1 Copyright 2012 by ASME

Proceedings of the ASME 2012 International Design Engineering Technical Conferences &
Computers and Information in Engineering Conference
IDETC/CIE 2012
August 12-15, 2012, Chicago, IL, USA
DETC 2012-71247
ADAPTIVE HELIOSTAT SOLAR ARRAYS USING SHAPE-OPTIMIZED COMPLIANT
MIRRORS


Li Meng
+

Department of Mechanical Engineering
Massachusetts Institute of Technology
Cambridge, MA 02139
lmeng@mit.edu
Amy M. Bilton
Department of Aeronautics and Astronautics
Massachusetts Institute of Technology
Cambridge, MA 02139
bilton@mit.edu


Zheng You
Department of Precision Instruments and
Mechanology,
Tsinghua University
10086 Beijing, China
yz-dpi@mail.tsinghua.edu.cn
Steven Dubowsky*
Department of Mechanical Engineering
Massachusetts Institute of Technology
Cambridge, MA 02139
dubowsky@mit.edu



+
Li Meng is a visiting-student of Department of Mechanical Engineering at MIT.
* Corresponding author, Email: dubowsky@mit.edu.
ABSTRACT
In a Solar Power Tower (SPT) system, the ideal shape of a
heliostat concentrator is a section of paraboloid which is a
function of the location in the array and the incidence sun
angle. This shape is difficult to achieve and limits the system
efficiency. A shape-optimized compliant (SOC) design of
parabolic heliostats is presented here to solve this problem. An
approximation of the ideal shape is suggested to use an
optimized stationary paraboloid shape which only varies with
heliostat location in the array. A compliant structure design is
proposed that to use a simple flat mirror with a two-
dimensional tailored stiffness profile to form the required
parabolic surface using adjustment mechanisms at each
corner. This design is validated by numerical simulations
including FEA tools, ray tracing, and classical nonlinear
optimization. The annual performance shows that the SOC
heliostat will substantially improve the efficiency and benefit
the SPT system.

1 INTRODUCTION

1.1 Background
Heliostats are mirrors used to collect solar energy for a
Solar Power Tower (SPT), a thermal power system producing
clean and renewable electricity from solar radiation [1]. By
deploying thousands of heliostats in a field, which redirect and
concentrate sunlight to a single receiver mounted on a high
tower, SPTs can achieve high temperature and power
conversion efficiency in thermal cycle. Due to their large scale,
they are potentially economically advantageous. Substantial
research has been done on heliostats and SPTs. The most
successful systems for both technology demonstration and
commercial use are Solar One/Two in the US [2] and PS10/20
in Spain (See Fig. 1) [3]. New concepts and technologies are
also being developed in other countries including Germany,
Israel, and China [4-5]. The current state-of-art SPTs produce
on the order of 10MW of power but unfortunately have low
efficiency and high lifetime cost. This cost is dominated by
their initial capital cost and land cost. Significant
improvements in system performance and cost reduction (by
50%) are required for SPTs to reach the price of conventional
fossil-fuel power [1]. Recently, volumetric receivers,
biomimetic heliostat field layouts, and new system concepts
2 Copyright 2012 by ASME
like beam-down/hillside SPTs are being researched to make
SPTs more practical [4, 6-7].


Figure 1. SOLAR POWER TOWER SYSTEM
PS10 AND PS20, SPAIN
1
.

A heliostat consists of one or more reflector surfaces, a
support structure, and a two-axis tracking device. Ideally, the
shape of the reflector surface should be a paraboloid section to
focus the majority of reflected light on the fixed receiver,
reducing sunlight falling on the mirror but not reaching the
receiver which is called optical spillage loss. This paraboloid
shape is a function of the mirror position in the field and the
sun angle that changes with time, season, and geographic
location. So, developing efficient heliostats is challenging in
both designing and manufacturing. Current heliostats are
expensive and constitute approximately 50% of the cost of a
typical SPT design [8-9].


