Visibility factor

View factors (including: irradiating numbers , form factors , angle ratios ) are used in calculating the radiation exchange between different surfaces for describing the mutual geometric "visibility", so the mutual position and orientation of the surfaces. The visibility factor assigned to the two surfaces 1 and 2 indicates which fraction of the total diffuse radiation emitted by surface 1 hits surface 2 directly. ${\displaystyle F_{12}}$

If the radiation-exchanging surfaces are black radiators or gray Lambert radiators (both radiator types are always diffuse radiators), the calculation of the exchanged radiation can be greatly simplified by using visibility factors. Visual factors can only be applied to non-diffusely radiating bodies in exceptional cases.

The calculation of visibility factors requires the ( analytical or numerical ) integration over the solid angles at which the surfaces see each other. Geometric relationships between the visual factors of the surfaces involved usually allow some of the sought-after visual factors to be derived from already known ones, thus avoiding some of the often complex integrations.

For numerous geometric arrangements of radiator surfaces, the associated visibility factors can be found in the relevant visibility factor catalogs as formulas or tables.

Radiation exchange

Basic photometric law

Two surfaces as mutual radiation partners in the basic photometric law

According to the photometric constitution of an infinitesimal surface element depends on a surface element transmitted radiated power from ${\displaystyle \mathrm {d} A_{1}}$ ${\displaystyle \mathrm {d} A_{2}}$ ${\displaystyle \mathrm {d} ^{2}\Phi _{12}}$

• the radiance emitted in the direction , ${\displaystyle \mathrm {d} A_{2}}$ ${\displaystyle L_{1}}$
• of the area sizes and , ${\displaystyle \mathrm {d} A_{1}}$ ${\displaystyle \mathrm {d} A_{2}}$
• of the angles and by which the surfaces are inclined with respect to their common connecting line, and ${\displaystyle \beta _{1}}$ ${\displaystyle \beta _{2}}$
• the mutual distance between the surfaces: ${\displaystyle r}$

${\displaystyle \mathrm {d} ^{2}\Phi _{12}={\frac {L_{1}\cdot \cos(\beta _{1})\,\cos(\beta _{2})\,\mathrm {d} A_{1}\,\mathrm {d} A_{2}}{r^{2}}}}$

In order to maintain the radiant power between finitely large areas and , the following must be integrated over both areas: ${\displaystyle A_{1}}$ ${\displaystyle A_{2}}$

${\displaystyle \Phi _{12}=\int _{A_{1}}\int _{A_{2}}{\frac {L_{1}\cdot \cos(\beta _{1})\,\cos(\beta _{2})\,\mathrm {d} A_{1}\,\mathrm {d} A_{2}}{r^{2}}}}$.

If one considers exclusively diffuse radiating surfaces with constant radiance everywhere, then the radiance is the same at all radiation locations and in all radiation directions and can be drawn as a constant in front of the integrals: ${\displaystyle L_{1}}$

${\displaystyle \Phi _{12}=L_{1}\cdot \int _{A_{1}}\int _{A_{2}}{\frac {\cos(\beta _{1})\,\cos(\beta _{2})\,\mathrm {d} A_{1}\,\mathrm {d} A_{2}}{r^{2}}}}$.

The integrals now only depend on the mutual geometric configuration of the surfaces involved.

Visibility factor

Furthermore, if one takes into account that the area radiating diffusely according to the assumption, i.e. in all directions of the half-space evenly with the radiance, alters the radiant power ${\displaystyle L_{1}}$ ${\displaystyle A_{1}}$

${\displaystyle \Phi _{1}=\pi L_{1}A_{1}}$

emits (see → Radiance ), then follows for the visibility factor between the two surfaces according to its definition:

${\displaystyle F_{12}:={\frac {\Phi _{12}}{\Phi _{1}}}={\frac {1}{\pi A_{1}}}\int _{A_{1}}\int _{A_{2}}{\frac {\cos(\beta _{1})\,\cos(\beta _{2})}{r^{2}}}\,\mathrm {d} A_{1}\,\mathrm {d} A_{2}}$

