View factor

Intensity of thermal radiation from the sun depends on view factor

In radiative heat transfer, a view factor, Template:Nowrap is the proportion of the radiation which leaves surface $A$ that strikes surface Template:Nowrap 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 Template:Em is conserved. Because of this, the sum of all view factors Template:Em a given surface, Template:Nowrap within the enclosure is unity as defined by the summation rule

$$\sum _{j=1}^n{F}_{S_i\to S_j}=1$$

where $n$ is the number of surfaces in the enclosure.Template:Rp Any enclosure with $n$ surfaces has a total $n^2$ view factors.

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

Confusion often arises when considering the radiation that Template:Em at a Template:Em surface. In that case, it generally does not make sense to sum view factors as view factor from Template:Mvar and view factor from Template:Mvar (above) are essentially different units. Template:Mvar may see 10% of Template:MvarTemplate:'s radiation and 50% of Template:MvarTemplate:'s radiation and 20% of Template:MvarTemplate:'s radiation, but without knowing how much each radiates, it does not even make sense to say that Template:Mvar receives 80% of the total radiation.

Reciprocity

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

$$A_i{F}_{i\to j}=A_j{F}_{j\to i}$$ where $A_i$ and $A_j$ are the areas of the two surfaces.Template:Rp

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, $F_{i\to i}=0.$Template:Rp

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

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. $$F_{1\to (2,3)}=F_{1\to 2}+F_{1\to 3}.$$

View factors of differential areas

File:View factor two differential areas illustration.svg

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 $\text{d}{A}_1$ and $\text{d}{A}_2$ at a distance Template:Mvar is given by:

$$d{F}_{1\to 2}=\frac{\cos {\theta }_1\cos {\theta }_2}{\pi s^2}\text{d}{A}_2$$

where ${\theta }_1$ and ${\theta }_2$ 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:Template:Rp $$F_{1\to 2}=\frac{1}{A_1}{\int }_{A_1}{\int }_{A_2}\frac{\cos {\theta }_1\cos {\theta }_2}{\pi s^2}\text{d}{A}_2\text{d}{A}_1.$$

Similarly the view factor $F_{2\to 1}$ is defined as the fraction of radiation that leaves $A_2$ and is intercepted by Template:Nowrap yielding the equation$$F_{2\to 1}=\frac{1}{A_2}{\int }_{A_1}{\int }_{A_2}\frac{\cos {\theta }_1\cos {\theta }_2}{\pi s^2}\text{d}{A}_2\text{d}{A}_1.$$

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:Template:Rp

Table 1: View factors for common infinite geometries

Geometry	Relation
Parallel plates of widths, $w_i,w_j$ with midlines connected by perpendicular of length $L$	$$F_{ij}=\frac{[(W_i+W_j{)}^2+4{]}^{1/2}-[(W_j-W_i{)}^2+4{]}^{1/2}}{2W_i}$$ where $W_i=w_i/L,W_j=w_j/L$
Inclined parallel plates at angle, Template:Nowrap of equal width, Template:Nowrap and a common edge	$$F_{ij}=1-sin(\frac{\alpha }{2})$$
Perpendicular plates of widths, $w_i,w_j$ with a common edge	$$F_{ij}=\frac{1+(w_j/w_i)-[1+(w_j/w_i{)}^2{]}^{1/2}}{2}$$
Three sided enclosure of widths, $w_i,w_j,w_k$	$$F_{ij}=\frac{w_i+w_j-w_k}{2w_i}$$

Nusselt analog

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 Template:Math and the element Template:Math can be obtained projecting the element Template:Math 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 Template:Math. The view factor is then equal to the differential area Template:Math times the proportion of the unit circle covered by this projection.

The projection onto the hemisphere, giving the solid angle subtended by Template:Math, takes care of the factors Template:Math and Template:Math; the projection onto the circle and the division by its area then takes care of the local factor Template:Math and the normalisation by Template: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. Template:Crossref 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.

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.