Optics of Solar Concentrators. Part III: Models of Light Collection of 3D-CPCs under Direct and Lambertian Irradiation

Transcription

1 International Journal of Optics and Applications 015, 5(3): 8-10 DOI: /j.optics Optics of Solar Concentrators. Part III: Models of Light Collection of 3D-CPCs under Direct and Labertian Irradiation Antonio Parretta 1,,*, Erica Cavallari 1 ENEA C.R. E. Cleentel, Via Martiri di Monte Sole 4, Bologna (BO), Italy Physics Departent, University of Ferrara, Via Saragat 1, Ferrara (FE), Italy Abstract The optical properties of noniaging solar concentrators irradiated in direct ode by diffused Labertian beas are investigated in detail adopting original siulation ethods. These ethods ere not liited to investigate useful properties for the practical application of the concentrators, but ere also used to study the as optical eleents ith specific transission, reflection and absorption characteristics. We have investigated, therefore, besides the flux transitted to the receiver, also the flux back reflected fro input aperture and that absorbed on the all of the concentrator. We have siulated the transission, reflection and absorption efficiencies, the average nuber of reflections of the transitted or reflected rays, their angular divergence and the distribution of flux on the receiver and on the internal all surface, as function of the angular divergence of the input bea and of the reflectivity of the internal all. The presented siulation ethods can be fruitfully applied to any other type of solar concentrator. Keyords Solar concentrator, Light ection, Optical siulation, Optical odeling, Noniaging optics 1. Introduction A revie of the theoretical odels of light irradiation and ection in solar concentrators (SC) as presented in the first part of this ork [1]. In the second part [] e presented the optical siulations of those odels applied to noniaging SC irradiated by direct and iated beas. The iportance of this irradiation lies in the fact that it represents the typical operating condition of a SC. We have chosen, for the optical siulations, a class of noniaging SC of the type 3D-CPC (Three-Diensional Copound Parabolic Concentrators) [3-9] to be used ainly as priary eleents of a concentrator syste. In this paper e extend the study of these SC by investigating diffused irradiation conditions, hich are ell represented by ertian beas (beas ith constant radiance) ith variable angular aperture. The results of our ork can be extended in this ay to CPC used as secondary eleents of concentration. Our interest on these SC is that they allo to reach very high concentration levels, coparable to the theoretical ones, and that their optical transission efficiency is quite constant ithin a defined angle of incidence of the input bea. A further advantage of these SC is that they operate * Corresponding author: parretta@fe.infn.it (Antonio Parretta) Published online at Copyright 015 Scientific & Acadeic Publishing. All Rights Reserved ith reflective surfaces that do not induce spectral dispersion of light. We have siulated the optical behavior of the noniaging SC by investigating its transission, reflection and absorption properties, and e have defined ne optical quantities. In particular, e have exained: i) the transitted flux in ters of the optical transission efficiency, average nuber of internal reflections of the transitted rays, spatial distribution of the flux density on the receiver, angular distribution of radiance at the exit aperture; ii) the flux reflected fro the input aperture in ters of reflection efficiency, angular divergence and average nuber of reflections; iii) the absorbed flux in ters of absorption efficiency and distribution of the absorbed flux on the internal all of the CPC. In this ay, all the optical features of the SC, considered as a generic optical eleent interacting ith a ertian bea, ere analyzed. The ethods of siulation and elaboration of the optical data presented in this ork can be considered as general tools for the analysis of any other optical device.. The Copound Parabolic Concentrator (CPC) The Copound Parabolic Concentrator (CPC) is a noniaging concentrator developed by R. Winston [3] to efficiently ect Cherenkov radiation in high energy experients. Since then, the noniaging concentrators have

2 International Journal of Optics and Applications 015, 5(3): been idely used to concentrate sunlight [4-18]. The CPC is a reflective concentrator ith a profile obtained by the cobination of to parabolas, and is characterized by a step-like transission efficiency alloing the efficient ection of light fro 0 to a axiu angle, the acceptance angle θ acc, here the suffix eans that the irradiation is done ith a iated bea. Fig. 1 shos the 3D-CPC used in this ork for the optical siulations. It is the sae that as used in the previous paper of this series []. It is characterized by a axiu divergence of rays at exit aperture equal to 90, and then only to independent paraeters are required for defining its shape. θ Y X CPC receiver Figure 1. Longitudinal section of the Copound Parabolic Concentrator (CPC) used in this ork for the optical siulations. The angle sin θ is the axiu angle of aperture of the input Labertian bea. The angle of acceptance of a parallel bea is θ acc = 5 We have chosen the acceptance angle at parallel bea, θ acc = 5, and the length L = 150. The focal length of the parabolic profile, f = 1.14, the radius of input aperture, a = 1.035, and the radius of output aperture, a = 1.05, becoe []: f = a'(1 + sin θ acc ) (1) Z characterized by a constant radiance L in and by a variable angular aperture θ. The total flux at input is then a function of θ and is expressed by [1]: Φ ( θ ) = L A π dθ sinθ cos θ =... (4) in in in in in in 0 = π Lin Ain θ... sin θ here θ in is the incident polar angle. In order to keep constant the radiance at input of the CPC, e have used in the siulations a nuber of input rays, that is an input flux, proportional to sin θ. 3. Analysis of the Transitted Flux 3.1. Analysis of the Total Flux Transitted to Receiver To study the flux transitted to the exit aperture of the CPC, e have closed the exit aperture by an ideal absorber, so all the rays reaching the aperture can be counted and analyzed by the siulation progra [19]. The flux transitted to the exit aperture is given by: θ Lin Ain d in in in dir in 0 Φ τ ( θ ) = π θ sinθ cos θ η ( θ ) (5) here ηdir ( θ in) is the transission efficiency of the CPC under direct and iated irradiation, and is a function of the incident polar angle θ [1, ] (see Fig.). in a' = a θ acc () sin L = ( a + a') ctgθ acc (3) Even if not so interesting for practical use (it is too long respect to its input diaeter), nevertheless the chosen canonical CPC is useful to be studied as representative of ideal noniaging concentrators characterized by a axiu exit angle of 90. In addition, the characterization ethods here described can be usefully applied to any other concentrator, ideal or not. The only change ade to the concentrator during the optical siulations as that of reflectivity of the internal all, R ; all the other added devices (absorbers, screens, etc.) ere external to the concentrator and ere used only as tools to iprove the knoledge of its optical properties. For the reflectivity of the internal all e have used only high values ( ); hen not specified, it is equal to 1 (no optical loss inside the concentrator). All the optical siulations ere carried out by using the TracePro ray-tracing softare of Labda Research [19]. The flux at input of the CPC is a Labertian flux dir in 3D-CPC calculated for three all reflectivities: R = 1.0, 0.9 and 0.8 Figure. Optical transission efficiency η ( R, θ ) of the The input ertian flux as siulated by sending to the input aperture a nuber of rays, each ith a constant flux of 1W, folloing the sin θ rule (see Eq. (4)), corresponding, in this siulation, to 50k rays at θ =10.

