Abstract

Solar tracking concentrators are optical systems that collect the solar energy flux either in a line or spot using reflective or refractive surfaces. The main problem with these surfaces is their manufacturing complexity, especially at large scales. In this paper, a line-to-spot solar tracking concentrator is proposed. Its configuration allows for a low-cost solar concentrator system. It consists of a parabolic trough collector (PTC) and a two-section PMMA Fresnel lens (FL), both mounted on a two-axis solar tracker. The function of the PTC is to reflect the incoming solar radiation toward a line. Then, the FL, which is placed near the focus, transforms this line into a spot by refraction. It was found that the system can achieve a concentration ratio of 100x and concentrate an average solar irradiance of 518.857W/m2 with an average transmittance of 0.855, taking into account the effect of the chromatic aberration.

© 2015 Optical Society of America

1. Introduction

The function of solar concentration is to deliver solar energy flux at high temperatures in a small well-defined area. This is done by interposing an optical body between the sun and the absorbing surface. The function of these bodies, typically parabolic mirrors or lenses, is to intercept the solar irradiance and concentrate it on a receiver. Depending on the amount of energy to collect, the concentration can be delivered with a line or spot focus.

The linear concentration is mainly done by parabolic trough collectors (PTC) in which the incoming solar radiation is concentrated into a line by tracking the solar movement around one axis. PTCs are able to concentrate the solar energy flux from 30 to 80x, reaching temperatures between 150°C and 400°C on the receiver [1,2]. They need to be mounted on a steel structure to maintain the geometry and to support the weight over high wind loads, which increases the cost of the entire unit. To reduce this structural cost, the PTC’s shape can be approximated by an array of mirrors placed at ground level, thereby reducing the structural demands, but increasing the tracking complexity, as every mirror requires an independent movement to reflect the beam radiation toward the line [3].

The major problem with linear concentrators is that it is difficult to achieve high concentrations. Therefore, the parabolic 2D profile from the PTC can be revolved around its optical axis, forming a 3D parabolic dish collector (PDC) that is able to concentrate the solar flux on a spot by tracking the sun in two axes. PDCs can concentrate the solar radiation from 1000 to 4000x, reaching temperatures between 750°C and 1400°C on the receiver [2]. Despite its performance, the 3D parabolic shape increases the system’s cost due to manufacturing complexity. Like the linear concentrators, its shape can be approximated by small flat mirrors, which reduces the reflective surface cost but increases the support structure complexity, as every mirror needs to be mounted in a different position with respect to the receiver.

A cheaper alternative to the reflective surfaces is to use a Fresnel lens collector (FL), which essentially is a chain of prisms that duplicates the slope of a section of a conventional planoconvex or aspheric lens, but without the material of its full body [5]. The main goal of the FL is to refract the sun’s rays and concentrate them in a line or spot focus, depending on whether it is an extruded or revolved profile. Unlike mirror concentrators, the bending of light on an FL follows Snell’s law, which occurs when the beam travels from a medium with a given refraction index to another with a different index. The main advantage of the FLs is that they are fabricated using a relatively inexpensive material, typically polymethylmethacrylate (PMMA). However, the manufacturing cost increases with the size, since large scale FLs must be manufactured in smaller sections and prism size must be small enough to concentrate the solar energy flux properly [7].

Innovative alternatives where a combination of refractive and reflective surfaces have been shown in the literature. Examples are exposed in [8, 9], where a two-stage collector transforms a linear concentration into several spots. The first stage is achieved by a one-axis PTC that concentrates the incoming solar radiation into a line. Then, the second stage occurs when an array of stationary nonimaging Compound Parabolic Concentrators (CPC) distributed along said line refracts it into several spots, one spot per CPC. Similar concepts are described in [10, 11], where, instead of being fixed, the CPCs have a secondary tracking axis. This tracking allows their inlet surfaces to be aligned to the direction of the linear concentration all year through, reducing the effect of the skew angle provoked by the position of the sun to the PTC. On one hand, the transformation of the line into a spot seems to be a novel approach to reduce the cost per m2 of the collecting surface. For this, a 2D reflective surface associated to a secondary refractive body replaces the 3D shape. On the other hand, a percentage of the concentrated line is lost due to the separation of the secondary bodies while considerable reflection losses occur due to multiple deviations of the solar beams inside them. Also, their weight can be considerable as they present a full-body shape instead of a FL configuration.

Taking into account the different kinds of solar concentrators previously described, in this paper, a novel line-to-spot solar tracking concentrator (LSTC) is proposed. It mainly consists of a PTC mirror and a two-section FL, both mounted on a two-axes solar tracker. The PTC concentrates 10x the solar irradiance in a line. Since the linear systems have a low energy flux concentration, a FL of two sections near the focus area transforms the line into a single spot, increasing the concentration ratio from 10 to 100x. Considering the incoming solar energy, the system is capable of concentrate an average solar irradiance of 518.857W/m2 with an average transmittance of 0.855. The LSTC’s configuration allows the 2D parabolic surface to emulate the function of the 3D surface by having a two-section FL. Because the 2D shape is easier to manufacture than the 3D, the unit cost is reduced. In addition, the two-section FL’s size is considerably reduced, as it remains close to the linear focus.

