In
radiometry
Radiometry is a set of techniques for measurement, measuring electromagnetic radiation, including visible light. Radiometric techniques in optics characterize the distribution of the radiation's power (physics), power in space, as opposed to phot ...
, radiosity is the
radiant flux
In radiometry, radiant flux or radiant power is the radiant energy emitted, reflected, transmitted, or received per unit time, and spectral flux or spectral power is the radiant flux per unit frequency or wavelength, depending on whether the Spec ...
leaving (emitted, reflected and transmitted by) a surface per unit area, and spectral radiosity is the radiosity of a surface per unit
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
or
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
, depending on whether the
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
is taken as a function of frequency or of wavelength. The
SI unit
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
of radiosity is the
watt
The watt (symbol: W) is the unit of power or radiant flux in the International System of Units (SI), equal to 1 joule per second or 1 kg⋅m2⋅s−3. It is used to quantify the rate of energy transfer. The watt is named after James Wa ...
per square metre (), while that of spectral radiosity in frequency is the watt per square metre per
hertz
The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that on ...
(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 (). The
CGS unit erg
The erg is a unit of energy equal to 10−7joules (100 nJ). It originated in the Centimetre–gram–second system of units (CGS). It has the symbol ''erg''. The erg is not an SI unit. Its name is derived from (), a Greek word meaning 'work' o ...
per square centimeter per second () is often used in
astronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
. Radiosity is often called
in branches of physics other than radiometry, but in radiometry this usage leads to confusion with
radiant intensity
In radiometry, radiant intensity is the radiant flux emitted, reflected, transmitted or received, per unit solid angle, and spectral intensity is the radiant intensity per unit frequency or wavelength, depending on whether the spectrum is taken ...
.
Mathematical definitions
Radiosity
Radiosity of a ''surface'', denoted ''J''
e ("e" for "energetic", to avoid confusion with
photometric quantities), is defined as
:
where
* ∂ is the
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
symbol
*
is the radiant flux ''leaving'' (emitted, reflected and transmitted)
*
is the area
*
is the ''emitted'' component of the radiosity of the surface, that is to say its
exitance
In radiometry, radiant exitance or radiant emittance is the radiant flux emitted by a surface per unit area, whereas spectral exitance or spectral emittance is the radiant exitance of a surface per unit frequency or wavelength, depending on whether ...
*
is the ''reflected'' component of the radiosity of the surface
*
is the ''transmitted'' component of the radiosity of the surface
For an ''
opaque
Opacity or opaque may refer to:
* Impediments to (especially, visible) light:
** Opacities, absorption coefficients
** Opacity (optics), property or degree of blocking the transmission of light
* Metaphors derived from literal optics:
** In lingu ...
'' surface, the ''transmitted'' component of radiosity ''J''
e,tr vanishes and only two components remain:
:
In
heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
, 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
:
where ''ν'' is the frequency.
Spectral radiosity in wavelength of a ''surface'', denoted ''J''
e,λ, is defined as
:
where ''λ'' is the wavelength.
Radiosity method
The radiosity of an ''opaque'',
gray
Grey (more common in British English) or gray (more common in American English) is an intermediate color between black and white. It is a neutral or achromatic color, meaning literally that it is "without color", because it can be composed o ...
and
diffuse
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
surface is given by
:
where
*''ε'' is the
emissivity
The emissivity of the surface of a material is its effectiveness in emitting energy as thermal radiation. Thermal radiation is electromagnetic radiation that most commonly includes both visible radiation (light) and infrared radiation, which is n ...
of that surface;
*σ is the
Stefan–Boltzmann constant
The Stefan–Boltzmann constant (also Stefan's constant), a physical constant denoted by the Greek letter ''σ'' (sigma), is the constant of proportionality in the Stefan–Boltzmann law: "the total intensity radiated over all wavelengths inc ...
;
*''T'' is the temperature of that surface;
*''E''
e is the
irradiance In radiometry, irradiance is the radiant flux ''received'' by a ''surface'' per unit area. The SI unit of irradiance is the watt per square metre (W⋅m−2). The CGS unit erg per square centimetre per second (erg⋅cm−2⋅s−1) is often used ...
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
In physics, radiation is the emission or transmission of energy in the form of waves or particles through space or through a material medium. This includes:
* ''electromagnetic radiation'', such as radio waves, microwaves, infrared, visi ...
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
In radiative heat transfer, a view factor, F_, is the proportion of the radiation which leaves surface A that strikes surface B. In a complex 'scene' there can be any number of different objects, which can be divided in turn into even more surfaces ...
'' 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'':
:
Now, employing the
reciprocity relation for view factors ''F''
''ji'' ''A''
''j'' = ''F''
''ij'' ''A''
''i'',
:
and substituting the irradiance into the equation for radiosity, produces
:
For an ''N'' surface enclosure, this summation for each surface will generate ''N''
linear equation
In mathematics, a linear equation is an equation that may be put in the form
a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficien ...
s with ''N'' unknown radiosities,
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
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
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:
:
Using the equation for radiosity ''J''
e,''i'' = ''ε''
''i''σ''T''
''i''4 + (1 − ''ε''
''i'')''E''
e,''i'', the irradiance can be eliminated from the above to obtain
:
where ''M''
e,''i''° is the
radiant exitance
In radiometry, radiant exitance or radiant emittance is the radiant flux emitted by a surface per unit area, whereas spectral exitance or spectral emittance is the radiant exitance of a surface per unit frequency or wavelength, depending on wheth ...
of a
black body
A black body or blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. The name "black body" is given because it absorbs all colors of light. 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
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
radiosity equations. To do this, the heat transfer at each surface is expressed as
:
where ''R''
''i'' = (1 − ''ε''
''i'')/(''A''
''i''''ε''
''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
An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sources, ...
''V'' = ''IR''.
Now performing a similar analysis for the heat transfer from surface ''i'' to surface ''j'',
:
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
An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sources, c ...
.
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
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
surface. In reality, however, the radiosity will have a
specular
Specular reflection, or regular reflection, is the mirror-like reflection of waves, such as light, from a surface.
The law of reflection states that a reflected ray of light emerges from the reflecting surface at the same angle to the surfac ...
component from the reflected
radiation
In physics, radiation is the emission or transmission of energy in the form of waves or particles through space or through a material medium. This includes:
* ''electromagnetic radiation'', such as radio waves, microwaves, infrared, visi ...
. So, the heat transfer between two surfaces relies on both the
view factor
In radiative heat transfer, a view factor, F_, is the proportion of the radiation which leaves surface A that strikes surface B. In a complex 'scene' there can be any number of different objects, which can be divided in turn into even more surfaces ...
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
In thermodynamics, an isothermal process is a type of thermodynamic process in which the temperature ''T'' of a system remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside thermal reservoir, and a ...
. 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.
[
]
SI radiometry units
See also
*Irradiance In radiometry, irradiance is the radiant flux ''received'' by a ''surface'' per unit area. The SI unit of irradiance is the watt per square metre (W⋅m−2). The CGS unit erg per square centimetre per second (erg⋅cm−2⋅s−1) is often used ...
*Radiant flux
In radiometry, radiant flux or radiant power is the radiant energy emitted, reflected, transmitted, or received per unit time, and spectral flux or spectral power is the radiant flux per unit frequency or wavelength, depending on whether the Spec ...
*Spectral flux density
In spectroscopy, spectral flux density is the quantity that describes the rate at which energy is transferred by electromagnetic radiation through a real or virtual surface, per unit surface area and per unit wavelength (or, equivalently, per unit ...
References
{{reflist
Physical quantities
Radiometry