A Cassegrainian solar concentrator, using a 7-meter diameter primary reflector, was analyzed in three forms: (1) an unmodified Cassegrainian, (2) the Ritchey-Chretien configuration, and (3) the unmodified Cassegrainian with a non-imaging tertiary reflector. Optical performance was not significantly improved with the Ritchey-Chretien; however, the tertiary resulted in significant improvement in intercept factor and optical efficiency. The effects of misalignment of the secondary and tertiary reflectors on the optical performance of the collector were also analyzed

Schwinkendorf, W. E.

Waterbury, S. S.

English

NASA Technical Reports Server

OPTICAL ANALYSIS OF CASSEGRAINIAN POINT FOCUS CONCENTRATORS
S. S. Waterbury and W. E. Schwinkendorf
The BDM Corporation
1801 Randolph Road S.E.
Albuquerque, New Mexico 87106
ABSTRACT
A Cassegrainian solar concentrator, using a 7-meter diameter primary
reflector, is analyzed in three forms: 1) an unmodified Cassegrainian,
2) the Ritchey-Chretien configuration, and 3) the unmodified Cassegrainian
with a nonimaging tertiary reflector. Performance was not significantly
improved with the Ritchey-Chretien; however, the tertiary resulted in sig-
nificant improvement in intercept factor and optical efficiency.
INTRODUCTION
The Cassegrainian optical configuration consists of a parabolic primary
reflector and a hyperbolic secondary mirror. A solar concentrator designed
using this configuration can benefit by allowing a more flexible receiver
design, since it no longer has to be supported at the primary focal point.
The main disadvantages of the Cassegrainian configuration are the additional
reflection and blocking due to the secondary.
In addition to the "true" Cassegrainian described above, a variant, referred
to as the Ritchey-Chretien, has been studied. The Ritchey-Chretien (R-C)
has a slightly hyperbolic primary with the secondary adjusted accordingly.
These modifications correct the system for the off-axis aberration referred
to as coma. Since the sun is not a point source, much of the incoming
insolation is off of the optical axis, causing coma. Elimination of coma
should decrease the overall spot size at the focal plane, increasing the
intercept factor for a given concentration ratio.
A nonimaging tertiary reflector, added at the focal point of the system, can
improve the optical performance of the Cassegrainian. The configuration
considered for this application is the hyperbolic flowline concentrator, as
described by Winston (1). This design has the advantage over other non-
imaging concentrators such as the compound elliptic concentrator (CEC), of
affecting only the edges of the beam, thus reducing the overall reflection
losses.
METHOD OF ANALYSIS
This study has used a Monte-Carlo ray trace computer program originally
developed by Honeywell (2). This code is modular in nature, and allows
modeling of any concentrator system by writing appropriate subroutines
describing the geometry of the system to be studied. It has the capability
to include the effects of RMS surface imperfections on system performance
and a finite sun size with nonuniform flux distribution.
38
https://ntrs.nasa.gov/search.jsp?R=19840020160 2020-03-22T10:06:49+00:00Z
The approach used in this study was to perform a parametric study on the
true Cassegrainian, compare the performance of the R-C to the true Casse-
grainian for selected parameters, and then analyze a tertiary reflector
added to the Cassegrainian system. Finally, an analysis was performed to
determine the sensitivity of the optical performance to misalignment of the
secondary and tertiary reflectors.
GEOMETRY
As stated previously, the
Cassegrainian consists of a
confocal parabola and hyper-
bola, as shown in figure 1.
The convex secondary reflec-
tor increases the focal
length of "the optical sys-
tem, and thus reduces the
angle of the extreme rays
reflected from the secondary
to the system focal plane.
Since the optical extent of
the system must be con-
served, an increased focal
length reduces the maximum
concentration ratio that can
be achieved. The theoret-
ical concentration ratio
for a system of axial sym-
metry is defined in equa-
tion 1.