2. Proposal

The LSTC is a solar tracking concentrator that combines reflection and refraction of the beam radiation to concentrate the solar energy flux in a small area. The concentration is accomplished in two steps. The first step is achieved by a two-axes PTC that reflects the incident rays of the sun. As the mirror has a 2D parabolic shape, every ray hitting its surface will be reflected toward a line (see Fig. 1(a)). The second step occurs when a FL transforms the linear concentration into a spot by refraction. This is done by a lens that has two surfaces: inner and outer, each with a specific profile (see Fig. 1(b)). The inner surface receives the rays of the sun divergent from the line and refracts them as parallel as possible to each other through the lens thickness toward the outer section, where they are deviated to a spot. The transformation from line to spot is possible due to the change in the cross-sectional direction from one section to another. On the one hand, the inner’s cross section travels along the v-axis, while on the other, the outer’s cross section rotates around the u-axis on the center of the lens.

 figure: Fig. 1

Fig. 1 (a) Line-to-spot solar tracking concentrator and (b) inner/outer section of the lens

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The cross section of both surfaces consists of an array of prisms, each with a different slope. The slope’s value depends on the distance R between the half-step s of the prism and the optical axis of the lens, the refraction index of the lens material nPMMA (1.49) and its focal length fin,out (see Fig. 2). The slopes can be found by using the Eq. proposed in [5], where

tanε=R/[nPMMA(R2+fin,out2)1/2fin,out]
The prism’s step s has the same value for the inner and outer array of the lens while the slope ε may be obtained from the Eq. (1) in both sections. In the case of the line-to-spot FL, a constant prism’s step is used. However, prisms of variable step as those shown in [6] may be considered. In both cases, the sun’s rays trajectory remains equal since the relation between ε and R is maintained.

 figure: Fig. 2

Fig. 2 Sun’s rays refraction through the two-section FL

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The values of fin,out, the focal lengths of the inner and outer section, are determined by their rim angle, which are established as ψin = 26.6° and ψout = 26.6°. Both are equal to the PTC’s rim angle, which also has a value of ψPTC = 26.6°. This indicates that ai = fi based on the next Eq.

tanψi=ai/(2fi)
The subscript i is replaced by the term PTC, in or out according to the required value. With these terms in mind, the rays’ trajectory through the system is explained in detail in the next section.

2.1. Trajectory of the sun’s rays

The path of the sun’s rays through the LSTC depends on the direction in which they arrive to the Earth, their reflection on the PTC and their refraction in the FL. Ideally, the sun’s rays have a coparallel direction to the Earth. However, the distance between the sun and the Earth and their relative sizes cause the incoming rays to strike the PTC with a half-angle θs = 0.275° [4], as seen in Fig. 1(a). This provokes the sun’s rays to be reflected on the PTC with the same half-angle, thus converging in an area close to the line.

Once this area is passed, the sun’s rays diverge toward the inner section of the lens, whose function is to refract them through the lens thickness toward the outer section. Ideally, the direction of the sun’s rays inside the lens thickness should be parallel to the u-axis so the outer section can concentrate them at the center of the spot. However, the half-angle causes the sun’s rays to be refracted at an angle γ to the u-axis (see Fig. 2).

The magnitud of γ depends on the divergence angle ω from the reflected sun’s ray to the normal Nin of the entrance surface of the prism from the inner section. When the prism is farther from the optical center, the ω angle is greater. This angle may be obtained using the Snell’s law formula [5]

nairsinω=nPMMAsinη
where nair is the refraction index of the air (1.0) and η is the refraction angle inside the lens thickness. Ideally, there is only one ray for every prism of the inner section. However, as a result of the divergence from the line more than one ray, each in a different direction, passes through the same prism and deviates at a different γ angle. Thus, the larger the s step of the prism, the greater the number of sun’s rays passing through it and their deviation from the u-axis.

To measure the deviation of the sun’s rays, γ, an optical simulation was held using three different steps (0.001m, 0.005m and 0.01m). The simulation was performed using a PTC (aPTC = 1.0m, lPTC = 1.0m, fPTC = 1.0m and ψPTC = 26.6°) and a PMMA inner section (ain = 0.01m, lin = 1.0m, fin = 0.01m, ψin = 26.6° and tFL = 0.005m). A ray’s source was placed at 90° with respect to the PTC’s aperture to emulate its position relative to the sun at a particular hour of the day. The source was programmed to simulate 1 × 104 rays of sunlight considering their half-angle (see Fig. 3).

 figure: Fig. 3

Fig. 3 (a) Ray trace simulation from the PTC to the inner section of the lens and (b) refraction of the sun’s rays through the inner section (outer section not shown)

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From the ray trace simulation, the γ deviation of the sun’s rays inside the lens thickness was obtained. This direction was calculated using the Eq.

tanγ=(rv2+rw2)1/2/ru
where ru, rv and rw are the vectorial components of the sun’s rays given by the simulation software. The γ deviation as a funcion of ain, given the previously described prism’s steps, is plotted in Fig. 4. The aim was to obtain the step in which the deviation has the lowest variability. It can be noted that the lower the step, the lower the range in which the sun’s rays deviate. For example, a lens with a s = 0.001m value has a deviation from 0.01° to 1.86°. On the other hand, a value of s = 0.01m deviates the rays of sunlight from 0.003° to 3.2° through the lens thickness. Therefore, an inner section with a small prism step will refract the sun’s rays closer to the u-axis toward the outer section, which will refract them closer to the center of the spot. Also, when the step has a small value or, in other words, tends to 0.0, a fewer number of rays pass through the prism, thus accepting a reduced ω range.

