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THE OPTICS OF THE SOLAR TOWER REFLECTOR
Akiba Segal and Michael Epstein
Solar Research Facilities Unit, Weizmann Institute of Science, P.O. Box 26, Rehovot, 76100, Israel,
Phone: +972 8 9342935, Fax: +972 8 9344117, E-mail: [email protected]
Abstract − The concept of the reflective solar tower is based on inverting the path of the solar rays
originating from a heliostat field to a solar receiver that can be placed on the ground. This system is based
on the property of a reflective quadric surface to reflect each ray oriented to one of its foci to its second
focus. Two shapes of surfaces can be used: one is concave − the ellipsoid, and the other is convex − the
hyperboloid (with two sheets). This study analyzes the achievable optical performances of these types of
reflectors and proves that the hyperboloidal surface is a superior reflector. The optics of the hyperboloid is
analyzed in detail.
1. INTRODUCTION
The experience gained during the last two decades in
developing applications of concentrated solar energy
shows that higher conversion efficiency of solar energy to
electricity can be achieved only at high temperatures
(more than 1100 K) (Segal and Epstein, 2000). At these
temperatures, the radiation emitted from the receiver at
the working temperature becomes the main mechanism of
thermal losses depending on the size of the receiver
aperture. It is obvious that, in order to increase the
receiver efficiency, the solar energy must be introduced
into the receiver at higher concentrations. To reach high
concentrations at the receiver aperture it is not sufficient
to improve the performance parameters of the primary
concentrator (the field of heliostats), but a secondary
concentration is often required. As any optical device,
the secondary concentrator has its own inherent losses,
which reduce the total optical efficiency. Introducing the
secondary concentrator has a substantial effect on the
optimal shape, size and arrangement of the primary
concentrator. The receiver coupled with its secondary
concentrator (receiver concentrator − RC) becomes a
combined unit, where both the thermal and the additional
optical losses have to be analyzed and considered (Segal
and Epstein, 1999a).
Other important losses in a thermal system are those
associated with the heat transport from the solar receiver
to the energy converter (e.g., a gas turbine). In order to
minimize these losses, the turbine is installed close to the
receiver. In usual large solar power plants, the receiver
and the turbine together with auxiliary equipment are
heavy burden to be supported on the top of a tower
structure. An alternative option is to invert the path of
the solar rays originating from a heliostat field in a way
that the solar receiver and the above equipment can be
placed on the ground. In order to carry out this optical
path inversion, a supplementary reflector has to be
installed. This causes the rays oriented to the aim point
of the field to be reflected down to the RC entrance
located near the ground. From an optical point of view,
only a reflective surface having two foci is capable of this
mission, namely, each ray that is oriented to one of its
foci (which coincides with the aim point of the heliostat
field) will be reflected to the second focus positioned at
the entrance plane of the RC. From mathematical point
of view, this surface is a quadric, namely, a hyperboloid
(with two sheets, of which only the upper one is used), or
an ellipsoid. A hyperboloidal mirror placed at a certain
height, lower than the aim point of the heliostats, can
achieve this goal. Similarly, an ellipsoidal mirror placed
at certain height above the aim point can provide
comparable results. The performances of these two types
of reflectors are compared later in this paper. The
comparison shows that the hyperboloidal surface is more
promising than the ellipsoidal one, and its optical analysis
is described in a detailed form.
2. MATHEMATICAL BACKGROUND
A ray of light can be described mathematically as a
straight line as follows:
(1)
where: (xi,yi,zi) are the coordinates of the origin of the ray
(on the surface of the heliostat) and (ux,uy,uz) are the
components of the ray's unit vector of direction. This ray
is intersected with the reflective surface (hyperboloidal or
ellipsoidal) having the following general equation:
where: xp=x–xo, yp=y–yo, zp=z–zo, subscript o represents
the quadric center and subscript c represents the center of
the secondary concentrator entrance, in the tower system
coordinates (x,y,z). The quadric semiaxes are a and b.
