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Citation: Fontani, D.; Sansoni, P.;
Francini, F.; Toni, F.; Jafrancesco, D.
Optical Raytracing Analysis of a
Scheffler Type Concentrator. Energies
2022,15, 260. https://doi.org/
10.3390/en15010260
Academic Editor: Chao Xu
Received: 28 October 2021
Accepted: 28 December 2021
Published: 31 December 2021
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energies
Article
Optical Raytracing Analysis of a Scheffler Type Concentrator
Daniela Fontani , Paola Sansoni * , Franco Francini, Francesco Toni and David Jafrancesco
National Research Council-National Institute of Optics (CNR-INO), 50125 Florence, Italy;
daniela.fontani@ino.cnr.it (D.F.); franco.francini@ino.cnr.it (F.F.); francesco.toni@ino.cnr.it (F.T.);
david.jafrancesco@ino.cnr.it (D.J.)
*Correspondence: paola.sansoni@ino.cnr.it
Abstract:
The Scheffler type concentrator is a curved metal reflector particularly suitable for solar
thermal systems with a receiver fixed to the ground. Its operating principle is to deform the reflector
throughout the year to optimize its performance in collecting sunlight. This study analyses the optical
performance of a Scheffler reflector during the year. A CAD software tool is utilized to reproduce
the mechanical deformations of a real Scheffler concentrator and the shape of the light spot on the
receiver is analyzed by means of raytracing simulations. The starting configuration is the equinoctial
paraboloid, which produces a point-like spot on the two equinox days only. On all other days of the
year, this paraboloid is deformed in a suitable way in order to keep the spot as small as possible, but,
even so, it is no longer a point-like spot. In the present work the simulated light distributions on the
receiver, generated by the paraboloids (deformed or original), are compared. The results confirm the
working principle of the Scheffler type concentrator and allow correctly sizing the receiver.
Keywords:
solar concentration; optical design; Scheffler concentrator; 3D CAD simulation; optical
analysis; thermal application
1. Introduction
Combating climate change needs to exploit, in the best possible way, solar radiation, a
famously renewable energy source.
Currently, solar energy is mainly exploited for electricity production, especially by
means of photovoltaic panels, but the production of thermal energy is also gaining interest
and efforts are currently being made to reduce the installation costs of such production.
All current systems to produce thermal energy consist of one or more mirrors (seldom
lenses) that redirect solar radiation towards a small target, but there are various types of
solar device, such as concentrating systems, point focus collectors, linear array; each one is
designed to be applied to a specific thermal energy production method.
It should be considered that, in choosing the system to be implemented, some fac-
tors are often preponderant, such as efficiency, complexity, costs of implementation and
maintenance.
Solar systems that use thermal receivers must necessarily use pipes to transport the
heat-transfer fluid. For this reason, it is advantageous to fix the receiver to the ground and
move only the concentrator mirror.
From the literature it is clear that the Scheffler-type configuration is a suitable configu-
ration for this purpose, especially considering its low production cost and its simplicity of
manufacturing, which is shown in [
1
], where is presented a comparison of some fixed-focus
parabolic systems, relying on the parameters that affect their efficiency.
Several articles describe how to design and build Scheffler mirrors, from either a
geometric-mechanical [2–4] or a mathematical [5] point of view.
In the literature there are various examples of applications, especially for desalination
purposes, solar cooking, agriculture [6], and for the production of electricity [7,8].
Energies 2022,15, 260. https://doi.org/10.3390/en15010260 https://www.mdpi.com/journal/energies
Energies 2022,15, 260 2 of 20
The Scheffler configuration utilizes an equatorial mount with a parabolic concentrator,
which must be mechanically deformed, daily, during the year to keep an acceptable con-
centration level of solar radiation. Often these deformations have manual regulation. The
reason for this deformation is well explained in the article by Panchal et al. [
7
], “The mirror
has to be occasionally titled about a perpendicular axis to compensate for the seasonal
variation in the sun’s declination.”
. . .
“A framework that supports the reflector includes a
mechanism that can be used to tilt it and also bend it appropriately. The mirror is never
exactly paraboloid, but it is always close enough towards the receiver” [
7
]. Ruelas et al.
in [
1
] report that “It also features a daily tracking mechanism based on an adjustable bar
and a central pivot that changes the reflector inclination to produce a maximum aperture
variation of
±
8%”. Rapp et al. in [
4
] realize an alternative configuration in which the
receiver is not on the floor, but is in a higher position with respect to the mirror, which is
placed almost horizontally. Reddy et al. in [
5
] explain that “Two telescopic clamps with
locks are provided at the top and bottom of the reflector
. . .
”. In order to refocus solar
radiation back at F, the two telescopic clamps are unlocked and the parabola is manually
tilted towards the sun’s new position” [
5
]. Islam et al. in [
8
] declare that “The paraboloid
shape is obtained by rearranging the mirrors in order to reflect the incident solar rays to a
common focus of the collector”. Vyas et al. in [
9
] state that “The dish uses automatic daily
tracking and manual seasonal tracking mechanism to ensure maximum optical efficiency
of the Scheffler dish solar collector”.
In order to allow easy manufacturing, this concentrator is often composed of several
elementary mirrors. Concerning the mosaic mirror, Reddy et al. in [
5
] affirm that “Mirror
size plays an important role in proper functioning of reflector surface. Smaller the mirror
size better will be the concentration due to better fitting into the reflector profile curvature
. . .
”. A code in MATLAB is developed, which requires square mirror size as an input for
a Scheffler reflector of given focal length and aperture area. The output of the code is the
total number of mirrors required and their arrangement over the reflector profile” [5].
Mechanical deformation is needed to limit the variation of the spot size on the receiver;
indeed, the parabola, the shape of which has been theoretically calculated for a determined
day of the year (very often the equinox), has to be adapted for the other days. So, a large
increase of the spot size is avoided, as it would inevitably lead to a lower concentration
ratio, which instead should be high, especially in the applications where high temperature
values are an issue. In fact, attempts have been published to control the concentration of
the system [9].
The size of the image also affects the size of both the receiver and the eventual sec-
ondary optics. Malwad et al. in [
10
] analyze three receiver versions: “In these experiments,
a receiver with three different types of arrangement was tested under identical solar ra-
diation condition
. . .
”. In this experiment, under average beam radiation 640 W/m
2
receiver temperature of 241
◦
C, 266
◦
C, 253
◦
C was achieved with first, second and third
arrangement. The average thermal efficiency of the receiver for first, second and third
arrangements has been calculated 57.71%, 60.93% and 68.03%, receiver cover with tube coil
increases the overall efficiency of the system” [
10
]. In this study, the receiver with the third
arrangement was found efficient, while, in [
11
], Ruelas et al. found that the more suitable
geometry for the receiver has an elliptical form. Kumar et al. in [
12
] analyze six different
shapes of receiver, though each with equal aperture and depth. The study was conducted
to examine convective heat losses. The results show that “among the six shapes analyzed,
the cone-cylindrical cavity is the most efficient in terms of solar flux utilization whereas
hemispherical cavity results in maximum heat loss” [12].
The aim of our work is to evaluate, by means of raytracing techniques, the spot on the
receiver as a function of the shape of the Scheffler concentrator, deformed to optimize its
performance in sunlight collection.
This analysis of a Scheffler receiver, with purely optical methodologies, represents a
novelty. Design methods can be found in the literature that are based only on mechanical
studies, but none of these employ raytracing methods developed for the design of optical
Energies 2022,15, 260 3 of 20
systems. The joint use of CAD3D software and commercial raytracing software makes it
possible to predict the shape and irradiance map of the light spot on the receiver. This
optical raytracing analysis is easily applicable to evaluating the use of a Scheffler-type
concentrator for a specific application.
The initial study is dedicated to a system with a continuous mirror because it is simpler
to design and serves to help in understanding the feasibility and performance of the system
under examination.
The transition to a mosaic mirror is strongly influenced by the sizing of the input open-
ing of the receiver; in fact, the minimum size of the spot is determined by the dimensions
of the elementary mirror. However, in this case, the surface where the mosaic of mirrors
will be allocated is a continuous one.