 figure: Fig. 4

Fig. 4 γ deviation of the sun’s rays through the lens thickness using (a) s = 0.001m, (b) s = 0.005m and (c) s = 0.01m

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2.1.1. Chromatic aberration

So far, a value of 1.49 was taken to be the PMMA refraction index. This value represents the interaction of the material with the yellow light of the solar spectrum at a wavelength of 589.2μm [5]. However, it is well known that sunlight is composed of several wavelengths, each with a specific solar spectral irradiance, as shown in Fig. 5(a). These wavelengths affect the trajectory of the sun’s rays through the FL, which, unlike the PTC, suffers from chromatic aberrations (CA). CAs arise from the fact that the lens refractive index depends on the wavelength of light [12]. This causes the separation of the sun’s rays into their different wavelengths once they are refracted by the inner and outer section of the lens. Following Snell’s law, shorter wavelengths are refracted farther away from the normal line of the prism’s entrance surface than longer wavelengths, thus limiting the concentration ratio on the spot due to light dispersion. The change of the refractive index as a function of the wavelength λ is approximated by the Hartmann’s dispersion formula [5]

nPMMA=1.4681+93.42/(λ1,235)
The relation between nPMMA and λ is shown in Fig. 5(b). It can be seen that the refraction index decreases when the wavelength increases in the range accepted by the PMMA (from 0.4047μm to 1.083μm [13]). The effect of the CA on the LSTC system is shown through a ray-trace simulation. The aim was to calculate the distribution of light on the spot’s area arising from the PTC and FL’s geometry and the CA’s effect. The simulation was performed using the previously described PTC and FL’s inner section. In this case, the outer section was included (aout = 1.005m, lout = 1.0m, fout = 1.005m and ψout = 26.6°). The prism’s step for both sections was established as s = 0.001m. The sun’s rays’ source was programmed to simulate the solar spectrum accepted by the PMMA divided into 8 different bandwidths (see Table 1). The average wavelength λm value of each bandwidth was taken to represent the full bandwidth itself. For each bandwidth, 1 × 104 rays were traced toward the PTC’s aperture, with their half-angle direction taken into account (see Fig. 6).

 figure: Fig. 5

Fig. 5 (a) Solar spectral irradiance using an air mass coefficient of 1.5 (AM1.5) [3]. For illustration purposes, the infrared bandwidth (from λ0 = 0.790 to λf = 1.083) accepted by the PMMA is shown (see Table 1). (b) Refractive index of PMMA as a function of the wavelength

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 figure: Fig. 6

Fig. 6 (a) Line-to-spot ray-trace simulation where the incoming sun’s rays are placed near the PTC for illustration purposes. (b) Focal plane close up

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Tables Icon

Table 1. Ray-traced solar spectrum bandwidths

The simulated image of the spot for each bandwidth is shown in Fig. 7. It is observed that the dispersion of sun’s rays in every bandwith (a–g) occupies an area of approximately 0.01m2 (0.1m × 0.1m), including the spot image that takes into account the full solar spectrum (h). The area occupied on the focus plane by said distribution determines the concentration ratio of the LSTC system. This ratio is achieved in two stages. The first stage is the linear concentration of the sun’s rays reflected from the PTC to the FL’s inner section. Considering both areas, 1.0m2 of the PTC and 0.1m2 of the FL, the first concentration stage has a value of 10x. The second stage is the transformation of this line into a spot based on lens refraction. With the area of the lens previously mentioned and the image that includes the full solar spectrum in Fig. 7(h), the second concentration stage has a value of 10x. Therefore, the resulting concentration ratio from the PTC to the spot is 100x. Although this ratio describes the area where the concentrated rays are distributed, it does not provide information about the solar radiation flux transmitted to the spot. To describe it, the reflectance of the PTC and the transmittance of the FL must be established. Both parameters are described in the next section.

 figure: Fig. 7

Fig. 7 (a–g) Solar spectrum bandwidth distribution on the spot and (h) spot image including all bandwidths. The colorbars represent the intensity of the energy in W/m2 distributed over the focal plane (see section 2.1.2 for further information)

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2.1.2. Reflectance and transmittance

The transmittance of the LSTC describes the ratio between the solar irradiance delivered by the sun to that concentrated on the spot. The total transmittance τ is the sum of the reflectance losses on the PTC and FL, which can be calculated by

τ=τPTCτFL
The reflectance loss of the PTC is taken to be ρPTC = 0.05, which results in a transmittance of τPTC = 1 − ρPTC = 0.95 of the solar radiation hitting its surface and reflected on its way to the lens. In contrast, the reflectance losses of the FL depend on the angles of incidence and refraction of the sun’s rays on every prism and the polarization of the solar radiation. According to [5], sunlight is considered to be unpolarized. However, the parallel and perpendicular reflection coefficients, which refer to the plane spanned by the incident and the normal surface, have to be calculated with the Fresnel Eqs.
rin=tan2(ηω)/tan2(η+ω)
rin=sin2(ηω)/sin2(η+ω)
Equations (78) only consider the passing of the sun’s rays from the air to the inner section of the lens (see Fig. 2). For the outer section, the reflection coefficients rout and rout can be calculated by replacing the η and ω angles with η′ and ω′, the incident angle of the sun’s rays on the prisms from the outer section and their refraction angle on the air, respectively. Therefore, the transmittance of the solar radiation through the lens may be obtained from
τFL=[1(rin+rin)/2][1(rout+rout)/2]
Equations (69) where programmed on the ray-trace simulation to obtain the LSTC’s total transmittance. The previously described number of rays were traced toward the PTC’S aperture from each bandwidth shown in Fig. 7. These result in a total of 7 × 104 rays from the source, given their half-angle. The spot’s plane vw was taken to be the transmittance’s measurement area. The results are shown in Fig. 8, where the transmittance from every bandwidth and full solar spectrum are illustrated. It can be observed that τ is distributed over the spot’s plane in every case, reaching its peak values on the central area. The maximum, minimum and average transmittances for each case are resumed in Table 2. By taking this values, the transmittance range for the full solar spectrum over the spot is obtained. Its range varies from τmin = 0.771 to τmax = 0.913, and, considering the transmittance distribution within this range, the average value results in τ̄ = 0.855.

