Fourier's Law of Heat Conduction
Fourier's Law: Q/t = -kA(ΔT/d).
Calculate heat flow through walls, pipes, and materials.
Includes R-value relationship and worked examples.
The Formula
Fourier's Law describes how heat flows through a solid material by conduction. The rate of heat transfer (Q/t, in watts) depends on three things: the material's thermal conductivity k, the cross-sectional area A through which heat flows, and the temperature gradient ΔT/d (temperature difference divided by thickness).
The negative sign indicates that heat flows from hot to cold — opposite to the direction of increasing temperature. In practical calculations, we usually drop the sign and just compute the magnitude.
Thermal conductivity k (W/m·K) is a material property. Copper: 401. Aluminum: 237. Steel: 50. Glass: 1.0. Brick: 0.7. Wood: 0.1–0.2. Fiberglass insulation: 0.04. Still air: 0.026.
In building science, this formula underlies the R-value system. R-value = d/k (in SI: m²·K/W). Higher R-value means better insulation. US R-values use imperial units (ft²·°F·h/BTU), where R-1 ≈ 0.176 m²·K/W.
For composite walls (multiple layers), total R-value = R₁ + R₂ + R₃ (resistances in series add). The heat flux is then Q/t = A × ΔT / R_total.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| Q/t | Rate of heat transfer (power) | Watts (W) |
| k | Thermal conductivity | W/(m·K) |
| A | Cross-sectional area | m² |
| ΔT | Temperature difference (hot − cold) | °C or K |
| d | Thickness of material | meters (m) |
Example 1
A brick wall (k = 0.7 W/m·K) is 0.2 m thick, 10 m² area, with 20°C inside and 0°C outside.
Q/t = 0.7 × 10 × (20/0.2) = 0.7 × 10 × 100
Q/t = 700 W (700 watts of heat lost through the wall)
Example 2
Adding 0.1 m of fiberglass insulation (k = 0.04) to the same wall. What is the new heat loss?
R_brick = 0.2/0.7 = 0.286 m²K/W; R_insulation = 0.1/0.04 = 2.5 m²K/W
R_total = 0.286 + 2.5 = 2.786; Q/t = 10 × 20 / 2.786
Q/t = 71.8 W (90% reduction in heat loss)
When to Use It
- Building energy audits and insulation specification
- Designing heat sinks for electronics cooling
- Industrial pipe insulation calculations
- Food storage and cold chain engineering