Figure 2. FLAT HELIOSTAT (LEFT
2
) AND
STRETCHED MEMBRANE HELIOSTAT (RIGHT
3
).

Various innovative methods have been developed for solar
concentrators like heliostats to reduce the cost and complexity
of fabricating precision shapes. In one solution, designers
divide the single surface into smaller flat or concave facets and
mount them onto a honeycomb-like support structure and align
each of them to the target (See Fig. 2 Left) [8]. Nonetheless, an
inevitable moderate focusing error still exists while the multi-

1
http://sine.ni.com/cs/app/doc/p/id/cs-11550
2
http://en.wikipedia.org/wiki/File:Heliostat.jpg
3
http://www.pveng.com/FEA/FEAGeneral/LargeDisplace/LargeDisplace.php
facet design introduces complexity considerably as the facet
number increases.
In another solution, researchers have developed methods
to form the mirror shape using compliant structures. Stretched
membrane heliostats (See Fig. 2 Right) [8-9] use controlled
pressure to adjust the reflector shape, but are complex and
imprecise. Other smart substructures with embedded actuators
and sensors are also developed to control compliant mirror
shape [10]; however for large heliostat arrays this approach
would require many more actuators than would be practical.
Other related research has studied forming parabolic trough
collectors and parabolic dishes by bending compliant structures
with optimized shapes and thicknesses [11-12]. However, these
works only consider simple one dimensional bending to form
parabolas using cable mechanisms. These works have not
solved the problem of bending plates simultaneously in two
directions to form paraboloids and develop a mechanism to
perform this bending.
Consequently, the limit on heliostat performance becomes
an important issue for SPT design. First of all, because of the
limited concentration ratio, C (the mirror area divided by the
projected receiver aperture area), of individual heliostats, more
mirrors are required to achieve higher temperature to improve
the thermal cycle efficiency. Second, in practical SPT systems,
the central receiver has a larger aperture area than the
individual heliostat surface or facet (e.g., C=0.047 for the
facets in Solar Two, [2]; C=0.73 for the heliostats in PS10,
[3]), which results in a dilemma for both mirror and receiver
design. To reduce the receiver aperture, smaller mirrors have
to be chosen, which increases the number of tracking devices.
To increase mirror area, larger receiver aperture has to be
designed, which causes both higher receiver cost and thermal
loss [4].


1.2 Objective and Approach
This research develops a new, low-cost, high efficiency
compliant heliostat concept, which will permit fewer mirrors
and a smaller central receiver. This concept could greatly
reduce costs and enable SPTs to be a more practical source of
clean energy.
In this approach, shape-optimized heliostats are formed
from flat sheets with individually tailored compliances, which
can be easily shipped to the site and assembled with
adjustment mechanisms to adapt in two dimensions as
required. The mirrors are deformed into a fixed surface of a
paraboloid section that optimizes the average heliostat
performance over the year. This shape is a function of heliostat
location in the field.
To evaluate the effectiveness of this approach, an optical
efficiency metric is defined given the concentration ratio and
applied to a conventional representative system with flat
mirrors. Then the compliant heliostat mirror is designed with
tailored surface stiffness in two dimensions that allows the
3 Copyright 2012 by ASME
surface to deform into a paraboloid section by simple manual
mechanisms, such as screws with some guidance, on the
corners. The proposed simplification (statically optimized
shape) is shown to be effective by comparing to the
theoretically ideal adapting mirror surface, and a flat heliostat
mirror using numerical simulation including the Finite
Element Analysis (FEA) method, optical ray tracing and
classical optimization.


2 PERFORMANCE METRIC AND DEVELOPMENTS
OF HELIOSTAT MIRROR MOTION

2.1 Reference System Assumptions
A virtual heliostat field is established here as a reference
case. The system is assumed to be located at latitude of 35N,
the approximate latitude of Los Angeles, California. A north-
facing cavity receiver is used to collect energy from the
heliostat field. The heliostat surface area is 1m
2
and the tower
height is 10m. The distance between the Sample Mirror and
receiver is 100m, as shown in Fig. 3.