The proposed analysis method allows predicting the shape of the light spot on the
receiver as a function of the deformations applied to the mirror.
The starting configuration is the paraboloid designed for the equinox, which produces,
on the receiver, a spot that is perfectly focused on that day only.
This article is divided into sections, which illustrate the development of the raytrac-
ing simulations that are combined to reproduce these mechanical deformations. These
studies employ two software packages: Zemax-OpticStudio, for raytracing analyses, and
SolidWorks, to reproduce the mechanical deformations of the actual paraboloid.
2. Optical Design of a Scheffler Concentrator
The Scheffler-type concentrator (Figures 1and 2) is the most cited in the literature for
applications with a receiver fixed to the ground. In theory there could be a paraboloid for
only each day of the year, with the axis directed at all times towards the sun and the focus
at the same point; the problem, here, is that this paraboloid should not only be rotated but
also translated to have the focus always in the same point, and the translation operation is
difficult, especially for objects of considerable size.
Energies 2022, 15, x FOR PEER REVIEW 3 of 21
systems. The joint use of CAD3D software and commercial raytracing software makes it
possible to predict the shape and irradiance map of the light spot on the receiver. This
optical raytracing analysis is easily applicable to evaluating the use of a Scheffler-type
concentrator for a specific application.
The initial study is dedicated to a system with a continuous mirror because it is sim-
pler to design and serves to help in understanding the feasibility and performance of the
system under examination.
The transition to a mosaic mirror is strongly influenced by the sizing of the input
opening of the receiver; in fact, the minimum size of the spot is determined by the dimen-
sions of the elementary mirror. However, in this case, the surface where the mosaic of
mirrors will be allocated is a continuous one.
The proposed analysis method allows predicting the shape of the light spot on the
receiver as a function of the deformations applied to the mirror.
The starting configuration is the paraboloid designed for the equinox, which pro-
duces, on the receiver, a spot that is perfectly focused on that day only.
This article is divided into sections, which illustrate the development of the raytrac-
ing simulations that are combined to reproduce these mechanical deformations. These
studies employ two software packages: Zemax-OpticStudio, for raytracing analyses, and
SolidWorks, to reproduce the mechanical deformations of the actual paraboloid.
2. Optical Design of a Scheffler Concentrator
The Scheffler-type concentrator (Figures 1 and 2) is the most cited in the literature for
applications with a receiver fixed to the ground. In theory there could be a paraboloid for
only each day of the year, with the axis directed at all times towards the sun and the focus
at the same point; the problem, here, is that this paraboloid should not only be rotated but
also translated to have the focus always in the same point, and the translation operation
is difficult, especially for objects of considerable size.
To avoid the translation of the paraboloid, one might consider using paraboloids of
different shapes that have a point on the surface and the focus in common. Some examples
are shown in Figure 2.
Since this option is not feasible, only one paraboloid is used, optimized for the equi-
nox but capable of deforming to approximate the paraboloids of different shapes that
should be used. The paraboloid, therefore, does not translate but merely rotates.
The construction principle of a Scheffler-type concentrator is illustrated in Figures 1
and 2. The different parabolas, which alternate throughout the year, lie around the point
P and the seasonal axis.
In all raytracing simulations the sun is considered to be at noon.
Figure 1. Parabolas calculated for the equinox (black) and for the summer solstice (orange). The
light beam has the direction it would have locally on 21 June. The figure shows the seasonal and
daily axes that pass through the point P common to the two parabolas.
Figure 1.
Parabolas calculated for the equinox (black) and for the summer solstice (orange). The light
beam has the direction it would have locally on 21 June. The figure shows the seasonal and daily
axes that pass through the point P common to the two parabolas.
To avoid the translation of the paraboloid, one might consider using paraboloids of
different shapes that have a point on the surface and the focus in common. Some examples
are shown in Figure 2.
Since this option is not feasible, only one paraboloid is used, optimized for the equinox
but capable of deforming to approximate the paraboloids of different shapes that should be
used. The paraboloid, therefore, does not translate but merely rotates.
Energies 2022,15, 260 4 of 20
Energies 2022, 15, x FOR PEER REVIEW 4 of 21
Figure 2. Side view of the beam concentrated by the parabolas calculated for: Equinox (a), winter
solstice (b), summer solstice (c), all passing through the same point P. Note that the distance PF
remains the same.
The drawings in Figure 2 show three parabolas whose shapes are calculated, respec-
tively, for the equinox (Figure 2a), the winter solstice (Figure 2b) and the summer solstice
(Figure 2c). The parabolas have a common point P; the rotation happens both around the
axis that connects the focus with the point P (daily rotation) and around the axis perpen-
dicular to the previous one and parallel to the ground (seasonal rotation). In reality, the
reflector is, physically, always the same, but its shape has to be continuously changed
throughout the year by deformation.
Calculation of the Reflector Profile for a Certain Day of the Year
Figure 3 clarifies how to calculate the reflector shape. If a section of the paraboloid
(on a plane that contains the axis) is considered, the equinoctial parabola passing through
the point P(x
P
; y
P
) and which has focus on F(x
F
; y
F
) is represented in red color in the figure.
Figure 2.
Side view of the beam concentrated by the parabolas calculated for: Equinox (
a
), winter
solstice (
b
), summer solstice (
c
), all passing through the same point P. Note that the distance PF
remains the same.
The construction principle of a Scheffler-type concentrator is illustrated in
Figures 1and 2
.
The different parabolas, which alternate throughout the year, lie around the point P and the
seasonal axis.
In all raytracing simulations the sun is considered to be at noon.
The drawings in Figure 2show three parabolas whose shapes are calculated, respec-
tively, for the equinox (Figure 2a), the winter solstice (Figure 2b) and the summer solstice
(Figure 2c). The parabolas have a common point P; the rotation happens both around
the axis that connects the focus with the point P (daily rotation) and around the axis per-
pendicular to the previous one and parallel to the ground (seasonal rotation). In reality,
the reflector is, physically, always the same, but its shape has to be continuously changed
throughout the year by deformation.
Calculation of the Reflector Profile for a Certain Day of the Year
Figure 3clarifies how to calculate the reflector shape. If a section of the paraboloid (on
a plane that contains the axis) is considered, the equinoctial parabola passing through the
point P(xP; yP) and which has focus on F(xF; yF) is represented in red color in the figure.
The generic equation of a parabola with vertical axis has the following expression:
y=ax2+ c (1)
Energies 2022,15, 260 5 of 20
Energies 2022, 15, x FOR PEER REVIEW 5 of 21
Figure 3. Scheme of the construction of the reflector (in red). The equinoctial parabola crossing P
focuses on F.
The generic equation of a parabola with vertical axis has the following expression:
y = a x
2
+ c (1)
The focus of this parabola has coordinates:
y
F
= (4 a c + 1)/4 a (2)
x
F
= 0 (3)
Therefore
4 a y
F
= 4 a c + 1 (4)
and by imposing the passage for P:
y
P
= a x
P2
+ c (5)
c = y
P
− a x
P2
(6)
Substituting Equation (6) into Equation (4)
4 a
2
x
P2
+ 4 a (y
F
− y
P
) − 1 = 0 (7)
The values of a are obtained by solving Equation (6). If a point at the same height of
F is chosen, with con y
F
= y
P
, the equation is further simplified. Once “a” is found, “c” is
computed from Equation (6).
All the other parabolas relative to the other days of the year must have a point lying
on the circle of radius d, which is the distance between F and P, as this is the location of
the parabolas that have the same focal distance from F. Starting from the value of d, the
parameters of the parabola of equation
y = a’ x
2
+ c’ (8)
are calculated for a certain day of the year by imposing that it passes through the point
P(x
P
; y
P
) and that it has its focus on the focus of the equinoctial parabola.
In fact, knowing the declination α for that day, the coordinates of the point P’(x’
P
; y’
P
)
are:
x’
P
= x
F
+ d cos(α) (9)
y’
P
= y
F
+ d sen(α) (10)
Figure 3.
