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Linear compound parabolic concentrators with cylindrical receivers are often combined with evacuated tubes along their focal length to suppress convective heat loss for use as thermal collectors. When investigating the optical efficiency of such collectors it is important to consider the reflection loss introduced by the evacuated tube particularly at large angles of incidence of light onto the CPC aperture. In this paper reflection losses are determined using ray-tracing as a function of the angle of incidence in both the longitudinal and transversal planes of a CPC. The reflection losses are found to be approximately constant except close to the maximum acceptance angle.
© 2015 Optical Society of America
School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, 115 Queensberry Street, Carlton, VIC 3053, Australia*Maja.Gajic@rmit.edu.au
1. Introduction
Compound parabolic collectors (CPCs) are non-imaging concentrators that ideally accept all rays within their acceptance half angle and concentrate by the maximum allowed for by phase space conservation [1]. These maximally concentrating collectors can be designed for various shaped receivers such as flat, fin and cylindrical [2]. In solar thermal collectors, high temperatures can be reached by concentrating solar radiation onto receivers and CPCs have been shown to be able to achieve this without tracking [3]. Of particular interest for thermal applications is the two dimensional CPC designed for a cylindrical absorber, also known as a CPC with a tubular receiver or involute shaped CPC. Some collector designs are not evacuated [4,5] however to suppress convective heat loss either an evacuated tube can be placed along the focal length of the CPC [6], the combination of reflecting walls and absorber can be evacuated [7] or the cavity can be sealed with a glass lid and evacuated tube [8].
Recently the number of different applications of CPCs has started to widen to include: solar energy for water purification [9] solar thermal cooling systems [10], methanol reforming [8] as well as building integrated water heaters [11]. To aid in the design of CPC collectors, ray-tracing simulations can be performed that help to predict optical performance. However to accurately simulate these types of systems several optical properties need to be known a priori.
A two dimensional CPC can be considered as trough like and this shape is ideal for thermal applications where an absorber can be placed along the focal length of the trough. A CPC concentrator in two dimensions as shown in Fig. 1 (a), with an entrance aperture d1 and absorber radius r has the geometric concentration ratio as shown in Eq. (1). is the acceptance half angle and ideally all rays within this angle will be transmitted to the cylindrical receiver by the CPC mirror walls. To minimize convective heat loss an evacuated tube can be placed around the absorber but this introduces reflection loss. From now on in this paper the term receiver will be used to describe the combination of absorbing tube (absorber) and evacuated glass envelope.
Fig. 1 (a) CPC with cylindrical receiver. (b) Longitudinal and transversal planes used for simulations.
The optical performance of thermal collectors is determined by the optical efficiency at normal incidence and the incidence angle modifier (IAM). The optical efficiency of a CPC without a glass cover is the product of and . Where <n> is the average number of reflections undergone by a ray before reaching the receiver [12],is the reflectance of the CPC mirrors and is the absorptance/transmittance product at the receiver at normal incidence [8], as shown in Eq. (2).
The IAM takes into account the change in optical efficiency as the angle of incidence of light varies due to the suns apparent motion through the sky. This includes the angular variation of:
- • mirror reflectance
- • average number of reflections undergone by rays off the CPC wall
- • absorptance of the absorber
- • gap losses due to a necessary gap between absorber and cusp of the CPC
- • angular dependence of reflectance of the evacuated tube
For translationally symmetric 2D collectors such as CPCs the incidence angle modifier can be approximated as the product of measurements in two planes, one transversal and one longitudinal, known as the biaxial approximation, described by Eq. (3) [13]. and are the angles made by the incoming sun vector projected onto the two planes shown in Fig. 1 (b). However this approximation has been shown to produce higher errors at larger incidence angles [14]. This paper investigates reflection loss from the glass envelope, a major component of the IAM, in the longitudinal and transversal planes.
