View factor
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.
, 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.
surface, no radiation can leave the surface and then hit it later, because radiation travels in straight lines. Hence, for convex surfaces,
For concave
surfaces, this doesn't apply, and so for concave surfaces
for view factors allows one to calculate
if one already knows 
. Using the areas of the two surfaces 
and 
,
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.
, 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
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
). But its main value now is essentially in building intuition.
textbooks.
, 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 convexConvex 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 theoremReciprocity 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 NusseltWilhelm 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.
See also
- RadiosityRadiosity (heat transfer)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 factorGebhart factorThe 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.
External links
A large number of 'standard' view factors can be calculated with the use of tables that are commonly provided in heat transferHeat transfer
Heat transfer is a discipline of thermal engineering that concerns the exchange of thermal energy from one physical system to another. Heat transfer is classified into various mechanisms, such as heat conduction, convection, thermal radiation, and phase-change transfer...
textbooks.
- list of view factors for specific geometry cases
- View3D, a computer program (FOSSFossFoss may refer toPeople*Foss , people with the last name Foss*Foss Shanahan , New Zealand diplomat*Foss Westcott , English bishop...
) for calculating view factors in 2D and 3D.