Applied Thermal Engineering 102 (2016) 695–700
Contents lists available at ScienceDirect
Applied Thermal Engineering
journal homepage: www.elsevier.com/locate/apthermeng
Research Paper
Design and optical analysis of the band-focus Fresnel lens solar
concentrator
Gang Wang a,⇑, Zeshao Chen b, Peng Hu b, Xiaofang Cheng b
a
b
School of Energy and Power Engineering, Northeast Dianli University, Jilin, Jilin 132012, China
University of Science and Technology of China, Hefei, Anhui 230027, China
h i g h l i g h t s
A new kind of band-focus Fresnel lens solar concentrator was proposed.
The design principle of band-focus Fresnel lens concentrator was given.
Optical analysis of this Fresnel lens concentrator showed a good concentrating uniformity.
a r t i c l e
i n f o
Article history:
Received 27 November 2015
Revised 5 April 2016
Accepted 8 April 2016
Available online 8 April 2016
Keywords:
Solar energy
Fresnel lens
Band-focus
CPV
Renewable energy
a b s t r a c t
Solar energy is one of the most promising renewable energies and meaningful for the sustainable development of energy source. In this paper, a new kind of band-focus Fresnel lens solar concentrator was proposed. The design principle of this solar concentrator was given and the spectral concentrating
performance was simulated by the means of Monte Carlo Ray Tracing Method (MCRT), which was compared with the linear Fresnel lens. The results show that both the spectral concentrating uniformity and
optical efficiency of the band-focus Fresnel lens were better than those of the linear one. Meanwhile several characteristic parameters of the band-focus Fresnel lens concentrator were analyzed under different
conditions and it can be drawn from the results that a high-ratio band-focus Fresnel lens concentrator
could increase the optical efficiency of a concentrating photovoltaic (CPV) system.
Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction
Solar energy is one of the most promising renewable energies
and meaningful for the sustainable development of energy source.
As the solar radiation energy flux density is low on the surface of
the earth, the spectral concentrating can be used in the solar
energy photovoltaic utilization process to reduce costs. Fresnel
lens is a common kind of spectral concentrator, which is used
not only in PV systems [1–3] but also in solar thermal systems
[4–6]. Traditional Fresnel lenses make lights focus to a point or a
line with wedges which are distributed on a plane or a curved surface [7,8]. When they are used in PV systems, the energy flux densities on solar cells are very non-uniform and that reduces the
efficiency of solar modules [9,10].
For this problem, Ryu et al. improved the point-focus Fresnel
lens and made rays through the new lens focus to a square which
had the same size as a solar cell [11]. That improves the spectral
⇑ Corresponding author.
E-mail address: kinggang009@163.com (G. Wang).
http://dx.doi.org/10.1016/j.applthermaleng.2016.04.030
1359-4311/Ó 2016 Elsevier Ltd. All rights reserved.
uniformity, but because of the design limitations, this new lens
must face the incident rays strictly when it is used in PV systems.
Otherwise, even a very small deviation could make the concentrating light deviate from solar cells and decrease the total efficiency
greatly. So a precise two-axis sun-tracker is necessary for this kind
of Fresnel lens.
This paper proposes a new kind of Fresnel lens concentrator,
based on the linear Fresnel lens, which could focus incident rays
to a uniform solar flux band. The optical simulation and analysis
of the band-focus Fresnel lens were carried out by the means of
Monte Carlo Ray Tracing Method, aiming at the investigation of
the spectral concentrating performance of this new concentrator.
2. Concentrator design
In order to make incident rays focus to a band which is presented in Figs. 1 and 2, the design principle was given as follows:
(1) The horizontal length of the Fresnel lens d was set to be odd
times of the width of a solar cell w, and the ratio N was equal to
d/w. The horizontal length was divided to N units. (2) In these N
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G. Wang et al. / Applied Thermal Engineering 102 (2016) 695–700
where sj and lj were the energy flux transmission efficiency and
tooth width of Wedge j, respectively. The ideal optical efficiency
of the lens was expressed as following:
g ¼ s1 s2
ð4Þ
The optical efficiency of lens under the condition which the rays
were not normal incident was analyzed later in this paper.
3. Method and calculation model
Monte Carlo Ray Tracing Method (MCRT) was employed for the
spectral concentrating simulation of the band-focus solar concentrator. It was a statistical method of tracking the random process
of a large number of beams. The calculation process of MCRT
was: Assuming that solar radiation energy was carried by a lot of
beams evenly, every beam would experience many optical processes including reflection, refraction, absorption and scattering.