The surface Eq. (2) has been written in a general way,
including the option to tilt the axis of the quadric surface
2c+
szryfx∂+
n
sc
u
;2
⋅−
=
/1
zyxf∂2
n
u
(see Section 3). The sign plus (+) in this equation
corresponds to the ellipsoid and minus (-) to the
hyperboloid. The coordinates of the point of intersection
between the ray and the reflector surface, (xs,ys,zs), are
obtained by solving the system of Eqs. (1) and (2) for the
parameter k. The normal at the point of the intersection
to the particular surface is calculated by solving the
following partial derivatives:
(3)
(a)
Finally, the reflected ray has the direction ur with the
components:
(4)
The intersection between the reflected ray and the
entrance plane of the RC is calculated by substituting in
the system of Eq. (1) the coordinates (xs,ys,zs) of the
intersection point on the reflector surface, the direction of
reflected rays ur and solving for k the equation zc=zs+kuzr,
zc being the z coordinate of the RC entrance. The
coordinates xc and yc in the entrance plane to the RC are
consequently calculated by Eq. (1).
Numerical results of the calculations are depicted later
for various practical cases.
3. THE PRINCIPLES OF A TOWER REFLECTOR
A tower reflector is an optical system comprises of a
quadric surface mirror (hyperboloidal or ellipsoidal),
where its upper focal point coincides with the aim point
of a heliostat field and its lower focal point is located at a
specified height, coinciding with the entrance plane of the
RC on the ground level (see Fig. 1, where only edge −
limb − rays from the sun have been drawn). This system
was proposed by Rabl (1976) and further investigated at
the Weizmann Institute of Science by Epstein and Segal
(1998), Kribus et al. (1997, 1998), Segal and Epstein
(1997, 1999a,b, 2000) and Yogev et al. (1998).
The beams from the heliostats are reflected downward
by this mirror situated before (if a hyperboloidal mirror is
used) or above (in the case of an ellipsoidal mirror) their
aim points. The optics of a tower reflector requires the
use of the RC if high concentrations are desired, because
the quadric surface mirror always magnifies the sun
image. The magnification is defined here as the ratio
between the image diameter at the second focal plane and
the image diameter at the aim point of the heliostat field.
This magnification is a function of the rapport of the
distances between the mirror vertex to the upper and to
(b)
Fig. 1. Principles of a solar tower reflector optics: (a)
hyperboloidal mirror; (b) ellipsoidal mirror.
the lower foci (Fig. 2 − M vs. f2/f1). Evidently, this
function is linear in a large range of the ratio f2/f1. In a
two-dimensional model, the linearity is absolute for the
entire range (Fig. 3) but, in a real three-dimensional
model, the linearity is perturbed by the image distortions
caused by the aberrations occurring at a smaller f2/f1 ratio.
The magnification depends also on the ratio between the
height of the aim point and the radius of the field. It
decreases as the radius of the field increases for the same
height of the aim point (Fig. 4).
Moving the hyperboloidal mirror down, towards the
lower focal point, or moving up the ellipsoidal mirror
causes the size of the image to be smaller. However, this
displacement results in two undesirable effects (Fig. 5).
In the case of hyperboloidal mirror, the size of the
reflector increases as its height above the heliostats is
decreased. The acceptance angle of the rays arriving to
the ground RC is also increased. The acceptance angle
determines the ability of the RC to concentrate the
radiation arriving at its entrance (Welford and Winston,
1989). In the ellipsoid case, the same optical behavior is
expected at the same ratio of f2/f1 as for the hyperboloid
case. One can define fh as the fractional position of the
vertex of the hyperboloid from the height of the aim point
(assuming that the lower focus of the hyperboloid is at
the heliostat level), and fe as the fractional position of the
ellipsoidal upper vertex to the aim point height.
4. COMPARISON BETWEEN ELLIPSOIDAL AND
HYPERBOLOIDAL MIRRORS
It was shown previously that the ellipsoidal mirror has a
substantial mechanical disadvantage: its position is
always above the aim point, while the hyperboloidal
mirror is below this point. In addition, as presented in
Section 3, the ellipsoidal mirror has also no optical
advantages. In order to determine definitely what type of
quadric mirror is preferable, the hyperboloidal or the
ellipsoidal, two real cases are analyzed: a small field (6.4
MW heat) and a large field (44 MW heat).
Fig. 2. Magnification M of image obtained at the RC
entrance plane vs. the ratio f2/f1 (three-dimensional case).
Fig. 3. Two-dimensional case: the magnification created
by the hyperboloidal or ellipsoidal reflector is identical
for a wide range of focus ratios.