Figure 3. REFERENCE HELIOSTAT FIELD.

2.2 Performance Metric
The performance metric or efficiency, q, of a heliostat
mirror is defined as the ratio of solar energy that falls on the
mirror to the reflected energy that reaches the receiver,


r
i
E
E
q = (1)

where: E
i
is the incidence solar radiation on the surface of the
heliostat.
E
r
is the energy reflected into the receiver aperture.
This metric is a function of the sun angle, u, concentration
ratio, C, and the mirror design. The atmospheric attenuation
and shading effects are neglected here, so

( ) , , mirror design C q q u = (2)

The sun angle is defined as the angle between incidence
and reflected sunlight on the mirror surface (See Fig. 4), which
can be assumed to be constant over the surface since the mirror
size (1m) is rather small in comparison to the receiver distance
(100m). The ratio of the projected heliostat area in the
direction of the sun to the surface area should be high for
practical SPTs [6]. This ratio, which is cos /2, is known as the
cosine efficiency. The mathematical development presented
here assumes a cosine efficiency greater than 75%.


Figure 4. THE LIGHT PATH OF HELIOSTAT.

Recall that the concentration ratio C is defined as the
mirror area divided by the projected receiver aperture area,
given by:

sec
m
r
A
C
A
o = (3)

where: A
m
is the mirror area.
A
r
is the receiver aperture area.
is the angle between normal of receiver plane and
reflected sunlight (See Fig. 4).
For a cavity receiver which has a plane aperture, the
term should be considered because the projected aperture size
will decrease from the mirror perspective. For a cylindrical
receiver, this term reduces because of the structure symmetry.
Therefore, C is a function of mirror location in the field as
shown in Fig. 5. This shows that the optimal mirror shape
requirements will vary throughout the field. For a given and
C, is only a result of mirror design.

4 Copyright 2012 by ASME

Figure 5. CONCENTRATION RATIO C (WITH
A
M
/A
R
=1) VARIATION OVER THE FIELD.

2.3 Flat Mirror Performance
For a flat mirror, the efficiency can be obtained as,

min ,1
co
2
1
s C
q
u


=
`

)
(4)

The detailed derivation can be found in the Appendix A.
With a cosine efficiency above 75%, this reveals the
performance of flat heliostat will decrease significantly with
increasing concentration ratio C.


2.4 Ideal Mirror Surface Determination and
Performance
To concentrate parallel sunlight to a small target, the
mirror surface has to be a section of paraboloid whose focal
point is the receiver and symmetry axis is parallel to the
sunlight as shown in Fig. 4. This surface is a function of
mirror location and sun angle. A temporary coordinate system,
XYZ, is introduced to deduce the surface in the body-fixed
mirror coordinate system XYZ.


Figure 6. COORDINATE SYSTEM.

Figure 6 shows the origin of XYZ placed at the center of
the heliostat; Z axis is parallel to sunlight; Y axis is
perpendicular to Z and within the plane of receiver-mirror-
sunlight; X axis constitute right-hand coordinate system with
Y and Z. In XYZ, the ideal parabolic surface is,


2 2
tan
4 2
x y
z y
f
u +
= + (5)


(1 cos )
2
R
f
u +
= (6)

where: f is focal length.
R is the receiver-mirror distance.
The body-fixed mirror coordinate system XYZ, in which Z
axis is the normal of mirror surface center, is obtained by
rotation from XYZ by /2 around the X axis.
By coordinate transformation the surface governing
equation in XYZ is,


2 2 2 2 2
cos sin sin 4 sec 0
2 2 2
x y z yz fz
u u u
u + + = (7)

Since the parabolic surface can theoretically concentrate
the sunlight into an infinitely small focal point, the efficiency
of ideal surface will be always 100% for any sun angle and
concentration ratio C.


