# Heat flux

Heat flux or thermal flux is the rate of heat energy transfer through a given surface per unit time. The SI derived unit of heat rate is joule per second, or watt. Heat flux density is the heat rate per unit area. In SI units, heat flux density is measured in [W/m2].[1] Heat rate is a scalar quantity, while heat flux is a vectorial quantity. To define the heat flux at a certain point in space, one takes the limiting case where the size of the surface becomes infinitesimally small.

Heat flux is often denoted ${\displaystyle {\overrightarrow {\phi _{q}}}}$, the subscript q specifying heat rate, as opposed to mass or momentum rate. Fourier's law is an important application of these concepts.

Heat flux ${\displaystyle {\overrightarrow {\phi _{q}}}}$ through a surface.

## Fourier's law

For most solids in usual conditions, heat is transported mainly by conduction and the heat flux is adequately described by Fourier's law.

### Fourier's law in one dimension

The heat flux associated with a temperature profile ${\displaystyle T(x)}$ in a material of thermal conductivity ${\displaystyle k}$ is given by

${\displaystyle \phi _{q}=-k{\frac {dT(x)}{dx}}}$

The negative sign shows that heat flux moves from higher temperature regions to lower temperature regions.

### Multi-dimensional extension

The multi-dimensional case is similar, the heat flux goes "down" the temperature gradient hence the negative sign:

${\displaystyle {\overrightarrow {q}}=-k{\nabla }T}$

where ${\displaystyle {\nabla }}$ is the gradient operator.

## Measuring heat flux

Main article: Heat flux sensor

The measurement of heat flux is most often done by measuring a temperature difference over a piece of material with known thermal conductivity. This method is analogous to a standard way to measure an electric current, where one measures the voltage drop over a known resistor.

## Relevance to science and engineering

One of the tools in a scientist's or engineer's toolbox is the energy balance. Such a balance can be set up for any physical system, from chemical reactors to living organisms, and generally takes the following form

${\displaystyle {\big .}{\frac {\partial E_{\mathrm {in} }}{\partial t}}-{\frac {\partial E_{\mathrm {out} }}{\partial t}}-{\frac {\partial E_{\mathrm {accumulated} }}{\partial t}}=0}$

where the three ${\displaystyle {\big .}{\frac {\partial E}{\partial t}}}$ terms stand for the time rate of change of respectively the total amount of incoming energy, the total amount of outgoing energy and the total amount of accumulated energy.

Now, if the only way the system exchanges energy with its surroundings is through heat transfer, the heat rate can be used to calculate the energy balance, since

${\displaystyle {\big .}{\frac {\partial E_{\mathrm {in} }}{\partial t}}-{\frac {\partial E_{\mathrm {out} }}{\partial t}}=\oint _{S}{\overrightarrow {\phi _{q}}}\cdot \,{\overrightarrow {dS}}}$

where we have integrated the heat flux density${\displaystyle {\overrightarrow {\phi _{q}}}}$ over the surface ${\displaystyle S}$ of the system.

In real-world applications one cannot know the exact heat flux at every point on the surface, but approximation schemes can be used to calculate the integral, for example Monte Carlo integration.