The relation between fe and fh for the same f2/f1 is
fe=fh/(2 fh–1). It can be seen that the optimal position
(maximum concentration at the receiver entrance) is
found between fh=0.75, which corresponds to fe=1.5, and
fh=0.7, where fe=1.75. In the case of the ellipsoid, this
means that the position of the reflective mirror is
considerable higher than the hyperboloidal mirror.
In the Northern Hemisphere, at typical latitude range of
25−35°, the heliostats are arranged in a northern field, or,
in the case of a large surrounding field, the northern part
is usually larger than the southern part. In these cases the
quadric axis must be tilted if a vertical position of the RC
is requested (Fig. 6). For a hyperboloidal mirror, this tilt
is made by moving slightly the lower focus in the
northern direction relative to the aim point. In the case of
the ellipsoid, its lower focus is moved to the south with
respect to the same aim point.
4.1 Small fields
For reasons of efficiency, a small heliostat field
producing typically less than 8−10 MW heat is usually
built in the Northern Hemisphere, as a northern field.
However, for a matter of simplification and first
approximation, a symmetrical circular field of 230
heliostats, 36 m2 each, having ideal performances (no
tracking errors and perfect surface) is considered. The
heliostats are arranged in a radial staggered array with a
field radius of 110 m. The height of the aim point is 75
m and the RC entrance is located 5 m above the
heliostats' level (a necessary space for the concentrator
height and the receiver). The power delivered to the aim
point is ~6.4 MW (at the design point − noon, March 21).
The comparative results for the two types of reflectors are
presented in Table 1 (see below). This configuration
should create equal size of the image of the concentrated
solar radiation at the entrance plane of the ground RC.
However, from Table 1 it can be seen that there are small
differences, especially in the large and small values of
f2/f1, where the effect of aberrations is more pronounced.
The dimensions of the upper mirror, as well as of the
entrance radius of the RC, which are depicted in Table 1,
were determined to allow 1% of the rays directed towards
the reflectors to be lost as spillage. Table 1 shows also
that the dimensions of the ellipsoidal mirror are larger for
all the focus ratios. In parallel, the smaller acceptance
angle obtained for the ellipsoidal mirror cannot
compensate the larger image, which is obtained with the
same ellipsoidal mirror at the RC entrance. The result is
that the exit from the RC (i.e., the receiver entrance
radius) is larger compared to the one in the corresponding
hyperboloid case. Consequently, the concentration at the
receiver entrance in the case of the ellipsoidal reflector is
smaller than in the case of the hyperboloid.
4.2 Large fields
A heliostat field having a reflective area of 64800 m2
(1800 heliostats, 36 m2 each), is conceived to illustrate
the difference between the two tower reflectors. The
optimum radial staggered arrangement of the heliostats
for this field is obtained according to the method
described by Segal and Epstein (1996).
Fig. 4. Magnification M as a function of ratio f2/f1 for
fields with various radii.
The resulting field has a shape of an ellipse with
semiaxes: 255 m in north-south direction and 243 m in
east-west direction.
The ellipse center is at 87 m north of the vertical
projection of the heliostats' aim point, located 125 m
above the ground. In order to comply with the constraint
of a vertical RC, the axis of the quadric is tilted as
described in the previous section. The tilt angle varies
with the position of the tower reflector relative to the aim
point. For the corresponding ellipsoidal reflector, its
position is determined, thus the same f2/f1, considered for
the hyperboloidal reflector, is obtained. In this situation,
the area of the ellipsoidal reflector is five times larger for
small f2/f1 values and 1.5 times larger for bigger f2/f1
values. Therefore, the criterion selected to compare these
reflectors is the surface area of the reflector, which
provides an equal average concentration at the exit from
the RC (i.e., at the receiver entrance). The results are
presented in Fig. 7.
Based on the results presented in this section, there is no
reason to prefer the ellipsoidal reflector. Therefore, in
the next section the hyperboloidal upper mirror is
selected for further investigations.
5. PERFORMANCE OF THE HYPERBOLOIDAL
TOWER REFLECTOR
An optimum position exists for a reflector that
maximizes the receiver efficiency. As illustrated in Fig.
8, the optimum situation occurs when the edge rays from
the heliostat field, after reflection from the tower mirror,
coincide exactly with the edge rays of the RC.