These integrals can for given surface pairs and one are executed once and for all and tabulated. ${\displaystyle A_{1}}$ ${\displaystyle A_{2}}$

Noting that the surface element of viewed from the solid angle spans (or similarly for ), then the integrals to simplify ${\displaystyle \mathrm {d} A_{2}}$ ${\displaystyle \mathrm {d} A_{1}}$ ${\displaystyle \mathrm {d} \Omega _{2}=\cos(\beta _{2})\,\mathrm {d} A_{2}/r^{2}}$ ${\displaystyle \mathrm {d} A_{1}}$

${\displaystyle {\begin{aligned}F_{12}&={\frac {1}{\pi A_{1}}}\int _{A_{1}}\int _{\Omega _{2}}\cos(\beta _{1})\,\mathrm {d} \Omega _{2}\;\mathrm {d} A_{1}\\&={\frac {1}{\pi A_{1}}}\int _{A_{2}}\int _{\Omega _{1}}\cos(\beta _{2})\,\mathrm {d} \Omega _{1}\;\mathrm {d} A_{2}\end{aligned}}}$

A visual factor is therefore essentially the integral over the solid angle spanned by one of the surfaces, weighted with the cosine of the angle of incidence on the other surface.

Two surfaces and have - even if they are at different distances from each other and are inclined to different degrees towards the line of sight - with regard to the same visual factor when viewed from the perspective they span the same solid angle and are seen at the same angle of incidence . ${\displaystyle \mathrm {d} A_{2}}$ ${\displaystyle \mathrm {d} A_{2}^{\prime }}$ ${\displaystyle \mathrm {d} A_{1}}$ ${\displaystyle \mathrm {d} A_{1}}$ ${\displaystyle \mathrm {d} A_{1}}$ ${\displaystyle \beta _{1}}$

Reciprocity relationship

If you swap the indices 1 and 2 in the definition equation of the visibility factor, you get the visibility factor for the radiation transport from 2 to 1 ( assuming that it is independent of location and direction ): ${\displaystyle F_{12}}$ ${\displaystyle L_{2}}$

${\displaystyle F_{21}:={\frac {\Phi _{21}}{\Phi _{2}}}={\frac {1}{\pi A_{2}}}\int _{A_{2}}\int _{A_{1}}{\frac {\cos(\beta _{2})\,\cos(\beta _{1})}{r^{2}}}\,\mathrm {d} A_{2}\,\mathrm {d} A_{1}}$.

The reciprocity relationship of the view factors follows from the comparison of the two equations:

${\displaystyle A_{1}\cdot F_{12}\,=\,A_{2}\cdot F_{21}}$

If one of the two visibility factors is known, this relationship allows the other to be determined immediately and without further calculation.

Additivity

An integral over an area can be broken down into a sum of integrals over its partial areas. Correspondingly, a visibility factor can also be broken down into a sum of partial visibility factors with regard to the partial areas. This can be advantageous if it is easier to integrate over the partial areas or if the simpler partial view factors can be taken from a table.

Sight factor algebra

If the radiation exchange takes place between surfaces which form a closed cavity (all assumed to be independent of location and direction), then the radiation balance for the surface follows ${\displaystyle n}$ ${\displaystyle L_{i}}$ ${\displaystyle i}$

${\displaystyle \Phi _{i1}+\Phi _{i2}+...+\Phi _{in}\,=\,\Phi _{i}}$

after division by the sum rule: ${\displaystyle \Phi _{i}}$

${\displaystyle \sum _{j=1}^{n}F_{ij}\,=\,1,\quad i=1,2,...n}$

The summand occurring in total describes the radiation exchange of the partial surface with itself. It is always zero for flat and convex surfaces, but can be different from zero for concave surfaces. ${\displaystyle F_{ii}}$ ${\displaystyle i}$

Overall, visibility factors occur in a cavity formed by partial areas . These do not necessarily all have to be determined individually by performing the integrals given above. Visibility factors can be determined by applying the sum rule to each of the faces. The reciprocity relationship provides further view factors. So there just remain ${\displaystyle n}$ ${\displaystyle n^{2}}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n(n-1)/2}$