3 84 Antonio Parretta et al.: Optics of Solar Concentrators. Part III: Models of Light Collection of 3D-CPCs under Direct and Labertian Irradiation Fig. 3 shos the nuber of the incident, transitted and reflected rays vs. θ, easured for a unitary all reflectivity. In this case e have no optical loss on the all of the CPC, but only loss due to the back reflected rays. We can see that, as long as θ < θacc, due to the square shape of the transission efficiency of the CPC (see Fig. ), all the input rays are transitted, hereas, at θ θ acc, the transitted rays stop groing and the reflected rays begin to appear and gro folloing the groth of the incident rays. As ell as e have defined a transission efficiency for a parallel bea at input of concentrator (see Fig. ), e define the transission efficiency for a ertian bea at input of concentrator, and call it direct ertian transittance, τ dir [1]. Figure 3. Nuber of incident, transitted and reflected rays vs. the angular aperture of the Labertian bea θ, calculated for a unitary all reflectivity Figure 4. Labertian transission efficiency of the CPC calculated as function of the angular aperture of the Labertian bea, θ, for a unitary all reflectivity This quantity refers to an angular aperture of the Labertian bea equal to 90. If the ertian bea at input is liited by the angular aperture θ, the above quantity becoes τdir ( θ ). We have added the ter "dir" to distinguish the Labertian bea that e send to the entrance aperture of the concentrator, fro the Labertian bea that e send to the exit aperture of the concentrator, and that e ill indicate by the ter "inv". The properties of a CPC concentrator irradiated by an inverse Labertian bea ill be discussed in a future paper of this series. The Labertian transission efficiency of the CPC is expressed [1] by the ratio beteen output and input flux: Φτ ( θ) τdir ( θ) = =... Φin( θ) θ (6)... = dθ sin cos ( ) in θin θin ηdir θin sin θ 0 The plot of τdir ( θ ), siulated for a unitary all reflectivity, is shon in Fig. 4. It is interesting to copare the behavior of τdir ( θ ) ith that of ηdir ( θ in) (see Fig.). We obtain a step-like transission efficiency curve also for a Labertian bea irradiation, ith the efficiency alost constant until about the acceptance angle ( θ acc = 5 ), but, differently fro the transission efficiency curve ηdir ( θ in), no the τdir ( θ ) curve decreases sloly at increasing θ. The reason is that, for θ > θacc, a constant portion of the input bea is alays ected, and the Labertian transission efficiency at these conditions can be expressed by (on the assuption that R =1.0): dir ( > acc ) =... θ dθin θin θin ηdir θin sin θ 0 θacc dθin θin θin ηdir θin sin θ 0 sin θacc sin θacc η dir θin sin θ sin θ τ θ θ... = sin cos ( )... (7)... sin cos ( ) ( ) In Eq. (7) e have put ηdir ( θin) 1, as it can be seen fro the curve of Fig. corresponding to R =1.0. Fig. 4 shos the Labertian transission efficiency of the CPC copared to the sin θ sin θ function. The perfect acc correspondence beteen the to curves hen θ > θacc is clearly evident. No e are able to define a ne optical quantity, the angular aperture of the input Labertian bea corresponding to the halving of the direct Labertian transittance at θ = 0. We call this angle the

4 International Journal of Optics and Applications 015, 5(3): Labertian acceptance angle, θ acc, indicated in Fig. 4 together ith θ acc irradiation. The quantity, the acceptance angle at parallel bea θ acc is iediately derived fro Eq. (7), and is about 7.1 for our CPC ith θ acc = 5.0 : θ acc -1-1 θacc = sin [ sin ] = sin [ sin 5 ] = 7.08 (8) Contrary to the parallel transission efficiency (Fig. ), the Labertian transission efficiency never reaches the zero because there is alays a portion of ertian bea that is transitted; in fact, it decreases reaching a iniu at θ = 90 equal to sin θ acc ( forθ acc = 5 ). Until no e have not yet analyzed the effect of R on the studied quantities. We do it starting ith the direct Labertian transittance, calculated for three all reflectivities: R =1.0, 0.9 and 0.8 (see Fig. 5). here N τ (R, θ ) represents the nuber of transitted rays. The average nuber of reflections has been calculated by using three pairs of all reflectivities: ( ); ( ); ( ). Figure 6. Average nuber of reflections experienced by the transitted rays, as function of the angular aperture of the Labertian bea, calculated for three pairs of all reflectivities: ( ); ( ); ( ) The results are reported in Fig. 6 and sho that the average nuber of reflections of the transitted rays is practically independent on the angular aperture of the Labertian bea and equal to. This result is in good agreeent ith hat as found [] by analyzing the average nuber of reflections of the transitted bea hen the CPC is irradiated ith a parallel bea at different polar angles respect to the optical axis. Figure 5. The Labertian transission efficiency calculated for three different all reflectivities: R =1.0; 0.9 and Average Nuber of Reflections of the Total Absorbed Rays The analysis of the three curves of τdir ( R, θ ) allos deriving the average nuber of reflections experienced by the transitted rays. In this regard e ill ake use of a forula siilar to Eq. (4) used in [], replacing η ( R, θ ) = η ( R, θ ) ith dir τ ( R, θ ): τ in dir in N τ ( R', R'', θ ) τdir ( R', θ) Nτ (R'', θ ) log τdir ( R'', θ) Nτ (R', θ ) (9) R' log R'' 3.3. Labertian Concentration Ratio We no introduce the quantity direct Labertian concentration ratio, C dir, defined in [1] as the ratio beteen the average output, or transitted, radiance and the constant input radiance. When the ertian bea at input has an angular aperture θ, the direct Labertian concentration ratio is indicated as Cdir ( θ ). We start calculating the average output radiance: Φτ ( θ) Lτ ( θ) = =... π Aout θ (10)... = L C dθ sinθ cos θ η ( θ ) in geo in in in dir in 0 and the direct ertian concentration ratio becoes: Lτ ( θ) Cdir ( θ) = =... L in θ geo dθin θin θin ηdir θin 0... = C sin cos ( ) =... (11) Φτ ( θ)... = C sin = ( ) sin Φ ( θ ) in geo θ τdir θ Cgeo θ

