View factor
In radiative heat transfer, a view factor,
, is the proportion of the radiation which leaves surface
In a complex 'scene' there can be any number of different objects, which can be divided in turn into even more surfaces and surface segments.
Radiation leaving a surface within an enclosure is conserved.
Because of this, the sum of all view factors from a given surface,
, within the enclosure is unity as defined by the summation rule where
For example, consider a case where two blobs with surfaces A and B are floating around in a cavity with surface C. All of the radiation that leaves A must either hit B or C, or if A is concave, it could hit A.
100% of the radiation leaving A is divided up among A, B, and C. Confusion often arises when considering the radiation that arrives at a target surface.
In that case, it generally does not make sense to sum view factors as view factor from A and view factor from B (above) are essentially different units.
The reciprocity relation for view factors allows one to calculate
[1]: 863 For a convex surface, no radiation can leave the surface and then hit it later, because radiation travels in straight lines.
Taking the limit of a small flat surface gives differential areas, the view factor of two differential areas of areas
are the angle between the surface normals and a ray between the two differential areas.
The view factor from a general surface
is defined as the fraction of radiation that leaves
The view factor is related to the etendue.
For complex geometries, the view factor integral equation defined above can be cumbersome to solve.
Solutions are often referenced from a table of theoretical geometries.
Common solutions are included in the following table:[1]: 865 where
A geometrical picture that can aid intuition about the view factor was developed by Wilhelm Nusselt, and is called the Nusselt analog.
The view factor between a differential element dAi and the element Aj can be obtained projecting the element Aj onto the surface of a unit hemisphere, and then projecting that in turn onto a unit circle around the point of interest in the plane of Ai.
The view factor is then equal to the differential area dAi times the proportion of the unit circle covered by this projection.
The projection onto the hemisphere, giving the solid angle subtended by Aj, takes care of the factors cos(θ2) and 1/r2; the projection onto the circle and the division by its area then takes care of the local factor cos(θ1) and the normalisation by π.
The Nusselt analog has on occasion been used to actually measure form factors for complicated surfaces, by photographing them through a suitable fish-eye lens.
But its main value now is essentially in building intuition.
A large number of 'standard' view factors can be calculated with the use of tables that are commonly provided in heat transfer textbooks.