Reflection losses inside the CPC with a cylindrical receiver occur as light crosses the air/glass boundary of the evacuated tube; these losses depend on the refractive index of the glass and the angle of incidence of light at the boundary. They can be described by Fresnel’s equations. For example for a flat piece of glass, reflection losses are constant until an angle of incidence of approximately 40°, after which they increase steeply [15]. However inside the CPC, light strikes the evacuated tube after one or more reflections and it is not straightforward to determine the angle of incidence of light striking the tube. Furthermore as the angle of light entering the CPC changes, the number of reflections and angle of incidence of light on the evacuated tube will also change. The values used for transmission/reflectance of glass tubes inside CPCs for performance modeling of these collectors by various researchers were reviewed and these values are summarized in Table 1. Except for [20], the values were not based on any experimental or simulation studies and resultantly the assumed values in the literature have a very large variation. The objective of this study is to determine by optical modeling the reflection losses from the evacuated tube CPC receiver over a range of incidence angle in both the transversal and the longitudinal planes.
Table 1. Summary of reflectance values used in literature.
2. Modeling reflection loss
LightTools version 8.0.0 is used to model optical performance of CPCs with cylindrical receivers. A utility function is used to create involute shaped CPCs based on the acceptance angle, radius of absorber and CPC length. The receiver is designed as shown in Fig. 2, as an absorber sitting concentrically within a glass tube with a vacuum gap between the two. To isolate the loss due to reflection, the gap between the absorber and glass envelope as well as the thickness of the glass is kept to a minimum, reflections will occur at both air/glass and air vacuum interfaces. The outer radius of the glass is equal to 20 mm with a thickness of 0.1 mm, the air gap is 0.1mm and the radius of the absorber is equal to 19.8 mm. In reality the glass will have a finite thickness that will affect the path length and potentially the angle at which rays strike the second glass/air interface, however we are assuming this will be negligible. This investigation consisted of several CPC design iterations with maximum half acceptance angles of: 20°, 30, 40°, 50° and 60°. The receiver radius was fixed so consequently the entrance aperture of the CPC was varied. The optical properties of the various components used in the model are shown in Table 2. By measuring the difference between the incident radiation at the entrance of the CPC and the absorbed radiation at the absorber, the losses due to reflection can be isolated.
Table 2. Optical properties of model
The Fresnel equations for unpolarized light used by LightTools were verified based on equations from [15]. A 10 cm x 10 cm x 1 cm block of glass with the same properties as used in the CPC model was created. Single rays were then traced at various angles of incidence and compared to equations in [15]. The Fresnel loss calculated theoretically matched closely (to within 0.89%) with values obtained by LightTools. The discrepancies can be explained due to LightTools terminating the rays when they fall below a threshold value of 0.01% of the power of the initial ray in addition to rounding errors.
Many ray tracing simulations were performed each with 40 million rays to keep the statistical errors associated with finite bin sizes on the receiver to no more than 1.1% for transversal simulations and no more than 5% for longitudinal ones. The sun source was modeled as a planar source with an airmass (AM) 1.5 spectral distribution. The reflection losses were obtained by incrementally increasing the angle of incidence from normal, in either plane. For longitudinal simulations, the end effects needed to be taken into account so the length of the CPC was set to two meters and perfectly reflecting mirrors were placed at either end.
3. Results and discussion
3.1 Reflection losses in two planes
It can be seen when considering losses in the transversal plane, shown in Fig. 3 that, for all acceptance angle CPCs, the curves have similar shapes and the reflection losses varied a small amount between 11% and 14%. Then as the maximum acceptance angle is approached the reflection losses rapidly increase as the angle of incidence that rays strike the evacuated tube increases. The reflection loss variation with longitudinal incidence angle, shown in Fig. 4, is independent of the acceptance angle. Shown in Fig. 5 is a comparison of the transmission of a 30° acceptance angle CPC and simulations performed by [21] for an evacuated tube with a cylindrical absorber (without a CPC). The inclusion of a CPC around the evacuated tube does not contribute to losses at lower longitudinal angles of incidence. At higher longitudinal angles of incidence the end effects of the CPC (radiation spilling due to inclusion of mirrors at ends) are adding to the optical losses, an effect which will not be present with an infinite CPC.
Fig. 5 Comparison of transmission of evacuated tube [21] and CPC with acceptance angle = 30° in the longitudinal plane.
It would be useful to be able to extract an average value of reflection loss however the optical behavior of a CPC collector is highly dependent on its orientation, location, tilt and azimuth. However it can be seen that in the longitudinal plane, up to an angle of approximately 40° the reflection loss is approximately constant. Similarly in the transversal plane the reflection loss is approximately constant until close to the acceptance angle.