5 86 Antonio Parretta et al.: Optics of Solar Concentrators. Part III: Models of Light Collection of 3D-CPCs under Direct and Labertian Irradiation Fig. 7 shos L τ ( θ ) calculated for three all reflectivities:, expressed in (W/ sr) and R =1.0, 0.9 and 0.8. then Eq. (11) becoes: opt ( θ) τdir ( θ) geo sin θ sin θacc geo C = C C (11 ) Considering that 1/ sin θ acc corresponds to the axiu value of the optical concentration ratio, equal to C [1], fro Eq. (11 ) e conclude that C ( θ ) 1. geo opt Figure 7. Average output radiance L τ ( θ ) of the transitted flux, calculated for three all reflectivities: 1.0, 0.9 and 0.8 dir The direct Labertian concentration ratio C ( θ ) calculated fro Eq. (11) is reported in Fig. 8. dir Figure 8. Direct ertian concentration ratio, C ( θ ) calculated for three all reflectivities: R = 1.0, 0.9 and 0.8 Fig. 8 shos that Cdir ( θ ) is alays 1, that is the average output radiance is alays saller than the input radiance, being equal to it only hen the internal all is an ideal irror and the input angular aperture is greater than the acceptance angle of the CPC (5 ). This can be easily deonstrated considering that, for θ > θacc, e have: dir acc τ ( θ ) sin θ sin θ 1 (1) 3.4. Analysis of the Flux Distribution on the Receiver To study the flux transitted to the exit aperture of the CPC, the exit aperture as closed by an ideal absorber. Fig. 9 shos the aps of the irradiance on the receiver as function of the angular aperture of the ertian bea at input, fro 1.0 to 6.0 ; the all reflectivity is 1.0. Contrary to the aps obtained by irradiating the CPC ith a parallel bea [], the aps of Fig. 9 are syetric respect to the optical axis, as consequence of the rotational syetry of both the CPC and the ertian bea. The optical siulations ere perfored by setting a nuber of input rays N in =100k for θ = 6.0, then Nin( θ ) is given by: N sin ( θ) in N in ( θ ) = (6 ) sin (6 ) (13) The cross-section profiles of the output irradiance on the receiver of the CPC, siulated for to all reflectivities, 1.0 and 0.9, are shon in Fig. 10. It can be seen that the flux is concentrated at first in the iddle of the receiver (θ = 1.0 ), then it is distributed in the central part (θ =.0 ); at θ = 3.0 the center is ipoverished, but again, starting fro θ = 3.5 e have an alost unifor distribution of the flux in the central part of the receiver, hich extends to the edge at increasing θ, to becoe unifor hen θ reaches a value equal to the acceptance angle (5 ). Then e can say that the flo affects at first the central part of the receiver and then the outer ring. In essence, the less inclined rays gather in the central part of the receiver, hile those ore inclined gather on its edge. We can see fro Fig. 10a, here the all reflectivity is unitary and e have no optical losses inside the CPC, that the output irradiance profile becoes flat as soon as the angular aperture θ exceeds the acceptance angle (5.0 ). By reducing R, the irradiance profiles are slightly soothed, because of the optical loss inside the CPC; at the sae tie, they becoe blunt at the edges, because the optical loss is higher there as an effect of the higher nuber of reflections, as e ill see shortly after.

6 International Journal of Optics and Applications 015, 5(3): a) b) c) d) e) f) Figure 9. Maps of output flux on the receiver for different angular apertures of the ertian bea at input: (e); 6.0 (f). Nuber of input rays: 100k at θ = 6. R = 1.0 θ = 1.0 (a);.0 (b); 3.0 (c); 4.0 (d); 5.0

