View factor

{{Short description|Proportion in thermal radiation}} {{Refimprove|date=December 2021}} thumb|Intensity of thermal radiation from the sun depends on view factorIn radiative heat transfer, a '''view factor''', {{nowrap|<math>F_{A \rarr B}</math>,}} is the proportion of the radiation which leaves surface <math>A</math> that strikes surface {{nowrap|<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''', '''angle factors''' or '''shape factors'''.

== Relations ==

=== Summation === Radiation leaving a surface {{em|within an enclosure}} is conserved. Because of this, the sum of all view factors {{em|from}} a given surface, {{nowrap|<math>S_i</math>,}} within the enclosure is unity as defined by the ''summation rule''

where <math>n</math> is the number of surfaces in the enclosure.<ref name=":0" />{{Rp|page=864}} Any enclosure with <math>n</math> surfaces has a total <math>n^2</math> view factors.

For example, consider a case where two blobs with surfaces {{mvar|A}} and {{mvar|B}} are floating around in a cavity with surface {{mvar|C}}. All of the radiation that leaves {{mvar|A}} must either hit {{mvar|B}} or {{mvar|C}}, or if {{mvar|A}} is concave, it could hit {{mvar|A}}. 100% of the radiation leaving {{mvar|A}} is divided up among {{mvar|A}}, {{mvar|B}}, and {{mvar|C}}.

Confusion often arises when considering the radiation that {{em|arrives}} at a {{em|target}} surface. In that case, it generally does not make sense to sum view factors as view factor from {{mvar|A}} and view factor from {{mvar|B}} (above) are essentially different units. {{mvar|C}} may see 10% of {{mvar|A}}{{'s}} radiation and 50% of {{mvar|B}}{{'s}} radiation and 20% of {{mvar|C}}{{'s}} radiation, but without knowing how much each radiates, it does not even make sense to say that {{mvar|C}} receives 80% of the total radiation.

=== Reciprocity === The ''reciprocity relation'' for view factors allows one to calculate <math>F_{i \rarr j}</math> if one already knows <math>F_{j \rarr i}</math> and is given as

<math display="block">A_i F_{i \rarr j} = A_j F_{j \rarr i}</math> where <math>A_i</math> and <math>A_j</math> are the areas of the two surfaces.<ref name=":0" />{{Rp|page=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, <math>F_{i \rarr i} = 0.</math><ref name=":0" />{{Rp|page=864}}

For concave surfaces, this doesn't apply, and so for concave surfaces <math>F_{i \rarr i} > 0.</math>

=== 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.<ref>{{cite book |last1= Cengel |first1= Yunus A. |last2= Ghajar |first2= Afshin J. |date= 2010 |title= Heat and Mass Transfer |edition= 4th |publisher= McGraw-Hill |isbn= 978-0-07-736664-3}}</ref> <math display="block">F_{1 \rarr (2,3)}=F_{1 \rarr 2}+F_{1\rarr 3}.</math>

== View factors of differential areas ==

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 areas of areas <math>\hbox{d}A_1</math> and <math>\hbox{d}A_2</math> at a distance {{mvar|s}} is given by:

<math display="block"> dF_{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:<ref name=":0">{{Cite book |title=Principles of Heat and Mass Transfer |date=2013 |publisher=Wiley |isbn=978-0-470-50197-9 |editor-last=Incropera |editor-first=Frank P. |edition=7. ed., international student version |location=Hoboken, New Jersey |editor-last2=DeWitt |editor-first2=David P. |editor-last3=Bergman |editor-first3=Theodore L. |editor-last4=Lavine |editor-first4=Adrienne S.}}</ref>{{Rp|page=862}} <math display="block"> 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>

Similarly the view factor <math>F_{2\rightarrow 1}</math> is defined as the fraction of radiation that leaves <math>A_2</math> and is intercepted by {{nowrap|<math>A_1</math>,}} yielding the equation<math display="block"> F_{2 \rarr 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.</math>

The view factor is related to the concept of 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:<ref name=":0" />{{Rp|page=865}} {| class="wikitable" |+Table 1: View factors for common infinite geometries ! scope="col" | Geometry ! scope="col" | Relation |- |Parallel plates of widths, <math>w_i, w_j</math> with midlines connected by perpendicular of length <math>L</math> |<math display="block">F_{ij}=\frac{[(W_i+W_j)^2+4]^{1/2}-[(W_j-W_i)^2+4]^{1/2}}{2W_i} </math> where <math display="inline">W_i=w_i/L,W_j=w_j/L</math> |- |Inclined parallel plates at angle, {{nowrap|<math>\alpha</math>,}} of equal width, {{nowrap|<math>w</math>,}} and a common edge |<math display="block">F_{ij}=1-sin(\frac{\alpha}{2})</math> |- |Perpendicular plates of widths, <math>w_i, w_j</math> with a common edge |<math display="block">F_{ij}=\frac{1+(w_j/w_i)-[1+(w_j/w_i)^2]^{1/2}}{2}</math> |- |Three sided enclosure of widths, <math>w_i, w_j, w_k</math> |<math display="block">F_{ij}=\frac{w_i+w_j-w_k}{2w_i}</math> |}

== Nusselt analog == 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 {{math|d''A''{{sub|i}}}} and the element {{math|''A''{{sub|j}}}} can be obtained projecting the element {{math|''A''{{sub|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 {{math|''A''{{sub|i}}}}. The view factor is then equal to the differential area {{math|d''A''{{sub|i}}}} times the proportion of the unit circle covered by this projection.

The projection onto the hemisphere, giving the solid angle subtended by {{math|''A''{{sub|j}}}}, takes care of the factors {{math|cos θ{{sub|2}}}} and {{math|1/''r''{{i sup|2}}}}; the projection onto the circle and the division by its area then takes care of the local factor {{math|cos θ{{sub|1}}}} and the normalisation by {{pi}}.

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>{{cite book |last1= Cohen |first1= Michael F. |last2= Wallace |first2= John R. |date= 1993 |title= Radiosity and Realistic Image Synthesis |publisher= Morgan Kaufmann |isbn= 0-12-178270-0 |p= 80}}</ref> {{crossref|(See also Hemispherical photography.)}} But its main value now is essentially in building intuition.

== See also == * 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 == {{Reflist}}

== 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.thermalradiation.net/indexCat.html List of view factors for specific geometry cases] * [https://view3d.sourceforge.net/ View3D], a computer program (FOSS) for calculating view factors in 2D and 3D.

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