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In [[radiative heat transfer]], a '''view factor''', <math>F_{A \rarr B}</math>, is the proportion of the radiation which leaves surface <math>A</math> that strikes surface <math>B</math>.

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 radiation leaving a surface is conserved, the sum of all view factors ''from'' a given surface, <math>S_i</math>, is [[1 (number)|unity]]:

:<math>\sum_{j=1}^n {F_{S_i \rarr S_j}} = 1</math>

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.

== Self-viewing surfaces ==
For a [[Convex function|convex]] surface, no radiation can leave the surface and then hit later, because radiation travels in straight lines. Hence, for convex surfaces, <math>F_{A \rarr A} = 0</math>

For [[concave function|concave]] surfaces, this doesn't apply, and so for concave surfaces <math>F_{A \rarr A} > 0</math>

== Superposition Rule ==
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.
:<math>F_{1 \rarr (2,3)}=F_{1 \rarr 2}+F_{1\rarr 3}</math> <ref>Heat and Mass Transfer, Yunus A. Cengel and Afshin J. Ghajar, 4th Edition</ref>

== Reciprocity ==
The [[reciprocity theorem]]{{dn|date=October 2012}} for view factors allows one to calculate <math>F_{B \rarr A}</math> if one already knows <math>F_{A \rarr B}</math>. Using the areas of the two surfaces <math>A_A</math> and <math>A_B</math>,

: <math>A_A F_{A \rarr B} = A_B F_{B \rarr A}</math>

== View factors of differential areas ==

[[File:view_factor_two_differential_areas_illustration.svg|thumb|150px|right|Two differential areas in arbitrary configuration]]

Taking the limit of a small flat surface gives differential areas, the view factor of two [[differential area]]s of areas <math>\hbox{d}A_1</math> and <math>\hbox{d}A_2</math> at a distance S is given by:

:<math>
F_{1 \rarr 2} = \frac{\cos\theta_1 \cos\theta_2}{\pi S^2}\hbox{d}A_2
</math>

where <math>\theta_1</math> and <math>\theta_2</math> are the angle between the surface normals and a ray between the two differential areas.

The view factor from a general surface <math>A_1</math> to another general surface <math>A_2</math> is given by:
:<math>
F_{1 \rarr 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
</math>

The view factor is related to the [[etendue#Conservation of etendue in free space|etendue]].

== 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 [http://cat.inist.fr/?aModele=afficheN&cpsidt=9207165 this article by G H Derrick].

== Nusselt analog ==
[[File:Nusselt analog.svg|thumb|150px|right|Nusselt analog: the projected solid angle]]

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 d''A''i and the element ''A''j can be obtained projecting the element ''A''j 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 ''A''i.
The view factor is then equal to the differential area d''A''i times the proportion of the unit circle covered by this projection.

The projection onto the hemisphere, giving the [[solid angle]] subtended by ''A''j, takes care of the factors cos(θ2) and 1/''r''2; 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]].<ref>Michael F. Cohen, John R. Wallace (1993), ''Radiosity and realistic image synthesis''. Morgan Kaufmann, ISBN 0-12-178270-0, p. [http://books.google.co.uk/books?id=7JiYl9m3Y6YC&pg=PA80 80]</ref> (see also [[Hemispherical photography]]). But its main value now is essentially in building intuition.

== See also ==
* [[Radiosity_(heat_transfer)|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.

== 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.
* [http://www.me.utexas.edu/~howell/tablecon.html list of view factors for specific geometry cases]
* [http://view3d.sourceforge.net/ View3D], a computer program ([[FOSS]]) for calculating view factors in 2D and 3D.

== References ==
{{Reflist}}

{{DEFAULTSORT:View Factor}}
[[Category:Heat transfer]]

Coherence theorem - Revision history

en>IkamusumeFan: /* Examples */ Use SVG instead of PNG

en>Byron.hawkins: Fixed a typo: the "s" was missing on "as".