Scheme of the construction of the reflector (in red). The equinoctial parabola crossing P
focuses on F.
The focus of this parabola has coordinates:
yF= (4 a c + 1)/4 a (2)
xF= 0 (3)
Therefore
4ayF=4ac+1 (4)
and by imposing the passage for P:
yP=axP2+ c (5)
c=yP−a xP2(6)
Substituting Equation (6) into Equation (4)
4 a2xP2+4a(yF−yP)−1 = 0 (7)
The values of a are obtained by solving Equation (6). If a point at the same height of F
is chosen, with con y
F
= y
P
, the equation is further simplified. Once “a” is found, “c” is
computed from Equation (6).
All the other parabolas relative to the other days of the year must have a point lying
on the circle of radius d, which is the distance between F and P, as this is the location of
the parabolas that have the same focal distance from F. Starting from the value of d, the
parameters of the parabola of equation
y=a0x2+ c0(8)
are calculated for a certain day of the year by imposing that it passes through the point
P(xP; yP) and that it has its focus on the focus of the equinoctial parabola.
In fact, knowing the declination
α
for that day, the coordinates of the point
P0(x0P; y0P) are:
x0
P= xF+ d cos(α) (9)
y0
P= yF+ d sen(α) (10)
the parameters “a
0
” and “c
0
” are calculated by solving the equation of a parabola that passes
through the point P(xP; yP) and with focus on F(0; yF).
Tables 1and 2show the values of “a, “c” and the radius of curvature Rc for two values
of the distance “d”: 2.7 m and 5 m. The value d = 2.7 m was obtained from an article by
Energies 2022,15, 260 6 of 20
A. Munir et al. [
2
]. The value d = 5 m comes from considerations originating from a request
to be able to concentrate a power of around 10,000 watts on the receiver.
Table 1. Parameters defining the parabola and radius of curvature for d = 2.7 m.
Summer Solstice
(21 June)
Equinox
(21 March; 22 September)
Winter Solstice
(21 December)
a 0.1324 0.1852 0.3080
c−1.8833 −1.3500 −0.8117
Rc (m) −3.7766 −2.7 −1.6234
Table 2. Parameters defining the parabola and radius of curvature for d = 5 m.
Summer Solstice
(21 June)
Equinox
(21 March; 22 September)
Winter Solstice
(21 December)
a 0.0715 0.100 0.1663
c−3.4969 −2.5 −1.5031
Rc (m) −6.9937 −5−3.0033
The parabolas defined by the parameters of Tables 1and 2have to be rotated by
an angle
β
around the common focus F(x
F
; y
F
). The rotated parabolas are now able to
concentrate the light rays coming from the
±β
directions into focus F because the optical
axis of the parabola has also been rotated by the same angle. Figure 4presents the parabola
profiles before and after rotation for the equinox, summer, and winter solstices.
The rotation formulas valid for each point of the parabola are:
x0= (x −xF) cos(β) + (y −yF) sen(β)+xF(11)
y0=−(x −xF) sen(β) + (y −yF) cos(β)+yF(12)
Energies 2022, 15, x FOR PEER REVIEW 6 of 21
the parameters “a’” and “c’” are calculated by solving the equation of a parabola that
passes through the point P(xP; yP) and with focus on F(0; yF).
Tables 1 and 2 show the values of “a, “c” and the radius of curvature Rc for two
values of the distance “d”: 2.7 m and 5 m. The value d = 2.7 m was obtained from an article
by A. Munir et al. [2]. The value d = 5 m comes from considerations originating from a
request to be able to concentrate a power of around 10,000 watts on the receiver.
Table 1. Parameters defining the parabola and radius of curvature for d = 2.7 m.
Summer Solstice
(21 June)
Equinox
(21 March; 22 September)
Winter Solstice
(21 December)
a 0.1324 0.1852 0.3080
c −1.8833 −1.3500 −0.8117
Rc (m) −3.7766 −2.7 −1.6234
Table 2. Parameters defining the parabola and radius of curvature for d = 5 m.
Summer Solstice
(21 June)
Equinox
(21 March; 22 September)
Winter Solstice
(21 December)
a 0.0715 0.100 0.1663
c −3.4969 −2.5 −1.5031
Rc (m) −6.9937 −5 −3.0033
The parabolas defined by the parameters of Tables 1 and 2 have to be rotated by an
angle β around the common focus F(xF; yF). The rotated parabolas are now able to con-
centrate the light rays coming from the +/− β directions into focus F because the optical
axis of the parabola has also been rotated by the same angle. Figure 4 presents the parab-
ola profiles before and after rotation for the equinox, summer, and winter solstices.
The rotation formulas valid for each point of the parabola are:
x’ = (x − xF) cos(β) + (y − yF) sen(β) + xF (11)
y’ = − (x − xF) sen(β) + (y − yF) cos(β) + yF (12)
(a) (b)
Figure 4. Profiles of the parabolas calculated for the equinox and the two solstices, (a) before rota-
tion and (b) after turning an angle β of +/− 23.5 degrees.
In the considered layout, the point P, which is common to all the parabolas, has the
same y coordinate as the focus.
Figure 4.
Profiles of the parabolas calculated for the equinox and the two solstices, (
a
) before rotation
and (b) after turning an angle βof ±23.5 degrees.
In the considered layout, the point P, which is common to all the parabolas, has the
same y coordinate as the focus.
If the equinoctial parabola were utilized on the other days of the year, some image
degradation would appear as an enlargement of the light spot on the receiver, as can be
seen from the comparison between Figures 5and 6.
Energies 2022,15, 260 7 of 20
Energies 2022, 15, x FOR PEER REVIEW 7 of 21
If the equinoctial parabola were utilized on the other days of the year, some image
degradation would appear as an enlargement of the light spot on the receiver, as can be
seen from the comparison between Figures 5 and 6.
Figure 5 refers to the equinoctial parabola with the sun at the equinox, while Figure
6 shows the widening of the light spot on the receiver when the equinoctial parabola is
used on the summer solstice. In this case, the simulation was obtained by rotating the
equinoctial parabola at the point indicated by the cross until the receiver was centered
again.
The sun is always considered to be at noon in the optical simulations.
The following optical simulation with the raytracing program (Zemax Optic Studio)
requires knowledge of the radii of curvature Rc indicated in Table 1, as well as the distance
d between the focus F and the center of rotation P.
Figure 5. Irradiance map on the receiver with the equinoctial parabola, calculated with the posi-
tion of the sun at the equinox (blue rays). The green rays represent the position of the sun at the
solstice. The irradiance values on the map are represented with an arbitrary scale.
Figure 6. Irradiance map on the receiver with the equinoctial parabola, calculated with the posi-
tion of the sun at the solstice. The parabola has been rotated around the point indicated by the
drawn cross until the receiver has been centered again. The position of the receiver is the same as
that shown in Figure 5. The irradiance values on the map are represented with an arbitrary scale.
In any case, it is mandatory to use a single reflector, because, due to its size and cost,
it cannot be replaced many times during the year. Consequently, the profile of the refer-
ence parabola (the equinoctial one) must be continuously modified during the year [2,3].
The drawback to this is that the final mirror profile is not easily controllable, because
the usual technique to adapt its shape is the deformation of the paraboloid by mechanical
methods.
Figure 5.
Irradiance map on the receiver with the equinoctial parabola, calculated with the position
of the sun at the equinox (blue rays). The green rays represent the position of the sun at the solstice.
The irradiance values on the map are represented with an arbitrary scale.
Energies 2022, 15, x FOR PEER REVIEW 7 of 21
If the equinoctial parabola were utilized on the other days of the year, some image
degradation would appear as an enlargement of the light spot on the receiver, as can be
seen from the comparison between Figures 5 and 6.
Figure 5 refers to the equinoctial parabola with the sun at the equinox, while Figure
6 shows the widening of the light spot on the receiver when the equinoctial parabola is
used on the summer solstice. In this case, the simulation was obtained by rotating the
equinoctial parabola at the point indicated by the cross until the receiver was centered
again.