Tables Icon

Table 2. Transmittances values of every bandwidth and full solar spectrum

 figure: Fig. 8

Fig. 8 Transmittance of the solar radiation irradiance over the spot’s area as a function of the (a) bandwidths and (b) full solar spectrum

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The distributed transmittance impacts on the amount of the solar energy to concentrate. Since the latter depends on the light’s wavelength of the solar spectrum, each bandwidth described on Table 1 delivers a different amount of energy to the spot. To calculate the concentrated energy, the irradiance delivered by the sun per bandwidth needs to be obtained. Then, such irradiance must be multiplied by the transmittance through the LSTC.

The first step is done by approximating the area under the curve of the solar spectral irradiance (from 0.405 to 1.083μm) by summing the bandwidth’s corresponding fraction, each one calculated by multiplying Δλ (the difference between λ0 and λf) of the bandwidth of interest, by the average spectral irradiance under such limits. For example, the infrared solar spectrum has an average energy of Īsun = 162.98W/m2λ = 0.293μm, = 556.2W/m2 * μm). The rest of the energy per bandwidth is shown in Table 3.

Tables Icon

Table 3. Average incident solar irradiance Īsun as a function of bandwidths

The second step is achieved when the corresponding average irradiance from the sun and the obtained transmittances from every bandwidth are multiplied as follows

Ispot=I¯sunτ
where Ispot is the delivered irradiance on the spot. The distribution of Ispot for each bandwidth over the focal plane is shown in Fig. 7. Since every bandwidth delivers a different range of concentrated irradiance, their corresponding average values Īspot are listed on Table 4. It can be seen that the minimum Īspot is delivered by the yellow light of the solar spectrum with a value of 17.272W/m2. Oppositely, the maximum irradiance corresponds to the infrared light with a value of 139.601W/m2. By summing the average irradiance of every bandwidth, the total irradiance over the spot is Īspot = 518.857W/m2.

Tables Icon

Table 4. Spot’s solar irradiance from every bandwidth and full solar spectrum

3. Conclusions and future work

A novel LSTC was proposed in which the solar energy flux is concentrated in two stages. The first one is achieved by a PTC that reflects the incoming solar radiation toward a line, which is then converted into a spot by refraction, using a two-section FL. The LSTC is mounted on a two-axes solar tracker to follow the position of the sun throughout the day. Since the PTC has a conventional geometry, this paper was focused on the design of the two-section FL. Its geometry consisted of an inner and outer section of PMMA. The inner section receives the sun’s rays coming from the line and refracts them in a pattern that tends to be parallel from one ray of sunlight to another toward the outer section, which concentrates them on a spot area. This concentration was made possible by changing the cross-sectional sweep direction of the inner and outer sections of the lens from extruded to revolved, respectively. The prism’s step of 0.001m for the inner section was found to be the value that allows the sun’s rays to be refracted in the previously mentioned pattern inside the lens thickness. This step was then used by the outer section to deflect the sun’s rays as close as possible to the center of the spot.

The performance of the LSTC was simulated on a ray-tracing software. The authors’ aim was to calculate the concentration ratio and the energy transmittance of the solar irradiance through the LSTC. For the concentration ratio calculation, several wavelength bandwidths were simulated toward a 1.0m2 PTC to measure the area occupied on the focal plane by the light dispersion. It was found that in the solar spectrum accepted by the PMMA (from 0.4047 to 1.083μm) the concentrated sun’s rays occupy an area of 0.01m2, which gives a concentration ratio of 100x. Furthermore, the energy transmittance over the spot was simulated considering a PTC’s reflectance of 0.95 and the reflection coefficients of the FL, which depends on the deviation of the sun’s rays through its body. It was found that, with an average transmittance of 0.855, it was possible to concentrate 518.857W/m2 average.

The main goal was to propose a simplified configuration for a solar tracking concentrator. On the one hand, the PTC’s shape is simpler than the PDC’s shape, thus reducing the manufacturing complexity and cost of the reflective surface. On the other, a two-section FL was used to raise the concentration ratio by transforming the line into a spot. As the FL is located near the linear focus, a small-scale lens is required.

Future research will be carried out to enhance the performance of the LSTC. So far, a low concentration ratio was achieved by the LSTC compared to the theoretical level reached by the 3D type concentrators. The LSTC’s ratio could be increased if the lens is located closer to the linear concentration. This will require a lens of a smaller area but a higher heat resistance material to withstand the line’s temperature without affecting the FL’s optical quality. Another aspect to be improved is the reduction of the focal distance caused by the two-stage concentration. This reduction can be achieved by using a PTC and a FL with a reduced focal length. However, the transmittance may be affected by high deviation angles.