${\displaystyle n^{2}-n-n(n-1)/2\,=\,n(n-1)/2}$

To determine visibility factors independently of each other. This number is reduced by the number of convex and flat surfaces for which is. ${\displaystyle F_{ii}=0}$

example

Consider a hollow space in the shape of a spherical shell, which is delimited by the inner spherical surface 1 and the outer spherical surface 2. To determine the view factors , , and . ${\displaystyle F_{11}}$ ${\displaystyle F_{12}}$ ${\displaystyle F_{21}}$ ${\displaystyle F_{22}}$

Since surface 1 is convex, it immediately follows . ${\displaystyle F_{11}=0}$

The sum rule applied to area 1 yields

${\displaystyle 1\,=\,F_{11}+F_{12}\,=\,0+F_{12}\,=\,F_{12}}$;

the entire radiation emitted by the inner surface 1 therefore falls on the outer surface 2.

It follows from the reciprocity relation

${\displaystyle F_{21}\,=\,{\frac {A_{1}}{A_{2}}}F_{12}\,=\,{\frac {A_{1}}{A_{2}}}\,=\,\left({\frac {r_{1}}{r_{2}}}\right)^{2}<1}$.

The sum rule applied to area 2 finally gives:

${\displaystyle F_{22}\,=\,1-F_{21}\,=\,1-{\frac {A_{1}}{A_{2}}}\,=\,1-\left({\frac {r_{1}}{r_{2}}}\right)^{2}}$.

${\displaystyle F_{22}}$ is not equal to zero, since surface 2 is concave and part of the radiation it emits hits itself again (at another point).

In this case with only a single view factor ( or ) remains to be determined from the geometric data of the cavity. In this example, this determination can even be made simply by calculating an algebraic term, that is, without executing the defining integral; however, this is not the normal case. ${\displaystyle n=2}$ ${\displaystyle F_{21}}$ ${\displaystyle F_{22}}$

application

The prerequisite for the use of visibility factors is that the radiance emanating from the surfaces involved is constant on each surface and is emitted independently of the direction (diffuse).

This prerequisite is met in particular if the surfaces involved are black bodies , each with a spatially constant temperature, since black bodies are inevitably also diffuse bodies. In this case, the radiation exchange is particularly easy to calculate, since each black partial area absorbs all incident radiation and no reflected radiation has to be taken into account.

If there are two black bodies 1 and 2 with a mutual visibility factor , the radiation power emanating from 1 and arriving at 2 is given by ${\displaystyle F_{12}}$

${\displaystyle \Phi _{12}\,=\,F_{12}\cdot \Phi _{1}}$

However, the radiation power emitted in all directions by the entire transmission surface is nothing other than the specific radiation of the radiator multiplied by the surface area , which in the case of a black radiator can be calculated using the Stefan-Boltzmann law : ${\displaystyle \Phi _{1}}$

${\displaystyle \Phi _{1}\,=\,\sigma \cdot A_{1}\cdot T_{1}^{4}}$.

${\displaystyle \sigma }$	:	Stefan-Boltzmann constant
${\displaystyle T_{1}}$	:	absolute temperature of surface 1

So it is

${\displaystyle \Phi _{12}\,=\,A_{1}\cdot F_{12}\cdot \sigma \,T_{1}^{4}}$

Since the receiver is also a black body with zero reflectance, all of the incident radiation is absorbed.

The self-emission of gray Lambert radiators with spatially constant temperature also fulfills the requirement of constant and diffuse radiance. Here, however, the radiance emitted by each sub-area is generally composed of the surface's own emission and a reflected portion of the radiation that arrives from the other sub-surfaces (and in turn contains both their own emissions and reflected portions). This requires the establishment of a correspondingly detailed system of equations (see e.g. radiosity ).

