Common symbols
${\displaystyle J_{\mathrm {e} }}$
Other units
erg·cm−2·s−1
In SI base unitsW·m−2
DimensionM T−3

## Mathematical definitions

Radiosity of a surface, denoted Je ("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 Je,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 Je,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 Je,λ, is defined as[3]

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

where λ is the wavelength.

The two radiosity components of an opaque surface.

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

Normally, Ee 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 Ee,ji Ai = Fji Aj Je,j where Fji 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 Ai:

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

Now, employing the reciprocity relation for view factors Fji Aj = Fij Ai,

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

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

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 at a surface can be determined by finding the difference between the incoming and outgoing energy:

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

Using the equation for radiosity Je,i = εiσTi4 + (1 − εi)Ee,i, the irradiance can be eliminated from the above to obtain

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

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

## 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 is expressed as

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

where Ri = (1 − εi)/(Aiεi) is the resistance of the surface.

Likewise, Me,i°Je,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 Rij = 1/(Ai Fij).

Because the above is between surfaces, Rij is the resistance of the space between the surfaces and Je,iJe,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, 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]

Quantity Unit Dimension Notes
Name Symbol[nb 1] Name Symbol Symbol
Radiant energy density we joule per cubic metre J/m3 ML−1T−2 Radiant energy per unit volume.
Radiant flux Φe[nb 2] watt W = J/s ML2T−3 Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power".
Spectral flux Φe,ν[nb 3] watt per hertz W/Hz ML2T−2 Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅nm−1.
Φe,λ[nb 4] watt per metre W/m MLT−3
Radiant intensity Ie,Ω[nb 5] watt per steradian W/sr ML2T−3 Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is a directional quantity.
Spectral intensity Ie,Ω,ν[nb 3] watt per steradian per hertz W⋅sr−1⋅Hz−1 ML2T−2 Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅nm−1. This is a directional quantity.
Ie,Ω,λ[nb 4] watt per steradian per metre W⋅sr−1⋅m−1 MLT−3
Radiance Le,Ω[nb 5] watt per steradian per square metre W⋅sr−1⋅m−2 MT−3 Radiant flux emitted, reflected, transmitted or received by a surface, per unit solid angle per unit projected area. This is a directional quantity. This is sometimes also confusingly called "intensity".
Spectral radiance Le,Ω,ν[nb 3] watt per steradian per square metre per hertz W⋅sr−1⋅m−2⋅Hz−1 MT−2 Radiance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. This is a directional quantity. This is sometimes also confusingly called "spectral intensity".
Le,Ω,λ[nb 4] watt per steradian per square metre, per metre W⋅sr−1⋅m−3 ML−1T−3
Flux density
Ee[nb 2] watt per square metre W/m2 MT−3 Radiant flux received by a surface per unit area. This is sometimes also confusingly called "intensity".
Spectral flux density
Ee,ν[nb 3] watt per square metre per hertz W⋅m−2⋅Hz−1 MT−2 Irradiance of a surface per unit frequency or wavelength. This is sometimes also confusingly called "spectral intensity". Non-SI units of spectral flux density include jansky (1 Jy = 10−26 W⋅m−2⋅Hz−1) and solar flux unit (1 sfu = 10−22 W⋅m−2⋅Hz−1 = 104 Jy).
Ee,λ[nb 4] watt per square metre, per metre W/m3 ML−1T−3
Radiosity Je[nb 2] watt per square metre W/m2 MT−3 Radiant flux leaving (emitted, reflected and transmitted by) a surface per unit area. This is sometimes also confusingly called "intensity".
Spectral radiosity Je,ν[nb 3] watt per square metre per hertz W⋅m−2⋅Hz−1 MT−2 Radiosity of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m−2⋅nm−1. This is sometimes also confusingly called "spectral intensity".
Je,λ[nb 4] watt per square metre, per metre W/m3 ML−1T−3
Radiant exitance Me[nb 2] watt per square metre W/m2 MT−3 Radiant flux emitted by a surface per unit area. This is the emitted component of radiosity. "Radiant emittance" is an old term for this quantity. This is sometimes also confusingly called "intensity".
Spectral exitance Me,ν[nb 3] watt per square metre per hertz W⋅m−2⋅Hz−1 MT−2 Radiant exitance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m−2⋅nm−1. "Spectral emittance" is an old term for this quantity. This is sometimes also confusingly called "spectral intensity".
Me,λ[nb 4] watt per square metre, per metre W/m3 ML−1T−3
Radiant exposure He joule per square metre J/m2 MT−2 Radiant energy received by a surface per unit area, or equivalently irradiance of a surface integrated over time of irradiation. This is sometimes also called "radiant fluence".
Spectral exposure He,ν[nb 3] joule per square metre per hertz J⋅m−2⋅Hz−1 MT−1 Radiant exposure of a surface per unit frequency or wavelength. The latter is commonly measured in J⋅m−2⋅nm−1. This is sometimes also called "spectral fluence".
He,λ[nb 4] joule per square metre, per metre J/m3 ML−1T−2
Hemispherical emissivity ε N/A 1 Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface.
Spectral hemispherical emissivity εν
or
ελ
N/A 1 Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface.
Directional emissivity εΩ N/A 1 Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface.
Spectral directional emissivity εΩ,ν
or
εΩ,λ
N/A 1 Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface.
Hemispherical absorptance A N/A 1 Radiant flux absorbed by a surface, divided by that received by that surface. This should not be confused with "absorbance".
Spectral hemispherical absorptance Aν
or
Aλ
N/A 1 Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance".
Directional absorptance AΩ N/A 1 Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance".
Spectral directional absorptance AΩ,ν
or
AΩ,λ
N/A 1 Spectral radiance absorbed by a surface, divided by the spectral radiance incident onto that surface. This should not be confused with "spectral absorbance".
Hemispherical reflectance R N/A 1 Radiant flux reflected by a surface, divided by that received by that surface.
Spectral hemispherical reflectance Rν
or
Rλ
N/A 1 Spectral flux reflected by a surface, divided by that received by that surface.
Directional reflectance RΩ N/A 1 Radiance reflected by a surface, divided by that received by that surface.
Spectral directional reflectance RΩ,ν
or
RΩ,λ
N/A 1 Spectral radiance reflected by a surface, divided by that received by that surface.
Hemispherical transmittance T N/A 1 Radiant flux transmitted by a surface, divided by that received by that surface.
Spectral hemispherical transmittance Tν
or
Tλ
N/A 1 Spectral flux transmitted by a surface, divided by that received by that surface.
Directional transmittance TΩ N/A 1 Radiance transmitted by a surface, divided by that received by that surface.
Spectral directional transmittance TΩ,ν
or
TΩ,λ
N/A 1 Spectral radiance transmitted by a surface, divided by that received by that surface.
Hemispherical attenuation coefficient μ reciprocal metre m−1 L−1 Radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral hemispherical attenuation coefficient μν
or
μλ
reciprocal metre m−1 L−1 Spectral radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Directional attenuation coefficient μΩ reciprocal metre m−1 L−1 Radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral directional attenuation coefficient μΩ,ν
or
μΩ,λ
reciprocal metre m−1 L−1 Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
1. ^ Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
2. Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance.
3. Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek)—not to be confused with suffix "v" (for "visual") indicating a photometric quantity.
4. Spectral quantities given per unit wavelength are denoted with suffix "λ" (Greek).
5. ^ a b Directional quantities are denoted with suffix "Ω" (Greek).