Areas of opportunity arise from this research, nevertheless it was proved that the transformation from line to spot of the sun’s rays was possible using a simplified PTC and FL geometry, which expands the possibility of using the free energy provided by the sun at a low cost.

Acknowledgments

The authors acknowledge Lambda Research for supporting research aimed at helping human development and for the licensed copy of TracePro, Autodesk for donating free software licenses to Tec de Monterrey students, Research Chair of Design and Innovation in Engineering: Solar Thermal Vehicle (STEV) for directly supporting this effort and Institute of Renewable Energy (IER-UNAM) through the Mexican Center of Innovation in Solar Energy (CeMIE-Sol) (Strategic Project 05 ``Development of solar thermal storage tanks''), through which it was possible to develop research and support the training of human resources at postgraduate level.

References and links

1. H. Price, E. Lüpfert, D. Kearney, E. Zarza, G. Cohen, R. Gee, and R. Mahoney, “Advances in Parabolic Trough Solar Power Technology,” J. Sol. Energy Eng. 124(2), 109–125 (2002). [CrossRef]  

2. F. Kreith and Y. Goswami, Handbook of Energy Efficiency and Renewable Energy (CRC, 2007). [CrossRef]  

3. J. Kreith and F. Kreider, Principles of Solar Engineering (Hemisphere Publishing Corporation, 1978).

4. J. Duffie and W. Beckman, Solar Engineering of Thermal Processes (John Wiley & Sons, Inc., 1991).

5. R. Leutz and A. Suzuki, Nonimaging Fresnel Lenses: Design and Performance of Solar Concentrators (Springer, 2001). [CrossRef]  

6. Sh. Klychev, “A Method to Calculate Fresnel Lenses,” J. Appl. Sol. Energy 49(1), 36–41 (2012).

7. W. Xie, Y. Dai, R. Wang, and K. Sumathy, “Concentrated solar energy applications using Fresnel lenses: a review,” J. Renew. Sustain. Energy Rev. 15(6), 2588–2606 (2011). [CrossRef]  

8. A. Mohr, T. Roth, and S. Glunz, “BICON: High concentration PV using one-axis tracking and silicon concentrator cells,” Prog. Photovolt. 14(7), 663–674 (2006). [CrossRef]  

9. M. Brunotte, A. Goetzberger, and U. Blieske, “Two-stage concentrator permitting concentration factors up to 300X with one-axis tracking,” J. Sol. Energy 56(3), 285–300 (1996). [CrossRef]  

10. T. Cooper, G. Ambrosetti, A. Pedretti, and A. Steinfeld, “Theory and design of line-to-point focus solar concentrators with tracking secondary optics,” J. Appl. Opt. 52(35), 8586–8616 (2013). [CrossRef]  

11. T. Cooper, M. Pravettoni, M. Cadruvi, G. Ambrosetti, and A. Steinfeld, “The effect of irradiance mismatch on a semi-dense array of triple-junction concentrator cells,” J. Sol. Energy Mater. Sol. Cells 116, 238–251 (2013). [CrossRef]  

12. E. Hecht, Optics, 4th ed., (Pearson Education, Inc., 2002).

13. M. Polyanskiy, “Optical Constants of Plastics,” (2015). http://refractiveindex.info/?shelf=3d&book=plastics&page=pmma

References

  • View by:

  1. H. Price, E. Lüpfert, D. Kearney, E. Zarza, G. Cohen, R. Gee, and R. Mahoney, “Advances in Parabolic Trough Solar Power Technology,” J. Sol. Energy Eng. 124(2), 109–125 (2002).
    [Crossref]
  2. F. Kreith and Y. Goswami, Handbook of Energy Efficiency and Renewable Energy (CRC, 2007).
    [Crossref]
  3. J. Kreith and F. Kreider, Principles of Solar Engineering (Hemisphere Publishing Corporation, 1978).
  4. J. Duffie and W. Beckman, Solar Engineering of Thermal Processes (John Wiley & Sons, Inc., 1991).
  5. R. Leutz and A. Suzuki, Nonimaging Fresnel Lenses: Design and Performance of Solar Concentrators (Springer, 2001).
    [Crossref]
  6. Sh. Klychev, “A Method to Calculate Fresnel Lenses,” J. Appl. Sol. Energy 49(1), 36–41 (2012).
  7. W. Xie, Y. Dai, R. Wang, and K. Sumathy, “Concentrated solar energy applications using Fresnel lenses: a review,” J. Renew. Sustain. Energy Rev. 15(6), 2588–2606 (2011).
    [Crossref]
  8. A. Mohr, T. Roth, and S. Glunz, “BICON: High concentration PV using one-axis tracking and silicon concentrator cells,” Prog. Photovolt. 14(7), 663–674 (2006).
    [Crossref]
  9. M. Brunotte, A. Goetzberger, and U. Blieske, “Two-stage concentrator permitting concentration factors up to 300X with one-axis tracking,” J. Sol. Energy 56(3), 285–300 (1996).
    [Crossref]
  10. T. Cooper, G. Ambrosetti, A. Pedretti, and A. Steinfeld, “Theory and design of line-to-point focus solar concentrators with tracking secondary optics,” J. Appl. Opt. 52(35), 8586–8616 (2013).
    [Crossref]
  11. T. Cooper, M. Pravettoni, M. Cadruvi, G. Ambrosetti, and A. Steinfeld, “The effect of irradiance mismatch on a semi-dense array of triple-junction concentrator cells,” J. Sol. Energy Mater. Sol. Cells 116, 238–251 (2013).
    [Crossref]
  12. E. Hecht, Optics, 4th ed., (Pearson Education, Inc., 2002).
  13. M. Polyanskiy, “Optical Constants of Plastics,” (2015). http://refractiveindex.info/?shelf=3d&book=plastics&page=pmma