3 COMPLIANT MECHANISM DESIGN
The ideal mirror shape is a function of mirror location and
sun angle. It varies continuously, but implementing shape
variation in real-time is difficult and costly. Here, an
approximate design called a Shape-Optimized Compliant
(SOC) heliostat is proposed. A SOC heliostat has a fixed
parabolic shape that is optimized for maximum annual average
performance, and thus is only a function of location. Since the
required surface deviation in Z-direction is relatively small
compared to the mirror size, it is proposed to implement the
shape by two-dimensional bending of a compliant surface.
With appropriate stiffness design, only two pairs of small
adjustment mechanisms are applied to the four cut corners of
the square sheet. Hence, the compliant heliostats can be easily
adjusted to the optimized shape during installation.


3.1 Analytical Design

3.1.1 Surface Curvature Requirements
The principal curvatures of a general paraboloid surface
can be obtained in the cylindrical coordinate system (, , z) as
5 Copyright 2012 by ASME
shown in Fig. 7. For a heliostat, they are functions of the
mirror location R and sun angle . The detailed mathematical
derivation is presented in Appendix B.


Figure 7. PARABOLOID AND ITS PRINCIPAL
CURVATURES ON A CYLINDRICAL COORDINATE
SYSTEM.


3 2
2
2
cos
1
2
2
2 (1 )
4
R
f
f

u
k

= =
+
(8)


2
2
1 1
2 cos
2 1
2
4
R
f
f

k
u

= =
+
(9)

Since R is much larger than the heliostat size, these
curvatures can be approximated as constants over the surface.
The body-fixed Y-axis aligns with the principal direction with
curvature of

while X-axis aligns with the other principal

direction of curvature

. Thus, the deflections in Z-direction

are given by,


2
2
16 cos
1
8
2
x
l
w l
R

k
u
= =
(10)


2
2
cos
2
1
1
8 6
y
l
w
R
l

u
k = = (11)

where: l is the length of the heliostat surface.
With R>>l and a cosine efficiency above 75%, the
maximum surface deflection is much smaller than the mirror
size,


( )
max ,
x y
w w l << (12)

(e.g., for the sample 1m
2
mirror in the reference case the
maximum deflection is about 1mm). This permits the
feasibility of mirror shaping using two-dimensional bending.


3.1.2 Mechanism Design
The mechanism for the compliant heliostat is shown in
Fig. 8. The mirror surface is a square and its diagonal lines
align with the principal directions in which the moments are
applied to form the curvatures. The corners of the surface are
equipped with adjustment mechanisms applying moments
toward the center. Because the adjustments are much smaller
than the mirror they can be regarded as concentrated moments
applied on the corners. To ensure the external loads are pure
moments, the M

adjustment mechanisms are free to move in

X-direction, while the M

adjustment mechanisms are free to

move in both the Y and Z directions. The entire structure is
fixed to its two-axis tracking system due to restrictions on Y-
axis and Z-axis by the X-track and the restriction on X-axis by
the Y-track.


Figure 8. OVERVIEW OF COMPLIANT
MECHANISM.

With this mechanism framework, the internal bending
moments of the mirror elements in XYZ coordinate system are
given by,

( )
x
m D

k vk = + (13)

( )
y
m D

k vk = + (14)

0
xy
m = (15)

6 Copyright 2012 by ASME

( )
3
2
12 1
Eh
D
v
=

(16)

where: D is the stiffness of the mirror element.
E is the Youngs modulus.
is the Poissons ratio of the material.
To achieve the paraboloid curvatures given in (8) (9), the
stiffness is tailored to a specific profile. It should be pointed
out that the calculation here neglects the singularity that
results from applying concentrated moment loads on the
corners. This will be considered later in this paper.


3.1.3 Tracking Demands
Since

and

are different, the two tracking axes for the

proposed heliostat are unique. Axis 1 is parallel to the line
between the receiver and the mirror center (See Fig. 9); Axis 2
is the body-fixed X-axis which is orthonormal to the plane
determined by the sun, the receiver, and the mirror center. This
tracking method, called spinning-elevation, is different from
the conventional azimuth-elevation heliostat tracking [13].


