## Abstract

We give the theoretical limit of concentration allowed by nonimaging optics for stationary solar concentrators after reviewing sun–earth geometry in direction cosine space. We then discuss the design principles that we follow to approach the maximum concentration along with examples including a hollow CPC trough, a dielectric CPC trough, and a 3D dielectric stationary solar concentrator which concentrates sun light four times (4x), eight hours per day year around.

© 2010 OSA

## 1. Introduction

Recently there has been a renewed interest in using optical concentrators in solar energy applications, such as concentrating photovoltaic (CPV) and concentrating solar thermal (CST). The motivation of CPV is to lower the system cost by replacing expensive photovoltaic cells with relatively inexpensive optics. Whereas in CST concentrating solar energy is necessary in order to achieve high temperature. A higher concentration is in general preferred. The maximum achievable concentration is subject to the sine-law of nonimaging optics [1], which relates the upper limit of the concentration to sine of the acceptance angle of the concentrator. The larger the concentration, the smaller must be the acceptance angle. Because the sun occupies a large portion of the sky over the year, high concentration systems often implement solar trackers such that concentrators are still able to “see” the sun all the time even though their acceptance angle is small. Another advantage of tracking the sun is that the input aperture of the concentrator is perpendicular to the incident rays so that the cosine factor of the incident solar irradiance is maximized. However, using trackers is not always practical. Precise solar trackers are sufficiently expensive that the added cost may not justify the cost saving from photovoltaic cells; applying trackers adds system complexity and makes installation difficult; moving mechanical parts have the potential of decreasing system reliability; trackers themselves consume electric energy; and there are applications, such as residential roof top, where implementing solar trackers may be problematic. Thus non-tracking solutions are always attractive, especially when high concentration is not critical.

Stationary solar concentrators have been explored previously [2,3]. The theoretical maximum concentration for a hollow concentrator (consists of a medium with refractive index n = 1) was typically determined between 1.5x and 2x depending on how many hours of sun light one wishes to collect per day. It is commonly accepted that by filling the hollow concentrator with a dielectric medium (with refractive index n>1) the theoretical concentration limit can be increased by n (for 2D concentrators) or n^{2} (for 3D concentrators) times. However the dielectric case has not been as thoroughly explored [5]. Except for heuristic arguments, it remains a question how high the concentration can be for a practical dielectric design. Moreover, any solar concentrators useful in real-world applications must conform to large-scale manufacturing techniques and stringent budget constraints. These set new constraints on possible optical designs, including but not limited to geometric shape, size and weight. This is especially relevant for the dielectric case as the amount of material used has a big impact on the processing time and product cost.

In this paper we first give a brief review of sun–earth geometry, following the beautiful exposition in [4]. Based on this, we calculate the theoretical limits to concentration allowed by nonimaging optics for stationary solar concentrators. Unlike the previous analysis commonly performed in real space, ours is carried out in direction cosine space on a polar plane (which will be defined shortly). We then use several examples to explain the design principles that we follow to approach the maximum concentration. The examples include a hollow CPC trough [6], a dielectric CPC trough [5], and a 3D dielectric stationary solar concentrator which is designed with practical considerations in mind and concentrates sun-light four times (4x), eight hours per day year around.

## 2. Sun-earth geometry in direction cosine space

Sun-earth geometry has been well-studied and detailed explanation can be found in [4,7]. Here we put together a summary before introducing direction cosine space. A Cartesian coordinate system can be set up fixed at any location on the earth as shown in Fig. 1
, where the x axis coincides with the polar axis, the y axis points west, and the z axis points to the sun at noon on equinox. We call the xy plane the polar plane. The direction of the sun over the year can then be characterized by a unit vector n_{s} pointing from the earth to the sun, with

Let L and M be the direction cosines corresponding to x and y, respectively. The direction of the sun can now be represented in the direction cosine space by a point$(\mathrm{sin}\delta ,\mathrm{cos}\delta \mathrm{sin}\omega )$. As shown in Fig. 2 , where the black circle is the unit circle, the sun’s directions during a day follows a straight line parallel to the M axis limited by the unit circle. The can be seen form Eq. (2), where the L coordinate of the sun, $\mathrm{sin}\delta ,$ is a constant for a given day (n constant). Another way to see this follows. Since the earth's motion through space is a fast rotation about its spin axis (period of one day) and a slow rotation about the sun (period of one year), and the spin axis is invariant, choosing one of the coordinate axes (in our case, the x axis) along the spin ensures that the direction cosine is nearly constant over a day. The two green lines ($L=\pm \mathrm{sin}23.45\xb0$), corresponding to the summer and winter solstice, set the boundaries along the L direction. For demonstration, we also plot the trajectories of the sun’s directions on the first day of each month by the vertical black lines in Fig. 2. A useful expression is:

