Radiosity (radiometry)
| Radiosity | |
|---|---|
Common symbols | <math>J_\mathrm{e}</math> |
| 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.<ref>FAIRsharing Team (2015). "Radiosity". Quantities, Units, Dimensions and dataTypes. doi:10.25504/FAIRsharing.d3pqw7. Retrieved 25 February 2021.</ref> 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<ref>Gan, Guohui (September 1994). "Numerical Method for a Full Assessment of Indoor Thermal Comfort". Indoor Air. 4 (3): 154–168. doi:10.1111/j.1600-0668.1994.t01-1-00004.x.</ref> 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 Je ("e" for "energetic", to avoid confusion with photometric quantities), is defined as<ref name="ISO_9288-1989">ISO 9288:1989 – Thermal insulation — Heat transfer by radiation — Physical quantities and definitions.</ref>
- <math>J_\mathrm{e} = \frac{\partial \Phi_\mathrm{e}}{\partial A} = J_\mathrm{e,em} + J_\mathrm{e,r} + J_\mathrm{e,tr},</math>
where
- ∂ is the partial derivative symbol
- <math>\Phi_e</math> is the radiant flux leaving (emitted, reflected and transmitted)
- <math>A</math> is the area
- <math>J_{e,em} = M_e</math> is the emitted component of the radiosity of the surface, that is to say its exitance
- <math>J_{e,r}</math> is the reflected component of the radiosity of the surface
- <math>J_{e,tr}</math> 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:
- <math>J_\mathrm{e} = M_\mathrm{e} + J_\mathrm{e,r}.</math>
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 Je,ν, is defined as<ref name="ISO_9288-1989" />
- <math>J_{\mathrm{e},\nu} = \frac{\partial J_\mathrm{e}}{\partial \nu},</math>
where ν is the frequency.
Spectral radiosity in wavelength of a surface, denoted Je,λ, is defined as<ref name="ISO_9288-1989" />
- <math>J_{\mathrm{e},\lambda} = \frac{\partial J_\mathrm{e}}{\partial \lambda},</math>
where λ is the wavelength.
Radiosity method
The radiosity of an opaque, gray and diffuse surface is given by
- <math>J_\mathrm{e} = M_\mathrm{e} + J_\mathrm{e,r} = \varepsilon \sigma T^4 + (1 - \varepsilon) E_\mathrm{e},</math>
where
- ε is the emissivity of that surface;
- σ is the Stefan–Boltzmann constant;
- T is the temperature of that surface;
- Ee is the irradiance of that surface.
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:
- <math>E_{\mathrm{e},i} = \frac{\sum_{j = 1}^N F_{ji}A_j J_{\mathrm{e},j}}{A_i}.</math>
Now, employing the reciprocity relation for view factors Fji Aj = Fij Ai,
- <math>E_{\mathrm{e},i} = \sum_{j = 1}^N F_{ij} J_{\mathrm{e},j},</math>
and substituting the irradiance into the equation for radiosity, produces
- <math>J_{\mathrm{e},i} = \varepsilon_i \sigma T_i^4 + (1 - \varepsilon_i)\sum_{j = 1}^N F_{ij} J_{\mathrm{e},j}.</math>
For an N surface enclosure, this summation for each surface will generate N linear equations with N unknown radiosities,<ref name="sparrow">Sparrow, Ephraim. M.; Cess, Robert D. (1978). Radiation heat transfer (3rd ed.). Washington, DC: Hemisphere. ISBN 9780070599109.</ref> 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 <math>\dot Q_i</math> at a surface can be determined by finding the difference between the incoming and outgoing energy:
- <math>\dot Q_i = A_i\left(J_{\mathrm{e},i} - E_{\mathrm{e},i}\right).</math>
Using the equation for radiosity Je,i = εiσTi4 + (1 − εi)Ee,i, the irradiance can be eliminated from the above to obtain
- <math>\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),</math>
where Me,i° is the radiant 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
- <math>\dot{Q_i} = \frac{M_{\mathrm{e},i}^\circ - J_{\mathrm{e},i}}{R_i},</math>
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,
- <math>\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}},</math>
where Rij = 1/(Ai Fij).