Fp
eJ>s
\
~=======~ R, __
Figure 1. Cassegrainian Geometry
(1)
where ~ is the entrance angle of the extreme rays and ~s ~s the sun angle.
As can be seen, ~ determines the maximum CR possible.
The variation in system focal
(Fp) and the eccentricity (e)(M), defined as
length is related to the primary focal length
of the secondary by the magnification factor
M =
The system focal length is the product of M and F. Likewise, the extreme
ray angle ~ is related to the primary rim angle. these relationships can be
used to determine M and the theoretical concentration ratio.
There are several factors that do not allow the theoretical concentration
ratio to be reached. In a perfect optical system (i.e., free from errors on
the reflector surfaces) the limi tat ions are caused by various aberrat ions.
39
It is possible to design an optical system to reduce or eliminate some of
these aberrations. One such design is the Ritchey-Chretien, which is cor-
rectedfor spherical aberration and coma. There have been several deri-
vations of the relationships required for this configuration. The one
chosen for this study is by Wetherell, et al (3). This development results
in a "sag" equation for the two surfaces as a function of the primary
reflector focal length, the system aperture area, the system focus position,
and the vertex-to-vertex spacing between the two reflectors.
Another major source of degradation in concentrator performance is imperfec-
tions on the reflector surfaces. They have the effect of increasing the
size of the cone of light reflected off of each surface.
It is advantageous from a cost standpoint to design a concentrator system
with large slope errors on the reflectors and a small primary rim angle.
Also, from an efficiency standpoint it is advantageous to reduce the size of
the secondary to reduce the blocking factor by increasing the spacing
between the primary and secondary reflectors. Unfortunately, these des ign
decisions tend to spread the beam of radiation impinging on the focal plane,
reducing the concentration ratio possible for a given intercept factor.
I-
::t:
Cl'
iii
::t:
>ex:
c(
~
ex:
w
I-
T
I
/
I
/
I
/
I
•I 1-28--1 I
r-2FH---!
Figure 2. Tertiary Geometry
To increase the concentration ratio
for a given intercept factor, a
hyperbolic tertiary reflector may be
added at the focal plane of the sys-
tem. The geometry for this non-
imaging concentrator is shown in
figure 2. One property of a hyper-
bola is that any ray directed at one
of the focal points will be
reflected towards the other focal
point. After an inf ini te number of
reflections, the ray will exit
through the bottom of the concentra-
tor. Rays that would intersect the
focal plane within the 2FH diameter
exit with correspondingly fewer
reflections, while those outside 2FH
would be rejected out the top of
the concentrator. The concentration
ratio of the tertiary is defined as:
CR =
(3)
For a given spot radius "FH", there exists a family of hyperbolas with vary-ing concentration ratios. The rule that governs the shape of the hyperbola
is:
-1 a
sin (r) < ~
H
40
(4)
This restriction is the limiting
possible for this concentrator.
the limit, the required height of
factor for the maximum concentration ratio
However, to intercept the entire beam at
the concentrator would be infinite.
There are two additional restrictions that are in effect for integrating the
tertiary into' the Cassegrainian design. They are:
The radius of the tertiary at the truncation height must not block any
rays that are reflected from the primary towards the secondary.
The radius of the tertiary, at the truncation height must intersect all
rays reflected from the secondary.
These two restrictions place a maximum and mlnlmum height restriction on the
tertiary, respectively. This, together with the required shape of the ter-
tiary for a given concentration ratio, limits the maximum concentration
ratio that can be attained.
RESULTS
Cassegrainian Only
A parametric study was performed for the Cassegranian concentrator.
Figure 3 illustrates the existance of an optimum ~/F for a particular geo-
metric concentration ratio. The optical efficiency is defined as the
product of the intercept factor and 1 minus the blocking factor.