This shows that the trajectory of the sun’s directions for fixed hours per day (i.e., ω is constant) belongs to an ellipse which has a major radius of 1 along the L direction and a minor radius sinω along the M direction. The red line in Fig. 2 shows such an ellipse corresponding to eight hours per day. Notice that only the portions inside the two green lines, i.e., the red solid lines, represent the sun’s directions. The unit circle itself is such a trajectory for twelve hours a day, which corresponds to sunrise and sunset at the equator. At other locations where the latitude ϕ (negative for south hemisphere and positive for north hemisphere) is nonzero, the sunrise/sunset trajectory becomes another ellipse with the major radius being 1 along the M direction, and minor radius cos(ϕ) along the L direction, or

This can be derived by making use of the angle ϕ made by the horizon and the polar plane. The sunrise/sunset ellipse at the latitude of 40 degree in the north hemisphere is shown in blue color in Fig. 2. Despite the appeared symmetric looking of the ellipse, the sun actually goes “behind” the polar plane when L > 0, in other words, the direction cosine along the z axis becomes negative. This corresponds to the fact that day time is longer than 12 hours in summer.## 3. Theoretical limits to concentration

We are discussing the theoretical maximum concentration allowed under the approximation of geometrical optics. One of the fundamental principles of nonimaging optics is the conservation of the generalized étendue [1,3], or

where in the case of a concentrator (x, y) and (x’, y’) are the coordinates for the input and output surfaces, (L,M) and (L’,M’) are the direction cosines of the input and output rays, and n and n’ are the refractive indices of the media at the input and output respectively. If the distributions of the direction cosines have no spatial dependence on the input and output surfaces and the refractive indices are constant, then the concentration of the system can be written asNow it becomes clear why it is advantageous to make plots in the direction cosine space such as Fig. 2. ∫ dLdM and ∫ dL’dM’ can be directly computed by the areas of the regions corresponding to the incident and output rays.

For example, consider a hollow stationary solar concentrator with n’ = n = 1. Assuming there is no limit on the direction of the output rays, then ∫ dL’dM’ is the area of the unit circle, equal to π. In order to collect the sun light 12 hours a day year-around, ∫ dLdM must includes the area enclosed by the unit circle and the two green lines in Fig. 2. The result is

where δ’_{s}is the modified declination on solstice, including the declination on solstice of 23.45 degrees and the half angular size of the sun which is 0.27 degree. Therefore the maximum achievable concentration is π / 1.56 = ~2. More generally, if our goal is to collect sun light for 2t hours per day, then the above equation becomes

An important way to further increase the theoretical maximum concentration is to use dielectric medium. If the refractive index of the medium is n’ = 1.5, then in theory the maximum concentration can be increased by a factor of n’^{2} = 2.25 using a 3D concentrator, so the limit is 2.3x2.25 = ~5.2.

## 4. Concentrator design

In this section we use several examples to discuss the design principles that we follow to approach the maximum concentration. In particular, we will show why an ideal 2D compound parabolic concentrator (CPC) or CPC trough falls short of maximum concentration, followed by introducing a dielectric stationary solar concentrator which not only has higher concentration but also is designed under practical considerations. We set the solar collection time be eight hours per day, which is sufficient to collect the majority of solar energy at most locations on the earth.

#### 4.1 Hollow CPC trough

The CPC trough is, by now a well-known design in nonimaging optics [6]. It’s an ideal concentrator in 2D, outperforming the conventional simple parabolic trough by a factor of two or more. If the acceptance angle of a hollow CPC trough is θ_{1} and no limitation is placed on the direction of the output rays, then its concentration C = 1/sin θ_{1} which is the maximum achievable in theory. Let the x axis be transverse to the trough and y axis along the trough (z axis is thus normal to the input surface). The acceptance angle in 3D follows from the fact that the CPC trough accepts all rays whose projected angle in the x, z plane are ≤ θ_{1}. This condition can be written as

In the LM plane this is the region inside an ellipse which has a major radius of 1 along the M direction and a minor radius sin θ_{1} along the L direction, such as the yellow ellipse in Fig. 3
, where an implication is that the trough is along east–west direction and is tilted such that the input surface faces the equator with tilt angle equal to the local latitude.

