22/12/2025
Law of Heat Conduction
Fourier's law is a fundamental principle in heat transfer that describes conduction.
Statement
The law of heat conduction, also known as Fourier's law (compare Fourier's heat equation), states that the rate of heat transfer through a material is proportional to the negative gradient in the temperature and to the area, at right angles to that gradient, through which the heat flows.
- Heat flux (q): The rate of heat transfer per unit area.
- Proportional to temperature gradient: q = -k ∇T
- q: heat flux (W/m²)
- k: thermal conductivity (W/(m·K))
- ∇T: temperature gradient (K/m)
Important Points
1. Direction of heat flow:
Heat flows from hot to cold (down the temperature gradient).
2. Thermal conductivity (k):
Material property; higher k means better conductor.
3. Steady-state conduction:
∇·(k∇T) = 0 (no internal heat generation).
Forms of Fourier's Law
1. 1D conduction: q = -k dT/dx
2. Cartesian coordinates: q = -k (∂T/∂x i + ∂T/∂y j + ∂T/∂z k)
3. Cylindrical/spherical coordinates: Adjust for geometry.
Applications
- Heat transfer analysis: Walls, pipes, electronics cooling.
- Insulation: Minimize heat loss/gain.
- Material processing: Thermal stresses, etc.
Example
- Plane wall: T1 at x=0, T2 at x=L. Then:
- q = k(T1-T2)/L
- Heat flux is constant (steady-state).
