Optical: systems and elements – Mirror – Plural mirrors or reflecting surfaces
Reexamination Certificate
2001-06-06
2002-10-22
Robinson, Mark A. (Department: 2872)
Optical: systems and elements
Mirror
Plural mirrors or reflecting surfaces
C359S869000, C126S684000
Reexamination Certificate
active
06467916
ABSTRACT:
BACKGROUND OF THE INVENTION
This invention relates to devices for the transmission of radiation, especially of light. In particular, it is a non-focussing reflector for the concentration of radiation such as sunlight at a desired region over a wide range of angles of incidence, and it is a non-focussing reflector for the reflection of radiation such as light over a relatively large solid angle.
A number of systems for passive or non-tracking concentration of solar energy have been produced in the past. Among such systems are those shown in U.S. Pat. Nos. 5,537,991; 3,957,041; 4,002,499; 4,003,638; 4,230,095; 4,387,961; 4,359,265 and 5,289,356, all of which are incorporated here by reference as if set forth fully. A common characteristic of these systems is the use of smooth surfaces to reflect light from the sun on a region to be heated. This is also true with most non-focussed reflectors, which are similar in construction such that it is appropriate to refer to the reflectors as light-transmission devices because it is immaterial whether the reflectors are concentrating radiation from a large solid angle of incidence or broadcasting radiation from a relatively small source to a relatively large solid angle.
Concentration of radiation is possible only if the projected solid angle of the radiation is increased. This requirement is the direct consequence of the law of conservation of the etendue, which is the phase space of radiation. Solar concentrators which achieve high concentration must track the sun; that is, they must continuously reorient in order to compensate for the apparent movement of the sun in an earth center (Ptolemaic) coordinate system. Reflectors, in contrast, are fixed in position for most lighting purposes. For tracking collectors the direction to the center of the sun is stationary with respect to their aperture. Such concentrators can achieve very high concentrations of about 45000 in air. Even higher concentrations have been achieved inside transparent media.
Tracking, however, is technically demanding because solar collectors are commonly fairly large and designing these systems for orientational mobility may add significantly to their cost. Moreover the absorber, which incorporates some heat transfer fluid as well as piping, also may need to be mobile. This is the motivation to study the concentration which can be achieved with stationary, non-tracking devices. The same principles apply when it is desired to deliver light or other radiant energy from a source to a relatively large solid angle.
First we derive the theoretical upper limits, without reference to any particular type of concentrators. We then focus on trough-type or linear systems, which posses a translational invariance along one direction. We show that trough systems are not ideal as stationary concentrators. For troughs more stringent upper limits apply.
The annual movement of the earth around the sun in a nearly circular orbit combined by the daily rotation around its axis which is inclined by the angle &dgr;, to the plane of its orbit (the ecliptic) accounts for an apparent movement of the sun in an earth-based coordinate system. We follow the same notation and use a coordinate system with one axis oriented horizontally East to West. The second axis points North to South, parallel to the axis of the earth, that is inclined with respect to the local horizontal direction by an angle equal to the latitude. The third axis, perpendicular to the other two, points toward the sun at noon, at equinox. This coordinate system corresponds to the common orientation of a stationary concentrator. A unit vector pointing in a certain direction is represented by its component k
E
along the E-W axis and its component k
N
along the N-S axis. The third component, k
H
is known from normalization. Area elements in the k
E
, k
N
space correspond to projected solar angle and can be used to assess concentration.
The apparent direction of the sun is given to a very good approximation by
k
N
=−sin &dgr;
o
cos(&ohgr;
y
t
)
k
E
=−{square root over (1−
k
N
2
)}sin(&ohgr;
d
(
t+T
))
where &ohgr;
y
=2&pgr;/year describes the yearly angular orbital movement, &ohgr;
d
=2&pgr;/day describes the angular daily rotation and t the time since equinox. The correction T comprises a constant offset, the time difference between nearest local noon and equinox, as well as time-dependent correction known as the equation of time, which is due to the deviation of the earth's orbit from a circular path. This correction varies slowly in the course of one year by a maximum of ±15 minutes. Its effects are negligible for the purpose of this work. The declination angle &dgr;
o
=23.45 degrees is the angle between the plane of the yearly orbit, the ecliptic, and the polar axis of rotation of the earth.
The movement of the sun is visualized in FIG.
4
. To a good approximation, the sun moves in the course of a day along a straight line k
N
≈const. parallel to the W-E axis. In the course of a year, the daily path oscillates between a maximum value at summer solstice and a minimum at winter solstice. This is indicated by the parallel lines which describe 36 sample days at equal time spacing over one year.
First we note that a surface oriented parallel to the axes chosen receives solar radiation only from inside the band
−sin(&dgr;
o
+&agr;
)≦k
N
≦sin(&dgr;
o
+&agr;
)
Here &agr;
=4.7 mrad is the half-angle subtended by the sun. It adds to the declination in order to account for rays from the rim of the solar disk.
A stationary concentrator which accepts radiation only from this band can achieve a maximum concentration, without rejecting any radiation, equal to the ratio of the area of the entire circle to the area of the band given by Eq. (2), that is
C
max
=
π
2
(
δ
o
+
α
∋
)
+
sin
(
2
δ
o
+
2
α
∋
)
≈
2.0
The value in Eq. (3) applies for an ideal device required to accept all rays. If we analyze
FIG. 1
, it is apparent that the radiation is not uniformly distributed within the band described by Eq. (2). The solar path spends more time near the extremes than in the center. We define the average relative radiance from a certain direction as the ratio of the radiance received from this direction to that constant radiance which we would receive from the same direction if the same power would be homogeneously distributed over all regions of the celestial sphere. This relative radiance is proportional to:
B
p
(
k
N
,
k
E
)
~
1
sin
2
δ
o
-
k
N
2
=
B
p
(
k
N
)
where d
t
denotes the time derivative and P the radiative power. In the numerator &ohgr;
y
&ohgr;
d
describes the frequency a region is visited, the root describes the intensity of a point source, proportional to the cosine of the incidence angle, or the ratio of solid angle to projected solid angle. The denominator accounts for the time the sun spends in an interval dk
N
dk
E
. Substituting the time derivatives
&LeftBracketingBar;
d
t
k
N
&RightBracketingBar;
=
sin
δ
o
ω
y
1
-
(
k
N
sin
δ
o
)
2
&LeftBracketingBar;
d
t
k
E
&RightBracketingBar;
=
ω
d
1
-
k
N
2
1
-
k
E
2
1
-
k
N
2
-
ω
d
1
-
k
E
2
-
k
N
2
into Eq. (4) yields for the relative intensity
B
p
(
k
N
,
k
E
)
~
1
sin
2
δ
o
-
k
N
2
=
B
p
(
k
N
)
.
In Eq. (6) we neglected the term proportional to
~
because w
y
because &ohgr;
y
<<&ohgr;
d
. Note that the radiance distribution does not depend on k
E
; it is constant along directions of equal latitude.
Consequently we dropped the dependence on k
E
. The decrease in speed in the W-E direction in the morning and evening is compensated by the cosine effect, whereas the decrease in speed in the S-N direction at the solstices is not. Equation (7) is strictly valid only in the limit of negligible size of the solar disk. This is a good approximation everywhere exce
Arch Development Corp.
Foley & Lardner
Rechtin Michael D.
Robinson Mark A.
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