Radiosity (heat transfer)

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Radiosity is a convenient quantity in optics and heat transfer that represents the total radiant flux density (i.e. power per area) of the radiation leaving a surface.^[1] Sometimes it is defined differently as the total radiant intensity leaving the surface,^[2] which has different units. 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

Realistically, the radiances from emitted radiation $L_{e}$ , and reflected radiation, $L_{r}$ , are both functions of angle from the surface. So, adding these together, the total radiance from the surface is defined as

$L_{{e+r}}(\theta ,\phi )={\frac {{\mathrm {d}}^{2}\Phi }{{\mathrm {d}}A\,{\mathrm {d}}{\omega }\cos \theta }}$ ^[3]

where $\Phi$ represents energy flux, $A$ is the surface area, and $\omega$ 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 radiance is integrated over a hemispherical surface enclosing the surface patch for all angles.

$J={\frac {{\mathrm d}\Phi }{{\mathrm d}A}}=\int _{\omega }{L_{{e+r}}(\theta ,\phi )\cos \theta \,{\mathrm d}\omega }$

Assuming a diffuse emitter and reflector, $L_{{e+r}}$ is constant with respect to the angle and the radiosity reduces to $J=\pi L_{{e+r}}$ . Furthermore, for a blackbody, $L_{r}=0$ and the radiosity reduces to $J=\pi L_{e}$ .^[4]

To generalize further, the radiosity can also be expressed as a function of the wavelength of the radiation – the spectral radiosity.

Radiosity method

A large clock tower and other buildings line a great river.

The radiosity $J$ , for a gray, diffuse surface, is the sum of the reflected and emitted irradiances. 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 irradiance 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 irradiance 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,^[5] 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.

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

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 $j$ ,

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

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.^[4]

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.^[5]

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
↑ It is the same equation as for the radiance
↑ 4.0 4.1 Yunus Cengel. Heat and Mass Transfer: A Practical Approach. McGraw Hill, Third Edition, 2007.
↑ 5.0 5.1 E.M. Sparrow and R.D. Cess. Radiation Heat Transfer. Hemisphere Publishing Corporation, 1978.

SI radiometry units
v
t
e
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
Radiant flux	Φ_e^{[nb 2]}	watt	W or J/s	M⋅L²⋅T⁻³	radiant energy per unit time, also called radiant power.
Spectral power	Φ_eλ^{[nb 2]}^{[nb 3]}	watt per metre	W⋅m⁻¹	M⋅L⋅T⁻³	radiant power per wavelength.
Radiant intensity	I_e	watt per steradian	W⋅sr⁻¹	M⋅L²⋅T⁻³	power per unit solid angle.
Spectral intensity	I_eλ^{[nb 3]}	watt per steradian per metre	W⋅sr⁻¹⋅m⁻¹	M⋅L⋅T⁻³	radiant intensity per wavelength.
Radiance	L_e	watt per steradian per square metre	W⋅sr⁻¹⋅m⁻²	M⋅T⁻³	power per unit solid angle per unit projected source area. confusingly called "intensity" in some other fields of study.
Spectral radiance	L_eλ^{[nb 3]} or L_eν^{[nb 4]}	watt per steradian per metre³ or watt per steradian per square metre per hertz	W⋅sr⁻¹⋅m⁻³ or W⋅sr⁻¹⋅m⁻²⋅Hz⁻¹	M⋅L⁻¹⋅T⁻³ or M⋅T⁻²	commonly measured in W⋅sr⁻¹⋅m⁻²⋅nm⁻¹ with surface area and either wavelength or frequency.
Irradiance	E_e^{[nb 2]}	watt per square metre	W⋅m⁻²	M⋅T⁻³	power incident on a surface, also called radiant flux density. sometimes confusingly called "intensity" as well.
Spectral irradiance	E_eλ^{[nb 3]} or E_eν^{[nb 4]}	watt per metre³ or watt per square metre per hertz	W⋅m⁻³ or W⋅m⁻²⋅Hz⁻¹	M⋅L⁻¹⋅T⁻³ or M⋅T⁻²	commonly measured in W⋅m⁻²⋅nm⁻¹ or 10⁻²² W⋅m⁻²⋅Hz⁻¹, known as solar flux unit.^{[nb 5]}
Radiant exitance / Radiant emittance	M_e^{[nb 2]}	watt per square metre	W⋅m⁻²	M⋅T⁻³	power emitted from a surface.
Spectral radiant exitance / Spectral radiant emittance	M_eλ^{[nb 3]} or M_eν^{[nb 4]}	watt per metre³ or watt per square metre per hertz	W⋅m⁻³ or W⋅m⁻²⋅Hz⁻¹	M⋅L⁻¹⋅T⁻³ or M⋅T⁻²	power emitted from a surface per unit wavelength or frequency.
Radiosity	J_e	watt per square metre	W⋅m⁻²	M⋅T⁻³	emitted plus reflected power leaving a surface.
Spectral radiosity	J_eλ^{[nb 3]}	watt per metre³	W⋅m⁻³	M⋅L⁻¹⋅T⁻³	emitted plus reflected power leaving a surface per unit wavelength
Radiant exposure	H_e	joule per square metre	J⋅m⁻²	M⋅T⁻²	also referred to as fluence
Radiant energy density	ω_e	joule per metre³	J⋅m⁻³	M⋅L⁻¹⋅T⁻²
See also: SI · Radiometry · Photometry

↑ Standards organizations recommend that radiometric quantities should be denoted with a suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
↑ 2.0 2.1 2.2 2.3 2.4 Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant emittance.
↑ 3.0 3.1 3.2 3.3 3.4 3.5 Spectral quantities given per unit wavelength are denoted with suffix "λ" (Greek) to indicate a spectral concentration. Spectral functions of wavelength are indicated by "(λ)" in parentheses instead, for example in spectral transmittance, reflectance and responsivity.
↑ 4.0 4.1 4.2 Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek)—not to be confused with the suffix "v" (for "visual") indicating a photometric quantity.
↑ NOAA / Space Weather Prediction Center includes a definition of the solar flux unit (SFU).

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