View factor

In radiative heat transfer, a view factor $F_{A\rightarrow B}$ is the proportion of all that radiation which leaves surface $A$ and strikes surface $B$.

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.

View factors are also sometimes known as configuration factors, form factors or shape factors.

Summation of view factors

Because all of the radiation leaving a surface is a fixed amount, one can add up all of the view factors from a given surface $S_{i}$, and they will always add up to one:

$\sum _{j=1...n}{F_{S_{i}\rightarrow S_{j}}}=1$

For example, consider a case where two blobs, 'A' and 'B' are floating around in a cavity 'C'. All the radiation that leaves surface A must either hit surface B or the cavity surface C, or if the surface A is concave, might again hit A. In terms of fractions, 100% of the radiation leaving surface A is divided up between surfaces A, B, and C, which is equivalent to the expression above.

Confusion often arises when considering the radiation that arrives at a target surface. In that case there, summation of view factors does not apply, because each 'incoming' view factor is a fraction of the radiation leaving some other surface: there is no reason for those fractions to add up to anything at all; in fact they can add up to less than one, or many times one, but this value has no significance.

Self-viewing surfaces

For a convex surface, no radiation can leave the surface and then hit it later, because radiation travels in straight lines. Hence, for convex surfaces, $F_{A\rightarrow A}=0$

For concave surfaces, this doesn't apply, and so for concave surfaces $F_{A\rightarrow A}>0$

Reciprocity

The reciprocity theorem for view factors allows one to calculate $F_{B\rightarrow A}$ if one already knows $F_{A\rightarrow B}$. Using the areas of the two surfaces $A_{A}$ and $A_{B}$,

$A_{A}F_{A\rightarrow B}=A_{B}F_{B\rightarrow A}$

View factors of differential areas

Two differential areas in arbitrary configuration

The most fundamental view factor is that of two differential areas. All other view factors may be calculated by integrating this view factor over the requested area.

The view factor of two differential areas of areas ${\hbox{d}}A_{1}$ and ${\hbox{d}}A_{2}$ at a distance S is given by:

$F_{1\rightarrow 2}={\frac {\cos \theta _{1}\cos \theta _{2}}{\pi S^{2}}}{\hbox{d}}A_{2}$

where $\theta _{1}$ and $\theta _{2}$ are the angle between the surface normals and a ray between the two differential areas.

Hottel's crossed string rule

The crossed string rule allows calculation of radiation transfer between opposite sides of a quadrilateral, and furthermore applies in some cases where there is partial obstruction between the objects. For a derivation and further details, see this article by G H Derrick.

See also

• Radiosity, a matrix calculation method for solving radiation transfer between a number of bodies.

External links

A large number of 'standard' view factors can be calculated with the use of tables that are commonly provided in heat transfer textbooks.

• list of view factors for specific geometry cases
• View3D, a computer program (FOSS) for calculating view factors in 2D and 3D.