View factor

# View factor

Discussion

Encyclopedia
In radiative heat transfer, a view factor, , is the proportion of all that radiation which leaves surface and strikes surface .

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, , is unity:

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 As radiation and 50% of Bs radiation and 20% of Cs 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
Convex function
In mathematics, a real-valued function f defined on an interval is called convex if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph is a convex set...

surface, no radiation can leave the surface and then hit it later, because radiation travels in straight lines. Hence, for convex surfaces,

For concave
Concave function
In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper convex.-Definition:...

surfaces, this doesn't apply, and so for concave surfaces

## 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.

## Reciprocity

The reciprocity theorem
Reciprocity theorem
Reciprocity theorem may refer to:*Quadratic reciprocity, a theorem about modular arithmetic**Cubic reciprocity**Quartic reciprocity**Artin reciprocity**Weil reciprocity for algebraic curves*Frobenius reciprocity theorem for group representations...

for view factors allows one to calculate if one already knows . Using the areas of the two surfaces and ,

## View factors of differential areas

Taking the limit of a small flat surface gives differential areas. View factors for arbitrary surfaces may be calculated by integrating differential view factors over the desired surfaces.

The view factor of two differential areas of areas and at a distance S is given by:

where and are the angle between the surface normals and a ray between the two differential areas.

The view factor is related to the 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 this article by G H Derrick.

## Nusselt analog

A geometrical picture that can aid intuition about the view factor was developed by Wilhelm Nusselt
Wilhelm Nusselt
Ernst Kraft Wilhelm Nußelt was a German engineer. Nusselt studied mechanical engineering at the Munich Technical University , where he got his doctorate in 1907...

, 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 a 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
Solid angle
The solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point...

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. (see also Hemispherical photography
Hemispherical photography
Hemispherical photography, also known as fisheye or canopy photography, is a technique to estimate solar radiation and characterize plant canopy geometry using photographs taken looking upward through an extreme wide-angle lens . Typically, the viewing angle approaches or equals 180-degrees, such...

). But its main value now is essentially in building intuition.

Radiosity is a convenient quantity in optics and heat transfer that represents the total radiation intensity leaving a surface. Radiosity accounts for two components: the radiation being emitted by the surface, and the radiation being reflected from the surface...

, a matrix calculation method for solving radiation transfer between a number of bodies.
• Gebhart factor
Gebhart factor
The Gebhart factors are used in radiative heat transfer, it is a means to describe the ratio of radiation absorbed by any other surface versus the total emitted radiation from given surface. As such, it becomes the radiation exchange factor between a number of surfaces...

, an expression to solve radiation transfer problems between any number of surfaces.