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.

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

The measurement of heat flux can be performed in a few different manners.

A commonly known, but often impractical, method is performed by measuring a temperature difference over a piece of material with a well-known thermal conductivity.

Usually this method is difficult to perform since the thermal resistance of the material being tested is often not known.

Using the thermal resistance, along with temperature measurements on either side of the material, heat flux can then be indirectly calculated.

The most common type of heat flux sensor is a differential temperature thermopile which operates on essentially the same principle as the first measurement method that was mentioned except it has the advantage in that the thermal resistance/conductivity does not need to be a known parameter.

However, differential thermopile heat flux sensors have to be calibrated in order to relate their output signals [μV] to heat flux values [W/(m2⋅K)].

Once the heat flux sensor is calibrated it can then be used to directly measure heat flux without requiring the rarely known value of thermal resistance or thermal conductivity.

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