2013 (2)

T. Cooper, G. Ambrosetti, A. Pedretti, and A. Steinfeld, “Theory and design of line-to-point focus solar concentrators with tracking secondary optics,” J. Appl. Opt. 52(35), 8586–8616 (2013).
[Crossref]

T. Cooper, M. Pravettoni, M. Cadruvi, G. Ambrosetti, and A. Steinfeld, “The effect of irradiance mismatch on a semi-dense array of triple-junction concentrator cells,” J. Sol. Energy Mater. Sol. Cells 116, 238–251 (2013).
[Crossref]

2012 (1)

Sh. Klychev, “A Method to Calculate Fresnel Lenses,” J. Appl. Sol. Energy 49(1), 36–41 (2012).

2011 (1)

W. Xie, Y. Dai, R. Wang, and K. Sumathy, “Concentrated solar energy applications using Fresnel lenses: a review,” J. Renew. Sustain. Energy Rev. 15(6), 2588–2606 (2011).
[Crossref]

2006 (1)

A. Mohr, T. Roth, and S. Glunz, “BICON: High concentration PV using one-axis tracking and silicon concentrator cells,” Prog. Photovolt. 14(7), 663–674 (2006).
[Crossref]

2002 (1)

H. Price, E. Lüpfert, D. Kearney, E. Zarza, G. Cohen, R. Gee, and R. Mahoney, “Advances in Parabolic Trough Solar Power Technology,” J. Sol. Energy Eng. 124(2), 109–125 (2002).
[Crossref]

1996 (1)

M. Brunotte, A. Goetzberger, and U. Blieske, “Two-stage concentrator permitting concentration factors up to 300X with one-axis tracking,” J. Sol. Energy 56(3), 285–300 (1996).
[Crossref]

Ambrosetti, G.

T. Cooper, G. Ambrosetti, A. Pedretti, and A. Steinfeld, “Theory and design of line-to-point focus solar concentrators with tracking secondary optics,” J. Appl. Opt. 52(35), 8586–8616 (2013).
[Crossref]

T. Cooper, M. Pravettoni, M. Cadruvi, G. Ambrosetti, and A. Steinfeld, “The effect of irradiance mismatch on a semi-dense array of triple-junction concentrator cells,” J. Sol. Energy Mater. Sol. Cells 116, 238–251 (2013).
[Crossref]

Beckman, W.

J. Duffie and W. Beckman, Solar Engineering of Thermal Processes (John Wiley & Sons, Inc., 1991).

Blieske, U.

M. Brunotte, A. Goetzberger, and U. Blieske, “Two-stage concentrator permitting concentration factors up to 300X with one-axis tracking,” J. Sol. Energy 56(3), 285–300 (1996).
[Crossref]

Brunotte, M.

M. Brunotte, A. Goetzberger, and U. Blieske, “Two-stage concentrator permitting concentration factors up to 300X with one-axis tracking,” J. Sol. Energy 56(3), 285–300 (1996).
[Crossref]

Cadruvi, M.

T. Cooper, M. Pravettoni, M. Cadruvi, G. Ambrosetti, and A. Steinfeld, “The effect of irradiance mismatch on a semi-dense array of triple-junction concentrator cells,” J. Sol. Energy Mater. Sol. Cells 116, 238–251 (2013).
[Crossref]

Cohen, G.

H. Price, E. Lüpfert, D. Kearney, E. Zarza, G. Cohen, R. Gee, and R. Mahoney, “Advances in Parabolic Trough Solar Power Technology,” J. Sol. Energy Eng. 124(2), 109–125 (2002).
[Crossref]

Cooper, T.

T. Cooper, M. Pravettoni, M. Cadruvi, G. Ambrosetti, and A. Steinfeld, “The effect of irradiance mismatch on a semi-dense array of triple-junction concentrator cells,” J. Sol. Energy Mater. Sol. Cells 116, 238–251 (2013).
[Crossref]

T. Cooper, G. Ambrosetti, A. Pedretti, and A. Steinfeld, “Theory and design of line-to-point focus solar concentrators with tracking secondary optics,” J. Appl. Opt. 52(35), 8586–8616 (2013).
[Crossref]

Dai, Y.

W. Xie, Y. Dai, R. Wang, and K. Sumathy, “Concentrated solar energy applications using Fresnel lenses: a review,” J. Renew. Sustain. Energy Rev. 15(6), 2588–2606 (2011).
[Crossref]

Duffie, J.

J. Duffie and W. Beckman, Solar Engineering of Thermal Processes (John Wiley & Sons, Inc., 1991).

Gee, R.

H. Price, E. Lüpfert, D. Kearney, E. Zarza, G. Cohen, R. Gee, and R. Mahoney, “Advances in Parabolic Trough Solar Power Technology,” J. Sol. Energy Eng. 124(2), 109–125 (2002).
[Crossref]

Glunz, S.