Figure 9. TRACKING AXIS DEMANDS.


3.1.4 Variation of Stiffness
As mentioned before, the stiffness of the compliant
heliostat surface has to be designed for the simple applied
moments to shape the surface into a paraboloid to reach high
optical performance. The equilibrium diagram is shown in Fig.
10. All the calculations use the small displacement
assumption.


Figure 10. FREE BODY DIAGRAM.
The applied moments are related to the internal moments
by,


2
( )
2
l
y
l y
y
M m dx

=
}
(17)


2
( )
2
l
x
l x
x
M m dy

=
}
(18)

In classical plate bending theory, the internal moments are
difficult to solve because of the singularity caused by the
asymmetrical concentrated load [14]. The calculations neglect
the challenges of the detailed corners modeling. However, here
they yield an approximate solution for the required stiffness.
By substituting (13) (14) into (17) (18), a set of integral
equations for the multi-variable stiffness function, D(x,y), is
obtained as,


( )
2
2
l
y
l
y
M
Ddx


k vk

| |

|
\ .
=
+
}
(19)


( )
2
2
l
x
l
x
M
Ddy


k vk

| |

|
\ .
=
+
}
(20)

The continuous solution for (19) (20) is difficult to find,
but a discontinuous approximate solution can be constructed,

( ) , 2
0
l
b x y
D x
c
others
y

s + s

(21)

where b, c are undetermined constants. So that,


2
2
2
2 ( )
2 2
l
y
l
y
cb y b
Ddx
l l
c y b y

| |

|
\ .
s

=

< s

}
(22)
7 Copyright 2012 by ASME


2
2
2
2 ( )
2 2
l
x
l
x
cb x b
Ddy
l l
c x b x

| |

|
\ .
s

=

< s

}
(23)

Substituting D to (17) (18), the external adjustment
moments are obtained as follows,


( )
( )
3
2
cos
1
2
2
2 6 1
2 cos
2
Ebh
M cb
R
R

u
k vk v
u
v
| |
|
~ + = +
|

|
|
\ .
(24)

( )
( )
3
2
cos
1
2
2
2 6 1
2 cos
2
Ebh
M cb
R
R

u
k vk v
u
v
| |
|
~ + = +
|

|
|
\ .
(25)

In this solution, the Eqn. (19) (20) are satisfied
everywhere except the corners. By definition of stiffness (16),
this solution can be implemented by varying the thickness of
the plate as shown in Fig. 11. The design consists of a square
frame with thickness of h, and a thin square center with
thickness of h/5. The adjustment mechanisms are located at the
corners of the frame. Note that since the thickness of the center
square of mirror is nominally zero, as shown in (21), it is set
here to h/5 to be able to support the reflective surface, such as a
Mylar sheet. This area will have a negligible effect on the
surface stiffness profile. This constructive stiffness design only
uses square sheets and two different thicknesses which can be
easily manufactured, for example, by lamination of three
layers.


Figure 11. VARIATION OF STIFFNESS (LEFT: TOP
VIEW; RIGHT: CROSS-SECTION VIEW).


3.2 FEA Validation and Numerical Optimization
For this study, a numerical simulation to verify the
heliostat mechanism design was developed using the
commercial Finite Element Analysis (FEA) software ADINA,
and synthesized with the optical ray tracing and optimization
process written in Matlab. The integrated simulation tool
includes:
- FEA: With the external moments as inputs, the
deformation and rotation of discrete elements are
generated over the mirror surface which is meshed to
resolve both thicknesses.
- Ray-tracing: The inputs include the sun angle,
receiver distance, and the surface information result
from FEA module. The normal direction for each
element is calculated using the rotation data. Then the
reflected points on focal plane located at receiver are
solved geometrically.
- Optimization: The objective is to minimize the optical
spillage loss that counts the elements by which the
reflected lights miss the receiver aperture. The
corresponding design variables are the adjustment
moments. This unconstrained nonlinear optimization
process is programmed using Newtons method using
the finite difference for gradient calculations.
The parameters for simulation include: distance R (100m);
sun angle (60); thickness h (5mm); width b (0.2m);
adjustment size (0.2m); Youngs modulus E (200GPa);
Poissons ratio (0.3); The FEA model contains 790 nodes.
The optimized deformation result is shown in Fig. 12
(Left) with the largest deviation w
max
shown in Tab. 1. The
error between optimized FEA deformation and the ideal
paraboloid surface is shown in Fig. 13 and in Tab. 1. The
simulation resulted surface is close to the analytical paraboloid
section especially in the center. The large errors occur on the
edges of the mirror and the transition points between different
thicknesses.


