# 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/m^{2}].^{[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 , the subscript *q* specifying *heat* rate, as opposed to *mass* or *momentum* rate. Fourier's law is an important application of these concepts.

## 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 in a material of thermal conductivity is given by

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:

where is the gradient operator.

## Measuring heat flux

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

where the three 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

where we have integrated the heat flux density over the surface 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.

## See also

- Radiant flux
- Eddy covariance flux (eddy correlation, eddy flux)
- Heat Notation
- Heat conduction
- Heat transfer
- Critical heat flux
- Latent heat flux
- Rate of heat flow
- Insolation
- Heat flux sensor
- Relativistic heat conduction

## References

- ↑ The NIST Reference on Constants, Units, and Uncertainty http://physics.nist.gov/cuu/Units/units.html