1.0.9
RIM ANGLE: 60 DEG
RIM ANGLE: 45 DEG
RIM ANGLE: 75 DEG
.8.7.6
CR" 1512.3
PRIMARY SURFACE ERROR: 2MRAD
SECONDARY SURFACE ERROR: 1MRAD
0.0 L..........._..L..----J._....l-_'---L_-'---L_.......--..J
.5
1.0
Figure 3. Optical Efficiency
Versus Z!F
ZlF
.8
.2
>(J
z
w
U~ .6
LL.
w
..J
~(J
~ .4
o
As was discussed earlier, the beam
incident on the focal plane spreads
as Z!F increases, reducing the
intercept factor for a given CR.
The shift of optimum optical effi-
ciency due to rim angle is caused
by the reduction in primary focal
length at higher rim angles, which
increases the angle of extreme rays
(~) reflected from the secondary.
This increase in ~ reduces the beam
size at the focal plane, according
to conservation of optical extent,
thereby increasing the intercept
factor for a given CR. Reducing
Z!F also results in an increase
in~, and therefore an increased
intercept factor, but at the
expense of an increased blocking
factor, reducing the optical effi-
ciency. Increasing the primary
surface error results in a shift of
the optimum Z!F to the left, in
essence trading increased block-
ing to obtain a higher optical
efficiency.
41
Table 1 lists the performance of the true Cassegrainian at maximum optical
efficiency and maximum intercept factor at a rim angle of 60 degrees for
various combinations of primary and secondary surface errors. Results for
45 and 75 degrees are not shown since performance at 45 degrees is low, and
75 degrees results in higher primary costs. The optical efficiency listed
assumes 100 percent reflectivity on the primary and secondary reflectors,
and 95 percent reflectivity on the tertiary.
Several results are apparent from this table. They are:
Primary surface errors (Op) affect the performance of the system much
more than errors on the secondary (as).
It is not possible to achieve the acceptance criteria of CR =1200 at
IF ~ .96 at maximum optical efficiency except for the case of op =2
mr.
A primary surface error of 8 mr will not produce an acceptable IF.
Table 2 lists the performance of the Ritchey-Chretien configuration in the
same format as above. Comparing the Ritchey-Chretian to the true Cassegran-
inian shows no significant improvement in optical efficiency. The major
reason for this is that the improvement resulting from elimination of coma
is masked by the effects of surface errors. Figure 4 shows the intensity
distribution for the true Cassegrainian and the R-C for a typical set of
parameters. The most noticeable difference between the two configurations
is the higher intensity of radiation in the center for the Ritchey-Chretien
due to the elimination of coma. If this is a desirable feature in the over-
all design of a concentrating system, then perhaps the Ritchey-Chretien
should be considered. If not, then there is little, if any advantage to
using the Ritchey-Chretien.
Cassegrainian with Tertiary Reflector
The main parameter required to integrate the tertiary into the Cassegrainian
design is the radius of the spot on the focal plane, FH. This parameter,
along with the desired concentration ratio, defines the required shape of
the tertiary. However, it is not necessarily advantageous to set FH equal
to the maximum radius of the spot, since this would require a relatively
tall, narrow concentrator. Examination of figure 4 reveals that the inten-
sity distribution is very close to a normal distribution. This fact can be
used to determine an appropriate FH. The procedure for determining FH is as
follows:
Determine the standard deviation of the flux distribution (00).
Calculate FH by determining the desired fraction of the total energy
available to be captured. These calculations have been performed for a
capture ratio of .995 and .975. It is apparent that as the capture
ratio decreases, the required tertiary height decreases.
Blockage of rays reflected from the primary reflector must also be avoided.
At high tertiary heights, this becomes a problem, and limits the concentra-
tion ratio of the system.
