OPTIMIZED 2-D SOLUTIONS FOR A LOW CONCENTRATION LINEAR NON-IMAGING FRESNEL LENS

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1 OPTIMIZED 2-D SOLUTIONS FOR A LOW CONCENTRATION LINEAR NON-IMAGING FRESNEL LENS Brian W. Raichle Department of Technology and Environmental Design raichlebw@appstate.edu James A. Russell Department of Technology and Environmental Design russellja@appstate.edu Gregory D. Norris Department of Technology and Environmental Design norrisgd@appstate.edu Kevin R. Howell Department of Technology and Environmental Design howellkr@appstate.edu ABSTRACT Non-imaging, non-tracking solar concentrating systems offer the potential for improved optical performance at a modest cost, especially considering the recent availability of inexpensive "printed" Fresnel lenses. However, high manufacturing set-up costs of these lenses make prototyping impractical. Computational modeling, therefore, is the optimization tool of choice. Fresnel lens technology offers exciting possibilities in photovoltaic, thermal, and daylighting applications. Designs which integrate these three technologies (PVTD), as well as ones which exploit the benefits of building integration and the concomitant system cost reductions, further increase the potential value of this technology. Linear Fresnel lens systems with simplified or no tracking can be designed to provide low concentration with uniform absorber irradiance. However, the optimization of such a system is a challenging problem. This research effort uses computational ray tracing to optimize the concentration and uniform illumination of a linear Fresnel lens on a flat absorber. Lens parameters considered include location, lens axis (NS/EW), radius of curvature, lens/absorber distance, absorber elevation angle, and absorber offset from the symmetry axis. Non-uniform radii of curvature will also be considered. Michalsky's sunpos code is used to generate hourly incident angles. This paper presents the results of a two dimensional optimization for a linear non-imaging Fresnel lens. Results of ray-tracing using non-paraxial optics and time- and cos( )-weighted incident radiation distributions will be shown. Time resolution is one minute. Time-lapse animation will also be shown. In addition, modeling implications and potential applications will be discussed. 1. INTRODUCTION Traditionally, tracking and concentrating solar systems have been utilized when high temperatures are necessary, such as solar thermal electric generation or process heat systems. These systems tend to employ imaging optics and therefore require precise tracking mechanisms and meticulous alignment. The use of non-imaging optics in solar harvesting systems, that is, non-tracking concentrators, has been studied for years, mainly focusing on the compound parabolic concentrator (CPC), a system identified as having complete radiation capture over a range of incident angles (1). The main advantages of non-imaging optics are a wider angular acceptance and non-tracking potential. 1

