Exergy analysis of solar radiation processes - Solar Energy Sciences and Engineering Applications - page 30

Exergy of arbitrary, non-polarized, black and uniform radiation, propagating
within solid angle 2π. The formula for such radiation is derived from (2.2.62) in
which formulae (2.2.53) and (2.2.54) are used with the interpretation explained by
formula (2.2.55):

σ
3 (3 T 4

T 0 − 4 T 0 T 3 ) b b = + (2.2.65)

To utilize formula (2.2.65) only the temperature of the black radiation is required.
The total exergy arriving at surface A can be calculated from formula (2.2.63).
It is noteworthy that equation (2.2.65) is identical to equation (2.2.40) derived for
the black emission. This similarity is a confirmation that exergy of black radiation is
equal to the exergy of emission of black surface. It is the consequence of the radiosity
of a black body being equal to its emission.
Exergy of non-polarized, uniform, black radiation propagating within solid angle
ω. The formula for such radiation is:

b b
π

b bω = cos β sin β dβ dϕ (2.2.66)

β ϕ

where solid angle ω has to be determined by the appropriate ranges of variation of the
flat angles β (declination) and φ (azimuth).
The magnitude ψ is defined as the ratio of exergy and energy of the same radiation.
Both the exergy and the energy are functions of state, thus they do not depend on any
geometrical configuration parameters. The angles β and ϕ do not have any geometrical
meaning but are only the coordinates determining the solid angle in which the spectrum
is considered.
For any arbitrary radiation the ratio ψ could be defined as the ratio of exergy
determined by formula (2.2.58) to the first term of the right hand side of this formula
which represents the radiation energy. Thus for a polarized radiation:

σT 0
3 π

cos β sin β dβ dϕ − ( L 0, λ ,min + L 0, λ ,max ) cos β sin β dβ dϕ

β ϕ β ϕ λ

ψ = 1 −

( i 0, λ ,min + i 0, λ ,max ) cos β sin β dβ dϕ

β ϕ λ

(2.2.67)

The exemplary values of ψ are discussed in this topic in examples 2.2.6.1 (for
non-polarized, uniform water vapor radiation propagating within a solid angle 2 π ),
and in example 2.4.1.2 (for non-polarized and uniform solar radiation).