Figure 12. DEFORMATION PLOT.

8 Copyright 2012 by ASME


Figure 13. SURFACE DEFLECTION ERROR.

Table 1 DEFLECTION RESULTS
Maximum
Deflection
Analytical (m) FEA+Optimization
(m)
wmax
1.064010
3
1.081410
3

Deflection
Error
Mean (m) Standard Deviation
(m)
| wFEAwAnalytical | 1.828410
5
1.052310
5

Figure 14 shows the ray tracing points on a 10 cm receiver
for the numerically optimized mirror. It can be seen that the
majority of the light falls in a center 5cm5cm region of the
receiver and the design provides an efficiency of 88.4% for a
receiver radius of 10cm and concentration ratio of 32. By
comparison, for the same conditions, the conventional flat
mirror heliostat has an efficiency of 4.17%, as given by Eqn.
(4). Although the efficiency of such compliant heliostat is
shown to be high, some small deviations exist between the
ideal shape for the SOC and the actual shape. First, the surface
determined by the optimization with the smallest deflection
error only has an efficiency of 85%, which reflects the
inaccuracy of predicting local curvatures. This can be
improved by integrating the ray tracing and efficiency
calculation into the optimization and setting the
comprehensive optimization objective, which however
consumes more computing time and increases the
programming complexity. Second, the external moments
predicted by analytical model are larger than the optimized
values. This is the result of the substantial assumptions and
approximation made in the analytical modeling.


Figure 14. REFLECTED POINTS ON FOCAL
PLANE.


4 PERFORMANCE OF SHAPE-OPTIMIZED
COMPLIANT HELIOSTATS
In this section, the annual average performance of the
shape-optimized compliant (SOC) heliostat is compared to the
flat mirror, the ideal stationary parabolic shaped mirror, and
the ideal time-dependent adaptive mirror.
The sample heliostat in the reference system is located at
99m (North) and 10m (East) with a distance of 100m to the
receiver as shown in Fig. 4. Neglecting any tracking errors, the
upper bound of sun angle for the sample mirror is 95.74,
which occurs when the sun rises in the morning, while the
lower bound is reached in the afternoon, varying by the season
from summer to winter with the range of [25.66, 71.77].
Therefore, the annual performance of heliostat is defined by
the mean of the efficiency distribution over the range of input
sun angles.
This paper develops a stationary compliant heliostat
whose shape is optimized corresponding to the annual average
performance. The optimization process used to determine the
shape is similar to section 3.2 with the objective changed to
maximize the annual energy collection. Meanwhile, diverse
concentration ratios, meaning various receiver sizes, are
investigated in this section to explore the feasibility of the
heliostat design.
The results (See Fig. 15) show that the SOC heliostat can
achieve much higher performance than the flat one, and
almost performs as well as the ideal case. Using 90% as a
standard of efficiency, the SOC can function with a
concentration ratio of 28 which results in a receiver radius of
10.7cm. With the same receiver, the flat mirror delivers less
than 10% of the light into the aperture. For the ideal stationary
parabolic mirror, the receiver size can be further reduced to
5.6cm and maintain a 90% efficiency. Such results suggest that
SOC heliostats are much more efficient than conventional flat
mirrors. They perform nearly as well as the ideal stationary
9 Copyright 2012 by ASME
parabolic mirrors, and even the ideal time-dependent adapting
ones, but are much simpler to implement.