Differential visibility factors

So far, visibility factors between finite surfaces have been dealt with. They were used to determine the radiant power transmitted from one of the surfaces to the other, measured in watts . In practice, however, differential surfaces often also occur, for example when the irradiance generated by a radiation source with the surface at a certain point with the infinitesimal surface , i.e. a power density measured in watts per square meter , is to be determined. ${\displaystyle A_{1}}$ ${\displaystyle \mathrm {d} A_{2}}$

First order differential surfaces are e.g. B. infinitesimally thin but finite or infinitely long strips, circular rings and the like. They often serve as a starting point for integrations over finite areas. Second order differential surfaces are infinitesimally small pieces of surface, as they have already been used, for example, in the basic photometric law.

In the notation, the respective index of the visibility factor can be used to indicate whether it is a finite or differential area (e.g. ). Visibility factors on a differential surface are themselves differential quantities (e.g. ). ${\displaystyle F_{d1\,2}}$ ${\displaystyle \mathrm {d} F_{1\,d2}}$

Two differential surfaces

The radiant power that the surface emits with the radiance in the half-space it overlooks is . The radiation power that hits the surface is given by the basic photometric law. The relationship of both is ${\displaystyle \mathrm {d} A_{1}}$ ${\displaystyle L_{1}}$ ${\displaystyle \pi \,L_{1}\,\mathrm {d} A_{1}}$ ${\displaystyle \mathrm {d} A_{2}}$

${\displaystyle \mathrm {d} F_{d1\,d2}\,=\,{\frac {L_{1}\cos(\beta _{1})\cos(\beta _{2})\,\mathrm {d} A_{1}\,\mathrm {d} A_{2}/r^{2}}{\pi \,L_{1}\,\mathrm {d} A_{1}}}\,=\,{\frac {\cos(\beta _{1})\cos(\beta _{2})}{\pi r^{2}}}\,\mathrm {d} A_{2}\,=\,{\frac {\cos(\beta _{1})\,\mathrm {d} \Omega _{2}}{\pi }}}$.

By comparing this expression with the expression for the reverse radiant flux (which results from interchanging the indices) one obtains the reciprocity relation

${\displaystyle \mathrm {d} A_{1}\cdot \mathrm {d} F_{d1\,d2}\,=\,\mathrm {d} A_{2}\cdot \mathrm {d} F_{d2\,d1}}$

A differential and a finite area

If the transmission area is differential, the emitted radiation power is again , while the basic photometric law now has to be integrated over the finite reception area : ${\displaystyle \mathrm {d} A_{1}}$ ${\displaystyle \pi \,L_{1}\,\mathrm {d} A_{1}}$ ${\displaystyle A_{2}}$

${\displaystyle F_{d1\,2}\,=\,{\frac {\int _{A_{2}}(L_{1}\cos(\beta _{1})\cos(\beta _{2})\,\mathrm {d} A_{1}/r^{2})\,\mathrm {d} A_{2}}{\pi \,L_{1}\,\mathrm {d} A_{1}}}\,=\,\int _{A_{2}}{\frac {\cos(\beta _{1})\cos(\beta _{2})}{\pi r^{2}}}\,\mathrm {d} A_{2}\,=\,\int _{A_{2}}{\frac {\cos(\beta _{1})\,\mathrm {d} \Omega _{2}}{\pi }}\,=\,\int _{A_{2}}\mathrm {d} F_{d1\,d2}}$.

If you look at the reverse flow of radiation, the transmission area is finite and it emits the radiation power . The basic photometric law is also to be integrated via : ${\displaystyle A_{2}}$ ${\displaystyle \textstyle \int _{A_{2}}\pi \,L_{2}\,\mathrm {d} A_{2}}$ ${\displaystyle A_{2}}$

${\displaystyle \mathrm {d} F_{2\,d1}\,=\,{\frac {\int _{A_{2}}(L_{2}\cos(\beta _{1})\cos(\beta _{2})\,\mathrm {d} A_{1}/r^{2})\,\mathrm {d} A_{2}}{\int _{A_{2}}\,\pi \,L_{2}\,\mathrm {d} A_{2}}}\,=\,{\frac {\mathrm {d} A_{1}}{A_{2}}}\int _{A_{2}}{\frac {\cos(\beta _{1})\cos(\beta _{2})}{\pi r^{2}}}\,\mathrm {d} A_{2}\,=\,{\frac {\mathrm {d} A_{1}}{A_{2}}}\int _{A_{2}}{\frac {\cos(\beta _{1})\,\mathrm {d} \Omega _{2}}{\pi }}\,=\,{\frac {\mathrm {d} A_{1}}{A_{2}}}\int _{A_{2}}\mathrm {d} F_{d1\,d2}}$.