The sun is always considered to be at noon in the optical simulations.
The following optical simulation with the raytracing program (Zemax Optic Studio)
requires knowledge of the radii of curvature Rc indicated in Table 1, as well as the distance
d between the focus F and the center of rotation P.
Figure 5. Irradiance map on the receiver with the equinoctial parabola, calculated with the posi-
tion of the sun at the equinox (blue rays). The green rays represent the position of the sun at the
solstice. The irradiance values on the map are represented with an arbitrary scale.
Figure 6. Irradiance map on the receiver with the equinoctial parabola, calculated with the posi-
tion of the sun at the solstice. The parabola has been rotated around the point indicated by the
drawn cross until the receiver has been centered again. The position of the receiver is the same as
that shown in Figure 5. The irradiance values on the map are represented with an arbitrary scale.
In any case, it is mandatory to use a single reflector, because, due to its size and cost,
it cannot be replaced many times during the year. Consequently, the profile of the refer-
ence parabola (the equinoctial one) must be continuously modified during the year [2,3].
The drawback to this is that the final mirror profile is not easily controllable, because
the usual technique to adapt its shape is the deformation of the paraboloid by mechanical
methods.
Figure 6.
Irradiance map on the receiver with the equinoctial parabola, calculated with the position
of the sun at the solstice. The parabola has been rotated around the point indicated by the drawn
cross until the receiver has been centered again. The position of the receiver is the same as that shown
in Figure 5. The irradiance values on the map are represented with an arbitrary scale.
Figure 5refers to the equinoctial parabola with the sun at the equinox, while Figure 6
shows the widening of the light spot on the receiver when the equinoctial parabola is used
on the summer solstice. In this case, the simulation was obtained by rotating the equinoctial
parabola at the point indicated by the cross until the receiver was centered again.
The sun is always considered to be at noon in the optical simulations.
The following optical simulation with the raytracing program (Zemax Optic Studio)
requires knowledge of the radii of curvature Rc indicated in Table 1, as well as the distance
d between the focus F and the center of rotation P.
In any case, it is mandatory to use a single reflector, because, due to its size and cost, it
cannot be replaced many times during the year. Consequently, the profile of the reference
parabola (the equinoctial one) must be continuously modified during the year [2,3].
The drawback to this is that the final mirror profile is not easily controllable, because
the usual technique to adapt its shape is the deformation of the paraboloid by mechani-
cal methods.
As previously discussed, after choosing the value of d, which represents the distance
of the point P from the focus F on the receiver, the parameters “a”, “c” and the radius of
curvature Rc of the paraboloid are calculated for certain days of the year. These Rc values
will be used to simulate the optical system and the irradiance map on the receiver with
a raytracing program (Zemax-OpticStudio). In Munir et al. [
2
,
3
], instead, it is defined a
priori that the paraboloid has an elliptical shape with a known area. Three support points
(pivot points) are then defined on this paraboloid, and they are used to rotate the structure
and to deform it with the help of two linear actuators. On the basis of this information, the
Energies 2022,15, 260 8 of 20
coefficients “a” and “c” are calculated; they are necessary to define the curvature of the
paraboloid and to simulate the behavior of the light rays with Zemax-OpticStudio.
3. Mechanical and Optical Simulation
In order to study the optical behavior of a simulated Scheffler concentrator in the
different operating conditions it was decided to deform the equinoctial paraboloid with the
Solidworks 3D CAD software.
For this purpose, the “Flex-bending” feature was used to reproduce the deformations
obtained by the lengthening or shortening of the two pistons. With this feature it is possible
to flex a body by defining a bend axis; one that remains fixed and around which the body
is bent, for which are also defined two trim planes that control what region of the body
is flexed. Therefore, the deformation is achieved by setting a desired angle (or radius of
curvature) between the two trim planes.
Table 3shows the values of the coefficients of the three paraboloids as described in the
Munir article [2]. The manufacturing data for the three paraboloids are:
collector area 8 m2
collector diameter 3.2 m
The values a and c in Table 3refer to Equation (1).
Table 3. Values of the coefficients of the three paraboloids in the Munir article [2].
Summer Solstice
(21 June)
Equinox
(21 March; 22 September)
Winter Solstice
(21 December)
a 0.28123 0.17447 0.12736
c 0.54394 0 0.53004
It is useful to specify, here, that the term parabola means the two-dimensional curve
based on the parameters of Table 3and Equation (8). Paraboloid, on the other hand,
indicates the rotation surface obtained from the parabolic profile and intersected with a
cylinder of suitable size. Therefore, the paraboloid is a three-dimensional object.
The actual manufactured mirror has the profile of the equinoctial equation (the equinoc-
tial paraboloid indicated as “Par_0”). The mirror is mounted on a structure shown in
Figure 7, which corresponds to Figure 9 at page 1499 of the article A. Munir et al./Solar
Energy 84 (2010) 1490–1502 [2].
In this structure there are three fixed points, indicated as Pivots A (two points) and B,
around which all the deformations take place. Figure 7presents the Scheffler concentrator
of the article by A. Munir [2].
The procedure can be outlined with the following steps:
Step 0—Original mirror: equinoctial paraboloid, indicated as “Par_0”.
Step 1—Rotation around Pivot B of the equinoctial paraboloid so that the local tangent
of the rotation point is the same as the parabola of the analyzed day. The rotated paraboloid
is indicated as “Par_1”.
Step 2—The action of the pistons in points C and D of the figure is reproduced by
flexing the paraboloid around Pivot B. Since the real system has three fixed points, while
the Solidworks function allows flexing only around one point, it was necessary to apply
two deformations for every paraboloid to correctly simulate the effective action of the
pistons on the mirror.
All the following figures presented in this section are based on the outputs of the CAD
software SolidWorks.
In the Solidworks simulations, shown below, the bend axis matches to point B in such
a way as to keep it stationary in two different directions: for the first deformation, the bend
axis is perpendicular to the plane X–Y (passing through B) and the trim planes correspond
to points C and D.
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Energies 2022, 15, x FOR PEER REVIEW 9 of 21
Figure 7. Scheffler concentrator shown in Figure 9 of the article A. Munir et al./Solar Energy 2010,
84, 1490–1502 [2].
The procedure can be outlined with the following steps:
Step 0—Original mirror: equinoctial paraboloid, indicated as “Par_0”.
Step 1—Rotation around Pivot B of the equinoctial paraboloid so that the local tan-
gent of the rotation point is the same as the parabola of the analyzed day. The rotated
paraboloid is indicated as “Par_1”.
Step 2—The action of the pistons in points C and D of the figure is reproduced by
flexing the paraboloid around Pivot B. Since the real system has three fixed points, while
the Solidworks function allows flexing only around one point, it was necessary to apply
two deformations for every paraboloid to correctly simulate the effective action of the
pistons on the mirror.
All the following figures presented in this section are based on the outputs of the
CAD software SolidWorks.
In the Solidworks simulations, shown below, the bend axis matches to point B in such
a way as to keep it stationary in two different directions: for the first deformation, the
bend axis is perpendicular to the plane X–Y (passing through B) and the trim planes cor-
respond to points C and D.
To complete the action of the pistons the equinoctial paraboloid was considered and
the distance between the paraboloid edge and Pivots A was measured (23.5 cm), as was
the distance established by the bars, visible in Figure 7. This distance is used for the second
deformation, in order to match the edges of the bars with the border of the shifted parab-
oloid, due to the first deformation. In this second deformation, the bend axis is coincident
with the tangent line of the parabola passing through B and results perpendicular to the
first bend axis, while two trim planes correspond to points S and S’.
Figure 7.
Scheffler concentrator shown in Figure 9 of the article A. Munir et al./Solar Energy 2010, 84,
1490–1502 [2].
