Heat current

A heat current is a kinetic exchange rate between molecules, relative to the material in which the kinesis occurs. It is defined as $\frac{dQ}{dt}$, where $Q$ is heat and $t$ is time.

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:

$\big. \frac{\Delta Q}{\Delta t} = -k A \frac{\Delta T}{\Delta x}$

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

$W = \sigma \cdot A \cdot T^4$

where the constant of proportionality $\sigma$ is the Stefan–Boltzmann constant, $A$ is the radiating surface area, and $T$ is temperature.

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.

See also

• Thermal conduction
• Thermal conductivity