7 88 Antonio Parretta et al.: Optics of Solar Concentrators. Part III: Models of Light Collection of 3D-CPCs under Direct and Labertian Irradiation a) b) Figure 10. Irradiance profiles of the flux at output of the CPC, irradiated by a ertian bea of different angular apertures θ. Wall reflectivity: R = 1.0 (a); R = 0.9 (b) 3.5. Average Nuber of Reflections of Locally Absorbed Rays As done for the total flux transitted to the receiver, e are able no to evaluate the average nuber of reflections experienced by the rays reaching a specific point on the receiver. We use therefore the sae Eq. (9), to be applied to the irradiance profiles of transitted flux, calculated for different values of the all reflectivity. The irradiance is expressed as: ER (, θ,x ), here x is the relative distance fro the optical axis of the point on the receiver. The average nuber of reflections N ( R', R'', θ,x ) calculated for a specific value of θ can be therefore obtained fro to siulations taken at different all reflectivities, as follos: N( R', R'', θ,x ) E( R', θ,x ) N( R'', θ) log E( R'', θ,x ) N( R', θ) (14) R' log R'' As the input bea is Labertian, Eq. (14) gives the average nuber of reflections that the input rays, incident at angles in the 0 θ interval, experience before reaching a point on the receiver. As e are interested to kno the average nuber of reflections done by the rays inclined in a narro range θ centered at a particular θ value, e have calculated ne irradiance profiles, ER (, θ,x ), obtained as: ER (, θ,x ) = ER (, θ',x )-ER (, θ'',x ) (15) here θ = θ' - θ'' >0, and fro these profiles e have obtained the searched result: N ( R', R'', θ,x )... E( R', θ,x ) N( R'', θ) log E( R'', θ ) N( R' θ)...,x, (16) R' log R'' Fig. 11 shos the curves of N ( R', R'', θ,x ) calculated for the five angular intervals: θ = 0-1 ; 1 - ; -3 ; 3-4 ; 4-5. The curves have a very interesting behavior. First of all, they overlap quite ell, shoing that the nuber of reflections of rays reaching a specific point at distance x, is only slightly dependent on the angular divergence at input. This is particularly true for iddle values of x, less for the points at the center and at the edge of the receiver. The second consideration to do is that N ( 1.0, 0.9, θ, x ) is lo at the center of the receiver, beteen about 1 and, and gros oving toards the edges, reaching values beteen about 3 and 5. This explains the stateent ade at the end of section 3.4. As N ( 1.0, 0.9, θ,x ) is not too uch dependent on θ, it is interesting to calculate its average respect to the different angular aperture intervals θ. The average quantity N ( 1.0, 0.9,x ) is shon in Fig. 1 (black curve). It is a very sooth curve, no longer containing the strong oscillations of the curves of Fig. 11.

8 International Journal of Optics and Applications 015, 5(3): seven. Figure 11. Average nuber of reflections of the rays incident at x relative distance fro the optical axis on the exit aperture of the 3D-CPC, irradiated by a ertian bea ith different values of angular divergence θ N ( 1.0, 0.9,x ) is ell fitted by the fourth degree polynoial (see red curve in Fig. 1): 4 N ( 1.0, 0.9,x ) = a+ bx + cx (17) ith a = 1.447; b = 0.11; c = Figure 1. Average nuber of reflections of the rays incident at x distance fro the optical axis on the exit aperture of the 3D-CPC, irradiated by a ertian bea ith angular divergence red curve is the fitting polynoial θ = 0-5 (black curve). The Fig. 13 (black curve) shos the average nuber of reflections N (R', R'', θ,x ) = N (1.0, 0.9,0-1,x ) siulated for an angular aperture of the ertian bea equal to 1. The curve shos that N is exactly 1.0 in the central part of the receiver and increases ith a step-like behavior oving toards the edge of the receiver, here the rays arrive after exactly 5 reflections. The rays at input are alost parallel to the optical axis, then the black curve of Fig. 13 is very siilar to the red curve obtained in [] hen a iated bea parallel to the optical axis irradiates the CPC. The difference beteen the to curves is very little: the red curve has a slightly larger zone ith one reflection, and the nuber of reflections for the rays on the edge is no Figure 13. Average nuber of reflections of the rays incident on the receiver at x distance fro the optical axis. The CPC is irradiated by a ertian bea ith angular divergence θ = 0-1 (black curve) and by a iated bea parallel to the optical axis of the CPC (red curve) 3.6. Angular Divergence of the Transitted Rays The study of the angular divergence of rays at output of the CPC is iportant to optiize the absorption properties of the receiver. In the practical use of a PV solar concentrator, for exaple, the receiver is not an ideal absorber, but a solar cell ith specific reflectance properties, that affect its light absorption capabilities in relation to the divergence of the incoing rays [0]. The flux absorbed by the solar cell can be expressed as: Φabs π Aout... π (18)... dθτ sin θτ cos θτ Lout ( θτ) (1 R( θτ)) 0 here Aout is the area of the cell, R( θ τ ) is the angle-resolved reflectance and L τ ( θ τ ) is the radiance of light transitted at exit angle θ τ by the CPC (the radiance is not function of the aziuthal angle φ τ because the syste input bea + CPC is rotationally syetric). In Eq. (18), in general R( θ τ ) gros ith θ τ [0], then it is desirable that L τ ( θ τ ) be not too high for high θ τ values. To check the angular distribution of radiance of light at the CPC receiver, e have irradiated the ideal CPC ( R = 1.0) by a ertian bea ith different values of the angular aperture θ, and the output flux has been ected by a heispherical absorber (radius R τ = 00 ) centered on the receiver (see Fig. 14). The rotationally syetric ap of the flux density on the internal screen surface, projected on a plane orthogonal to the optical axis (see Fig. 16) and the corresponding radial profile, siulated at θ =1, are shon in Fig. 15.

9 90 Antonio Parretta et al.: Optics of Solar Concentrators. Part III: Models of Light Collection of 3D-CPCs under Direct and Labertian Irradiation Figure 14. Schee of the CPC irradiated by a ertian bea ith 5 angular aperture, θ = 5. The transitted rays are ected by a heispherical screen ith a radius R τ = 00. Soe of rays (green color) are back reflected and attenuated because the all reflectivity is R = 0.9. Most of rays are transitted and ipact on the spherical screen; the red rays are less attenuated (loer nuber of reflections inside the CPC), hereas the green rays are ore attenuated (higher nuber of reflections inside the CPC) a) b) Figure 15. Map (a) and corresponding radial profile (b) of the flux density (irradiance) (in W/ ) produced on the screen, projected over a plane orthogonal to the optical axis. The input is a ertian bea ith θ = 1 Figure 16. Scheatic representation of the transission of light to the spherical screen, the production of a flux density (irradiance) E( θ τ ) on its internal all and the projection of this irradiance, Epro( θ τ ), on a plane orthogonal to the optical axis If E( θ τ ) (W/ ) is the irradiance on the screen surface, Epro ( θ τ ) (W/ ) is its projection on the orthogonal plane and I τ ( θ τ ) (W/sr) is the radiant intensity, the radiance ( θ ) (W/ sr) of the transitted light becoes: L τ τ τ out R... = Epro( θτ ) A out τ θτ τ θτ I ( ) R E( ) L(θ τ τ) = = =... A cosθ A cosθ τ out τ (19) here A out is the area of the output aperture of the CPC and R τ is the screen radius. Fro Eq. (19) e see that the profile of E pro(θ τ ), that reported in Fig. 15b, is the sae of the radiance L τ ( θ τ ) ( Rτ A out is a constant factor), and then the flux ap of Fig. 15 is qualitatively the ap of the transitted radiance. The profile of L τ ( θ τ ) is orth of being analyzed in detail. Fig. 17 shos the effect of the angular divergence θ on the angular distribution of radiance of the transitted bea, for a all reflectivity R = 1.0 (no optical loss by absorption). The nuber of rays incident at input aperture are a function of θ and follos thesin θ rule, hich, in this case, corresponds to 50k at θ = 6. The angular divergence θ as varied fro 1 to 10 ith 1 steps, but e sho in Fig. 17 only the first six diagras, because the polar profiles of L τ ( θ, θ τ ) do not change significantly forθ > 6, apart fro sall fluctuations at near θ τ = 0. The radiance profiles sho a strong central peak at θ < θacc values. This peak is ainly produced by rays