The comparison of the two visual factors obtained in this way provides the reciprocity relationship

${\displaystyle \mathrm {d} A_{1}\,F_{d1\,2}\,=\,A_{2}\,\mathrm {d} F_{2\,d1}}$

Visibility factors between a differential and a finite area are often easier to determine than visibility factors between two finite areas, since instead of a double integral only an integral over an area has to be carried out.

example

What fraction of the heat radiation of the ground reaches a viewer whose lower half of the field of vision is occupied by the ground?

This example shows the application of the reciprocity relationship between a differential and a finite surface.

A flat, diffuse radiation source with constant radiance fills half of the field of view of the point of view . The resulting irradiance at the point must be determined . Think, for example, of a point on a building wall, the lower half of the field of view is occupied by the heat radiating ground. How much of the heat radiated from the earth in all directions arrives at this point? ${\displaystyle A_{2}}$ ${\displaystyle L_{2}}$ ${\displaystyle \mathrm {d} A_{1}}$ ${\displaystyle B_{1}}$ ${\displaystyle \mathrm {d} A_{1}}$

The radiation source is , the radiation receiver is . It therefore seems obvious to solve the problem using the visibility factor : ${\displaystyle A_{2}}$ ${\displaystyle \mathrm {d} A_{1}}$ ${\displaystyle \mathrm {d} F_{2\,d1}}$

The total radiant power emitted by the surface in all directions is ${\displaystyle A_{2}}$

${\displaystyle \Phi _{2}\,=\,\pi \,L_{2}A_{2}}$.

The on incident radiation power results from the fact by using the view factor of to : ${\displaystyle \mathrm {d} A_{1}}$ ${\displaystyle \mathrm {d} \Phi _{2\,d1}}$ ${\displaystyle A_{2}}$ ${\displaystyle \mathrm {d} A_{1}}$

${\displaystyle \mathrm {d} \Phi _{2\,d1}=\Phi _{2}\cdot \mathrm {d} F_{2\,d1}}$.

When calculating this visibility factor, however, the infinitely extended transmission area would have to be integrated, whereby the emission angle aimed in the direction of the receiving point and the angle of incidence at the receiving point would have to be determined for each surface element : ${\displaystyle A_{2}}$ ${\displaystyle \mathrm {d} A_{2}}$ ${\displaystyle \beta _{2}}$ ${\displaystyle \beta _{1}}$

${\displaystyle \mathrm {d} F_{2\,d1}\,=\,{\frac {\mathrm {d} A_{1}}{A_{2}}}\int _{A_{2}}{\frac {\cos(\beta _{1})\cos(\beta _{2})}{\pi r^{2}}}\,\mathrm {d} A_{2}}$.

The occurrence of differential quantities is also not conducive to clarity. Determining the reversed visibility factor is simpler and, above all, more descriptive than calculating a differential visibility factor. This is (see previous section)

${\displaystyle F_{d1\,2}\,=\,\int _{A_{2}}{\frac {\cos(\beta _{1})\cos(\beta _{2})}{\pi r^{2}}}\,\mathrm {d} A_{2}\,=\,{\frac {1}{\pi }}\,\int _{\Omega _{2}}\cos(\beta _{1})\,\mathrm {d} \Omega _{2}}$,

so that only the integral weighted with the cosine of the angle of incidence has to be carried over the solid angle that the surface spans when viewed from the point . Apart from the integral, there are no differential quantities. ${\displaystyle \Omega _{2}}$ ${\displaystyle A_{2}}$ ${\displaystyle \mathrm {d} A_{1}}$

The calculation is simple: the entire field of view spans the solid angle , an integral over this solid angle, weighted with the cosine of the angle of incidence, has the value . Here is only to be integrated over half the field of view, so the present integral has the value and it is ${\displaystyle 2\pi }$ ${\displaystyle \pi }$ ${\displaystyle \pi /2}$

${\displaystyle F_{d1\,2}\,=\,{\frac {1}{\pi }}\cdot {\frac {\pi }{2}}\,=\,{\frac {1}{2}}}$.

