Environmental Engineering Reference
In-Depth Information
The suns radius
R
s
¼
0.696
10
6
km, and its distance
D
es
about 93 million miles
10
8
km) from earth make its angular radius seen from the earth very small,
about 0.266
. We can infer the total power density at the suns surface as
P
(1.496
2
m
2
10
7
W
m
2
¼
1366 W
=
ð
D
es
=
R
s
Þ
¼
6
:
312
=
;
ð
5
:
2
Þ
where the geometric ratio of concentration is 46 200. It is this power density that
corresponds to
s
SB
T
4
, with
T
2
p
5
k
4
/(15
h
3
c
2
)
10
8
¼
5973, where
s
SB
¼
¼
5.67
W/ m
2
K
4
.
The concept of the black body radiator is a surface that emits the power density
equally into all directions. A hemisphere centered above a black body emitter will
receive equal energy on each point of its surface. This can be called a solid angle 2
p
.
(In reverse, a black surface will absorb equal energy from directed beams originating
at each point on the hemispherical surface.)
The model for a black body radiator is a small opening into an enclosed cavity at
temperature
T
, such as a pizza oven glowing red on the inside. The opening into the
cavity acts as a perfectly black absorber because a light ray entering the small opening
has no chance of coming out, independent of the direction it took while entering the
cavity. Inside the equilibrium cavity photons propagate in all directions, and any one
of these can come out through the hole. The light emitted from the hole is directed
equally into each angular range de
ned by polar and azimuthal angles d
'
at
'
and d
at
. The sun acts as a black body radiator, and many types of surfaces act to a good
approximation as black body absorbers and radiators.
Let us imagine a small area of black body surface, with angle from the surface
normal taken as
. The sun is at a particular angular location
and represents an angular diameter of 0.53
. How can we design an optical system
that will maximally concentrate the suns rays onto this small area of black body
surface? The problem is equivalent to designing an optical system that will take all of
the light emitted by the black body surface (into all 2
p
radians) and focus it into the
angular range defined by the sun. An example of an optical system that can approach
this is illustrated in Figure 5.1.
A concentrator system is perhaps more commonly based on a parabolic mirror as
the primary element, and a system of this type has demonstrated a concentration
ratio 84 000. Literally, this focused light is more intense than the surface of the sun,
and would have intensity 1366
and azimuthal angle
'
114.7MW/m
2
84 000
¼
if operated above the
atmosphere.
The sunlight reaching earth is diminished by speci
c absorption from molecules
in the atmosphere and by Rayleigh scattering of the sort that make the sky blue. Peak
illumination at the earth surface is 1000W/m
2
, in round numbers. The details of the
spectrum are more important for semiconductor devices, and for this purpose the
spectral power expressed per unit energy is preferable. Figure 5.2 shows the clear day
noontime spectrum for the northern hemisphere, a standard spectrum called AM
1.5, meaning 1.5 air mass traversal, at 48
from the vertical. Most of this intensity is
direct light, with a small diffuse contribution from the Rayleigh scattering. Most of
this intensity can be concentrated as described above. On a cloudy day, the light
intensity may not be too different, but the light certainly is diffuse and cannot be