42
TABLE 1. CASSEGRAINIAN PERFORMANCE
Rim Angle: 60°
0p =4mr Os =2mr
CR IF nmax ~/F r(cm; P(kW»)'(* IFmax ~/F no P(kW)
1200 0.9 0.77 0.7 10.2 30 0.98 0.55 0.58 22.5
1500 0.85 0.73 0.675 9.0 28 0.96 0.55 0.55 22
0p =8mr Os =2mr
CR IF nmax ~/F r(cm) P(kW) I Fmax ~/F Do P(kW)
1200 0.69 0.54 0.65 10.2 20.7 0.79 0.55 0.45 17.5
1500 0.63 0.49 0.65 9.0 18.8 0.73 0.55 0.42 16.5
0p =4mr Os =1mr
CR IF T)max ~/F r(cm) P(kW) I Fmax ~/F no P(kW)
1200 0.90 0.77 0.70 10.2 30.0 0.98 0.55 0.58 22.5
1500 0.89 0.73 0.675 9.0 28.5 0.965 0.55 0.57 22.5
0p =2mr Os = Imr
CR IF nmax ~/F r(cm) P(kW) IFmax ~/F no P(kW)
1200 0.98 0.90 0.75 10.2 34.4 0.997 0.7 0.87 33.4
1500 0.96 0.87 0.75 9.0 34.0 0.99 0.65 0.81 31.0
* receiver radiusr =
** . receiver plane in kilowattsp =power enterlng the
TABLE 2. RITCHEY-CHRETIEN PERFORMANCE
Rim Angle: 60°
0p =4mr Os =2mr
CR IF nmax ~/F r(cm) P(kW) I Fmax ~/F Do P(kW)
1200 .91 0.77 .70 10.2 29.5 .94 .65 .71 27.3
1500 .87 0.73 .70 9.0 28.0 .90 .65 .68 26.0
0p = 2mr as =1mr
CR IF nmax ~/F r(cm) P(kW) I Fmax ~/F no P(kW)
1200 .986 0.89 .75 10.2 34.4 .998 .70 .85 32.9
1500 .967 0.88 .75 9.0 33.7 .993 .70 .85 32.7
43
600L.----l2L....--.J----1.--.L--=~!B:_
4 6 8 10 12
RECEIVER RADIUS (em)
CASSEGRAINIAN
RITCHEY' - CHRETIEN
RIM ANGLE: 60 DEGREES
ZlF = .76
PRIMARY SURFACE ERROR: 2mr
SECONDARY SURFACE ERROR: 1mr
Comparison of Intensity
Distributions
3
1
2
4
Figure 4.
Results for the Cassegrainian with
the tertiary reflector are listed
in table 3 for a reduced secondary
diameter. The most significant
resul t is that the optical effi-
ciency does not peak as a function
of ~/F. The tertiary reflector
redirects the beam to the desired
receiver aperture regardless of
the size of the beam on the ter-
tiary. The major penalty for
redirecting a large beam is a tall
tertiary.
Analysis of the radiation on the
secondary reflector indicates that
the diameter can be reduced with a
small gain in performance. This
is because the flux of reflected
energy on the outer ring of the
secondary is very low, and by not
intercepting that energy, it is
possible to decrease the blocking
factor. Reducing the secondary
diameter also benefits the ter-
tiary re£lecter design. Since the
"source" for the radiation that
the tertiary intercepts is now
smaller, the height of the ter-
tiary can be reduced for a given
concentration ratio, or con-
versely, a higher concentration
ratio can be achieved for a given
height.
TABLE 3. CASSEGRAINIAN PLUS TERTIARY PERFORMANCE
Rim Angle: 60° ap =4mr
Secondary Diameter =1.96 m
as =2mr aT =2mr ~/F =0.75
Capture Ratio: .995
..£!L r(cm) IF nmax
1200 10.1 .98 .89
1300 9.7 .98 .89
1500 9.0
2200 7.5
Capture Ratio: .975
..£!L IF nmax
1200 .97 .88
1300 .97 .88
1500 .96 .87
2200 .96 .87
"(~T =height of the tertiary
ZT(cm)*
99.0
114.0
ZT(cm)
42
47
60
118
P(kW)
34.4
34.4
P(kW)
33.7
33.7
33.6
33.3
44
Comparing the maximum efficiencies for the Cassegrainian only and the Cas-
segralnlan plus tertiary yields a 17 percent increase at CR =1200 and a
19 percent increase at CR = 1500. These increases in efficiency result from
increasing the intercept factor. Additionally, it is possible to increase
the concentration ratio without significantly degrading the efficiency when
using the tertiary.
