Intensity (heat transfer)

In the field of heat transfer, intensity of radiation [math]\displaystyle{ I }[/math] is a measure of the distribution of radiant heat flux per unit area and solid angle, in a particular direction, defined according to

[math]\displaystyle{ dq = I\, d\omega\, \cos \theta\, dA }[/math]

where

* [math]\displaystyle{ dA }[/math] is the infinitesimal source area
* [math]\displaystyle{ dq }[/math] is the outgoing heat transfer from the area [math]\displaystyle{ dA }[/math]
* [math]\displaystyle{ d\omega }[/math] is the solid angle subtended by the infinitesimal 'target' (or 'aperture') area [math]\displaystyle{ dA_a }[/math]
* [math]\displaystyle{ \theta }[/math] is the angle between the source area normal vector and the line-of-sight between the source and the target areas.

Typical units of intensity are W·m−2·sr−1.

Intensity can sometimes be called radiance, especially in other fields of study.

The emissive power of a surface can be determined by integrating the intensity of emitted radiation over a hemisphere surrounding the surface:

[math]\displaystyle{ q = \int_{\phi=0}^{2\pi} \int_{\theta=0}^{\pi/2} I \cos \theta \sin \theta d\theta d\phi }[/math]

For diffuse emitters, the emitted radiation intensity is the same in all directions, with the result that

[math]\displaystyle{ E = \pi I }[/math]

The factor [math]\displaystyle{ \pi }[/math] (which really should have the units of steradians) is a result of the fact that intensity is defined to exclude the effect of reduced view factor at large values [math]\displaystyle{ \theta }[/math]; note that the solid angle corresponding to a hemisphere is equal to [math]\displaystyle{ 2\pi }[/math] steradians.

Spectral intensity [math]\displaystyle{ I_\lambda }[/math] is the corresponding spectral measurement of intensity; in other words, the intensity as a function of wavelength.

## See also

* Non-ionising radiation
* Emissivity
* Radiant intensity

## References

* Lienhard and Lienhard, A heat transfer textbook, 5th Ed, 2019 (available for free online)
* J P Holman,  Heat Transfer 9th Ed, McGraw Hill, 2002.
* F. P. Incropera and D. P. DeWitt, Fundamentals of Heat and Mass Transfer, 4th Ed, Wiley, 1996.

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