2 A Fresnel lens is another optical system that has been used in non-imaging optical systems. Fresnel lens technology has recently evolved to the point where arbitrary focal length lenses can be produced at low cost as a thin film, opening up new application possibilities. Some of these exciting new applications include building integration for day lighting or active solar, and concentrated PV or solar thermal. Custom, thin film Fresnel lenses can potentially enhance a wide range of systems. In this paper, we report on a methodology to characterize and optimize the ability of Fresnel lens-based non-imaging optical systems to collect radiation from the sun on an annual basis. This methodology considers the varying incident angles due to the sun s motion and geometrical factors of the lens system. 2. REVIEW OF PREVIOUS STUDIES The literature is full of studies of the performance of tracking optical systems. A recent review by Xie (2) of papers describing recent improvements in concentrated solar energy applications of Fresnel lenses lists over 30 papers since 2002, but only 3 of these deal with non-imaging, linear lenses the geometry that has non-tracking potential. A variety of enhancements for Fresnel lens system performance have been studied, including curved lenses (see 3, for example) and secondary concentrators (see 4, for example). 3. METHODOLOGY The object of this project is to develop a methodology that facilitates computational optimization of nonimaging optical solar systems. This modeling effort is novel in several ways. First, an exact optical treatment was followed. Second, very high temporal, and therefore, spatial resolution was made tractable by use of matrix weighting factors indexed by sun elevation and azimuth angles. Cos( ) and time weighting matrices are presented here, but any spatially-dependent parameters that effect performance of solar collecting systems can be treated in the same way. All code was written in C The system An infinitely long linear Fresnel lens was modeled. In other words, a single solar angle was considered that being the angle perpendicular to the lens symmetry axis. For a N/S lens axis, only the sun s azimuth angle was considered; for an E/W lens axis, only the sun s elevation angle was considered. This method does not take into account incident angle modifiers of any glazing or absorbers, and can provide no information about the incident vector at the absorber. Intersection locations of refracted rays on an absorber located below and parallel to the lens were accumulated. 3.2 Incident Radiation Time Ordered Michalsky's sunpos code (5) was used to generate solar altitude and azimuth angles for each minute of the year. A one-minute time basis was chosen to provide reasonable resolution. For example, a one-hour time basis provides only around 10 sun locations per day, while a one-minute time base provides around 600 sun locations per day. Currently, all radiation is assumed to be direct beam, but this is not a fundamental limitation. Sequentially, for each calculated sun position, some number of incident rays (e.g. 1,000) were uniformly distributed across the lens. The intersection of the transmitted rays and the absorber plane was calculated and an absorber histogram was accumulated. This method maintains the temporal ordering of the rays, and animations were generated to aid in the optimization process. 3.3 Incident Radiation Matrix Method Computationally, refracting 1,000 or so rays for each of the 525,600 minutes of the year is necessary to achieve high spatial resolution, but is rather tedious. A method was needed to preserve this high resolution, but speed up the calculations to allow optimization. The method that was devised is essentially a change of basis from temporal to spatial. Rather than sequentially treating each of the 525,600 minutes in a year for each run, two weighting matrices were calculated for each degree of azimuth (between 90⁰ and 270⁰) and each degree of elevation (from 0⁰ to 90⁰). A time matrix was generated that describes the time spent in each 1⁰ 1⁰ angle bin, and a cos( ) matrix was generated that allows the calculation of the component of radiation normal to the plane of aperture. The component equation is from Duffy and Beckman (6). Each matrix has 16,200 cells not a trivial number but a reduction in over 30 times from the number of minutes in a year. Further, these matrices need to be generated only once per latitude (for time matrix) and lens/absorber tilt (for cos( ) matrix). As an example, at 30⁰ latitude, the time matrix element for 170⁰ azimuth and 60⁰ elevation is , representing the fraction of the year the sun spends at this location. For a surface tilted at 30⁰, the cos( ) matrix element for 170⁰ azimuth and 60⁰ elevation is Figure 1 shows relative cos( ) weightings for surfaces with a tilt angle = 0 at various latitudes. The cos( ) weighting 2

3 factors are accumulated in sun elevation angle bins; in other words the cos( ) weighting terms for all sun azimuth angles at a particular sun elevation angle are summed. This is conceptually equivalent to summing across a typical sun path chart. elevation bins. These are independent of surface tilt. A peak in energy corresponding to the high summer sun is apparent. Fig. 3: Cumulative time weighting factors as a function of sun elevation at various latitudes. Fig. 1: Cumulative cos( ) weighting factors as a function of sun elevation for flat surfaces at various latitudes. Figure 2 shows a similar summation of cos( ) weightings for surfaces with a tilt angle = latitude at various latitudes. The enhancement in energy collection is most pronounced at lower sun angles at the higher latitudes. The obvious knee at low sun elevation angle in both Figures 1 and 2 corresponds to the noontime sun elevation at the winter solstice the lowest noontime elevation angle. Finally, Figure 4 shows the product of cos( ) time for various surface tilt angles at a latitude of 30. The most obvious behavior is the dramatic peak in energy, in this case at around 37⁰. Additional runs at various latitudes should reveal the trend. Clearly a flat surface underperforms tilted surfaces; however, all tilt angles between latitude ± 15 have comparable performance. Graphs with a finer tilt angle resolution will be generated to further investigate this behavior. Fig. 2: Cumulative cos( ) weighting factors as a function of sun elevation for surfaces tilted at latitude at various latitudes. Figure 3 shows a similar summation of time weightings at various latitudes. Still, the weights are accumulated in sun Fig. 4: Cumulative cos( ) time weighting for various surface tilt angles at 30 latitude. This matrix method can easily be extended to incorporate any spatially varying quantities that affect the performance of the solar collection system, including attenuation in the atmosphere and diffuse radiation. For example, common 3