Whether these optical processes occurred or not were controlled
by random numbers. By tracking the propagation processes of
these beams, the energy flux density distribution on the radiation
absorbing surface could be obtained [13].
Assuming that incident rays distributed evenly on the incident
plane of the band-focus Fresnel lens and this plane was considered
as the emitting surface of sampling beams, the probability model
of the emitting point (x0, y0, z0) was:
Fig. 1. Spectral concentration effect of band-focus Fresnel lens.
8
>
< x0 ¼ xL RX xL =2
y0 ¼ f
>
:
z0 ¼ zL RZ
Fig. 2. Design of the horizontal orientation of band-focus Fresnel lens.
units, the center one was made as a plane without wedges. Other
units all have the same number of wedges and the angles u of
every wedge which were in the same unit should be the same.
If the index of refraction of the lens material was n, the angle u
of a wedge could be calculated as following:
sin c
u ¼ arctan
n cos c
ð1Þ
where c was the light deviation angle of this wedge, which can be
seen in Fig. 2. When the wedge angles of the Fresnel lens were
given, the overall size of the lens would be obtained according to
the size of solar cells and the focal length f. In this paper, the size
of solar cells was assumed to be 25 mm 125 mm, f would be
changed according to different conditions.
The ideal optical efficiency was an important characteristics
parameter of the band-focus Fresnel lens. When the light was normal incident, the energy flux transmission efficiency should be
[12]:
s1 ¼
4n
ð2Þ
ðn þ 1Þ2
As shown in Eq. (2), the energy flux transmission efficiency was
only related to the index of refraction of lens material when the
light was normal incident.
The energy flux transmission efficiency of the light output surface of the band-focus Fresnel lens was equal to the weighted average of the energy flux transmission efficiencies of all wedges and
the center plane [12]:
s2 ¼
M
X
sj lj þ ws1
1
!,
d
ð3Þ
ð5Þ
where RX ; RZ 2 ð0; 1Þ were the random numbers of x axis and y axis
respectively, and xL, zL and f were the x axis length, y axis length and
focal length of the band-focus Fresnel lens respectively. The nonparallel angle of solar radiation was 320 , so the incident rays was
considered as a light cone with ha = 160 . In this cone, the solar
energy distribution followed the Lambert law which was that the
directional radiation intensities were the same. So we could obtain
the zenith angle and circumferential angle perpendicular to the
incident light direction as follows [14]:
h ¼ arcsin
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Rh sinðh2a Þ
u ¼ 2pRu
ð6Þ
ð7Þ
where Rh ; Ru 2 ð0; 1Þ were the random numbers of cone angle and
circumference along the emitting direction. The direction vector of
the normal incident light P was:
P ¼ ðsin h sin u; cos h; sin h cos uÞ
ð8Þ
and the direction vector of the refraction light A was:
A ¼ P þ CN
ð9Þ
where N was the normal direction vector on the incident point of
the refraction surface, and C was a constant. If the incident angle
was a, the refraction angle was b, C would be:
C ¼ n cos b cos a
ð10Þ
When the rays were not normal incident, the direction vector P⁄
could be obtained by using a transformation matrix U:
P ¼ UP
ð11Þ
where transformation matrix was:
2
u11
6
U ¼ 4 u21
u31
u12
u13
3
u22
7
u23 5
u32
u33
ð12Þ
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G. Wang et al. / Applied Thermal Engineering 102 (2016) 695–700
8
u11
>
>
>
>
>
u12
>
>
>
>
>
u13
>
>
>
>
>
>
< u21
u22
>
>
>
> u23
>
>
>
>
> u31
>
>
>
>
> u32
>
>
:
u33
¼ cos hy cos hz
¼ sin hx sin hy cos hy þ cos hx sin hz
focus Fresnel lens were both better than those of the linear one
under the same condition.
¼ sin hx sin hz cos hx sin hy cos hz
4.2. Research on the relationship of CR, CR0 , f and g
¼ cos hy sin hz
¼ cos hx cos hz sin hx sin hy sin hz
ð13Þ
¼ cos hx sin hy sin hz þ sin hx cos hz
¼ sin hy
¼ sin hx cos hy
¼ cos hx cos hy
hx, hy and hz were the three angles between the incident light and
three axes, respectively. By this way, incident rays, refraction rays
and output rays could be all ascertained, and the energy flux density
distribution on the absorbing surface could be also calculated.
