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Fourier's Law of Heat Conduction
Relates the rate of heat conduction through a material to its thermal conductivity and the temperature gradient across it.
Interpretation
q = −k(dT/dx). Heat flux due to temperature gradient. k is thermal conductivity. Used in steady‑state heat transfer analysis. Foundation of thermal design.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| q | Heat flux | W/m2 |
| k | Thermal conductivity | W/m.K |
| dT/dx | Temperature gradient | K/m |
What it means
This is identical to id=56. Fourier’s law is the basic equation for conduction heat transfer. It states that the heat flux (q) is proportional to the negative temperature gradient. The thermal conductivity k is a material property that measures the ability to conduct heat. This law is used to compute heat loss through walls, insulate pipes, cool electronics, and design heat exchangers. In solids, heat conduction is due to lattice vibrations and free electrons. Understanding Fourier’s law is crucial for engineers in building, chemical, mechanical, and aerospace disciplines to manage energy and maintain safe operating temperatures.
Worked example
Fourier's Law – Two Examples
Real‑World| Parameter | Value |
|---|---|
| k | 401 W/m·K |
| dT/dx | −100 K/m |
| Parameter | Value |
|---|---|
| k | 0.04 W/m·K |
| dT/dx | −20 K/m |
Common mistakes
- Same as ID 56 – see that entry.
- Note: For multi‑dimensional heat flow, use the full heat equation.
Applications
Fourier's law (q = −k·dT/dx) is the fundamental equation for conductive heat transfer. It is used in design of thermal insulation, heat exchangers, and thermal management systems. Engineers determine heat flux, temperature gradients, and thermal conductivity. The law is critical for sizing cooling systems in electronics, for designing furnaces and ovens, and for evaluating building energy efficiency. By applying Fourier's law, engineers can predict temperature profiles, ensure safe operating temperatures, and optimise the thermal performance of devices and systems. It is also the basis for thermal conductivity measurement techniques.
- Design of heat sinks, heat spreaders, and thermal interface materials
- Insulation design for buildings, pipes, and cold chains
- Thermal modelling of power electronics and LED lighting
- Furnace and reactor thermal design
- Thermal property characterisation of materials
Frequently Asked Questions
Fourier's law states that the heat flux is proportional to the temperature gradient: q = –k · ∇T. For one dimension, q = –k · dT/dx. It is the fundamental equation for heat conduction.
k is a measure of a material's ability to conduct heat. High k (e.g., copper ~400 W/m·K) indicates good conduction; low k (e.g., air ~0.025) indicates good insulation.
Steady‑state: temperature does not change with time (dT/dt=0). Fourier's law directly gives the heat flux. Transient: temperature changes with time, requiring the heat equation: ρ·c_p·∂T/∂t = ∇·(k∇T) + Q_gen.
For layers in series, the heat flux is constant through all layers. The total thermal resistance is the sum of individual resistances (L/kA). The heat flux is q = (T₁ – T₂) / Σ(L_i/k_i).
- Metals: 50‑400 W/m·K.
- Ceramics: 2‑10 W/m·K.
- Polymers: 0.1‑1 W/m·K.
- Insulators: 0.01‑0.05 W/m·K.
For metals, k generally decreases with increasing temperature (electron scattering). For insulators, k often increases with temperature (phonon transport). At low temperatures, k can peak.
The heat equation combines Fourier's law with energy conservation: ρ·c_p·∂T/∂t = k·∇²T + Q_gen. It is the governing equation for transient conduction.
- Using a constant k when it varies with temperature.
- Ignoring the negative sign.
- Applying steady‑state to transient problems.
- Not including heat generation.
Using steady‑state methods (guarded hot plate) or transient methods (laser flash). The method depends on the material and temperature range.
- Designing thermal insulation.
- Heat exchanger design.
- Electronic cooling.
- Thermal stress analysis.