Because the above is between surfaces, Rij is the resistance of the space between the surfaces and Je,i − Je,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.<ref name=sparrow/>
SI radiometry units
| Quantity | Unit | Dimension | Notes | ||
|---|---|---|---|---|---|
| Name | Symbol<ref group="nb" name="note-suffix-e">Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.</ref> | Name | Symbol | ||
| Radiant energy | Qe<ref group="nb" name="note-alternative-symbol-radiometric">Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance.</ref> | joule | J | M⋅L2⋅T−2 | Energy of electromagnetic radiation. |
| Radiant energy density | we | joule per cubic metre | J/m3 | M⋅L−1⋅T−2 | Radiant energy per unit volume. |
| Radiant flux | Φe<ref group="nb" name="note-alternative-symbol-radiometric"></ref> | watt | W = J/s | M⋅L2⋅T−3 | Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power", and called luminosity in Astronomy. |
| Spectral flux | Φe,ν<ref group="nb" name="note-suffix-nu">Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek letter nu, not to be confused with a letter "v", indicating a photometric quantity.)</ref> | watt per hertz | W/Hz | M⋅L2⋅T −2 | Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅nm−1. |
| Φe,λ<ref group="nb" name="note-suffix-lambda">Spectral quantities given per unit wavelength are denoted with suffix "λ".</ref> | watt per metre | W/m | M⋅L⋅T−3 | ||
| Radiant intensity | Ie,Ω<ref group="nb" name="note-suffix-omega">Directional quantities are denoted with suffix "Ω".</ref> | 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 | Ie,Ω,ν<ref group="nb" name="note-suffix-nu"></ref> | watt per steradian per hertz | W⋅sr−1⋅Hz−1 | M⋅L2⋅T−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<ref group="nb" name="note-suffix-lambda"></ref> | watt per steradian per metre | W⋅sr−1⋅m−1 | M⋅L⋅T−3 | ||
| Radiance | Le,Ω<ref group="nb" name="note-suffix-omega"></ref> | 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 Specific intensity |
Le,Ω,ν<ref group="nb" name="note-suffix-nu"></ref> | watt per steradian per square metre per hertz | W⋅sr−1⋅m−2⋅Hz−1 | M⋅T−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<ref group="nb" name="note-suffix-lambda"></ref> | watt per steradian per square metre, per metre | W⋅sr−1⋅m−3 | M⋅L−1⋅T−3 | ||
| Irradiance Flux density |
Ee<ref group="nb" name="note-alternative-symbol-radiometric"></ref> | 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 |
Ee,ν<ref group="nb" name="note-suffix-nu"></ref> | watt per square metre per hertz | W⋅m−2⋅Hz−1 | M⋅T−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<ref group="nb" name="note-suffix-lambda"></ref> | watt per square metre, per metre | W/m3 | M⋅L−1⋅T−3 | ||
| Radiosity | Je<ref group="nb" name="note-alternative-symbol-radiometric"></ref> | 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 | Je,ν<ref group="nb" name="note-suffix-nu"></ref> | watt per square metre per hertz | W⋅m−2⋅Hz−1 | M⋅T−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<ref group="nb" name="note-suffix-lambda"></ref> | watt per square metre, per metre | W/m3 | M⋅L−1⋅T−3 | ||
| Radiant exitance | Me<ref group="nb" name="note-alternative-symbol-radiometric"></ref> | 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 | Me,ν<ref group="nb" name="note-suffix-nu"></ref> | watt per square metre per hertz | W⋅m−2⋅Hz−1 | M⋅T−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<ref group="nb" name="note-suffix-lambda"></ref> | watt per square metre, per metre | W/m3 | M⋅L−1⋅T−3 | ||
| Radiant exposure | He | 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 | He,ν<ref group="nb" name="note-suffix-nu"></ref> | joule per square metre per hertz | J⋅m−2⋅Hz−1 | M⋅T−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<ref group="nb" name="note-suffix-lambda"></ref> | joule per square metre, per metre | J/m3 | M⋅L−1⋅T−2 | ||
| See also: | |||||