4 sky models, including isotropic, circumsolar, or horizon brightening, can be parameterized and then used to generate an additional weighting matrix to simply include in the product. 3.4 Optics The previous section described the method in which incident rays were created. This section describes how the optics of the Fresnel lens system was handled. The exact, non-paraxial matrix method of Nussbaum and Phillips was followed (7). A Fresnel lens was simulated by assigning an upper radius of curvature of, a lens thickness of 0, and varying the lower radius of curvature. Facet angles were not considered, so in fact this method is valid for any lens system. Results for normal incidents agreed with the Fresnel lens equation of Leutz and Suzuki (8) A fixed number of rays were introduced to the lens. A histogram of hits on the absorber, weighted by the product of the time matrix and the cos( ) matrix histogram (rather than unity), was computed. This matrix methodology is generalizable. 3.5 Analysis The system s concentration ratio is calculated by where the As represents the widths (or areas in 2D) of the absorber and lens. Absorber width is a parameter that can be varied by the experimentalist. Fig. 5: An example absorber output. The rectangular histogram results from a lens aperture 50 bins wide. The broader distribution results from refraction. 4. PRELIMINARY RESULTS The model includes a Fresnel lens 50 units wide located at a height of 100 units above an absorber. An initial radius of curvature of 200 units (f = 100 units) was used. 4.1 Converging Versus Diverging Radius of Curvature The sign of the radius of curvature (and therefore focal length) was changed independently for each side of the lens. Figure 6 shows the result of these calculations for 30 latitude. The trend shown was seen in all other latitudes. A representative absorber output is shown in Figure 5. Two histograms are shown: the histogram from a no lens system, and the histogram resulting from the lens of interest. Each histogram represents the sum of weighted hits rather than simply the number of hits. In this case the absorber width is taken to be 50 bins due to the broad peak, and the calculated concentration ratio is Varied parameters include latitude, surface tilt, radius of curvature, and lens/absorber separation. Preliminary optimization considers only simple functions of radii of curvature, but any R(x) could be modeled. Fresnel lenses of arbitrary R are now physically possible to construct, and this capability is perhaps the most compelling justification of this optical analysis. Fig. 6: Comparison of variations of positive and negative radii of curvature on both sides of the lens As can be seen here, the configuration of Positive (south) and Negative (north) radii of curvature results in the highest annual solar concentration. 4

5 4.2 Incremented Radius of Curvature Values The radius of curvature was then made positive on both sides of the lens kept equal and was incremented by 40 over a wide range of values. Figure 7 shows that a radius of curvature of 240 (f slightly greater than lens/absorber distance) produces the highest annual solar concentration. A similar pattern emerges for each curvature value as well as the evidence that the location of highest concentration along the absorber is in fact dependent on the radii of curvatures of the lens when applied in this way. Changing the radius of curvature of the lens is shown to have a greater effect on the solar concentration than other geometrical parameters studied. It is observed that a tilt angle of 45 produces the highest annual solar concentration when used at a latitude of 30. A tilt angle of latitude + 15 degrees maximized concentration for all latitudes tested. 5. SUMMARY AND CONCLUSIONS A computational method was developed that allows the optimization of Fresnel lens parameters for solar energy collection. This methodology considers the time and geometrical weighting of sun position. Progress was made toward optimization, but additional parameters need to be systematically investigated with a finer resolution. The final optimization step will be to identify the ideal radius of curvature as a function of distance from the lens symmetry axis. In the future, the matrix method will be extended to include other solar radiation effects that depend on sun angle, for example diffuse beam irradiance and sky absorption. Finally, this method will be validated with experimentation. 6. REFERENCES Fig. 7: Comparison of annual solar concentration with incremented radii of curvatures 4.3 Latitude Versus Tilt-Angle In addition to the above parameters, the effect of tilt angle was investigated. Results for a latitude of 30 are shown in Figure 8. As before, the observed trend was seen for all tested latitudes. (1) Welford, W.T. and Winston, R. (1978). The Optics of Nonimaging Concentrators: Light and Solar Energy. Waltham, MA: Academic Press (2) Xei, W.T., Dai, Y.J, Wang, R.Z., and Sumathy, K. (2011). Concentrated solar energy applications using Fresnel lenses: A review, Renewable and Sustainable Energy Reviews, 15, (3) Leutz, R., Suzuki, A, Akisawa, A., and Kashiwagi, T. (1999) Design of a nonimaging Fresnel lens for solar concentration, Solar Energy, 65, (4) Collares-Pereira, M. (1979) High temperature collector with optimal concentration: Non-focusing Fresnel lens with secondary concentrator, Solar Energy, 23, (5) Michalsky, J.J. (1988) The Astronomical almanac s algorithm for approximate solar position ( ), Solar Energy, (40) Fig. 8: Comparison of the annual solar concentration as a function of tilt angles at a latitude of 30. (6) Duffy, J.A. and Beckman W.A.. (2006). Solar Engineering of Thermal Processes, 3 rd ed. Hoboken, NJ: John Wiley and Sons 5

6 (7) Nussbaum, A. and Phillips, R.A.. (1976). Contemporary Optics fof Scientists and Engineers, Solid State Physical Electronics Series. Englewood Cliffs, NJ: Prentice Hall (8) Leutz, R. and Suzuki, A. (2001). Nonimaging Fresnel Lenses: Design and Performance of Solar Concentrators, Springer Series in Optical Sciences. New York: Springer 6