View factor
View factors (also: view factors, form factors, angular relationships ) are used in the calculation of radiation exchange between different surfaces to describe the mutual geometric " visibility ", ie, the relative position and orientation of surfaces. The two surfaces 1 and 2 associated view factor F12 specifies what fraction of the area of 1 products diffusely emitted radiation impinges directly on surface 2.
If it is in the radiation -exchanging surfaces around black radiator or gray Lambert radiator, then the calculation of the exchanged radiation by use of summary measures can be greatly simplified. On non- diffusely radiating body view factors can be applied only in exceptional cases.
The calculation of view factors requires ( analytical or numerical ) integration over the solid angle under which the surfaces see each other. Geometrical relationships between the visual factors of the involved surfaces usually allow to derive some of the required view factors from already known, thus bypassing a portion of the often complex integrations.
- 4.1 Example
- 6.1 Two differential surfaces
- 6.2 A differential and a finite surface
- 6.3 Example
Radiation exchange
Photometric Basic Law
According to the photometric Basic Law the power transmitted by an infinitesimal surface element on a surface element radiation power depends
- Of the votes in the direction of beam density,
- Of the area sizes and,
- Of the angles, and to which the faces are inclined relative to their common connection line, and
- On the mutual distance of surfaces:
To the radiant power between finite spaces and to obtain, must be integrated over both surfaces:
Considering only diffusely emitting surfaces with everywhere constant radiance, the radiance is at all Ausstrahlorten and the same in all Ausstrahlrichtungen and can be considered as constant in front of the integrals:
The integrals now depend only on the mutual geometrical configuration of the surfaces involved.
View factor
Considering further that the in all directions in the hemisphere evenly radiant, according to condition the beam density surface total radiation power
Outputs (see radiance ), then it follows for the view factor between two surfaces:
These integrals can be performed and a tabulated once and for all for a given area and pairs.
Considering that a surface element of the solid angle of view subtending (or analogously for ), then the integrals simplify
A view factor is thus essentially the integral of the one of the surfaces spanned by solid angle, weighted with the cosine of the incidence angle on the other surface.
Two areas and have - regardless of their distance - the same view factor regarding when viewed from the span of the same solid angle and have the same angle of incidence.
Reziprozitätsbeziehung
One exchanges in the definition equation of the view factor, the indices 1 and 2, we obtain the view factor for the radiative transfer from 2 to 1 ( of location and multi-directional provided ):
By comparing the two equations, the Reziprozitätsbeziehung perspective factors follows:
If one of the view factors are known, allows this relationship to determine immediately and without further account of the other.
Additivity
An integral over a surface can be decomposed into a sum of integrals over the sub-areas. Accordingly, a visual factor can be decomposed to a target area in a total of view factors faces. This may be of advantage when it can be integrated more easily over the surfaces or part of the simpler part view factors of a table can be removed.
View factor algebra
Find the radiation exchange between surfaces instead of forming a closed cavity ( assumed to be the location-and direction-independent ) follows, as from the radiation balance of the surface
By dividing by the sum rule:
The occurring in the sum summand describes the radiation exchange of the part surface with himself, he is for flat and convex surfaces always zero, for concave surfaces but be nonzero.
In a cavity formed of patches occur on a total view factors. These need not necessarily be determined individually by performing all of the above integrals. View factors can be determined by the sum rule is applied to each of the subareas. The Reziprozitätsbeziehung provides further summary measures. There remain therefore only
View factors to determine independently. This number is still reduced by the number of convex and flat patches, is for.
Example
Consider a spherical shell -like cavity which is delimited by the inner spherical surface 1 and the outer spherical surface 2. To determine the view factors, and.
Since area 1 is convex, it follows immediately.
The applied on surface 1 sum rule provides
Thus, the entire output from the inner surface of one radiation incident on the outer surface 2
From Reziprozitätsbeziehung follows
The applied on surface 2 sum rule finally yields:
Is not equal to zero because surface 2 is concave and part of its issuing of radiation strikes again ( elsewhere) to itself.
In this case, therefore, only a single view of factor (or) from the geometric data of the cavity is to be determined. This determination can be, in this example, even without carrying out the defining integral; however, this is not the normal case dar.
Application
Is a prerequisite for the application of summary measures that emanating from the participating surfaces beam density on each surface is constant and isotropic ( diffuse) is emitted.
This condition is in particular satisfied if the involved surfaces are blackbody radiators with spatially constant temperature, respectively, as blackbody radiators are necessarily diffuse emitters. In this case, the radiation exchange is particularly easy to calculate, since each sub-area Black absorbs all incident radiation and no reflected radiation is taken into account.
If two black radiator 1 and 2 with the mutual visibility factor before, the one of outgoing and incoming at 2 radiation power is given by
However, now the output from the entire transducer face in all directions radiating power nothing but the surface multiplied with the emittance of the radiator, which can be calculated in the case of a black body with the help of the Stefan- Boltzmann law:
It is therefore
Since the receiver is also a black body with the reflectance is zero, the total incident radiation is absorbed.
