Radiosity (radiometry)

This article is about the thermodynamic quantity. For the rendering algorithm, see Radiosity (3D computer graphics).

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. In heat transfer, combining these two factors into one radiosity term helps in determining the net energy exchange between multiple surfaces.

Definition

Consider a unit of surface area. Realistically, the intensity from emitted radiation , $I_{e}$, and the intensity of reflected radiation, $I_{r}$, are both functions of angle from the surface. So, adding these together, the intensity of the total radiation leaving the surface is defined as

$I_{e+r}(\theta ,\phi )={\frac {d{\Phi }}{d\omega \cos \theta }}$

where $d{\Phi }$ represents energy flux and $d\omega$ is the solid angle. The $\cos \theta$ term accounts for the projected area of the surface at an angle. Now, to find the radiosity, the flux is integrated over a hemispherical surface enclosing the unit area for all angles. To generalize further, the radiosity should also be expressed as a function of the wavelength of the radiation .

$J(\theta ,\phi ,\lambda )=\int \limits _{\lambda }{\int \limits _{S}{I_{e+r}(\theta ,\phi ,\lambda )\cos \theta d\omega d\lambda }}$

Assuming a diffuse emitter and reflector, $I_{e+r}$ is constant and the radiosity reduces to $J=\pi I_{e+r}$ over all wavelengths. Furthermore, for a blackbody, $I_{r}=0$ and the radiosity reduces to $J=\pi I_{e}$ .

Radiosity Method

The radiosity $J$, for a gray, diffuse surface, is the sum of the reflected and emitted intensities. Or,

$J=\epsilon \sigma T^{4}+(1-\epsilon )H\,\!$

where $\epsilon \sigma T$ is the gray body radiation due to temperature $T$, and $H$ is the incident radiation. Normally, $H$ is the unknown variable and will depend on the surrounding surfaces. So, if some surface $i$ is being hit by radiation from some other surface $j$, then the radiation energy incident on surface $i$ is $H_{ji}=F_{ji}A_{j}J_{j}$. So, the incident intensity is the sum of radiation from all other surfaces per unit surface of area $A_{i}$.

$H_{i}={\frac {\sum _{j=1}^{N}{(F_{ji}A_{j}J_{j})}}{A_{i}}}$

$F_{ji}$ is the view factor, or shape factor, from surface $j$ to surface $i$. Now, employing the reciprocity relation,

$H_{i}=\sum _{j=1}^{N}{F_{ij}J_{j}}$

and substituting the incident intensity into the original equation for radiosity, produces

$J_{i}=\epsilon \sigma T^{4}+(1-\epsilon )\sum _{j=1}^{N}{F_{ij}J_{j}}$

For an $N$ surface enclosure, this summation for each surface will generate $N$ linear equations with $N$ unknown radiosities . For an enclosure with only a few surfaces, this can be done by hand. But, for a room with many surfaces, linear algebra and a computer are necessary.

Once the radiosities have been calculated, the net heat transfer at a surface can be determined by finding the difference between the incoming and outgoing energy.

${\dot {Q_{i}}}=A_{i}(J_{i}-H_{i})$

Using the equation for radiosity, $J=\epsilon \sigma T^{4}+(1-\epsilon )H$, the incident radiation, $H$, can be eliminated from the above to obtain

${\dot {Q_{i}}}={\frac {A_{i}\epsilon _{i}}{1-\epsilon _{i}}}(\sigma T_{i}^{4}-J_{i})$

Circuit Analogy

For an enclosure consisting of only a few surfaces, it is often easier to represent the system with an analogous circuit rather than solve the set of linear radiosity equations. To do this, the heat transfer at each surface, $i$, is expressed as

${\dot {Q_{i}}}={\frac {A_{i}\epsilon _{i}}{1-\epsilon _{i}}}(E_{bi}-J_{i})={\frac {E_{bi}-J_{i}}{R_{i}}}\qquad {\text{where}}\quad R_{i}={\frac {1-\epsilon _{i}}{A_{i}\epsilon _{i}}}$

and $R_{i}$ is known as the surface resistance. Likewise, $(E_{bi}-J_{i})$ is the blackbody radiation minus the radiosity and serves as the 'potential difference.' These quantities are formulated to resemble those from an electrical circuit $V=IR$.

Now performing a similar analysis for the heat transfer from surface $i$ to surface $j$,

${\dot {Q_{ij}}}=A_{i}F_{ij}(J_{i}-J_{j})={\frac {J_{i}-J_{j}}{R_{i}j}}\qquad {\text{where}}\quad R_{ij}={\frac {1}{A_{i}F_{ij}}}$

Because the above is between surfaces, $R_{ij}$ is known as the space resistance and $(J_{i}-J_{j})$ serves as the potential difference.

Combining the surface elements and space elements, a circuit is formed. The heat transfer is found by using the appropriate potential difference and equivalent resistances, similar to the process used in analyzing electrical circuits .

Other Methods

In the radiosity method and circuit analogy, several assumptions were made to simplify the model. The most significant is that the surface is a diffuse emitter. In such a case, the radiosity does not depend on the angle of incidence of reflecting radiation and this information is lost on a diffuse surface. In reality, however, the radiosity will have a specular component from the reflected radiation . So, the heat transfer between two surfaces relies on both the view factor and the angle of reflected radiation.

It was also assumed that the surface is a gray body and that its emissivity is independent of radiation wavelength. However, if the range of wavelengths of incident and emitted radiation is large, this will not be the case. In such an application, the radiosity must be calculated mono chromatically and then integrated over the range of radiation wavelengths.

Yet another assumption is that the surfaces are isothermal. If they are not, then the radiosity will vary as a function of position along the surface. However, this problem is solved by simply subdividing the surface into smaller elements until the desired accuracy is obtained .

See also

• Irradiance
• Radiant flux