Overall it results

${\displaystyle B_{1}\,=\,{\frac {\mathrm {d} \Phi _{2\,d1}}{\mathrm {d} A_{1}}}\,=\,{\frac {\Phi _{2}\cdot \mathrm {d} F_{2\,d1}}{\mathrm {d} A_{1}}}\,=\,{\frac {\Phi _{2}\cdot {\frac {\mathrm {d} A_{1}}{A_{2}}}F_{d1\,2}}{\mathrm {d} A_{1}}}\,=\,{\frac {\Phi _{2}\cdot F_{d1\,2}}{A_{2}}}\,=\,{\frac {\pi \,L_{2}A_{2}\cdot {\frac {1}{2}}}{A_{2}}}\,=\,{\frac {\pi \,L_{2}}{2}}}$.

If one looks at the specific radiation of the ground instead of the radiance , then the simple relationship emerges ${\displaystyle L_{2}}$ ${\displaystyle M_{2}=\pi \,L_{2}}$

${\displaystyle B_{1}\,=\,{\frac {M_{2}}{2}}}$.

For example, if the warm ground radiates with a specific radiation of 400 W / m², it generates an irradiance of 200 W / m² on the facade.

In general

${\displaystyle B_{1}\,=\,{\frac {\Phi _{2}}{A_{2}}}F_{d1\,2}\,=\,M_{2}\,F_{d1\,2}}$.

This relationship is often required if the irradiance generated by planar radiation sources at a receiving point is to be determined. The formula gets along (although the reception point is a differential surface) without unreporting differential quantities if it is formulated using the inverse visibility factor.

illustration

The fisheye diagram shows a schematic scene with the ground (green), sky (blue) and a building (gray).

The following situation is considered as an example for the geometric illustration of visibility factors: To investigate the nocturnal condensation on a facade, the thermal radiation incident on this point at night from the surroundings must be calculated for a given point on its surface. The environment consists of the ground in the lower half of the field of view, the sky in the upper half, and a neighboring building in the field of view.

The adjacent diagram shows this environment from the point of view. Shown is the field of the point in a fish eye -like projection , the ground is green, the sky blue and the building (perspective distortion) Gray located. Lines at the same angle of incidence are also drawn, from 0 ° in the center of the image to 90 ° at the circular edge of the field of view.

The field of view spans the solid angle . Without the building, both the earth and the sky would each occupy the solid angle and would have a visibility factor of ½ (see example above). However, the building spans the solid angle (determined by numerical integration), ${\displaystyle 2\pi }$ ${\displaystyle \pi }$ ${\displaystyle \Omega =1{,}21}$

${\displaystyle \Omega _{2}\,=\,\int _{\Omega _{2}}\mathrm {d} \Omega _{2}\,=\,1{,}21}$.

so that the solid angle 2.92 remains for the partially covered earth and the solid angle 2.16 remains for the sky.

To determine the respective visibility factors, the integrals must be repeated over the solid angles, but now with the cosine of the angle of incidence as an additional weighting. The middle part of the building lies in the center of the field of view (angle of incidence ≈0 °) and therefore receives the weight ≈1, but the outer parts are seen at larger angles of incidence and are therefore slightly more weighted; the weighted integral has the value 0.99: ${\displaystyle \Omega _{g2}}$

${\displaystyle \Omega _{g2}\,=\,\int _{\Omega _{2}}\cos(\beta _{1})\,\mathrm {d} \Omega _{2}\,=\,0{,}99}$.

