Fourier's law

Fourier's law relates thermal (heat) conduction to the temperature gradient. Thermal conduction is the transfer of internal energy by microscopic collisions of atoms or molecules and the movement of electrons within a body. The rate at which energy is conducted as heat between two bodies is a function of the temperature gradient between the two bodies. Heat spontaneously flows from a hotter to a colder source.

Fourier's law in a single dimension is

${\dot{q}}_x=-k\frac{dT}{dx}$

(1)

where

• ${\dot{q}}_x$ is the heat flux, which is expressed as energy per unit area per unit time.
• $k$ is the thermal conductivity. The dimension is area per unit time, so typical units for expressing it would be m²/s.
• $T$ is the temperature.
• $x$ is position, the dimension of which is length.

In two or more dimensions we use $\mathrm{\nabla }$, or the gradient operator. Independent of the coordinate system, Fourier's law is expressed as

$\dot{\mathbf{q}}=-k\mathrm{\nabla }T$

(2)

where $\dot{\mathbf{q}}$ denotes the heat flux vector.

Integral Form of Fourier's Law

If you integrate the single dimension form, shown above, over the materials surface area S you get the integral form:

$\frac{\partial Q}{\partial t}=-k\underset{S}{\overset{}{\text{∯}}}\mathrm{\nabla }T\cdot dS$

(3)

where

• $\frac{\partial Q}{\partial t}$ is the heat transferred per unit time in W
• $dS$ is the oriented surface area element in ${\mathbf{m}}^{\mathbf{2}}$