Effects of Misalignment
There are three loss mechanisms that affect the performance of the tertiary
reflector. They are:
Rejection of rays through the inlet aperture.
Absorption of energy caused by multiple reflections.
Non-interception of radiation by the inlet aperture.
Another loss mechanism, introduced by the reduction of the secondary diam-
eter, is non-interception of radiation by the secondary reflector.
1.00
ox = 0.0254
1.501.00.50
ox = RADIAL MISALIGNMENT 1m)
DZ =AXIAL MISALIGNMENT 1m)
......
..........
......... ........
"''''',
..~ DZ=O
...',,~... DZ = +.0254
'DZ = -.0254
.50 '------'-----'------'
0.00
.70
.60
.90
>() .80
Z
w(j
ii:
II.
W
-'~()
f
o
Figure 5. Effect of Misalignment
on Optical Efficiency
ROTATIONAL MISALIGNMENT OF SECONDARY (DEGI
Figures 5 and 6 show the effects
of multiple misalignments on
optical efficiency and intercept
factor. The criteria chosen for
determining the maximum amount
of misalignment permissible was
to set the minimum optical effi-
ciency equal to the maximum
efficiency attainable without
the tertiary with perfect align-
ment. This is an arbitrary
decision, although it does give
a rational bound on system effi-
ciency. Using this criteria,
the maximum combined misalign-
ment is:
Rotational misalignment has the
largest' effect on increased
loss, followed by axial align-
ment. Positive axial misalign-
ment reduces the percentage of
radiation not intercepted by the
secondary. Perhaps the second-
ary diameter could be increased,
resulting in less sensitivity to
axial misalignment.
Axial: + .0254m ( 1")
-
Radial: + .0254m (1" )
-
Rotational: + .5°
-
45
ox =0.0264
.70
.60
CONCLUSIONS
The Ritchey-Chretien modification
does not significantly improve the
performance of the Cassegrainian
configuration due to the presence
of surface slope errors that
largely mask improvements produced
by the elimination of coma.
The Cassegrainian concentrator is a
viable system for the 7-meter dish
studied •
This results in a decrease in opti-
cal efficiency and intercept factor
of approximately 11 percent, such
that the minimum optical efficiency
is 77 percent, with an intercept
factor of 85.5 percent.
1.601.00
DX • RADIAL MISALIGNMENT 1m)
DZ • AXIAL MISALIGNMENT 1m)
.60'-__....... ....__--'
0.00
a:
~
Ii:
w(,)
a:
i
ROTATIONAL MISALIGNMENT OF SECONDARY (DEG)
Figure 6. Effect of Misalignment
on Intercept Factor
A non-imaging tertiary reflector
significantly improves the optical
performance of the Cassegrainian,
and should be integrated into the
design.
REFERENCES
1. Winston, R., Welford, W. T. "Geometrical Vector Flux and Some New Non-
imaging Concentrators", J. Opt. Soc. Am., Vol. 69, No.4, April 1979.
2. Survey: Mirrors and Lenses and Their Required Surface Accuracy.
Honeywell, Inc. Minneapolis, HN, DOE/CS/35348-T1, January 1980.
3. Wetherell, W. B., Rimmer, M. P. "General Analysis of Aplanatic Casse-
grain, Gregorian, and Schwartzchild Telescopes", Applied Optics,
Vol. 11, No. 12, December 1972.
46


Optical Analysis of Cassegrainian Point Focus Concentrators

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