Heaven and earth extend to the edge of the field of vision, where they are heavily weighed down because of the flat angle of incidence. The weighted integral for the earth only has the value 1.38, that for the sky 0.77.

Division of the weighted integrals by yields the view factors. The visibility factor for the building is 0.32: ${\displaystyle \pi }$

${\displaystyle F_{d1\,2}\,=\,{\frac {1}{\pi }}\,\int _{\Omega _{2}}\cos(\beta _{1})\,\mathrm {d} \Omega _{2}\,=\,0{,}32}$.

The visibility factors for earth and sky are 0.44 and 0.24, respectively. The sum of all visibility factors is 1, as required by the sum rule.

The geometric irradiation conditions are thus recorded. For a concrete example it is assumed that buildings, earth and sky are gray emitters with a common temperature of 20 ° C, but the emissivities , and . Reflections are neglected. According to the Stefan-Boltzmann law, the respective specific emissions are M _G = 356 W / m², M _E = 398 W / m² and M _H = 314 W / m². The irradiance of the starting point results in B = 0.32 × 356 + 0.44 × 398 + 0.24 × 314 W / m² = 364 W / m². ${\displaystyle \varepsilon _{G}=0{,}85}$ ${\displaystyle \varepsilon _{E}=0{,}95}$ ${\displaystyle \varepsilon _{H}=0{,}75}$

If the neighboring building were not there, the irradiance would only be B '= 0.5 × 398 + 0.5 × 314 W / m² = 356 W / m².

If the examined facade were oriented in a slightly different direction so that the neighboring building was closer to the edge of the field of view, the solid angle spanned by this building would remain unchanged, but its visibility factor would decrease because it would be in more heavily weighted parts of the field of view.

literature

• HD Baehr, K. Stephan: Heat and mass transfer. 5th edition, Springer, Berlin 2006, ISBN 978-3-540-32334-1 , chap. 5: thermal radiation.
• R. Siegel, JR Howell, J. Lohrengel: Heat transfer through radiation. Part 2: Radiation exchange between surfaces and in enclosures. Springer, Berlin / Heidelberg / New York 1991, ISBN 3-540-52710-9 , chap. 2: Radiation exchange between black isothermal surfaces.
• B. Glück: Radiant heating - theory and practice. CF Müller, Karlsruhe 1982, ISBN 3-7880-7157-5 , chap. 5: Radiation figures, PDF (detailed examples).
• R. Siegel, JR Howell: Thermal Radiation Heat Transfer. 4th edition, Taylor & Francis, New York / London 2002, ISBN 1-56032-839-8 , Chapter 5: Configuration Factors for Surfaces Transferring Uniform Diffuse Radiation.

Web links

• John R. Howell: A Catalog of Radiation Heat Transfer Configuration Factors - extensive tabulation of visibility factors

Individual evidence

1. ↑ ^{a b} H.D. Baehr, K. Stephan: Heat and mass transfer. 5th edition, Springer, Berlin / Heidelberg / New York 2006, ISBN 978-3-540-32334-1 , p. 637
2. ↑ ${\displaystyle \textstyle \int _{\mathrm {Halbraum} }\mathrm {d} \Omega \,=\,\int _{0}^{\pi /2}\int _{0}^{2\pi }\sin(\beta )\,\mathrm {d} \beta \,\mathrm {d} \varphi \,=\,2\pi \int _{0}^{\pi /2}\sin(\beta )\,\mathrm {d} \beta \,=\,2\pi }$
3. ↑ ${\displaystyle \textstyle \int _{\mathrm {Halbraum} }\cos(\beta )\mathrm {d} \Omega \,=\,\int _{0}^{\pi /2}\int _{0}^{2\pi }\cos(\beta )\sin(\beta )\,\mathrm {d} \beta \,\mathrm {d} \varphi \,=\,2\pi \int _{0}^{\pi /2}{\frac {1}{2}}\sin(2\beta )\mathrm {d} \beta \,=\,\pi \int _{0}^{\pi }\sin(\eta )\,{\frac {\mathrm {d} \eta }{2}}\,=\,\pi }$