Figure 15. PERFORMANCE RESULTS.


5 CONCLUSION
This paper presents a low-cost, high efficiency, location-
based adaptive heliostat mirror. The design uses a tailored
compliant surface with simple adjustment mechanisms to
achieve the desired parabolic shape. The shape is optimized to
achieve the maximum annual efficiency of transferring solar
radiation to the receiver with a two-axis spinning-elevation
mode tracking device. This is a stationary simplification of the
ideal time-dependent adapting shape. These adaptive mirrors
can be configured in the field during assembly and adjusted as
required.
The results of numerical analysis show that the shape-
optimized compliant heliostats can be substantially more
effective than flat mirrors. They also perform closely to the
ideal fixed parabolic mirrors. These low-cost adaptive
heliostats will permit smaller receivers and larger mirrors for
Solar Power Towers to improve system efficiency and
economic benefits.
ACKNOWLEDGMENTS
The authors would like to thank the King Fahd University
of Petroleum and Minerals in Dhahran, Saudi Arabia, for
funding the research reported in this paper through the Center
for Clean Water and Clean Energy at MIT and KFUPM. The
authors would also like to thank the Chinese Scholarship
Council for the support of Li Meng for his research in MIT.
The authors would also like to thank Leah Kelley, Elizabeth
Reed, and Aditya Bhujle for their generous help during the
development of this work.
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2011. A Novel Approach for Designing Parabolic Mirrors
Using Optimized Compliant Bands. Proceedings of the
ASME 2011 International Design Engineering Technical
Conferences & Computers and Information in
Engineering Conference, August 28-31, 2011,
Washington, DC, USA.
[12] Li, L. and Dubowsky, S., 2011. A New Design Approach
for Solar Concentrating Parabolic Dish Based on
Optimized Flexible Petals. Journal of Mechanisms and
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K., Karni, J., Buck, R., Pfahl, A., and Bligh, T. P., 2004.
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APPENDIX A: OPTICAL EFFICIENCY OF FLAT
MIRRORS
For a flat mirror, the total solar energy captured is,

10 Copyright 2012 by ASME
cos
2
i m
E GA
u
= (26)
where: G is the solar constant.
The reflected energy flux on the receiver is,

cos
r
D G o = (27)
So the energy radiated on the receiver is,


cos
2
cos
cos
2
2 cos
m
r r
r
m
m r
A
GA cos A
E
A
GA cos A
u
o
o
u
u
o

<

>

(28)

Thus the efficiency for flat mirror is,
min ,1
co
2
1
s C
q
u


=
`

)
(29)
APPENDIX B: PRINCIPAL CURVATURES OF A
PARABOLOID
Geometrically, the paraboloid F (See Fig. 7) is a surface of
revolution which can be parameterized as,
( )
2
, ( cos , sin , )
4 f

= F (30)
For any point PeF, the tangent plane is spanned by
tangent vectors F

and F

:

(cos , sin , )
2 f

= F (31)
( sin , cos , 0)

= F (32)

The shape operator is,


3 2
2
2
2
2
1
0
2 (1 )
4
1
0
2 1
4
P
f
f
S
f
f

(
(
(
+
(
=
(
(
(
+
(
(

(33)

Since S
P
is a diagonal matrix, the principal curvatures of
F at any point are equal to those entries located on principal
diagonal with principal directions parallel to F

and F

respectively.

3 2
2
2
1
2 (1 )
4
f
f

=
+
(34)

2
2
1
2 1
4
f
f

=
+
(35)

For a given mirror location R and sun angle , can be
determined as,

sin R u = (36)

with the expression (6) of focal length f the two curvatures of
heliostat surface are,


3 2
2
2
cos
1
2
2
2 (1 )
4
R
f
f

u
k

= =
+
(37)

2
2
1 1
2 cos
2 1
2
4
R
f
f

k
u

= =
+
(38)