Exergy analysis of solar radiation processes - Solar Energy Sciences and Engineering Applications - page 60

Using equations (b), (c), (e), (g) and (i) in (2.4.27):

⎡
⎣

⎤
⎦

T 0 1

L 2

R 2

T a −

−

= −

η E ε a 2 (2.4.29)

T 4 R 2
L 2

Using formulae (a)-(i) in (2.4.28), the exergy conversion efficiency of solar
radiation into heat can be determined as follows:

2 T 0 ) L 2 T 4 T 0 − ( T a −

−

T a − T 0

R 2

η B = 3 ε a (2.4.30)

T a

3 T 4 T 0 − 4 T 0 T 3 +

The larger the ratio L / R , the smaller are both efficiencies. The increasing emissivity
ε a of the absorbing surface will increase the conversion efficiencies.
The larger the T a , the smaller is the energy efficiency η E , however, the exergy con-
version efficiency η B is at maximum. The optimal temperature T a , opt can be calculated
based on (2.4.30) from the condition:

∂η B
∂T a =

0 (2.4.31)

For example, if the solar radiation is considered at ε a = 1, T 0 = 300 K, T =

363 K (90 ◦ C). If the

6000 K, R = 6 . 955 · 10 8 m and L = 1 . 495 · 10 11 m, then T a , opt ≈

383 K (110 ◦ C).
The T a optimum, at the unchanged exergy b ω of solar radiation, results from the
fact that with increasing T a , which increases the heat quality ( b q ), the amount of this
heat decreases. The emissivity value ε a does not appear in equation (2.4.31) so this
emissivity has no effect on the optimal temperature T a , opt .
The universal traveling of the human population motivates considering the envi-
ronment temperature in a wide range, theoretically for 0 < T 0 < T . This aspect is
shown in Figure 2.4.11. With a decreasing environment temperature T 0 , the optimal
temperature T a , opt of the absorbing surface continuously diminishes and the exergy
conversion efficiency η B grows approaching 80% for T 0 →

environment temperature drops to T 0 = 273 K then T a , opt ≈

0. At the same time the

Carnot efficiency, η C , a = 0.

For further analysis of solar radiation conversion, the energy and exergy balances
equations for the absorbing surface (Fig. 2.4.10) are:

1 − T 0 /T a , also grows and reaches 100% for T 0 →

e ω − (1 − ε a ) e ω + e 0 = e a + q (k)

b ω −

−

ε a ) b ω +

= + +

(1 b 0 b a b q δb (l)

To calculate exergy loss in equation (l) some entropy formulae has to be used.
Entropy s ω of the solar radiation arriving at the absorbing surface:

s R 2
L 2

s ω = (m)