Multiband radiosity method
The multiband radiosity method calculates infrared exchange by integrating wavelength-dependent radiative quantities over spectral bands.
The radiosity method, also referred to as the net-radiation method or Oppenheim method, calculates thermal radiative exchange, including the effects of multiple reflections. It introduces an extra unknown for each computation point in the radiative enclosure. This extra unknown is often referred to as a radiosity node or an Oppenheim element. The derivation of the radiosity method begins with the definition of radiosity, which is the total energy flux leaving a surface. Radiosity is the sum of emitted and reflected fluxes.
For a given wavelength, the outgoing radiative power density from element k is:
where:
- qo,k is the outgoing radiative power density from element k.
- qI,k is the incoming radiative power density to element k.
- εk(λ,T) is the wavelength and temperature-dependent emissivity.
- P is the blackbody power spectrum.
- Tk is the temperature of element k.
The infrared spectrum is represented as a set of G energy bands, where each band g covers the wavelength interval from λg−1 to λg. Within each band, the spectral outgoing and incoming power densities are integrated over the wavelength range to obtain band-averaged radiative quantities.
The band-integrated outgoing and incoming power densities are defined as:
The band-averaged emissivity is defined as:
The proportion of power emitted by a blackbody at temperature T into band g is defined as:
The wavelength-dependent emissivity is approximated by a band-averaged emissivity weighted by the blackbody spectrum:
This approximation is equivalent to the approximation used with one energy band in the Oppenheim method.
Substituting the definitions and approximation into the band-integrated equations gives the band-averaged equations:
The incoming radiative power density is the sum of contributions from all other elements in the radiative enclosure:
where Fk-j is the view factor from element k to element j. Using the reciprocity theorem, this expression becomes:
At this stage, the method assumes that the view factors are wavelength independent and does not modify them to reflect wavelength-dependent specularity. With this assumption:
Substituting this equation into the band-averaged equation gives:
The heat balance on element k can be written as:
Rearranging the band-averaged equations to isolate the radiosity term on the left-hand side and substituting the result into the heat balance equation yield.
where the band Oppenheim temperature is defined by:
And the band compensation terms Skg are computed during the iteration. Similarly, the heat balance for the original radiating element is:
Both heat balance equations for the Oppenheim and original elements can be directly integrated into the system of nonlinear equations that describe the thermal model.