To complete the action of the pistons the equinoctial paraboloid was considered
and the distance between the paraboloid edge and Pivots A was measured (23.5 cm), as
was the distance established by the bars, visible in Figure 7. This distance is used for
the second deformation, in order to match the edges of the bars with the border of the
shifted paraboloid, due to the first deformation. In this second deformation, the bend
axis is coincident with the tangent line of the parabola passing through B and results
perpendicular to the first bend axis, while two trim planes correspond to points S and S0.
The paraboloid after the second bending is indicated as “Par_2”.
In the raytracing simulations the sun is always considered to be at noon.
The changes made on the equinoctial paraboloid (Par_0) to adapt it for the summer
solstice are shown as an example.
The profiles (on the X–Y plane) of the ideal summer solstice parabola (in red, in
Figure 8) and the rotated equinoctial parabola (in yellow, in Figure 8) were traced, both
calculated from the values of Table 3.
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Energies 2022, 15, x FOR PEER REVIEW 10 of 21
The paraboloid after the second bending is indicated as “Par_2”.
In the raytracing simulations the sun is always considered to be at noon.
The changes made on the equinoctial paraboloid (Par_0) to adapt it for the summer
solstice are shown as an example.
The profiles (on the X–Y plane) of the ideal summer solstice parabola (in red, in Fig-
ure 8) and the rotated equinoctial parabola (in yellow, in Figure 8) were traced, both cal-
culated from the values of Table 3.
The angle of rotation of the equinoctial parabola, in order to follow, as accurately as
possible, the profile of the ideal summer solstice parabola, is 11.75 deg around Pivot B.
The two parabolas are shown in Figure 8, while the two paraboloids are shown in Figure
9.
Figure 8. Comparison between the correct summer solstice parabola (in red) and the rotated equi-
noctial parabola (in yellow).
To compare the two paraboloids, a reference solstice paraboloid with an area equal
to that of the equinoctial paraboloid was used, maintaining the same longitudinal diame-
ter.
Figure 9. Comparison between the paraboloid of the ideal summer solstice (in red) (Par_0s) and
the rotated equinoctial paraboloid (in yellow) (Par_1s).
To simulate the actual deformation, we act directly on the paraboloid to modify its
entire surface and not just on the profile of the generating parabola, starting with the first
deformation. In Figure 10 are evidenced points C and D of the trim planes.
Figure 8.
Comparison between the correct summer solstice parabola (in red) and the rotated equinoc-
tial parabola (in yellow).
The angle of rotation of the equinoctial parabola, in order to follow, as accurately as
possible, the profile of the ideal summer solstice parabola, is 11.75 deg around Pivot B. The
two parabolas are shown in Figure 8, while the two paraboloids are shown in Figure 9.
Energies 2022, 15, x FOR PEER REVIEW 10 of 21
The paraboloid after the second bending is indicated as “Par_2”.
In the raytracing simulations the sun is always considered to be at noon.
The changes made on the equinoctial paraboloid (Par_0) to adapt it for the summer
solstice are shown as an example.
The profiles (on the X–Y plane) of the ideal summer solstice parabola (in red, in Fig-
ure 8) and the rotated equinoctial parabola (in yellow, in Figure 8) were traced, both cal-
culated from the values of Table 3.
The angle of rotation of the equinoctial parabola, in order to follow, as accurately as
possible, the profile of the ideal summer solstice parabola, is 11.75 deg around Pivot B.
The two parabolas are shown in Figure 8, while the two paraboloids are shown in Figure
9.
Figure 8. Comparison between the correct summer solstice parabola (in red) and the rotated equi-
noctial parabola (in yellow).
To compare the two paraboloids, a reference solstice paraboloid with an area equal
to that of the equinoctial paraboloid was used, maintaining the same longitudinal diame-
ter.
Figure 9. Comparison between the paraboloid of the ideal summer solstice (in red) (Par_0s) and
the rotated equinoctial paraboloid (in yellow) (Par_1s).
To simulate the actual deformation, we act directly on the paraboloid to modify its
entire surface and not just on the profile of the generating parabola, starting with the first
deformation. In Figure 10 are evidenced points C and D of the trim planes.
Figure 9.
Comparison between the paraboloid of the ideal summer solstice (in red) (Par_0s) and the
rotated equinoctial paraboloid (in yellow) (Par_1s).
To compare the two paraboloids, a reference solstice paraboloid with an area equal to
that of the equinoctial paraboloid was used, maintaining the same longitudinal diameter.
To simulate the actual deformation, we act directly on the paraboloid to modify its
entire surface and not just on the profile of the generating parabola, starting with the first
deformation. In Figure 10 are evidenced points C and D of the trim planes.
In this way it was possible to improve the overlap of the rotated equinoctial paraboloid
profile with that of the paraboloid calculated for the summer solstice (ideal), the bending
obtained is 5 degrees wide with a radius of curvature of 36.52 m. The results are shown
in Figure 11. The yellow paraboloid, now, in some areas is more adherent to the reference
paraboloid for the solstice (in red). However, the sides of the paraboloid are far from ideal.
Energies 2022,15, 260 11 of 20
Energies 2022, 15, x FOR PEER REVIEW 11 of 21
Figure 10. Diagram of the first bending. Point B is stationary and the action has been taken on
points C and D.
In this way it was possible to improve the overlap of the rotated equinoctial parabo-
loid profile with that of the paraboloid calculated for the summer solstice (ideal), the bend-
ing obtained is 5 degrees wide with a radius of curvature of 36.52 m. The results are shown
in Figure 11. The yellow paraboloid, now, in some areas is more adherent to the reference
paraboloid for the solstice (in red). However, the sides of the paraboloid are far from ideal.
Figure 11. Comparison between the paraboloid of the ideal summer solstice (Par_0s, in red) and
the equinoctial paraboloid rotated and flexed by holding B fixed and acting on its points C and D
(Par_2s, in yellow).
As explained before, for a correct simulation of the problem the two Pivots A must
also remain in position. Figure 12 shows points S, S’, A and B. Since the two points are
anchored to the edge of the paraboloid with two bars, the measured distance between
Pivot A and edge of the paraboloid must remain constant. In particular, the constraints
considered for the second deformation are the position on the X–Y plane of Pivots A and
the length of the bars. The result of this movement is shown in Figure 13.
Figure 10.
Diagram of the first bending. Point B is stationary and the action has been taken on points
C and D.
Energies 2022, 15, x FOR PEER REVIEW 11 of 21
Figure 10. Diagram of the first bending. Point B is stationary and the action has been taken on
points C and D.
In this way it was possible to improve the overlap of the rotated equinoctial parabo-
loid profile with that of the paraboloid calculated for the summer solstice (ideal), the bend-
ing obtained is 5 degrees wide with a radius of curvature of 36.52 m. The results are shown
in Figure 11. The yellow paraboloid, now, in some areas is more adherent to the reference
paraboloid for the solstice (in red). However, the sides of the paraboloid are far from ideal.
Figure 11. Comparison between the paraboloid of the ideal summer solstice (Par_0s, in red) and
the equinoctial paraboloid rotated and flexed by holding B fixed and acting on its points C and D
(Par_2s, in yellow).
As explained before, for a correct simulation of the problem the two Pivots A must
also remain in position. Figure 12 shows points S, S’, A and B. Since the two points are
anchored to the edge of the paraboloid with two bars, the measured distance between
Pivot A and edge of the paraboloid must remain constant. In particular, the constraints
considered for the second deformation are the position on the X–Y plane of Pivots A and
the length of the bars. The result of this movement is shown in Figure 13.
Figure 11.
Comparison between the paraboloid of the ideal summer solstice (Par_0s, in red) and the
equinoctial paraboloid rotated and flexed by holding B fixed and acting on its points C and D (Par_2s,
in yellow).
As explained before, for a correct simulation of the problem the two Pivots A must
also remain in position. Figure 12 shows points S, S
0
, A and B. Since the two points are
anchored to the edge of the paraboloid with two bars, the measured distance between
Pivot A and edge of the paraboloid must remain constant. In particular, the constraints
considered for the second deformation are the position on the X–Y plane of Pivots A and
the length of the bars. The result of this movement is shown in Figure 13.
