# Heat current

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For conduction, heat current is defined by Fourier's law as

${\frac {\partial Q}{\partial t}}=-k\oint _{S}{{\overrightarrow {\nabla }}T\cdot \,{\overrightarrow {dS}}}$ where

${\big .}{\frac {\partial Q}{\partial t}}{\big .}$ is the amount of heat transferred per unit time [W] and
${\overrightarrow {dS}}$ is an oriented surface area element [m2]

The above differential equation, when integrated for a homogeneous material of 1-D geometry between two endpoints at constant temperature, gives the heat flow rate as:

${\big .}{\frac {\Delta Q}{\Delta t}}=-kA{\frac {\Delta T}{\Delta x}}$ where

A is the cross-sectional surface area,
$\Delta T$ is the temperature difference between the ends,
$\Delta x$ is the distance between the ends.

For thermal radiation, heat current is defined as

$W=\sigma \cdot A\cdot T^{4}$ Heat current can also be thought of as the total phonon distribution multiplied by the energy of one phonon, times the group velocity of the phonons. The phonon distribution of a particular phonon mode is given by the Bose-Einstein factor, which is dependent on temperature and phonon energy.