The condition of constant and diffuse radiance is also satisfied by Gray Lambert radiator. Here is, however, in general, the output of each sub-area radiance composed of the self-emission of the surface and a reflected portion of that radiation it receives from the other faces here ( and in turn both their own issues and reflected components contains ). This requires the preparation according to detailed systems of equations (see, eg radiosity ).
Differential visual factors
To date, summary measures were treated between finite surfaces. In practice, however, often occur and differential areas, such as when the intensity of radiation generated by a radiation source at a given point, that is, measured in Watts per square meter power density to be determined.
Differential surfaces are first-order example but infinitesimally thin finite or infinite long strips, circular rings and the like. They often serve as a starting point for integrations over finite surfaces. Differential second-order surfaces are infinitesimally small area, just as they have already been used up.
In the notation can be marked on the index of the view factor, whether it is a finite or differential surface (eg). View factors on a differential surface are themselves differential variables (eg ).
Two differential surfaces
The radiation power, which emits the area in the surveyed her half-space is that on the surface radiation striking power of which is given by the photometric Basic Law. The relationship is both
By comparing this expression with the expression for the inverse radiation flux ( obtained by interchanging the indices), one obtains the Reziprozitätsbeziehung
A differential and a finite surface
Is the transmission area differential, so the emitted radiation output is in turn while the photometric fundamental law must be integrated over the finite reception area:
Considering the reverse radiation flux, the transmission area is finite, and it emits the radiation power. The photometric Basic Law is in turn integrated over:
The comparison of the two summary measures thus obtained provides the Reziprozitätsbeziehung
View factors between a differential and a finite area are often easier to detect than view factors between two finite areas, since only one integral must be carried out over an area rather than a double integral.
Example
An area radiation source with constant emittance fill one half of the visual field of the Aufpunkts. To determine the irradiance at point. Consider, for example at a point on a vertical building facade, the lower half of the visual field of heat is taken radiating ground.
The total area of emitted in all directions radiating power is. The to incident radiation power is given by and can be calculated taking into account the prescribed Einstrahlgeometrie as. However, instead of integrating over the entire surface and all their Ausstrahlrichtungen, required here to determine the visibility factor, it is easier to consider the opposite view factor. It is
So that only the integral is to be run over the solid angle subtending seen the face of point. The entire field on the space spanned angle an integral over this solid angle, weighted with the cosine of the angle of incidence has a value. Here is to integrate only over half the visual field, ie, the present integral has the value and it is
Overall,
Beam say for example the warm soil with a specific charisma of 400 W / m², it produces on the facade and an irradiation intensity of 200 W / m².
Illustration
As an example, the geometric illustration, consider the following situation: To investigate the nocturnal condensation is a facade for a given point on the surface at night from the environment incident to this point heat radiation to be calculated. The environment consists of ground in the lower half of the visual field, the sky in the upper half, and an in- field of the neighboring building.
The diagram shows this environment from the perspective of the reference point. Shown the field of the point is in a fish eye -like projection, the earth is green, the sky blue and the building ( perspectively distorted) drawn in gray. Also shown are lines of constant incident angle of 0 ° at the center to 90 ° at the circular edge of the field.
The field of view spans the space angle 2π on. Without the building both earth and heaven would take each case the solid angle π and had the view factor ½ ( see example above ). However, the building spans the space angle Ω = 1.21 on ( determined by numerical integration ),
So remain for the partially hidden earth, the solid angle and 2.92 for the sky, the solid angle 2.16.
To determine the respective view factors, however, the integrals over the solid angle must be repeated, now with the cosine of the incidence angle as an additional weighting. The middle part of the building is in the center of the visual field ( incident angle ≈ 0 °) and therefore receives the weight ≈ 1, the outer parts are, however, seen among larger angles of incidence and therefore abgewichtet something more; the weighted integral Ωg2 has the value 0.99:
Heaven and earth extend to the edge of the field, where they are strongly abgewichtet because of the shallow angle of incidence. The weighted integral for the Earth only has the value 1.38, the. 0.77 for the sky
Division of weighted integrals by π provides the view factors. For the building, there is the view factor of 0.32:
The summary measures for earth and heaven are 0.44 and 0.24. The sum of all factors view is 1, as required by the sum rule.
Thus, the geometric Einstrahlverhältnisse are covered. For a concrete example, assume that building, earth and sky Gray bodies with the common temperature of 20 ° C, but the emissivities eG = 0.85, εE = 0.95 and εH = 0.75 are. Reflections are neglected. According to the Stefan- Boltzmann law amount to the respective specific emanations MW = 356 W / m², ME = 398 W / m² and MH = 314 W / m². The irradiance of the Aufpunkts is found to be B = 0.32 × 356 0.44 × 398 0.24 × 314 W / m² = 364 W / m².
If not present, the neighboring building, there would result only the irradiance B '= 0.5 × 398 0.5 × 314 W / m² = 356 W / m².
If the examined facade aligned in a slightly different direction, so that the neighboring building would be closer to the edge of the field, the plane defined by this building solid angle would remain the same, his view factor would take off but since it was in more abgewichteten parts of the visual field.