10 International Journal of Optics and Applications 015, 5(3): traveling close to the optical axis and crossing undisturbed the CPC ithout reflections on the internal all. At θ < θ acc, e find a dead zone large 15, ith no rays, folloed by a large band of radiance extending fro about 15 to 60, that takes, near the acceptance angle, the characteristic shape of a butterfly or dragonfly. The dead zone is very siilar to that occurring ith irradiation by a iated bea parallel to the optical axis []. In that ork, e could explain that the dead zone upper liit (about 15 ) as due to those rays ipacting on the CPC all at about 7 far fro the axis, and coing out ith a divergence of about 15 ; this divergence is a loer liit, since aay ore fro the axis, these rays undergo a second reflection that akes the diverge even ore, and therefore akes it ipossible exit angles of less than about 15. Despite being the irradiation Labertian, instead of parallel, this phenoenon is aintained, because the ertian bea contains incoing parallel rays parallel or nearly parallel to the optical axis. When the input angular divergence θ reaches the acceptance angle (5 ), the radiance profile is quite flat up to 75 (the sall fluctuations depend only on the liited nuber of incident rays) and keeps alost equal up to θ = 10. The sae profile of transitted radiance as achieved by increasing θ. Fig. 18a shos, for exaple, the transitted radiance profile obtained at θ = 90. The siulation as carried out ith a flux of 400k rays (400k W) and a processing tie of about 1 hours; the nuber of rays, hoever, as less than ould have been necessary to satisfy the sin θ rule, hich assures a constant radiance at input: ~ 4.5M rays and ~130 hours of processing tie. The consequence is a loss in the signal-to-noise ratio, as it can be seen in Fig. 18a. Apart fro the lo signal-to-noise ratio, the siulation at θ = 90 has not changed significantly the radiance profile found at 6 (see Fig. 17f) (e kno in fact fro [], section 4., that input rays tilted ore than 5.8 have no chances to reach the CPC exit opening): a quite flat profile at ~ W/ sr fro 0 to 75, folloed by a decay up to ~ W/ sr at 90. Despite not flat up to 90, this is effectively the profile obtained ith a ertian bea. This as verified by irradiating the heispherical screen directly ith a ertian bea, after reoving the CPC. The result is the profile of Fig. 18b, here the intensity drops to 50% of axiu at the exit angle of 90. The evolution of the radiance profiles as function of θ can be folloed also transforing the radiance polar diagras of Fig. 17 into Cartesian diagras and overlapping the (see Fig. 19). I added also the θ =90 profile obtained at a reduced processing tie (~11 ties), distinguishable for the lo signal-to-noise ratio. Apart fro the radiance peak near θ τ = 0 that e have already discussed, e see that the radiance profile onotonically gros in the θτ ~15-80 interval at increasing θ fro 1 to 4. A significant change in the radiance profile happens in the θ = 4-5 interval, here e observe the filling of the dead zone at lo θ τ values, as ell as of the region beteen ~0 and ~75 ; the interval reains unfilled as explained before. (a) (Scale: 1E9) (b) (Scale: 3E9) (c) (Scale: 3.75E9) (d) (d) (Scale: 3.5E9)

11 9 Antonio Parretta et al.: Optics of Solar Concentrators. Part III: Models of Light Collection of 3D-CPCs under Direct and Labertian Irradiation (e) (Scale: 4E9) (f) (Scale: 4E10) Figure 17. Polar representation of the distribution of the radiance of the transitted flux, hen the CPC is irradiated ith a ertian bea ith angular divergence θ = 1 (a); (b); 3 (c); 4 (d); 5 (e); 6 (f). For the angular divergence θ = 6 (f), We have used 500k rays at input, 10 ties ore than required to follo the sin θ rule. The scale of radiance is expressed in W/ sr and is reported near each figure. The all reflectivity is constant and equal to 1.0 Figure 19. Cartesian representation of the distribution of the radiance of transitted flux, hen the CPC is irradiated ith a ertian bea ith angular divergence θ =1 ; ; 3 ; 4 ; 5 ; 6 ; 90. The all reflectance is constant and equal to Average Nuber of Reflections of Transitted Rays We have previously analyzed the nuber of reflections ade by the transitted rays as function of θ, both as an average on the total flux, and as function on the distance, fro the optical axis, of the ipact point on the receiver. No e look at the average nuber of reflections ade by rays transitted as function of the exit angle θ τ fro the CPC, at different values of the input ertian divergence θ. For this purpose, e have repeated the siulations of radiance shon in Fig. 17 ith a different all reflectance, R = 0.9. (a) (Scale: 4.5E9) (b) (Scale: 3.5E9) Figure 18. Polar radiance of the transitted flux by the CPC irradiated ith a ertian bea. (a) Angular divergence θ = 90, 400k rays at input, ~ 11 ties less than required to follo the sin θ rule. (b) Angular divergence θ = 90, 500k rays projected on the heispherical screen directly, ithout the presence of the CPC. The all reflectivity is 1.0. The scale of radiance is expressed in W/ sr a) (Scale: 4.0) b) (Scale: 3.5)

