Exergy analysis of solar radiation processes - Solar Energy Sciences and Engineering Applications - page 22

The algebraic overall entropy growth due to simultaneous emission and
absorption of heat taking place at surface A:

q
T +

=− s − s 0 (2.2.38)

where q/T is the decrease of entropy of heat source at temperature T , and the entropy

s , J/(m 2 K), of emission density

−

4
3 σT 3

s = (2.2.39)

is used for determination of s and s 0 , respectively for surface A and A 0 .

The exergy loss δb is determined by the Gouya-Stodola law (2.2.10).
Making use of (2.2.37), (2.2.38) and (2.2.39) in equation (2.2.36), after some
rearranging and using (2.2.34), the formula for the exergy of emission density b , W/m 2 ,
of a perfectly gray surface of emissivity ε is obtained:

−

= ε σ

3 (3 T 4

T 0 − 4 T 0 T 3 ) b + (2.2.40)

The mathematical analysis of formula (2.2.40) (Petela (1964)), reveals first of all
that exergy b is always nonnegative and it has the lowest value zero when T

T 0 . The
exergy b reaches also zero if the considered surface is white (i.e. perfectly reflecting,
ε

=

0).
Keeping in mind formula (2.2.33), it follows that the exergy of emission of black
surface ( ε

=

1), for the environment temperature approaching absolute zero, becomes
equal to the energy of emission:

=

= σT 4

lim
T 0 →

( b b ) = e b (2.2.41)

0

It could be noticed that the characteristic term in brackets of formula (2.2.31),
appearing also in formula (2.2.40) was derived by Petela (1964) from the consideration
without using the Stefan-Boltzmann law (2.2.32). The obtained equations (2.2.31) as
well as (2.2.40) can be recognized as independent of equation (2.2.32). Therefore, the
energy of emission e can be interpreted as the particular case of the exergy of this
emission at the theoretical condition T 0 0, or other words, the Stefan-Boltzmann

law is a particular case of the emission exergy law expressed by formula (2.2.31).
As the surface temperature T approaches absolute zero the exergy of emission
expressed by formula (2.2.40) approaches the finite value:

=

ε σ

3 T o

lim
T

( b ) = (2.2.42)

→ 0

Based on equation (2.2.40), Figure 2.2.5 illustrates the exergy b b (solid thick line)
of emission density of a black surface ( ε = 1) at the constant value of the environment

temperature T 0 = 300 K. For comparison, the energy e b (solid thin line) of emission

density according to equation (2.2.32) is presented. For a sufficiently small temperature
T the exergy of black emission is larger than energy of such emission. The dashed line