# View factor

In radiative heat transfer, a view factor, ${\displaystyle F_{A\rightarrow B}}$, is the proportion of the radiation which leaves surface ${\displaystyle A}$ that strikes surface ${\displaystyle B}$. 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, angle factors or shape factors.

## Relations

### Summation

Radiation leaving a surface within an enclosure is conserved. Because of this, the sum of all view factors from a given surface, ${\displaystyle S_{i}}$, within the enclosure is unity as defined by the summation rule

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

where ${\displaystyle n}$ is the number of surfaces in the enclosure.[1]: 864  Any enclosure with ${\displaystyle n}$ surfaces has a total ${\displaystyle n^{2}}$ view factors.

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. C may see 10% of A 's radiation and 50% of B 's radiation and 20% of C 's radiation, but without knowing how much each radiates, it does not even make sense to say that C receives 80% of the total radiation.

### Reciprocity

The reciprocity relation for view factors allows one to calculate ${\displaystyle F_{i\rightarrow j}}$ if one already knows ${\displaystyle F_{j\rightarrow i}}$ and is given as

${\displaystyle A_{i}F_{i\rightarrow j}=A_{j}F_{j\rightarrow i}}$

where ${\displaystyle A_{i}}$ and ${\displaystyle A_{j}}$ are the areas of the two surfaces.[1]: 863

### Self-viewing

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, ${\displaystyle F_{i\rightarrow i}=0.}$[1]: 864

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

### Superposition

The superposition rule (or summation rule) is useful when a certain geometry is not available with given charts or graphs. The superposition rule allows us to express the geometry that is being sought using the sum or difference of geometries that are known.

${\displaystyle F_{1\rightarrow (2,3)}=F_{1\rightarrow 2}+F_{1\rightarrow 3}.}$ [2]

## View factors of differential areas

Taking the limit of a small flat surface gives differential areas, the view factor of two differential areas of areas ${\displaystyle {\hbox{d}}A_{1}}$ and ${\displaystyle {\hbox{d}}A_{2}}$ at a distance s is given by:

${\displaystyle dF_{1\rightarrow 2}={\frac {\cos \theta _{1}\cos \theta _{2}}{\pi s^{2}}}{\hbox{d}}A_{2}}$

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

The view factor from a general surface ${\displaystyle A_{1}}$ to another general surface ${\displaystyle A_{2}}$ is given by:[1]: 862

${\displaystyle F_{1\rightarrow 2}={\frac {1}{A_{1}}}\int _{A_{1}}\int _{A_{2}}{\frac {\cos \theta _{1}\cos \theta _{2}}{\pi s^{2}}}\,{\hbox{d}}A_{2}\,{\hbox{d}}A_{1}.}$

Similarly the view factor ${\displaystyle F_{2\rightarrow 1}}$is defined as the fraction of radiation that leaves ${\displaystyle A_{2}}$ and is intercepted by ${\displaystyle A_{1}}$, yielding the equation

${\displaystyle F_{2\rightarrow 1}={\frac {1}{A_{2}}}\int _{A_{1}}\int _{A_{2}}{\frac {\cos \theta _{1}\cos \theta _{2}}{\pi s^{2}}}\,{\hbox{d}}A_{2}\,{\hbox{d}}A_{1}.}$
The view factor is related to the etendue.

## Example solutions

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

Table 1: View factors for common geometries
Geometry Relation
Parallel plates of widths, ${\displaystyle w_{i},w_{j}}$ with midlines connected by perpendicular of length ${\displaystyle L}$
${\displaystyle F_{ij}={\frac {[(W_{i}+W_{j})^{2}+4]^{1/2}-[(W_{j}-W_{i})^{2}+4]^{1/2}}{2W_{i}}}}$

where ${\textstyle W_{i}=w_{i}/L,W_{j}=w_{j}/L}$

Inclined parallel plates at angle, ${\displaystyle \alpha }$, of equal width, ${\displaystyle w}$, and a common edge
${\displaystyle F_{ij}=1-sin({\frac {\alpha }{2}})}$
Perpendicular plates of widths, ${\displaystyle w_{i},w_{j}}$ with a common edge
${\displaystyle F_{ij}={\frac {1+(w_{j}/w_{i})-[1+(w_{j}/w_{i})^{2}]^{1/2}}{2}}}$
Three sided enclosure of widths, ${\displaystyle w_{i},w_{j},w_{k}}$
${\displaystyle F_{ij}={\frac {w_{i}+w_{j}-w_{k}}{2w_{i}}}}$

## Nusselt analog

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.[3] (see also Hemispherical photography). But its main value now is essentially in building intuition.

• Radiosity, a matrix calculation method for solving radiation transfer between a number of bodies.
• Gebhart factor, an expression to solve radiation transfer problems between any number of surfaces.

## References

1. Incropera, Frank P.; DeWitt, David P.; Bergman, Theodore L.; Lavine, Adrienne S., eds. (2013). Principles of heat and mass transfer (7. ed., international student version ed.). Hoboken, NJ: Wiley. ISBN 978-0-470-50197-9.
2. ^ Heat and Mass Transfer, Yunus A. Cengel and Afshin J. Ghajar, 4th Edition
3. ^ Michael F. Cohen, John R. Wallace (1993), Radiosity and realistic image synthesis. Morgan Kaufmann, ISBN 0-12-178270-0, p. 80