Now, the paraboloid has assumed the shape most similar to the real one; it is useful
to note that the two paraboloids are now better overlapped, and the edges are very close
together (Figure 14). The bending obtained with the second deformation is equal to
−9 degrees with a radius of curvature of −18.91 m.
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Energies 2022, 15, x FOR PEER REVIEW 12 of 21
Figure 12. Parabolas calculated for the equinox (black) and for the summer solstice (orange). The
light beam has the direction it would have locally on 21 June. The figure shows the seasonal and
daily axes that pass through the point P common to the two parabolas.
(a) (b)
Figure 13. Lateral view of the paraboloid, before the second bending (a) and after (b). Note that
the edge of the paraboloid is in the correct position, at the end of the bar (b).
Now, the paraboloid has assumed the shape most similar to the real one; it is useful
to note that the two paraboloids are now better overlapped, and the edges are very close
together (Figure 14). The bending obtained with the second deformation is equal to −9
degrees with a radius of curvature of −18.91 m.
Figure 12.
Parabolas calculated for the equinox (black) and for the summer solstice (orange). The
light beam has the direction it would have locally on 21 June. The figure shows the seasonal and
daily axes that pass through the point P common to the two parabolas.
Energies 2022, 15, x FOR PEER REVIEW 12 of 21
Figure 12. Parabolas calculated for the equinox (black) and for the summer solstice (orange). The
light beam has the direction it would have locally on 21 June. The figure shows the seasonal and
daily axes that pass through the point P common to the two parabolas.
(a) (b)
Figure 13. Lateral view of the paraboloid, before the second bending (a) and after (b). Note that
the edge of the paraboloid is in the correct position, at the end of the bar (b).
Now, the paraboloid has assumed the shape most similar to the real one; it is useful
to note that the two paraboloids are now better overlapped, and the edges are very close
together (Figure 14). The bending obtained with the second deformation is equal to −9
degrees with a radius of curvature of −18.91 m.
Figure 13. Lateral view of the paraboloid, before the second bending (a) and after (b). Note that the
edge of the paraboloid is in the correct position, at the end of the bar (b).
Energies 2022, 15, x FOR PEER REVIEW 13 of 21
Figure 14. Comparison between the paraboloid of the ideal summer solstice (Par_0s, in red) and
the equinoctial paraboloid rotated and flexed twice (Par_2s, in yellow).
Tables 4–6 summarize the manufacturing data for each paraboloid. The tables also
report the coordinates of the extreme points C and D and the areas of the paraboloids
designed from the parameters of Table 3. The sign of the rotation angle is positive for
counterclockwise rotations and negative for clockwise rotations. The positions of points
C and D of Par_2s and Par_2w are particularly useful for obtaining the deformation that
reaches the optimal configuration, from an optical point of view, as explained in Section
4.2.
Table 4. Data of the paraboloids used for the simulations for the equinox.
Name Description
Rotation
[°]
Bending
C-D [°]
Bending
S-S’ [°]
Position of
Point C (x;y)
[m]
Position of
Point D (x;y)
[m]
Area [m
2
]
Par_0
Paraboloid based on the parabola
for the equinox, whose parame-
ters are in Table 3.
original - - 4.06; 2.88 1.32; 0.31 8.28
Table 5. Data of the paraboloids used for the simulations for the summer solstice.
Name Description
Rotation
[°]
Bending
C-D [°]
Bending
S-S’ [°]
Position of
Point C (x;y)
[m]
Position of
Point D (x;y)
[m]
Area [m
2
]
Par_0s
Paraboloid based on the parabola
for summer solstice, whose param-
eters are in Table 3.
original - - 4.37; 2.49 1.18; 0.57 8.27
Par_1s Par_0 rotated around B −11.75 - - 4.36; 2.56 1.16; 0.60
Par_2s Par_1s bent twice −11.75 5.0 −9.0 4.40; 2.52 1.17; 0.57
Figure 14.
Comparison between the paraboloid of the ideal summer solstice (Par_0s, in red) and the
equinoctial paraboloid rotated and flexed twice (Par_2s, in yellow).
Energies 2022,15, 260 13 of 20
Tables 4–6summarize the manufacturing data for each paraboloid. The tables also
report the coordinates of the extreme points C and D and the areas of the paraboloids
designed from the parameters of Table 3. The sign of the rotation angle is positive for
counterclockwise rotations and negative for clockwise rotations. The positions of points
C and D of Par_2s and Par_2w are particularly useful for obtaining the deformation that
reaches the optimal configuration, from an optical point of view, as explained in Section 4.2.
Table 4. Data of the paraboloids used for the simulations for the equinox.
Name Description Rotation [◦]Bending
C-D [◦]
Bending
S-S0[◦]
Position of
Point C (x;y)
[m]
Position of
Point D (x;y)
[m]
Area [m2]
Par_0
Paraboloid based
on the parabola for
the equinox,
whose parameters
are in Table 3.
original - - 4.06; 2.88 1.32; 0.31 8.28
Table 5. Data of the paraboloids used for the simulations for the summer solstice.
Name Description Rotation [◦]Bending
C-D [◦]
Bending
S-S0[◦]
Position of
Point C (x;y)
[m]
Position of
Point D (x;y)
[m]
Area [m2]
Par_0s
Paraboloid based
on the parabola for
summer solstice,
whose parameters
are in Table 3.
original - - 4.37; 2.49 1.18; 0.57 8.27
Par_1s Par_0 rotated
around B −11.75 - - 4.36; 2.56 1.16; 0.60
Par_2s Par_1s bent twice −11.75 5.0 −9.0 4.40; 2.52 1.17; 0.57
Table 6. Data of the paraboloids used for the simulations for the winter solstice.
Name Description Rotation [◦]Bending
C-D [◦]
Bending
S-S0[◦]
Position of
Point C (x;y)
[m]
Position of
Point D (x;y)
[m]
Area [m2]
Par_0w
Paraboloid based
on the parabola for
the winter solstice,
whose parameters
are in Table 3.
original - - 3.67; 3.19 1.53; 0.06 8.36
Par_1w Par_0 rotated
around B 11.75 - - 3.70; 3.13 1.55; 0.05
Par_2w Par_1w bent twice 11.75 5.0 −8.3 3.65; 3.14 1.53; 0.07
Precise control and quantification of the mechanical deformations of the Scheffler
reflector are essential for the correct operation of the Scheffler reflector to adapt it to
every season. However, the innovative part consists in the raytracing study, which allows
understanding the effects of these deformations on the image on the receiver.
4. Irradiance Maps on the Receiver
In order to evaluate the optical performance of the Scheffler concentrator, various
simulations were performed using the Zemax-OpticStudio software. In particular, these
Energies 2022,15, 260 14 of 20
raytracing simulations provide the irradiance maps of the solar radiation focused on the
receiver. In the following figures the incoherent irradiance is reported in Watt/cm2.
The first series of simulations check the spot produced by the paraboloid after only
step 1, i.e., the rotation around Pivot B of the equinoctial paraboloid (Par_0).
The second series of simulations considers the paraboloid produced by the bending; it
can be considered as the real deformed paraboloid.
The figures below show the irradiance figures on the receiver calculated for noon, so
they all have symmetry along the axes.
Considering the actual movement of the paraboloid during the day, it rotates around
the center (0,0) with a clockwise angle equal to −15 degrees for each hour.
In order to evaluate the functioning of the system, a large target (1 m
×
1 m), was
employed, on which the irradiation maps and the average size of the spot were calculated.
The target was placed so that the map of paraboloid 1 at the equinox was centered as seen
in Figure 15.
Energies 2022, 15, x FOR PEER REVIEW 14 of 21
Table 6. Data of the paraboloids used for the simulations for the winter solstice.