A. Mohr, T. Roth, and S. Glunz, “BICON: High concentration PV using one-axis tracking and silicon concentrator cells,” Prog. Photovolt. 14(7), 663–674 (2006).
[Crossref]

Goetzberger, A.

M. Brunotte, A. Goetzberger, and U. Blieske, “Two-stage concentrator permitting concentration factors up to 300X with one-axis tracking,” J. Sol. Energy 56(3), 285–300 (1996).
[Crossref]

Goswami, Y.

F. Kreith and Y. Goswami, Handbook of Energy Efficiency and Renewable Energy (CRC, 2007).
[Crossref]

Hecht, E.

E. Hecht, Optics, 4th ed., (Pearson Education, Inc., 2002).

Kearney, D.

H. Price, E. Lüpfert, D. Kearney, E. Zarza, G. Cohen, R. Gee, and R. Mahoney, “Advances in Parabolic Trough Solar Power Technology,” J. Sol. Energy Eng. 124(2), 109–125 (2002).
[Crossref]

Klychev, Sh.

Sh. Klychev, “A Method to Calculate Fresnel Lenses,” J. Appl. Sol. Energy 49(1), 36–41 (2012).

Kreider, F.

J. Kreith and F. Kreider, Principles of Solar Engineering (Hemisphere Publishing Corporation, 1978).

Kreith, F.

F. Kreith and Y. Goswami, Handbook of Energy Efficiency and Renewable Energy (CRC, 2007).
[Crossref]

Kreith, J.

J. Kreith and F. Kreider, Principles of Solar Engineering (Hemisphere Publishing Corporation, 1978).

Leutz, R.

R. Leutz and A. Suzuki, Nonimaging Fresnel Lenses: Design and Performance of Solar Concentrators (Springer, 2001).
[Crossref]

Lüpfert, E.

H. Price, E. Lüpfert, D. Kearney, E. Zarza, G. Cohen, R. Gee, and R. Mahoney, “Advances in Parabolic Trough Solar Power Technology,” J. Sol. Energy Eng. 124(2), 109–125 (2002).
[Crossref]

Mahoney, R.

H. Price, E. Lüpfert, D. Kearney, E. Zarza, G. Cohen, R. Gee, and R. Mahoney, “Advances in Parabolic Trough Solar Power Technology,” J. Sol. Energy Eng. 124(2), 109–125 (2002).
[Crossref]

Mohr, A.

A. Mohr, T. Roth, and S. Glunz, “BICON: High concentration PV using one-axis tracking and silicon concentrator cells,” Prog. Photovolt. 14(7), 663–674 (2006).
[Crossref]

Pedretti, A.

T. Cooper, G. Ambrosetti, A. Pedretti, and A. Steinfeld, “Theory and design of line-to-point focus solar concentrators with tracking secondary optics,” J. Appl. Opt. 52(35), 8586–8616 (2013).
[Crossref]

Pravettoni, M.

T. Cooper, M. Pravettoni, M. Cadruvi, G. Ambrosetti, and A. Steinfeld, “The effect of irradiance mismatch on a semi-dense array of triple-junction concentrator cells,” J. Sol. Energy Mater. Sol. Cells 116, 238–251 (2013).
[Crossref]

Price, H.

H. Price, E. Lüpfert, D. Kearney, E. Zarza, G. Cohen, R. Gee, and R. Mahoney, “Advances in Parabolic Trough Solar Power Technology,” J. Sol. Energy Eng. 124(2), 109–125 (2002).
[Crossref]

Roth, T.

A. Mohr, T. Roth, and S. Glunz, “BICON: High concentration PV using one-axis tracking and silicon concentrator cells,” Prog. Photovolt. 14(7), 663–674 (2006).
[Crossref]

Steinfeld, A.

T. Cooper, G. Ambrosetti, A. Pedretti, and A. Steinfeld, “Theory and design of line-to-point focus solar concentrators with tracking secondary optics,” J. Appl. Opt. 52(35), 8586–8616 (2013).
[Crossref]

T. Cooper, M. Pravettoni, M. Cadruvi, G. Ambrosetti, and A. Steinfeld, “The effect of irradiance mismatch on a semi-dense array of triple-junction concentrator cells,” J. Sol. Energy Mater. Sol. Cells 116, 238–251 (2013).
[Crossref]

Sumathy, K.

W. Xie, Y. Dai, R. Wang, and K. Sumathy, “Concentrated solar energy applications using Fresnel lenses: a review,” J. Renew. Sustain. Energy Rev. 15(6), 2588–2606 (2011).
[Crossref]

Suzuki, A.

R. Leutz and A. Suzuki, Nonimaging Fresnel Lenses: Design and Performance of Solar Concentrators (Springer, 2001).
[Crossref]

Wang, R.

W. Xie, Y. Dai, R. Wang, and K. Sumathy, “Concentrated solar energy applications using Fresnel lenses: a review,” J. Renew. Sustain. Energy Rev. 15(6), 2588–2606 (2011).
[Crossref]

Xie, W.

W. Xie, Y. Dai, R. Wang, and K. Sumathy, “Concentrated solar energy applications using Fresnel lenses: a review,” J. Renew. Sustain. Energy Rev. 15(6), 2588–2606 (2011).
[Crossref]

Zarza, E.