Mathematically, a new concept can be introduced:
transformation by reflection from a quadric surface. The
optimum is achieved when the volume in phase space of
the rays leaving the heliostats is equal to the volume in
phase space of the reflected rays arrived to the RC.
Figure 9a shows the acceptance angle, the entrance radius
of the RC and its exit radius, as a function of the reflector
Fig. 5.
Dependence of the reflector surface and
acceptance angle of the position of the tower reflector.
position, fh, for the specific example of the small field
comprises 230 heliostats, 36 m2 each (Segal and Epstein,
1999a). The maximum average concentration level at the
receiver aperture is obtained when the ratio between the
distance from the apex of the mirror to the lower focus
and the distance between the two foci is equal (in this
particular case) to 0.71 (Fig. 2). Nevertheless, as shown
below, the highest concentration is only an indication of
the maximum efficiency.
Fig. 7. The area of the ellipsoidal/hyperboloidal tower
reflector as a function of the concentration level at the
receiver entrance.
Fig. 6. Concept of tilting the axis of the tower reflector
in the case of the asymmetrical surrounding field, where
the northern part is larger. Tilting is required if there is a
condition for a vertical RC. The units of the axes are in
"tower height".
This position does not automatically provide the
optimum for the receiver efficiency. Only a detailed
search for optimization, as presented by Segal and
Epstein (2000), will give the true position that assures
higher receiver efficiency.
Figure 9b shows the situation where two different aim
points of the heliostat field are considered. For the higher
aim point, the acceptance angles are smaller for the same
hyperboloidal mirror height, and as a result, higher
concentrations can be achieved at the receiver. However,
in order to find the optimum position of the reflector, its
height is varied according to a certain strategy. For each
position, the optimal geometry of the RC is established
following the method described by Segal and Epstein
(1999b). In connection with the variation of the position
of the hypoboloidal mirror, a particular case is worth
mentioning: when the mirror is positioned at exactly
equal distance between the aim point and the entrance
plane of the RC. In this case, the hyperboloid becomes a
flat mirror: the magnification is 1. That means that the
aim point image is exactly reflected to the RC entrance.
This option is not practical because the size of the mirror
is to half of the heliostat field area and therefore, this
choice is disregarded.
The energy flux distribution of the sun image at the
entrance plane of the RC (the lower focal plane) has a
peak at the center (Fig. 10). As one moves radially from
the center to the edges of the image, the flux decreases.
Attempting to capture the entire image into a single RC
will results in diluting the average flux at its entrance.
This is disadvantageous from a thermodynamic point of
view. In addition, since the reflective tower magnifies
the sun image at the RC entrance plane, a single
concentrator capable of collecting most of the energy will
have a large opening and the losses by reradiation will be
high. Therefore, the optimization method, as described in
the next section, is aimed at reducing the size of the RC
opening, in spite of some increasing of the spillage
losses. A possible solution was proposed by Ries et al.
(1995) for exploiting the edge part of the image for lower
temperature applications or for preheating purposes,
while the central portion of the image can be used for the
topping of the thermodynamic cycle. This approach can
increase the total efficiency of a power conversion
system.
Fig. 8. Optimum matching of the edge rays from the
heliostat field and the RC.
Figure 11 illustrates the flux distribution at the
concentrator entrance plane and the power that can be
absorbed by the receiver as a function of the distance
from the center, for the two positions of the
hyperboloidal mirror, as depicted in Fig. 5a and 5b
(fh=0.9 and fh=0.6, respectively). As expected, when the
reflector is in a lower position, the flux decreases sharply
with the distance from the center of the image.
%
b
P
e
d
(o
w
a
b
r)se
r
kW
ux(lm
/
u
a
m
sd
r(o
e
in
r+
4
2
1
0
c
),(R
M
S
H
m
C
cpc
=
t)Flol+
8
6
1
R
2
0
4
)(H
p
C
cS
N
g
oh
rlm
e
irlier
(a)
Fig. 11. Flux and power absorbed vs. collector radius.
(b)
Fig. 9. Variation of acceptance angle, collector radius
and exit radius from the concentrator vs. the hyperboloid
position: (a) the highest concentration in the receiver is
obtained for fh=0.73; (b) two aim points - the higher aim
point gives the higher concentration in the receiver.
