Formula & Calculator
Fourier's Law of Conduction
Heat flux is proportional to the negative temperature gradient through a material.
Interpretation
q = −k(dT/dx). Heat flux due to conduction. k is thermal conductivity. Describes steady‑state heat transfer. Used in thermal design, insulation, and electronic cooling.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| q | Heat flux | W/m² |
| k | Thermal conductivity | W/m·K |
| dT/dx | Temperature gradient | K/m |
What it means
Fourier’s law of heat conduction states that the heat flux (q) – the rate of heat transfer per unit area – is proportional to the negative temperature gradient (dT/dx) in the direction of heat flow. The proportionality constant is the material’s thermal conductivity (k). The negative sign indicates that heat flows from hotter to colder regions. This law is fundamental for analysing steady‑state conduction in solids. It is used to design thermal insulation, heat exchangers, electronic cooling systems, and building envelopes. In combination with the energy balance, it leads to the heat equation. Engineers use Fourier’s law to calculate heat losses, determine insulation thickness, and predict temperature distributions. Materials with high k (e.g., metals) are good conductors; those with low k (e.g., foams) are insulators. Understanding this law is essential for thermal management in engineering and for energy efficiency.
Worked example
Fourier's Law of Conduction – Two Examples
Real‑World| Parameter | Value |
|---|---|
| k | 0.04 W/m·K |
| dT/dx | −20 K/m |
| Parameter | Value |
|---|---|
| k | 401 W/m·K |
| dT/dx | −100 K/m |
Common mistakes
- Negative sign: Heat flows from hot to cold – the negative sign indicates that the flux is in the direction of decreasing temperature.
- Thermal conductivity k: Material‑dependent and often temperature‑dependent – use the value at the mean temperature.
- Temperature gradient dT/dx: The rate of change of temperature with distance – in K/m (or °C/m).
- Units: q in W/m², k in W/(m·K), gradient in K/m – ensure consistency.
- Steady‑state: This law applies to steady‑state conduction (no accumulation). For transient, use the heat equation.
Applications
Fourier's law of conduction, q = −k·(dT/dx), describes the rate of heat transfer by conduction through a material, where q is the heat flux (W/m²), k is the thermal conductivity, and dT/dx is the temperature gradient. This law is fundamental to thermal design in electronics, buildings, and industrial processes. Mechanical engineers use it to design heat sinks, insulators, and heat exchangers. In building engineering, it guides insulation thickness for energy efficiency. In materials science, it helps characterise the thermal properties of new materials. By applying Fourier's law, engineers can predict temperature distributions, size cooling systems, and optimise thermal management to ensure safety and performance in electronic devices, engines, and energy systems.
- Thermal design of electronics (heat sinks, thermal interface materials)
- Insulation design for buildings, pipes, and cryogenic systems
- Heat exchanger performance analysis and sizing
- Thermal management in automotive and aerospace systems
- Materials characterisation and thermal conductivity measurement
Frequently Asked Questions
Fourier's law states that the heat flux (heat transfer per unit area per unit time) is proportional to the negative temperature gradient: q = –k · dT/dx, where q is heat flux (W/m²), k is thermal conductivity (W/m·K), and dT/dx is the temperature gradient. The negative sign indicates heat flows from hot to cold.
Thermal conductivity is a material property that measures its ability to conduct heat. High k means the material is a good heat conductor (e.g., copper ~400 W/m·K), low k means it is an insulator (e.g., foam ~0.03 W/m·K). It depends on the material's atomic structure and temperature.
Steady‑state conduction occurs when the temperature at each point does not change with time (dT/dt = 0). Fourier's law applies directly. Transient conduction occurs when temperatures change over time, and the heat equation must be used: ρ·c_p·∂T/∂t = k·∇²T + Q_gen.
For a plane wall of thickness L with temperatures T₁ and T₂ on the two surfaces, the heat flux is q = –k · (T₂ – T₁) / L. The heat rate is Q = q·A. This is the basis for designing insulation and heat exchangers.
- Metals: 50–400 W/m·K (silver ~430, copper ~400).
- Water: ~0.6 W/m·K.
- Building materials: 0.1–1 W/m·K.
- Insulators: 0.01–0.05 W/m·K.
- Air: ~0.025 W/m·K.
For metals, k generally decreases with increasing temperature because electron scattering increases. For insulators, k often increases with temperature due to increased phonon transport. At very low temperatures, k may show complex behaviour.
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 and is solved with appropriate boundary conditions.
For anisotropic materials (e.g., composites, layered structures), thermal conductivity is a tensor. Fourier's law becomes q_i = –k_ij·∂T/∂x_j. The heat equation then involves directional derivatives.
- Using a constant k when it varies with temperature.
- Ignoring the negative sign.
- Applying the steady‑state form to transient problems.
- Not accounting for heat generation (e.g., in electrical resistors or nuclear fuel).
- Sizing insulation in buildings and refrigeration.
- Design of heat exchangers.
- Thermal management of electronics.
- Predicting temperature profiles in manufacturing processes (e.g., casting, welding).