12 International Journal of Optics and Applications 015, 5(3): c) (Scale: 3.5) d) (Scale: 3.0) e) (Scale: 3.0) f) (Scale: 3.0) Figure 0. Polar aps of the average nuber of reflections of the transitted rays as function of the exit angle θ τ at different values of the ertian angular divergence θ : 1 (a); (b); 3 (c); 4 (d); 5 (e); 6 (f) The average nuber of reflections as obtained by applying the folloing equation, ith R ' = 1.0 and R '' = 0.9: Lτ(R', θ, θτ) log Lτ(R'', θ, θτ) N ( R', R'', θ, θτ ) (0) R' log R'' here L τ (R, θ, θ τ ) is the radiance of the flux transitted at angle θ τ. The nubers of output rays do not appear as they ere the sae at the to all reflectivities. Fig. 0 shos the polar distribution of N ( θ, θτ ) for angular divergence values θ = 1 6, ith 1 steps. There are any interesting features of the polar diagras of N ( θ, θτ ) to highlight. First of all they are copletely developed at θ = 5 = θ acc, as it as for the radiance aps (Fig. 17). Then e notice that N ( θ, θτ ) = 0 for exit rays close to the optical axis ( θ τ 7 ); these rays do not touch the CPC all. Then, fro θ τ ~7 to θ τ ~, N ( θ, θτ ) is exactly 1. At higher values of θ τ, N ( θ, θτ ) gros foring a lobo centered at ~45, hich is narro at lo θ values, ith a peak at N ax 4 for θ = 1 and N ax = 3 for θ = -3. The lobo then idens occupying the hole range fro ~45 to 90. At θ = 1, the rays exit the CPC at θ τ = 90 after one reflection; by increasing θ, the reflections increase up to.75. One last consideration to do is about the value of N ( θ, θτ ). Whereas the nuber of reflections of a ray ust be exactly an integer, this rarely happens to a ixture of rays, because they have not the sae characteristics (sae distance x at input aperture and sae θ ). Exceptions are N ( θ, θτ ) = 1 at all θ hen θτ ~7 (see Figs. 0a-f), or at θ = 1 hen θ τ = 90 (see Figs. 0a,b), or again N ( θ, θτ ) = 3 at θ = 3 hen θ τ ~ 45 (see Figs. 0b,c). The diagras of radiance of Fig. 17 and those of the average nuber of reflections of Fig. 0 do not clarify, therefore, hat are, at the origin, the rays causing a particular radiance or nuber of reflections profile. To iprove the analysis of the average nuber of reflections of the transitted rays, e divided the θ interval (0-5 ) in five saller intervals θ = (0-1 ); (1 - ); ( -3 ); (3-4 ); (4-5 ), ith θ = θ'' θ'. The forula used to calculate the nuber of reflections is: N ( R', R'', θ, θ ) =... τ Lτ(R', θ'', θτ) Lτ(R', θ', θτ) log Lτ(R'', θ'', θτ) Lτ(R'', θ', θτ)... = R' log R'' (1)

13 94 Antonio Parretta et al.: Optics of Solar Concentrators. Part III: Models of Light Collection of 3D-CPCs under Direct and Labertian Irradiation 4. Analysis of the Reflected Flux Figure 1. Cartesian representation of the average nuber of reflections of the transitted rays for an input ertian bea ith angular divergence in the intervals: θ =(0-1 ) (black); (1 - ) (red); ( -3 ) (blue); (3-4 ) (dark cyan); (4-5 ) (agenta) The radiance profiles Lτ( R, θ, θτ) are those previously siulated ith R ' = 1.0 and R '' = 0.9. Eq. (1) only requires subtraction of radiance profiles ith different R values. The obtained profiles of N ( 1.0,0.9, θ, θτ ) are reported, in Cartesian representation, in Fig. 1, hich allos to distinguish better the transitted rays. Fig. 1 confirs that N = 0 for θτ <7.5 and that N = 1 for θτ = 7.5. We note also that the various N profiles consist of bands ith the axiu on integer values of N, as desired. The position of these bands ( θ b ) and their axiu value N b are reported in Tab. 1, as function of θ Optical Reflection Efficiency Let us consider at first the total reflected flux: ρ in dir in α τ Φ =Φ ρ ( R, θ ) =Φ Φ Φ = =Φin 1 αdir ( R, θ) τdir ( R, θ) here Φ α and () Φ τ are the total absorbed and transitted dir dir fluxes, respectively, and ρ ( R, θ ), α ( R, θ ), τdir ( R, θ ) are the ertian reflection, absorption and transission efficiencies, respectively. The total reflected flux can be expressed as function of the radiance Lρ( R, θ, θ ρ) of the reflected light: π Φ (, θ ) =... (3) ρ R = A π dθ sin θ cos θ L ( R, θ, θ ) in ρ ρ ρ ρ ρ 0 here θ ρ is the polar angle of the reflected ray. To siulate the reflection properties of the CPC, e have adopted a schee siilar to that used to easure the transitted light (see Figs. 14, 16), adding a heispherical screen ith R ρ = 1000 radius and ideal absorbance, able to gather all the reflected light fro the CPC. The CPC protrudes out of the screen and the center of input aperture eets that of the screen (see Fig. ). Table 1. Features of N profiles. The bands θb are reported as function of the θ interval N b, ith angular position θ ( ) N b θ b ( ) 4 (3.8) (.9) (3.1) (.9) 61 (.1) 90 Figure. Schee of the 3D-CPC irradiated by a ertian bea ith angular divergence θ. The reflected flux is easured by an absorbing heispherical screen of radius R ρ = The CPC of the figure is not ideal ( R = 0.9), as consequence, the incident rays (red color) are attenuated after reflection (green color). The output of the CPC has been left open, so the transitted rays are visible. Most of the transitted rays are attenuated (green color); the red bea on the z-axis is ade of rays crossing undisturbed the CPC

