View factor

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|$S\_i$,}} within the enclosure is unity as defined by the summation rule

$$\backslash sum\_\{j=1\}^n\; \{F\_\{S\_i\; \backslash rarr\; S\_j\}\}\; =\; 1$$

where $n$ is the number of surfaces in the enclosure.{{Rp|page=864}} Any enclosure with $n$ surfaces has a total $n^2$ 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.

The reciprocity relation for view factors allows one to calculate $F\_\{i\; \backslash rarr\; j\}$ if one already knows $F\_\{j\; \backslash rarr\; i\}$ and is given as

$$A\_i\; F\_\{i\; \backslash rarr\; j\}\; =\; A\_j\; F\_\{j\; \backslash rarr\; i\}$$

where $A\_i$ and $A\_j$ are the areas of the two surfaces.{{Rp|page=863}}

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, $F\_\{i\; \backslash rarr\; i\}\; =\; 0.${{Rp|page=864}}

For concave surfaces, this doesn't apply, and so for concave surfaces $F\_\{i\; \backslash rarr\; i\}\; >\; 0.$

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.{{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}}

$$F\_\{1\; \backslash rarr\; (2,3)\}=F\_\{1\; \backslash rarr\; 2\}+F\_\{1\backslash rarr\; 3\}.$$

File:view_factor_two_differential_areas_illustration.svg

Taking the limit of a small flat surface gives differential areas, the view factor of two differential areas of areas $\backslash hbox\{d\}A\_1$ and $\backslash hbox\{d\}A\_2$ at a distance {{mvar|s}} is given by:

dF_{1 \rarr 2} = \frac{\cos\theta_1 \cos\theta_2}{\pi s^2}\hbox{d}A_2

where $\backslash theta\_1$ and $\backslash theta\_2$ are the angle between the surface normals and a ray between the two differential areas.

The view factor from a general surface $A\_1$ to another general surface $A\_2$ is given by:{{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.}}{{Rp|page=862}}

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.

Similarly the view factor $F\_\{2\backslash rightarrow\; 1\}$ is defined as the fraction of radiation that leaves $A\_2$ and is intercepted by {{nowrap|$A\_1$,}} yielding the equation$$$$

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.

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:{{Rp|page=865}}

File:Nusselt analog.svg

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|dA{{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|dA{{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.{{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}} {{crossref|(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.

{{Reflist}}

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]
• [http://view3d.sourceforge.net/ View3D], a computer program (FOSS) for calculating view factors in 2D and 3D.

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Category:Heat transfer