To use the CPC trough as a stationary solar concentrator, this region must be sufficiently large to include the area corresponding to sun’s directions during the desired collecting time period. One immediate conclusion is that it’s impossible to get twelve hours per day using hollow stationary CPV trough. At the time the CPC ellipse covers the area between the two green solstice lines and the circle, the ellipse overlaps the circle, meaning there is no concentration at all. For eight hours’ collection per day, the smallest ellipse is shown in Fig. 3, where the ellipse (yellow) goes just through the four corners of the area covered by the sun (the intersecting points by the solstice lines (green) and the eight-hour lines (red)). This ellipse corresponds to the one with the smallest acceptance angle among all the eligible hollow stationary CPC troughs, and the acceptance angle θ_{1} can be solved analytically by Eq. (2), (4) and (11), yielding

For eight hours per day, this gives θ_{1} = ~42 degrees. Its concentration C = 1 / sin(θ_{1}) = ~1.5, about 65% of the theoretical maximum of 2.3. The fact that the concentration falls short of the theoretical maximum can be visually verified by the oversized area of the CPC ellipse with respect to the sun’s coverage. Based on Eq. (7), the maximum concentration is only achieved when the two regions coincide.

While a concentration of 1.5x is relatively small, it is sufficient to enable some CST applications with medium temperature up to 200° C, such as solar cooling and process heat. However higher concentration is required for applications such as CPV and high temperature CST.

#### 4.2 Dielectric CPC trough

A well-known method of increasing the concentration of a hollow stationary CPC is to fill it with a dielectric medium which has a refractive index n > 1. One would then expect that the concentration of the CPC is increased by a factor of n since the trough has a 2D structure. Here we will show that the effective concentration is increased more than just n times, due to a subtle factor which we believe has not hitherto received sufficient attention.

We plot the angular acceptance of the dielectric CPC trough in the direction cosine space. The analysis follows section 5.2 in [3]. Inside the dielectric the acceptance region is an ellipse like the hollow CPC. At the input interface, Snell’s law is

where (L, M) and (L_{n}, M

_{n}) are the direction cosines just outside and inside the dielectric medium, respectively. Therefore the acceptance region outside the dielectric medium is a portion of a scaled-up ellipse (scale factor n) intersected by the unit circle. The analytic form of this scaled-up ellipse is:

_{1}is the acceptance angle of the dielectric CPC trough. Figure 4 shows an example of this ellipse in blue with n = 1.5 being assumed.

Figure 5
shows the ellipse (in blue) corresponding to a dielectric stationary CPC trough designed for collecting sun light eight hours per day. The design principle is the same as the hollow case, i.e., let the ellipse cover the area corresponding to the sun’s directions. The acceptance angle θ_{1} can be solved analytically by Eq. (2), (4) and (14), yielding

For eight hours per day, this requires θ_{1} = ~29 degrees. Its concentration C = n / sin (θ_{1}) = ~3.2. Recall that the concentration of the hollow CPC is C_{hollow} = 1.5. So this is an increase of 3.2/1.5 = 2.1 times, more than just n ( = 1.5) times as commonly expected for 2D concentrators. This may seem contradictory to the principles of nonimaging optics, but it is not. Although the trough is a 2D structure in the geometrical sense, it’s been used as a 3D concentrator for concentrating sun light without tracking. So the theoretical maximum concentration can be n^{2} times higher instead of just n times. The direct reason for this extra enhancement is related to the increased acceptance of the skew rays, a property first derived analytically by R. S. Scharlack [8], This can be readily visualized in direction cosine space as shown in Fig. 4 (from Fig. 5.3 of Ref [9].), where the relative width of the acceptance region when M ≠ 0 can be seen to be larger for the dielectric CPC (blue) than the hollow CPC (yellow). As a consequence, the shape of the acceptance area of the dielectric CPC is a better fit to the solar motion (Fig. 5).

#### 4.3 Concentrators with 3D structures

As discussed before, the ultimate concentration limit for eight hours per day is ~5.2 (n = 1.5). This requires an ideal 3D dielectric concentrator whose acceptance in the LM plane matches exactly to the sun’s directions. With this in mind, we find that there is still room for further increase of concentration for the dielectric CPC trough, especially along the trough direction as indicated by Fig. 5, where the blue ellipse exceeds the sun’s region by a considerable amount in the M direction. The oversized part can be identified as two regions, region 1: the part between the red line and the unit circle, and region 2: the part outside the unit circle. Region 1 corresponds to the sun light beyond eight hours per day. Region 2 is unphysical in the LM space and corresponds to the nonexistent rays whose angle exceeds the critical angle inside the dielectric medium. Limited by its 2D structure, the CPC trough has to include these rays in the output étendue space, causing a waste of concentration. To regain the concentration, a 3D structure is necessary.