14 International Journal of Optics and Applications 015, 5(3): The optical reflection efficiency is defined as the ratio of the output reflected flux to the input flux: ρ Φ ( R, θ ) = dir ρ ( R, θ ) Φ in (4) Fig. 3 shos the curve of reflection efficiency calculated for R = 1.0 (black curve), condition for hich ρ dir is equal to the ratio beteen the nuber of the back-reflected rays to the input rays, because of the absence of optical loss inside the CPC. The reflection efficiency is zero belo the acceptance angle, as all the rays are transitted; then it appears in correspondence of θ acc (see red curve). Figure 3. Labertian reflection efficiency ρ ( R, θ ) of the 3D-CPC calculated for dir factor of 10) dir R = 1.0 and 0.9 all reflectivity. A portion of the curve of ρ (1.0, θ ) is also shon vs. θ /10 (x axis scaled by a The groth of ρdir ( R, θ ) for R =1.0 is easily obtained considering that it is siply the copleent to 1 of the transission efficiency τdir ( R, θ ) hen θ θ acc (see Eq. (7)). We have therefore: sin θacc dir (1.0, ) 1 dir (1.0, ) 1 sin θ ρ θ = τ θ = (5) The evolution of ρdir (1.0, θ ) vs. θ is quite different fro that of ηρ (1.0, θ in), the reflection efficiency relative to a parallel bea []; in that case it as groing rapidly before the acceptance angle, reaching half of its axiu just in correspondence of it. With a ertian bea, ρdir (1.0, θ ) is groing ore sloly (see Fig. 3) because a part of the bea, that corresponding to θ =, is alays transitted. Fro Eq. (5) e see that the θ acc liit of ρdir (1.0, θ ), for θ = 90, is: 1 sin θ The reflection efficiency data relative to R = 0.9 are shon in Fig. 3 (blue curve). While the curve of ρdir (1.0, θ ) is alays groing, the curve of ρdir (0.9, θ ) reaches a axiu at θ 14 and then decreases don to 16% for θ = 90, being strongly liited by the absorbance of light on the CPC all. acc 4.. Angular Divergence of the Reflected Rays The study of the angular divergence of back reflected rays fro input aperture has not a practical relevance as it has in the case of the transitted rays, but it is a useful exercise to apply also to the back reflected rays concepts of the theory of solar concentrators. To siplify the discussion, e liit ourselves to consider an ideal CPC ( R =1). As already seen discussing the transitted flux, the TracePro softare produces on the ecting screen a flux ap corresponding to the irradiance on the screen all projected on a plane orthogonal to the z axis. Apart fro a diensional constant factor, equal to R ρ / Ain, ith R ρ = 1000 radius of the screen and A in input aperture of the CPC, this ap is equivalent to that of the radiance of light back reflected by the input aperture, as it has been deonstrated in Eq. (19) for the transitted flux. The plot of these aps is not necessary, because they are syetric ith respect to the optical axis, and then they give the sae inforation of the profiles of their cross sections. Figure 4. Profiles of the reflected radiance vs. the exit reflection angle θ ρ, siulated for soe values of θ fro 5 to 0. The envelope profile gives a reversed band due to the loss by transitted rays and its FWHM x 4.5. The all reflectivity is 1.0 A Cartesian representation of the reflected radiance siulated for soe values of θ θ acc up to θ ρ = 0 is shon in Fig. 4. The reflected radiance appears at θ acc = 5 and then gros in intensity reaching a axiu of about 3.x10 9 (W/ sr) at around 1. Keeping constant in

15 96 Antonio Parretta et al.: Optics of Solar Concentrators. Part III: Models of Light Collection of 3D-CPCs under Direct and Labertian Irradiation intensity, the radiance band expands then in ters of angular divergence. The envelope of all profiles corresponds to the radiance profile at θ = 90, and is characterized by a depression in the center, caused by the loss by transitted rays. This profile, if overturned, for a band of 3.x10 9 (W/ sr) intensity and a idth FWHM x 4.7. This band definitely has to do ith the issing transitted rays, but it is unexpected that it is equal to x 4.5 instead of being x θ acc = x 5. The siulations ere ade throughout the angular range fro 5 to 90. In Fig. 5 the center of the radiance bands θ p and their full idth at half axiu θ = θ θ1 is reported as function of θ. The evolution of θ p is perfectly linear and is defined by: θ ( R, θ ) = θ (1.0, θ ) = θ (6) p p θ is different; after a slo rising up to The evolution of 10, the behavior becoes linear and is defined by the function: θ(, θ ) = θ(1.0, θ ) = θ (7) R 1) and cross the sae port, the input aperture of area A in, then e have: A in sin θin Ain sin θout θin θout = = (30) n θ θ n A A Figure 6. Schee of a generic concentrator ith the three ain paraeters for the input and output apertures: index of refraction, area and angular divergence The polar diagras of the angular distribution of reflected radiance are reported in Fig. 7 and are distributed throughout the angular range fro 5 to 90. The upper value of polar angle of the reflected light is: θ θ = θp + = θ 0. θ (8) a) (Scale: 1.4E8) b) (Scale: 3.5E9) Figure 5. Center of the radiance bands axiu θ as function of θ θ p and their full idth at half The axiu value of the exit angle, θ, therefore, is alost equal to the axiu entrance angle, θ. This result is a direct consequence of the Liouville theore [6], establishing the invariance of the generalized étendue, the volue occupied by the syste in the phase space, as expressed by Eq. (9) (see also Fig. 6): n A sin θ = ( n ') A ' sin θ ' = const (9) In our case the incident and reflected rays are in air (n, n = c) (Scale: 3.5E9) d) (Scale: 3.5E9)

