Radiosity (radiometry)

Radiosity
Common symbols	${\displaystyle J_{\mathrm {e} }}$
SI unit	W·m−2
Other units	erg·cm−2·s−1
Dimension	M T−3

In radiometry, radiosity is the radiant flux leaving (emitted, reflected and transmitted by) a surface per unit area, and spectral radiosity is the radiosity of a surface per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength.^[1] The SI unit of radiosity is the watt per square metre (W/m²), while that of spectral radiosity in frequency is the watt per square metre per hertz (W·m⁻²·Hz⁻¹) and that of spectral radiosity in wavelength is the watt per square metre per metre (W·m⁻³)—commonly the watt per square metre per nanometre (W·m⁻²·nm⁻¹). The CGS unit erg per square centimeter per second (erg·cm⁻²·s⁻¹) is often used in astronomy. Radiosity is often called intensity^[2] in branches of physics other than radiometry, but in radiometry this usage leads to confusion with radiant intensity.

Radiosity of a surface, denoted J_e ("e" for "energetic", to avoid confusion with photometric quantities), is defined as^[3]

${\displaystyle J_{\mathrm {e} }={\frac {\partial \Phi _{\mathrm {e} }}{\partial A}}=J_{\mathrm {e,em} }+J_{\mathrm {e,r} }+J_{\mathrm {e,tr} },}$

where

• ∂ is the partial derivative symbol
• ${\displaystyle \Phi _{e}}$ is the radiant flux leaving (emitted, reflected and transmitted)
• ${\displaystyle A}$ is the area
• ${\displaystyle J_{e,em}=M_{e}}$ is the emitted component of the radiosity of the surface, that is to say its exitance
• ${\displaystyle J_{e,r}}$ is the reflected component of the radiosity of the surface
• ${\displaystyle J_{e,tr}}$ is the transmitted component of the radiosity of the surface

For an opaque surface, the transmitted component of radiosity J_e,tr vanishes and only two components remain:

${\displaystyle J_{\mathrm {e} }=M_{\mathrm {e} }+J_{\mathrm {e,r} }.}$

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

Spectral radiosity in frequency of a surface, denoted J_e,ν, is defined as^[3]

${\displaystyle J_{\mathrm {e} ,\nu }={\frac {\partial J_{\mathrm {e} }}{\partial \nu }},}$

where ν is the frequency.

Spectral radiosity in wavelength of a surface, denoted J_e,λ, is defined as^[3]

${\displaystyle J_{\mathrm {e} ,\lambda }={\frac {\partial J_{\mathrm {e} }}{\partial \lambda }},}$

where λ is the wavelength.

The radiosity of an opaque, gray and diffuse surface is given by

${\displaystyle J_{\mathrm {e} }=M_{\mathrm {e} }+J_{\mathrm {e,r} }=\varepsilon \sigma T^{4}+(1-\varepsilon )E_{\mathrm {e} },}$

where

• ε is the emissivity of that surface;
• σ is the Stefan–Boltzmann constant;
• T is the temperature of that surface;
• E_e is the irradiance of that surface.

Normally, E_e 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 E_e,ji A_i = F_ji A_j J_e,j where F_ji is the view factor or shape factor, from surface j to surface i. So, the irradiance of surface i is the sum of radiation energy from all other surfaces per unit surface of area A_i:

${\displaystyle E_{\mathrm {e} ,i}={\frac {\sum _{j=1}^{N}F_{ji}A_{j}J_{\mathrm {e} ,j}}{A_{i}}}.}$

Now, employing the reciprocity relation for view factors F_ji A_j = F_ij A_i,

${\displaystyle E_{\mathrm {e} ,i}=\sum _{j=1}^{N}F_{ij}J_{\mathrm {e} ,j},}$

and substituting the irradiance into the equation for radiosity, produces

${\displaystyle J_{\mathrm {e} ,i}=\varepsilon _{i}\sigma T_{i}^{4}+(1-\varepsilon _{i})\sum _{j=1}^{N}F_{ij}J_{\mathrm {e} ,j}.}$

For an N surface enclosure, this summation for each surface will generate N linear equations with N unknown radiosities,^[4] and N unknown temperatures. 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 ${\displaystyle {\dot {Q}}_{i}}$ at a surface can be determined by finding the difference between the incoming and outgoing energy:

${\displaystyle {\dot {Q}}_{i}=A_{i}\left(J_{\mathrm {e} ,i}-E_{\mathrm {e} ,i}\right).}$

Using the equation for radiosity J_e,i = ε_iσT_i⁴ + (1 − ε_i)E_e,i, the irradiance can be eliminated from the above to obtain

${\displaystyle {\dot {Q}}_{i}={\frac {A_{i}\varepsilon _{i}}{1-\varepsilon _{i}}}\left(\sigma T_{i}^{4}-J_{\mathrm {e} ,i}\right)={\frac {A_{i}\varepsilon _{i}}{1-\varepsilon _{i}}}\left(M_{\mathrm {e} ,i}^{\circ }-J_{\mathrm {e} ,i}\right),}$

where M_e,i° is the radiant exitance of a black body.

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 is expressed as

${\displaystyle {\dot {Q_{i}}}={\frac {M_{\mathrm {e} ,i}^{\circ }-J_{\mathrm {e} ,i}}{R_{i}}},}$

where R_i = (1 − ε_i)/(A_iε_i) is the resistance of the surface.

Likewise, M_e,i^° − J_e,i is the blackbody exitance 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,

${\displaystyle {\dot {Q}}_{ij}=A_{i}F_{ij}(J_{\mathrm {e} ,i}-J_{\mathrm {e} ,j})={\frac {J_{\mathrm {e} ,i}-J_{\mathrm {e} ,j}}{R_{ij}}},}$

where R_ij = 1/(A_i F_ij).

Because the above is between surfaces, R_ij is the resistance of the space between the surfaces and J_e,i − J_e,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.

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, that is to say its emissivity is independent of radiation frequency or wavelength. However, if the range of radiation spectrum is large, this will not be the case. In such an application, the radiosity must be calculated spectrally and then integrated over the range of radiation spectrum.

Yet another assumption is that the surface is isothermal. If it is 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.^[4]

• Irradiance
• Radiant flux
• Spectral flux density