Unfortunately in general there is no known prescription to design an ideal 3D concentrator for a given shape in LM space. Here we take a practical strategy. We simply combine another CPC trough with the current one in such way that the two troughs are perpendicular with each other and their output surfaces are aligned. We then take the overlapping part of the two troughs as our new 3D concentrator (referring to Fig. 6 ). This kind of structure has been mentioned in [6]. The idea is to use the added CPC to regain the otherwise wasted concentration along the M direction. The result won’t be ideal because the CPC is usually designed for meridian rays and the behavior of the skew rays can only known by a ray tracing afterwards. Nevertheless, at least part of the wasted concentration shall be saved. We will show one of such design below.

#### 4.4 Practical issues

A disadvantage of CPC in its original form is that the parabolic surface is very tall compared to the scale of the collecting aperture, requiring a lot of material to build. The problem is even worse for the dielectric CPC because the inside must be filled with solid material. A heavy and bulky concentrator is not only hard to install, but also cost more in raw material and manufacturing. A truncated CPC gives up very little concentration in exchange for a considerable reduction of the height of the parabolic surface [10]. Such a tradeoff is often preferred by economy.

There is another tradeoff between concentration and cost. In its simplest form, the dielectric CPC may require reflective coating on the parabolic surfaces near the bottom, requiring an extra manufacturing process and material cost. This can be eliminated by using the CPC with total internal reflection (TIR) [5]. Again, the price paid is a very slight reduction in concentration.

#### 4.5 A 4X Stationary concentrator

Following the above design principles, we designed a 4X stationary concentrator as shown in Fig. 6, where the units of the dimensions are in mm. The two pairs of walls belong to two truncated CPCs with TIR, with their acceptance angle being 29 degree and 60 degree respectively. No reflective coating is needed due to TIR design. The size of the output aperture is chosen such that PV cell strips can be attached for low concentration photovoltaic application. The height of the concentrator is 9 mm and the volume is 511 mm^{3}. If we use PMMA (density = ~1200 Kg/m^{3}) as the dielectric medium, the concentrator weights about 0.61g each piece, or 7.6 Kg/m^{2} for densely packed arrays.

The optical performance is evaluated by a ray tracing program with a source model which simulates the light rays from the sun eight hours per day for a whole year. The source implements the standard AM1.5G solar spectrum and takes into account the cosine effect of the incident solar irradiance. As an average about 95% of the incident solar energy is collected if we don’t consider the Fresnel loss on the top surface. Out of the uncollected solar energy, about 1% is absorbed by PMMA, the rest 4% is due to ray rejection as a consequence of the non-ideal design of the concentrator. The number of rays that are rejected is a trade-off with the concentration. One extreme case is the trough structure with which the ray rejection is zero, but the concentration goes down to about 3. With the top surface Fresnel loss included, the efficiency of the 4x concentrator becomes 91%.

The light weight and low profile of this concentrator make it well-suitable for large volume manufacture process such as molding. Alternatively, one pair of walls can be shaped by extrusion, followed by shaping the other pair by precise cutting or other tooling. This may be more economic because extrusion costs less than molding.

In an elegant paper Yablonovitch and Cody [11] suggest a 4 n^{2 }enhancement in a solar cell device. Their factor 4 is distinct from ours and would provide an enhancement of a factor 4 *in addition to *our four-fold concentration under the conditions envisioned in their paper. However, the n^{2 }factor in both papers has a common origin.

## 5. Conclusion

Stationary concentrator is attractive because of no requirement of trackers. The maximum concentration is subject to the laws of nonimaging optics and is limited by sun-earth geometry. In direction cosine space the maximum concentration is equal to the ratio of the area of the unit circle to the area corresponding to sun’s directions over the intended collection time duration, multiplied by the square of the refractive index of the medium. The design principle is thus to maximally match the acceptance of concentrators to the sun’s directions in direction cosine space and to use media with high refractive indices, although the best solution may be the result of a tradeoff of concentration with practical constrains such as cost, weight, and size.

## Acknowledgments

We thank project DEDALOS for contributing to support of this research.

## References and Links

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