet?
Water conducts about 15 times better than still air, and the mineral fiber’s trapped air is displaced, so the effective k can rise tenfold. Keep insulation dry and use a vapor barrier on cold pipes.
Do I use the inside or outside area for a pipe?
Use the resistance formula R′ = ln(r₂ ÷ r₁) ÷ (2πk) per meter of pipe; it needs no area choice. If you use an overall U, state which area it is based on.
How accurate is the one-dimensional method?
Good for large flat walls and long pipes. Corners, windows, bridges and supports add heat flow in two or three dimensions: use a heat flow program for the detail.
References
- Incropera, F. P., et al., Fundamentals of Heat and Mass Transfer, Wiley (conduction, composite walls, critical radius).
- ISO 6946 (building components: thermal resistance and U-value) and ISO 10456 (hygrothermal properties).
- ASHRAE Handbook – Fundamentals, Chapter 25–26 (heat transfer, thermal and water vapor properties).
- ASTM C680 (heat gain or loss from insulated pipe and equipment) and C177.
Last reviewed: October 2026. Conductivities and film coefficients are typical.