H. Price, E. Lüpfert, D. Kearney, E. Zarza, G. Cohen, R. Gee, and R. Mahoney, “Advances in Parabolic Trough Solar Power Technology,” J. Sol. Energy Eng. 124(2), 109–125 (2002).
[Crossref]

J. Appl. Opt. (1)

T. Cooper, G. Ambrosetti, A. Pedretti, and A. Steinfeld, “Theory and design of line-to-point focus solar concentrators with tracking secondary optics,” J. Appl. Opt. 52(35), 8586–8616 (2013).
[Crossref]

J. Appl. Sol. Energy (1)

Sh. Klychev, “A Method to Calculate Fresnel Lenses,” J. Appl. Sol. Energy 49(1), 36–41 (2012).

J. Renew. Sustain. Energy Rev. (1)

W. Xie, Y. Dai, R. Wang, and K. Sumathy, “Concentrated solar energy applications using Fresnel lenses: a review,” J. Renew. Sustain. Energy Rev. 15(6), 2588–2606 (2011).
[Crossref]

J. Sol. Energy (1)

M. Brunotte, A. Goetzberger, and U. Blieske, “Two-stage concentrator permitting concentration factors up to 300X with one-axis tracking,” J. Sol. Energy 56(3), 285–300 (1996).
[Crossref]

J. Sol. Energy Eng. (1)

H. Price, E. Lüpfert, D. Kearney, E. Zarza, G. Cohen, R. Gee, and R. Mahoney, “Advances in Parabolic Trough Solar Power Technology,” J. Sol. Energy Eng. 124(2), 109–125 (2002).
[Crossref]

J. Sol. Energy Mater. Sol. Cells (1)

T. Cooper, M. Pravettoni, M. Cadruvi, G. Ambrosetti, and A. Steinfeld, “The effect of irradiance mismatch on a semi-dense array of triple-junction concentrator cells,” J. Sol. Energy Mater. Sol. Cells 116, 238–251 (2013).
[Crossref]

Prog. Photovolt. (1)

A. Mohr, T. Roth, and S. Glunz, “BICON: High concentration PV using one-axis tracking and silicon concentrator cells,” Prog. Photovolt. 14(7), 663–674 (2006).
[Crossref]

Other (6)

E. Hecht, Optics, 4th ed., (Pearson Education, Inc., 2002).

M. Polyanskiy, “Optical Constants of Plastics,” (2015). http://refractiveindex.info/?shelf=3d&book=plastics&page=pmma

F. Kreith and Y. Goswami, Handbook of Energy Efficiency and Renewable Energy (CRC, 2007).
[Crossref]

J. Kreith and F. Kreider, Principles of Solar Engineering (Hemisphere Publishing Corporation, 1978).

J. Duffie and W. Beckman, Solar Engineering of Thermal Processes (John Wiley & Sons, Inc., 1991).

R. Leutz and A. Suzuki, Nonimaging Fresnel Lenses: Design and Performance of Solar Concentrators (Springer, 2001).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 (a) Line-to-spot solar tracking concentrator and (b) inner/outer section of the lens
Fig. 2
Fig. 2 Sun’s rays refraction through the two-section FL
Fig. 3
Fig. 3 (a) Ray trace simulation from the PTC to the inner section of the lens and (b) refraction of the sun’s rays through the inner section (outer section not shown)
Fig. 4
Fig. 4 γ deviation of the sun’s rays through the lens thickness using (a) s = 0.001m, (b) s = 0.005m and (c) s = 0.01m
Fig. 5
Fig. 5 (a) Solar spectral irradiance using an air mass coefficient of 1.5 (AM1.5) [3]. For illustration purposes, the infrared bandwidth (from λ0 = 0.790 to λf = 1.083) accepted by the PMMA is shown (see Table 1). (b) Refractive index of PMMA as a function of the wavelength
Fig. 6
Fig. 6 (a) Line-to-spot ray-trace simulation where the incoming sun’s rays are placed near the PTC for illustration purposes. (b) Focal plane close up
Fig. 7
Fig. 7 (a–g) Solar spectrum bandwidth distribution on the spot and (h) spot image including all bandwidths. The colorbars represent the intensity of the energy in W/m2 distributed over the focal plane (see section 2.1.2 for further information)
Fig. 8
Fig. 8 Transmittance of the solar radiation irradiance over the spot’s area as a function of the (a) bandwidths and (b) full solar spectrum

Tables (4)

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Table 1 Ray-traced solar spectrum bandwidths

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Table 2 Transmittances values of every bandwidth and full solar spectrum

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Table 3 Average incident solar irradiance Īsun as a function of bandwidths

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Table 4 Spot’s solar irradiance from every bandwidth and full solar spectrum

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

tan ε = R / [ n PMMA ( R 2 + f in , out 2 ) 1 / 2 f in , out ]
tan ψ i = a i / ( 2 f i )
n air sin ω = n PMMA sin η
tan γ = ( r v 2 + r w 2 ) 1 / 2 / r u
n PMMA = 1.4681 + 93.42 / ( λ 1 , 235 )
τ = τ PTC τ FL
r in = tan 2 ( η ω ) / tan 2 ( η + ω )
r in = sin 2 ( η ω ) / sin 2 ( η + ω )
τ FL = [ 1 ( r in + r in ) / 2 ] [ 1 ( r out + r out ) / 2 ]
I spot = I ¯ sun τ

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