Conduction transfers thermal energy through matter because neighboring regions have different temperatures.

In a solid, microscopic interactions carry energy from hotter regions toward colder ones without requiring bulk motion of the material.

Fourier's law

For one-dimensional conduction through area $A$,

$$\dot Q=-kA\frac{dT}{dx},$$

where $k$ is the thermal conductivity.

The minus sign states that heat flows down the temperature gradient.

Thermal conductivity

A large $k$ means a material conducts heat readily. Metals often have relatively high thermal conductivity, while gases and insulating solids tend to have lower values.

Plane wall

For steady one-dimensional conduction through a plane wall of thickness $L$ with constant $k$,

$$\dot Q=kA\frac{T_1-T_2}{L}.$$

This can be written using a thermal resistance

$$R_{th}=\frac{L}{kA},\qquad \dot Q=\frac{\Delta T}{R_{th}}.$$