Radiosity (radiometry)

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.

Mathematical definitions

Radiosity

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

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;
Φ_e is the radiant flux leaving (emitted, reflected and transmitted);
A is the area;
J_e,em = M_e is the emitted component of the radiosity of the surface, that is to say its exitance;
J_e,r is the reflected component of the radiosity of the surface;
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:

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

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

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]

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

where λ is the wavelength.

Radiosity method

The two radiosity components of an opaque surface.

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

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:

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,

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

and substituting the irradiance into the equation for radiosity, produces

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:

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

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

{\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 M_e,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

{\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,

{\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.

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]

SI radiometry units

**SI radiometry units**
Quantity		Unit		Dimension	Notes
Name	Symbol^{[nb 1]}	Name	Symbol	Symbol	Notes
Radiant energy	Q_e^{[nb 2]}	joule	J	M⋅L²⋅T⁻²	Energy of electromagnetic radiation.
Radiant energy density	w_e	joule per cubic metre	J/m³	M⋅L⁻¹⋅T⁻²	Radiant energy per unit volume.
Radiant flux	Φ_e^{[nb 2]}	watt	W or J/s	M⋅L²⋅T⁻³	Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power".
Spectral flux	Φ_e,ν^{[nb 3]} or Φ_e,λ^{[nb 4]}	watt per hertz or watt per metre	W/Hz or W/m	M⋅L²⋅T⁻² or M⋅L⋅T⁻³	Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅sr⁻¹⋅m⁻²⋅nm⁻¹.
Radiant intensity	I_e,Ω^{[nb 5]}	watt per steradian	W/sr	M⋅L²⋅T⁻³	Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is a directional quantity.
Spectral intensity	I_e,Ω,ν^{[nb 3]} or I_e,Ω,λ^{[nb 4]}	watt per steradian per hertz or watt per steradian per metre	W⋅sr⁻¹⋅Hz⁻¹ or W⋅sr⁻¹⋅m⁻¹	M⋅L²⋅T⁻² or M⋅L⋅T⁻³	Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W⋅sr⁻¹⋅nm⁻¹. This is a directional quantity.
Radiance	L_e,Ω^{[nb 5]}	watt per steradian per square metre	W⋅sr⁻¹⋅m⁻²	M⋅T⁻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	L_e,Ω,ν^{[nb 3]} or L_e,Ω,λ^{[nb 4]}	watt per steradian per square metre per hertz or watt per steradian per square metre, per metre	W⋅sr⁻¹⋅m⁻²⋅Hz⁻¹ or W⋅sr⁻¹⋅m⁻³	M⋅T⁻² or M⋅L⁻¹⋅T⁻³	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".
Irradiance Flux density	E_e^{[nb 2]}	watt per square metre	W/m²	M⋅T⁻³	Radiant flux received by a surface per unit area. This is sometimes also confusingly called "intensity".
Spectral irradiance Spectral flux density	E_e,ν^{[nb 3]} or E_e,λ^{[nb 4]}	watt per square metre per hertz or watt per square metre, per metre	W⋅m⁻²⋅Hz⁻¹ or W/m³	M⋅T⁻² or M⋅L⁻¹⋅T⁻³	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 = 10⁻²⁶ W⋅m⁻²⋅Hz⁻¹ and solar flux unit (1SFU = 10⁻²² W⋅m⁻²⋅Hz⁻¹=10⁴Jy).
Radiosity	J_e^{[nb 2]}	watt per square metre	W/m²	M⋅T⁻³	Radiant flux leaving (emitted, reflected and transmitted by) a surface per unit area. This is sometimes also confusingly called "intensity".
Spectral radiosity	J_e,ν^{[nb 3]} or J_e,λ^{[nb 4]}	watt per square metre per hertz or watt per square metre, per metre	W⋅m⁻²⋅Hz⁻¹ or W/m³	M⋅T⁻² or M⋅L⁻¹⋅T⁻³	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".
Radiant exitance	M_e^{[nb 2]}	watt per square metre	W/m²	M⋅T⁻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	M_e,ν^{[nb 3]} or M_e,λ^{[nb 4]}	watt per square metre per hertz or watt per square metre, per metre	W⋅m⁻²⋅Hz⁻¹ or W/m³	M⋅T⁻² or M⋅L⁻¹⋅T⁻³	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".
Radiant exposure	H_e	joule per square metre	J/m²	M⋅T⁻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	H_e,ν^{[nb 3]} or H_e,λ^{[nb 4]}	joule per square metre per hertz or joule per square metre, per metre	J⋅m⁻²⋅Hz⁻¹ or J/m³	M⋅T⁻¹ or M⋅L⁻¹⋅T⁻²	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".
Hemispherical emissivity	ε			1	Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface.
Spectral hemispherical emissivity	ε_ν or ε_λ			1	Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface.
Directional emissivity	ε_Ω			1	Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface.
Spectral directional emissivity	ε_Ω,ν or ε_Ω,λ			1	Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface.
Hemispherical absorptance	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	A_ν or A_λ			1	Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance".
Directional absorptance	A_Ω			1	Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance".
Spectral directional absorptance	A_Ω,ν or A_Ω,λ			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			1	Radiant flux reflected by a surface, divided by that received by that surface.
Spectral hemispherical reflectance	R_ν or R_λ			1	Spectral flux reflected by a surface, divided by that received by that surface.
Directional reflectance	R_Ω			1	Radiance reflected by a surface, divided by that received by that surface.
Spectral directional reflectance	R_Ω,ν or R_Ω,λ			1	Spectral radiance reflected by a surface, divided by that received by that surface.
Hemispherical transmittance	T			1	Radiant flux transmitted by a surface, divided by that received by that surface.
Spectral hemispherical transmittance	T_ν or T_λ			1	Spectral flux transmitted by a surface, divided by that received by that surface.
Directional transmittance	T_Ω			1	Radiance transmitted by a surface, divided by that received by that surface.
Spectral directional transmittance	T_Ω,ν or T_Ω,λ			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	μ_ν or μ_λ	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	μ_Ω	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	μ_Ω,ν or μ_Ω,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.
See also: SI · Radiometry · Photometry

↑ Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
1 2 3 4 5 Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance.
1 2 3 4 5 6 7 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.
1 2 3 4 5 6 7 Spectral quantities given per unit wavelength are denoted with suffix "λ" (Greek).
1 2 Directional quantities are denoted with suffix "Ω" (Greek).

References

↑ http://www.qudt.org/qudt/owl/1.0.0/quantity/Instances.html#Radiosity. Accessed 2013-10-08.
↑ Gan, Guohui, Numerical Method for a Full Assessment of Indoor Thermal Confort, Indoor Air Journal, 1994,4:154-158
1 2 3 "Thermal insulation — Heat transfer by radiation — Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved 2015-03-15.
1 2 E.M. Sparrow and R.D. Cess. Radiation Heat Transfer. Hemisphere Publishing Corporation, 1978.

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