4. Spectral concentrating simulation and analysis
4.1. Concentrating performance simulation
It is assumed that the base thickness of the band-focus Fresnel
lens was 2 mm, the horizontal length of the lens d was 225 mm,
the vertical length was 125 mm, the tooth width of every wedge
was 1 mm, f was 100 mm, theoretical spectral concentrating ratio
CR was 9, the lens material was PMMA, n was 1.4935; Meanwhile,
a linear Fresnel lens model, which had the same material and
parameters except the wedge angles, was established to compare
with the band-focus lens. The energy flux density of incident rays
was assumed to be 1000 W/m2 and the number of the incident rays
was 106. The width and length of the receiving surface were 25 mm
and 125 mm, which was the same as the size of solar cells. The
absorbility of the incident surface was assumed to be 100%.
The light density distribution performance contrast of the two
kinds of Fresnel lenses is presented in Fig. 3. Fig. 3(a) is the concentrating performance of the linear Fresnel lens and Fig. 3(b) is the
one of the band-focus lens. It can be seen from Fig. 3 that both
the vertical light flux density distributions of the two lenses were
uniform, but the horizontal light flux density distribution of the
band-focus lens was much more uniform than the linear one. The
horizontal light flux density distribution contrast of the two lenses
is presented in Fig. 4, in which red1 dotted lines represent the linear
Fresnel lens and black solid lines represent the band-focus one.
The natural logarithm of horizontal light flux density distributions on the receiving surface under different focal lengths and
light deviation angles is presented in Fig. 5. It can be seen that different spectral concentrating ratios for the same light deviation
angle c had close spectral concentrating uniformities. When f
was fixed, the uniformity under a small CR was better than the
one under a big CR. Therefore, the band-focus Fresnel lens concentrator could obtain a better spectral concentrating uniformity by
increasing the focal length and spectral concentrating ratio
properly.
In this paper, when the focal length was 100 mm, the actual
concentrating parameters of the linear and band-focus Fresnel lens
under different theoretical spectral concentrating ratios were calculated, and the results are presented in Tables 1 and 2. Here, Imax,
Imin and Ia were the maximal, minimal and average light flux density on the receiving surface respectively, CR0 was the actual spectral concentrating ratio, and g0 was the actual optical efficiency.
Results in Tables 1 and 2 show that with CR increasing, g0
decreased. And the optical efficiency and uniformity of the band1
For interpretation of color in Fig. 4, the reader is referred to the web version of
this article.
For band-focus Fresnel lens with different focal lengths, the
relationships of theoretical spectral concentrating ratios, actual
spectral concentrating ratios and ideal optical efficiencies were calculated, and results are presented in Figs. 6 and 7.
As shown in Fig. 6, the maximal theoretical spectral concentrating ratio increased with the focal length increasing. Thus a highratio Fresnel lens should have a big focal length.
According to Fig. 7, when f was fixed, g decreased as the spectral
concentrating ratio increased. And the decreasing rate decreased if
f was increased. So it can be drawn from Fig. 7 that a high-ratio
Fresnel lens could make better use of solar energy and increase
the optical efficiency of the PV system.
4.3. Effect on the optical efficiency under different incident angles
The analysis of the ideal optical efficiency of band-focus Fresnel
lens above was under the condition which the rays were normal
incident, and that needed a precise two-axis sun-tracking. For
the one-axis sun-tracking (The horizontal direction of the lens
was along the north–south and the vertical direction was along
the east–west.), assuming that the north–south tracking was precise, there was an incident angle, which was not equal to 0°, in
the surface perpendicular to the top surface of the Fresnel lens.
The incident angle was b1 and the refraction angle under the top
surface was b2, then the incident angle on the wedge surface could
be expressed as:
b3 ¼ arccosðcos u cos b2 Þ
ð14Þ
and the final exit angle of output ray was:
b4 ¼ arcsinðn sin b3 Þ
ð15Þ
Assuming as follows:
8
bþ
>
>
>
<
b
>
b0þ
>
>
: 0
b
¼ b1 þ b2
¼ b1 b2
¼ b3 þ b4
ð16Þ
¼ b3 b4
the light flux transmission efficiency of the top surface of the lens
was:
"
#,
2
tan2 ðb Þ sin ðb Þ
2
T1 ¼ 1 þ
tan2 ðbþ Þ sin2 ðbþ Þ
ð17Þ
and the light flux transmission efficiency of every wedge bottom
surface was:
"
#,
2
tan2 ðb0 Þ sin ðb0 Þ
T2 ¼ 1 þ
2
tan2 ðb0þ Þ sin2 ðb0þ Þ
ð18Þ
Using Eqs. (17) and (18), the theoretical optical efficiencies of
the lens under different incident angles could be calculated. For a
band-focus Fresnel lens whose f was equal to 100 mm and CR
was equal to 9, the calculation result of the relationship between
b1 and g is presented in Fig. 8.