To further investigate this reflection loss it is worth considering at what angles of incidence (AOI) rays strike the evacuated tube. In order to calculate this distribution the ray data including the ray power, direction, and location at which the ray strikes was obtained from LightTools. The AOI of each ray was calculated as shown in Eq. (4):
The mean AOI striking the tube was calculated as shown in Eq. (5):
i is the ray number that varies from 1 to m, is the direction of the ith ray, is the normal vector of the cylinder surface at the location the ray strikes and Pi is the power associated with the ith ray. The angle of incidence spans from normal to 90° and is divided into a series of one degree intervals. Due to the fact that rays can be reflected multiple times individual ray power also varies.
A CPC with acceptance angle of 30° was investigated by varying the angle of incidence of light entering the CPC aperture, for three different longitudinal angles (0°,15° and 23°) and the transversal angle was increased from normal, up until the maximum half acceptance angle of the CPC. Figure 6 shows the variation of the distribution of AOI on the tube at longitudinal angle = 0. It can be seen that for lower angles of incidence on the aperture, rays strike the tube with a wide distribution of angles and a peak around 30°. At 24° a sharp cut-off appears as rays start to strike the evacuated tube with a narrow distribution of angles. After 27° the rays are increasingly striking at a large angle of incidence until at 29° they strike the evacuated tube almost tangentially resulting in large reflection loss.
Fig. 6 Distribution of angles of incidence of light striking the receiver of a CPC with acceptance angle = 30° (longitudinal angle = 0). Starting from normal incidence, top left, the angle of light entering the CPC aperture was increased in the transversal plane.
Figure 7 shows spatial distributions of rays along the tube for the same case as Fig. 6 (longitudinal angle = 0). It can be seen that for lower angles of incidence into the CPC aperture rays are striking the tube both directly and after reflecting off the CPC walls. In this way the CPC walls acts to redistribute rays around the tube. For high angles of incidence onto the aperture it is evident the rays strike the tube in one location and the AOI distribution has narrowed.
Fig. 7 Spatial distribution of angles of incidence of light striking the receiver of a CPC with acceptance angle = 30°. Starting from normal incidence, top left, the angle of light entering the CPC aperture was varied in the transversal plane. The longitudinal angle was set to zero.
Figure 8 shows how little the reflection losses change as the longitudinal angle is increased and Fig. 9 shows the variation in average AOI on the tube for the different longitudinal angles. While there is an increase of about 10° in the average, this is not large enough to increase the reflection loss significantly. This suggests that if the CPC operates within the range of angles that show constant reflection loss in the transversal and longitudinal plane the total reflection loss will be approximately constant, in these simulations it is around 12-14% which is certainly significant.
4. Conclusion
The reflection loss from evacuated tubes inside CPCs with different acceptance angles is investigated as a function of incidence angle in the longitudinal and transversal planes. It is found that longitudinally the shape of the CPC did not contribute to reflection loss and in the transversal plane the reflection loss was approximately constant until the acceptance angle was approached. The large losses close to the acceptance angle slightly lower the real acceptance angle of the collector. On analyzing the spatial and angular distributions of incident rays on the evacuated tube, it became evident that the CPC acts to spread out the angles of incidence of light striking the evacuated tube but the mean angle of incidence at any longitudinal angle does not vary significantly until the acceptance angle is approached. As the mean angle of incidence does not vary greatly, neither does the reflection loss. As the optical behavior of a CPC collector is highly dependent on orientation, location, tilt and azimuth, the presented methodology and results would guide a collector designer in making appropriate choice of collector orientation in a way to minimize reflection losses from the evacuated tube. New improvements in anti-reflective coatings have shown promising results in reducing reflection losses over broadband AM 1.5 spectrum [22]. Such anti-reflective coatings are a promising option to reduce reflection losses from glass tubes inside CPC receivers.
Acknowledgments
This research was undertaken as part of the Micro Urban Solar Integrated Concentrators (MUSIC) project that is supported by the Australia Renewable Energy Agency (ARENA). M.G. would like thank Mr Mali Maliyagoda from Synopsys' Optical Solutions for his valuable support with LightTools.
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