Also, the partition between the size of the central RC
and the peripheral preheaters, as well as their acceptance
angles, are subject to an optimization search aimed at
maximizing the efficiency of the entire system (Segal and
Epstein, 1999a,b, 2000).
The criterion for this
optimization can be the maximization of the receiver
efficiency, or, if economic parameters are introduced, the
minimization of the cost of kilowatt thermal power
absorbed by the receiver.
In either case, the
concentration at the receiver entrance is only an
indicating parameter for better efficiency but not the
ultimate criterion for optimization. As an example,
receiver efficiencies are shown in Fig. 12 for the specific
case of the small field, previously analyzed, for three
aperture radii (one of them, Ropt, is the optimal radius).
All parameters remain the same, except for the RC
acceptance angle. Various average fluxes are obtained at
the receiver entrance. It is evident from these curves that
the maximum efficiency does not coincide with the
maximum average flux (concentration) and therefore, the
concentration at the receiver entrance is not the accurate
parameter for its efficiency, although represents a good
indication about this efficiency.
6. OPTIMIZATION OF A TOWER REFLECTOR
Fig. 10. Energy flux distribution and power collected at
the RC entrance a small field example.
As a result, the radii of the central collector and
preheaters are relatively small, compared with the
situation where the hyperboloid is in a higher position.
But this is not the only parameter that determines the
hyperboloid position. A large reflector area means a
lower average (and peak) flux on the reflector itself.
As previously mentioned, the criterion for optimizing
the tower reflector and the RC for a particular heliostat
field (already arranged in an optimal manner) is done by
minimizing the cost of this part of the optical system per
unit of power absorbed in the receiver. The total cost of
the optical system (not including the heliostats) can be
written as follows:
(5)
where: CM is the cost of the hyperboloidal mirror (a
function of its surface, Sm, and of its height Hm), Ccpc is
the cost of the receiver concentrator (which depends of its
surface, Sc), Nper and Cper are, respectively, the number of
the peripheral concentrators and their cost. Various
parameters have opposite tendencies.
For example, increasing the height of the reflector
decreases the size of the mirror (and implicitly, its cost)
but, in parallel, the cost of the tower is increased. Also,
as was described in Section 3, the image at the RC
entrance increases. This affects the cost of the RC (and
the thermal efficiency of the receiver). Other possibility
for decreasing the mirror area (and its cost) is to assume a
priori some acceptable spillage. A similar situation
happens if some spillage around the RC is allowed.
Fig. 12. Receiver efficiency at various average fluxes for
three aperture radii (Ropt − optimum radius of the
aperture).
Table 1. Comparison between hyperboloidal and ellipsoidal mirrors for the example of a small field
fh
f2/f1
Mirror
height
Accept.
angle
Mirror
area
Mirror
radius
Conc.
(kW/m2)
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.86
2.33
3.00
4.00
5.67
9.00
19.00
50.5
54.0
57.5
61.0
64.5
68.0
71.5
32.0
26.0
20.5
16.0
12.5
10.5
12.5
2904
1958
1328
829
458
204
57
31.0
25.4
21.0
16.4
12.2
8.1
4.2
3805
4033
4106
3917
3265
1848
277
2.17
1.75
1.50
1.33
1.21
1.13
1.06
1.86
2.33
3.00
4.00
5.67
9.00
19.00
156.7
127.5
110.0
98.3
90.0
83.8
78.9
32.0
22.5
17.5
15.0
11.5
9.5
10.5
19100
5938
2738
1462
612
249
59
75.7
44.0
29.9
21.7
14.1
8.9
4.3
2476
3199
4041
3606
3178
2059
360
CPC
Coll.
exit
radius
(99%)
Hyperboloidal mirror
1.39
0.73
1.63
0.71
2.01
0.70
2.62
0.72
3.65
0.79
5.77
1.05
12.54
2.71
Ellipsoidal mirror
1.71
0.91
1.62
0.62
1.83
0.55
2.56
0.66
3.50
0.70
5.73
0.95
12.46
2.27
7. CONCLUSIONS
The study performed in this work demonstrates that the
hyperboloidal reflector is superior to the ellipsoidal one
for a reflective solar tower. The optics of such a tower
reflector is analyzed and advantages and disadvantages of
this system are presented in detail.
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