Name Description
Rotation
[°]
Bending
C-D [°]
Bending
S-S’ [°]
Position of
Point C (x;y)
[m]
Position of
Point D (x;y)
[m]
Area [m
2
]
Par_0w
Paraboloid based on the parabola
for the winter solstice, whose pa-
rameters are in Table 3.
original - - 3.67; 3.19 1.53; 0.06 8.36
Par_1w Par_0 rotated around B 11.75 - - 3.70; 3.13 1.55; 0.05
Par_2w Par_1w bent twice 11.75 5.0 −8.3 3.65; 3.14 1.53; 0.07
Precise control and quantification of the mechanical deformations of the Scheffler re-
flector are essential for the correct operation of the Scheffler reflector to adapt it to every
season. However, the innovative part consists in the raytracing study, which allows un-
derstanding the effects of these deformations on the image on the receiver.
4. Irradiance Maps on the Receiver
In order to evaluate the optical performance of the Scheffler concentrator, various
simulations were performed using the Zemax-OpticStudio software. In particular, these
raytracing simulations provide the irradiance maps of the solar radiation focused on the
receiver. In the following figures the incoherent irradiance is reported in Watt/cm
2
.
The first series of simulations check the spot produced by the paraboloid after only
step 1, i.e., the rotation around Pivot B of the equinoctial paraboloid (Par_0).
The second series of simulations considers the paraboloid produced by the bending;
it can be considered as the real deformed paraboloid.
The figures below show the irradiance figures on the receiver calculated for noon, so
they all have symmetry along the axes.
Considering the actual movement of the paraboloid during the day, it rotates around
the center (0,0) with a clockwise angle equal to −15 degrees for each hour.
In order to evaluate the functioning of the system, a large target (1 m × 1 m), was
employed, on which the irradiation maps and the average size of the spot were calculated.
The target was placed so that the map of paraboloid 1 at the equinox was centered as seen
in Figure 15.
Figure 15. Irradiance map for the equinoctial paraboloid at the equinox (Par_0).
4.1. Rotation around Pivot B
The collectors and the corresponding irradiance maps on the receiver are presented
in Figures 16–19. Figures 16 and 18 show the paraboloids calculated by rotating the equi-
noctial paraboloid in the summer solstice and winter solstice positions, corresponding to
Figure 15. Irradiance map for the equinoctial paraboloid at the equinox (Par_0).
4.1. Rotation around Pivot B
The collectors and the corresponding irradiance maps on the receiver are presented in
Figures 16–19. Figures 16 and 18 show the paraboloids calculated by rotating the equinoctial
paraboloid in the summer solstice and winter solstice positions, corresponding to the Par_1s
and Par_1w of Tables 5and 6. Figures 17 and 19 show the respective irradiance maps.
Energies 2022, 15, x FOR PEER REVIEW 15 of 21
the Par_1s and Par_1w of Tables 5 and 6. Figures 17 and 19 show the respective irradiance
maps.
In Figure 15 the beam is a small point, while in Figures 17 and 19 the beam deforms
and widens considerably, confirming that the same paraboloid cannot be used for all days
of the year.
Figure 16. Side view of the Par_0 (reference equinox, in yellow), Par_0s (summer solstice, in red)
and Par_1s (Par_0 rotated as indicated by the arrow, in green).
Figure 17. Irradiance map for the equinoctial paraboloid rotated at the summer solstice Par_1s.
Figure 16.
Side view of the Par_0 (reference equinox, in yellow), Par_0s (summer solstice, in red) and
Par_1s (Par_0 rotated as indicated by the arrow, in green).
Energies 2022,15, 260 15 of 20
Energies 2022, 15, x FOR PEER REVIEW 15 of 21
the Par_1s and Par_1w of Tables 5 and 6. Figures 17 and 19 show the respective irradiance
maps.
In Figure 15 the beam is a small point, while in Figures 17 and 19 the beam deforms
and widens considerably, confirming that the same paraboloid cannot be used for all days
of the year.
Figure 16. Side view of the Par_0 (reference equinox, in yellow), Par_0s (summer solstice, in red)
and Par_1s (Par_0 rotated as indicated by the arrow, in green).
Figure 17. Irradiance map for the equinoctial paraboloid rotated at the summer solstice Par_1s.
Figure 17. Irradiance map for the equinoctial paraboloid rotated at the summer solstice Par_1s.
Energies 2022, 15, x FOR PEER REVIEW 15 of 21
the Par_1s and Par_1w of Tables 5 and 6. Figures 17 and 19 show the respective irradiance
maps.
In Figure 15 the beam is a small point, while in Figures 17 and 19 the beam deforms
and widens considerably, confirming that the same paraboloid cannot be used for all days
of the year.
Figure 16. Side view of the Par_0 (reference equinox, in yellow), Par_0s (summer solstice, in red)
and Par_1s (Par_0 rotated as indicated by the arrow, in green).
Figure 17. Irradiance map for the equinoctial paraboloid rotated at the summer solstice Par_1s.
Figure 18.
Side view of the Par_0 (reference equinox, in yellow), Par_0w (winter solstice, in blue) and
Par_1w (Par_0 rotated as indicated by the arrow, in pink).
Energies 2022, 15, x FOR PEER REVIEW 16 of 21
Figure 18. Side view of the Par_0 (reference equinox, in yellow), Par_0w (winter solstice, in blue)
and Par_1w (Par_0 rotated as indicated by the arrow, in pink).
Figure 19. Irradiance map for the equinoctial paraboloid rotated to the position of the winter sol-
stice Par_1w.
4.2. Bending
Here, the paraboloids rotated in the previous step are bent. The bending decreases
the beam size along the x direction. The collectors and the corresponding irradiance maps
on the receiver are presented in Figures 20–23. Figures 20 and 22 report the side view of
the real paraboloids Par_2s and Par_2w for the summer and winter solstices, respectively.
The corresponding spots on the target are in Figures 21 and 23.
The dimensions of the beam have decreased in both directions X and Y, although
they are not as small as the focus in Figure 15.
Figure 20. Side view of Par_2s (green) superimposed to Par_0s (red), for the summer solstice.
Figure 19.
Irradiance map for the equinoctial paraboloid rotated to the position of the winter solstice
Par_1w.
In Figure 15 the beam is a small point, while in Figures 17 and 19 the beam deforms
and widens considerably, confirming that the same paraboloid cannot be used for all days
of the year.
4.2. Bending
Here, the paraboloids rotated in the previous step are bent. The bending decreases the
beam size along the x direction. The collectors and the corresponding irradiance maps on
the receiver are presented in Figures 20–23. Figures 20 and 22 report the side view of the
real paraboloids Par_2s and Par_2w for the summer and winter solstices, respectively. The
corresponding spots on the target are in Figures 21 and 23.
Energies 2022,15, 260 16 of 20
Energies 2022, 15, x FOR PEER REVIEW 16 of 21
Figure 18. Side view of the Par_0 (reference equinox, in yellow), Par_0w (winter solstice, in blue)
and Par_1w (Par_0 rotated as indicated by the arrow, in pink).
Figure 19. Irradiance map for the equinoctial paraboloid rotated to the position of the winter sol-
stice Par_1w.
4.2. Bending
Here, the paraboloids rotated in the previous step are bent. The bending decreases
the beam size along the x direction. The collectors and the corresponding irradiance maps
on the receiver are presented in Figures 20–23. Figures 20 and 22 report the side view of
the real paraboloids Par_2s and Par_2w for the summer and winter solstices, respectively.
The corresponding spots on the target are in Figures 21 and 23.
The dimensions of the beam have decreased in both directions X and Y, although
they are not as small as the focus in Figure 15.
Figure 20. Side view of Par_2s (green) superimposed to Par_0s (red), for the summer solstice.
Figure 20. Side view of Par_2s (green) superimposed to Par_0s (red), for the summer solstice.
Energies 2022, 15, x FOR PEER REVIEW 17 of 21
Figure 21. Irradiance map for Par_2s.
Figure 22. Side view of Par_2w (pink) superimposed to Par_0w (blue), for the winter solstice.
Figure 23. Irradiance map for Par_2w.
The spot is smaller than that obtained with the previous procedure (Figures 17 and
19). The more relevant difference is that the spots generated by parabolas Par_2w and
Par_2s are not centered on the origin; this could be a problem in the case of a physical
receiver, because the beam risks falling outside the receiver area.