According to Fig. 8, when b1 was between 0° and 26.5°, the optical efficiency of the lens was high (bigger than 90%) and had small
changes with b1 increasing. When b1 was just past 26.5°, the optical efficiency decreased sharply, which was caused by the total
reflection of the light in the Fresnel lens. Different lens materials
had different critical incident angles, past which the optical effi-
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G. Wang et al. / Applied Thermal Engineering 102 (2016) 695–700
Fig. 3. The concentrating performance contrast: (a) linear Fresnel lens; (b) band-focus Fresnel lens.
Table 1
Results of the concentrating simulation of the linear Fresnel lens.
CR
Imax (W m2)
Ia (W m2)
Imin (W m2)
CR0
g0 (%)
3
5
7
9
5955.06
7454.72
9860.23
9959.88
2574.5
4276.7
5585.2
5956.4
676.78
745.39
833.20
840.15
2.627
4.364
5.699
6.078
87.57
87.28
81.41
67.53
Table 2
Results of the concentrating simulation of the band-focus Fresnel lens.
CR
Imax (W m2)
Ia (W m2)
Imax (W m2)
CR0
g0 (%)
3
5
7
9
2735.45
4389.91
5766.12
6410.23
2663.6
4321.8
5688.9
6095.6
1498.34
2273.08
2917.80
3150.14
2.72
4.41
5.81
6.22
90.67
88.20
83.00
69.11
Fig. 4. Light flux density distribution contrast.
Fig. 6. The relationship of CR, CR0 and f.
Fig. 5. Horizontal light flux density distributions on the receiving surface under
different f and c.
ciency decreased sharply. In Fig. 8, when b1 increased to 60°, the
optical efficiency decreased to about 43%.
For another case of the one-axis sun-tracking, which was the
east–west tracking was precise and the incident angle b1 in the
north–south direction increased from 0°, the concentrating light
on the focal plane would deviate slowly. When b1 increased to a
certain value, the output rays of different units of the lens were
already not able to coincide completely, which made the concentrating light turn wider and non-uniform, and the output rays of
some units (for example the outermost units) even separate from
G. Wang et al. / Applied Thermal Engineering 102 (2016) 695–700
Fig. 7. The relationship of CR, g and f.
699
Fig. 10. Horizontal light flux density distributions under different incident angles.
significance of this concentrator. Assuming the base thickness of
the lens was 2 mm, the relationship between b1 and concentrating
light deviation quantity (Dd) on solar cells under different focal
lengths was calculated and the result is presented in Fig. 9. For
the band-focus Fresnel lens whose focal length was 500 mm and
CR was 35, the horizontal light flux density distributions of the
receiving surface under different incident angles are shown in
Fig. 10.
According to Figs. 9 and 10, the concentrating light deviation
quantity increased with f increasing. Though the concentrating
light on solar cells would deviate and turn non-uniform, if the incident angle was controlled to be very small (for instance, b1 was
smaller than 0.5°), the deviation quantity and concentrating light
uniformity changes caused by the north–south tracking error
would be ignored and the normal use of the CPV system would also
be not influenced. That means this band-focus Fresnel lens solar
concentrator had a more simple sun-tracking requirement than
the Fresnel lens concentrator proposed by Ryu [11].
Fig. 8. The relationship between the incident angle and optical efficiency.
5. Conclusions
In this paper, for the solar energy application, the band-focus
Fresnel lens solar concentrator was proposed. The design principle
was given and the concentrating performance investigation was
carried out by MCRT simulation and theoretical analysis. The spectral concentrating simulation results show that both the spectral
concentrating uniformity and optical efficiency of the band-focus
Fresnel lens were better than those of the linear lens. Meanwhile,
the relationships of several key parameters of the band-focus Fresnel lens were analyzed and the results indicate that a high-ratio
band-focus Fresnel lens could make use of solar energy more effectively and increase the optical efficiency of the CPV system. The
effect on the optical efficiency under different incident angles
was also studied. The analysis results show that even a precise
one-axis would meet the application requirement of the solar concentrator and very a tiny north–south tracking error would not
influence the concentrating performance of this CPV system.
Acknowledgement
Fig. 9. The relationship between the incident angle and concentrating light
deviation quantity.
The authors appreciate the support of the Natural Science Foundation of China (Grant No. 51376167).
the concentrating light. As b1 increased even bigger, the output
rays of the outermost units would disappear because of the total
reflection and the concentrating light also turn more
non-uniform, which the band-focus Fresnel lens lost the original
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