Tables 7–9 show the data of the simulations carried out. For each configuration,
whose data are shown in Tables 4–6, Tables 7–9 report the concentrated power (peak
Figure 21. Irradiance map for Par_2s.
Energies 2022, 15, x FOR PEER REVIEW 17 of 21
Figure 21. Irradiance map for Par_2s.
Figure 22. Side view of Par_2w (pink) superimposed to Par_0w (blue), for the winter solstice.
Figure 23. Irradiance map for Par_2w.
The spot is smaller than that obtained with the previous procedure (Figures 17 and
19). The more relevant difference is that the spots generated by parabolas Par_2w and
Par_2s are not centered on the origin; this could be a problem in the case of a physical
receiver, because the beam risks falling outside the receiver area.
Tables 7–9 show the data of the simulations carried out. For each configuration,
whose data are shown in Tables 4–6, Tables 7–9 report the concentrated power (peak
Figure 22. Side view of Par_2w (pink) superimposed to Par_0w (blue), for the winter solstice.
The dimensions of the beam have decreased in both directions X and Y, although they
are not as small as the focus in Figure 15.
The spot is smaller than that obtained with the previous procedure (Figures 17 and 19).
The more relevant difference is that the spots generated by parabolas Par_2w and Par_2s
are not centered on the origin; this could be a problem in the case of a physical receiver,
because the beam risks falling outside the receiver area.
Energies 2022,15, 260 17 of 20
Energies 2022, 15, x FOR PEER REVIEW 17 of 21
Figure 21. Irradiance map for Par_2s.
Figure 22. Side view of Par_2w (pink) superimposed to Par_0w (blue), for the winter solstice.
Figure 23. Irradiance map for Par_2w.
The spot is smaller than that obtained with the previous procedure (Figures 17 and
19). The more relevant difference is that the spots generated by parabolas Par_2w and
Par_2s are not centered on the origin; this could be a problem in the case of a physical
receiver, because the beam risks falling outside the receiver area.
Tables 7–9 show the data of the simulations carried out. For each configuration,
whose data are shown in Tables 4–6, Tables 7–9 report the concentrated power (peak
Figure 23. Irradiance map for Par_2w.
Tables 7–9show the data of the simulations carried out. For each configuration, whose
data are shown in Tables 4–6, Tables 7–9report the concentrated power (peak irradiance
and total power) and the coordinates of the center of the spot on the target (the root–mean–
square of the spot dimensions).
Table 7. Numerical results of the simulations for the equinox.
Name Description Peak Irradiance
[Watts/cm2]
Total Power
[Watts]
Par_0
Paraboloid based on the
parabola for the equinox, whose
parameters are in Table 3.
1288.90 5889.0
Table 8. Numerical results of the simulations for the summer solstice.
Name Description Peak Irradiance
[Watts/cm2]
Total Power
[Watts]
RMS of Spot Size
in X [mm]
RMS of Spot Size
in Y [mm]
Par_0s
Paraboloid based on the
parabola for summer
solstice, whose parameters
are in Table 3.
1131.9 4659.4
Par_1s Par_0 rotated around B 9.0489 4644.3
Par_2s Par_1s bent twice 151.25 4648.4 40.27 20.55
Table 9. Numerical results of the simulations for the winter solstice.
Name Description Peak Irradiance
[Watts/cm2]
Total Power
[Watts]
RMS of Spot size
in X [mm]
RMS of Spot size
in Y [mm]
Par_0w
Paraboloid based on the
parabola for the winter
solstice, whose parameters
are in Table 3.
935.58 4614.1
Par_1w Par_0 rotated around B 11.804 4595.1
Par_2w Par_1w bent twice 79.286 4609.0 64.58 34.07
For these simulations, the power of the source is set to 1 sun = 1000 W/m
2
and the
mirrors are considered ideal, so no losses are computed.
The paraboloid configurations, generated from the ideal parabolas, concentrate a spot
on the target with highest peaks (elevated power) and smallest dimensions (focused image),
as shown in Figure 16.
The tables show only the RMS of the spot size for the two final configurations for the
summer and winter solstices, as they are the only ones corresponding to real situations and
they are necessary to define the size of the receiver.
Energies 2022,15, 260 18 of 20
This study, with purely optical methodologies, to design a continuous mirror Scheffler
system serves to help understand its feasibility for a specific application. The raytracing
software furnishes the shape and irradiance map of the light spot on the receiver obtainable
with different Scheffler-type reflectors. The various versions of the Scheffler reflector
(experiencing mechanical deformations) have been compared. The illustrations in Section 3
evidence the differences from a mechanical the point of view. In Section 4, the light spot on
the receiver is analyzed by an optical point of view. These optical analyses can be used to
estimate the applicability of a Scheffler-type system for a particular configuration or in a
defined system.
5. Sizing the Receiver
All previous efforts have been intended to set the size of the thermal system receiver.
Considering that the spot has an asymmetrical shape, the size of the receiver must be as
large as the major axis of the focused beam.
The previous analyses demonstrate that the worst case is the winter solstice: in this
case the receiver, in order to collect more than 99% of the energy, must have a radius of at
least three times the maximum RMS value of the spot size. Therefore, the receiver radius
is 194 mm in order to intercept the beam all year round, as is shown in Figure 24, which
shows the spot calculated for noon. In Figure 25 there is a rotation of 30 degrees, which
corresponds to 10 a.m. or 2 p.m.
Energies 2022, 15, x FOR PEER REVIEW 19 of 21
least three times the maximum RMS value of the spot size. Therefore, the receiver radius
is 194 mm in order to intercept the beam all year round, as is shown in Figure 24, which
shows the spot calculated for noon. In Figure 25 there is a rotation of 30 degrees, which
corresponds to 10 a.m. or 2 p.m.
In a previous study, using combined optical and mechanical simulations, Ruelas et
al. [11] conclude that “When performing a ray tracing simulation and analyses of the ge-
ometric of solar image in the receiver for comparison with the infrared image, we found
that the more suitable geometry for the receiver has an elliptical form”. However, they do
not consider that during the day the reflector is rotated, so the image also rotates around
the center of the receiver. Hence, the optimal shape is circular, with a radius correspond-
ing to the major dimension (three times the maximum RMS value of the spot size).
Figure 24. Spot on a receiver with a 194-mm radius for the winter solstice.
.
Figure 25. Spot of the paraboloid rotated by 30° (corresponding to 10 o’clock or 14 o’clock).
The size of the beam also affects the concentration ratio of the system.
Given that the aperture of the receiver is often determined by the configuration at the
equinox, it appears that the actual losses at the solstices are greater than estimated. An
accurate analysis of the dimensions of the concentrated beam, even for different positions
(up to the solstices), can lead to considering the use of larger dimensions for the aperture
and, possibly, also for the receiver. However, this causes the concentration ratio to de-
crease. For applications that instead require a certain concentration ratio, the option of
adding a secondary optic should be considered. In this case, the operation is not easy;
since this secondary optic is close to the focus of the system, it must withstand considera-
ble power densities, even if it is hit only by the external part of the concentrated beam (in
green, in Figures 24 and 25).
Figure 24. Spot on a receiver with a 194-mm radius for the winter solstice.
Energies 2022, 15, x FOR PEER REVIEW 19 of 21
least three times the maximum RMS value of the spot size. Therefore, the receiver radius
is 194 mm in order to intercept the beam all year round, as is shown in Figure 24, which
shows the spot calculated for noon. In Figure 25 there is a rotation of 30 degrees, which
corresponds to 10 a.m. or 2 p.m.
In a previous study, using combined optical and mechanical simulations, Ruelas et
al. [11] conclude that “When performing a ray tracing simulation and analyses of the ge-
ometric of solar image in the receiver for comparison with the infrared image, we found
that the more suitable geometry for the receiver has an elliptical form”. However, they do
not consider that during the day the reflector is rotated, so the image also rotates around
the center of the receiver. Hence, the optimal shape is circular, with a radius correspond-
ing to the major dimension (three