16 International Journal of Optics and Applications 015, 5(3): As it can be seen in Fig. 7, the reflected light is a bea ith the shape of a hollo cone, including an internal all ith sei-opening θ acc, and an external all ith sei-opening equal to the angular aperture of the ertian bea θ. Cross sections of these beas are then rings ith the inner circle groing, at increasing θ, fro 5 to 1, and the outer circle fro 5 to 90. e) (Scale: 3.5E9) f) (Scale: 3.5E9) g) (Scale: 3.5E9) h) (Scale: 3.5E9) 4.3. Average Nuber of Reflections of the Total Reflected Rays To derive the average nuber of reflections of the total reflected flux, N ρ ( R',R'', θ), it is sufficient to analyze the ertian reflectance ρdir ( R, θ ) at to all reflectivities and using the forula: N ρ ( R',R'', θ ) ρ ( ', ) ( ) dir R θ N ρ R'', θ log ρdir ( R'', θ) Nρ ( R', θ ) (31) R' log R'' here N ρ (R, θ ) is the nuber of reflected rays ected by the screen. The reflection efficiency functions used are those calculated at R =1.0 and R =0.9 (see Fig. 3). Fig. 8 shos the curve of N ρ (1.0,0.9, θ ), obtained applying Eq. (31) to the pair of reflectivities (1.0; 0.9). The average nuber of reflections is four at near the acceptance angle (5 ) and then gros onotonically reaching up to 16 reflections for θ = 90. i) (Scale: 3.5E9) l) (Scale: 3.5E9) Figure 7. Polar diagras of the angular distribution of the reflected radiance, siulated for different values of the angular aperture of the ertian bea at input: θ =5 (a); 10 (b); 0 (c); 30 (d); 40 (e); 50 (f); 60 (g); 70 (h); 80 (i); 90 (l). The scale of radiance is expressed in W/ sr Figure 8. Average nuber of internal reflections of the back reflected rays, siulated by applying Eq. (31) for the pair of values of internal all reflectivity: ( R ', R '' ) = (1.0; 0.9) 4.4. Average Nuber of Reflections Associated to the Reflected Radiance As e have done ith the transitted flux, the nuber of

17 98 Antonio Parretta et al.: Optics of Solar Concentrators. Part III: Models of Light Collection of 3D-CPCs under Direct and Labertian Irradiation internal reflections of the reflected rays, as function of the exit angle θ ρ at different values of the angular aperture θ, can be calculated by siulating the radiance shon in Fig. 7 ith a different all reflectance, R = 0.9. The average nuber of reflections N ρ( R', R'', θ, θρ ), ith R ' =1.0 and R '' =0.9, as obtained by applying the folloing equation: N ( R', R'', θ, θ )... ρ L log L... ρ (R', θ, θ ) N (R'', θ) (R'', θ, θ ) N (R', θ) R' log R'' ρ ρ ρ ρ ρ ρ (3) here L ρ (R, θ, θ ρ ) is the radiance of the flux reflected at θ ρ angle and N ρ (R, θ ) is the nuber of total reflected rays. The average nuber of internal reflections of the rays reflected at θ ρ angle is shon in Fig. 9a for θ = 5, 6, 8, 10, 1, 18, 0, and in Fig. 9b for θ = 30, 40, 50, 60, 70, 80, 90. It is interesting to note that, apart fro the curve corresponding to θ = 5, all other curves have a characteristic V shape ith a iniu of N ρ equal to 5 on the direction of the optical axis of the CPC, and a axiu of N ρ that gros at increasing θ and falling approxiately at θρ θ. The results of Fig. 9 are very plausible: the higher the angular divergence of the incoing bea, the greater the nuber of reflections that the rays experience ithin the CPC, the greater the exit angle of the rays that ake the axiu nuber of internal reflections. It is also interesting to note that, for exit angles θ ρ < θ, all the curves sho an increasing trend of N ρ( θρ) that is alost linear, particularly for high values of θ (see Fig. 9b). An exception to hat has been said so far akes the curve corresponding to θ = 5. It shos a constant trend N ρ( θρ) = 5 for θ ρ < θ ( θ acc ), but then decreases ith increasing of θ ρ over the value of θ, as do all the other curves. With regard to the values of N ρ( θ, θρ), they range fro about 5 to about 30 and gro at groing θ, as illustrated in Fig. 8. Finally, Fig. 30 shos to exaples of polar representation of N ρ( θ, θρ) for θ = 5 and 90. a) b) Figure 9. Cartesian representation of the average nuber of internal reflections of the back reflected rays, as function of exit angle θ ρ, siulated for ( R ', R '' ) = (1.0; 0.9). Angular aperture of input bea: (a) 5, 6, 8, 10, 1, 18, 0 ; (b) 30, 40, 50, 60, 70, 80, 90 a) (Scale: 5.5) b) (Scale: 35) N ρ ρ for θ = 5 Figure 30. Polar representation of (1.0,0.9, θ, θ ) (a) and 90 (b)

18 International Journal of Optics and Applications 015, 5(3): Analysis of the Absorbed Flux 5.1. Optical Absorption Efficiency Fro the data of transission efficiency and reflection efficiency e iediately derive the absorption efficiency by the expression [1]: dir dir dir α ( R, θ ) = 1 τ ( R, θ ) ρ ( R, θ )(33) The absorption efficiency αdir ( R, θ ), calculated for the all reflectivities R = 0.9 and 0.8, is shon in Fig. 31. The siulation ith R = 1.0 is useless in this case, as it ould give a systeatic zero absorption efficiency. For θ < θacc = 5, the absorption of light is due to the internal reflections of ainly the transitted rays, these reflections being about, as e see in Fig. 6. dir calculated for to all reflectivities: R = 0.9 and 0.8 Figure 31. Absorption efficiency α ( R, θ ) of the 3D-CPC As a result, the absorption of the incident flux ill be x10% hen R = 0.9, and x0% hen R = 0.8, as it can be seen in Fig. 31. For θ > θacc, the absorption of light inside the CPC increases due to the contribution given by the back reflected rays, hose average nuber of internal reflections increases fro 4 to 16 at increasing θ fro 4 to 90 (see Fig. 8). 5.. Distribution of the Absorbed Flux Here e study ho the absorbed flux is distributed inside the CPC. At this purpose a value of all reflectivity R <1 as selected. After each irradiation, by selecting the internal all of the CPC, the siulation progra produces a ap of the absorbed flux, projected on the x/y plane orthogonal to the optical axis z. In this ay, the ap is an annulus ith outer radius that of input aperture, a = 1.035, and ith inner radius that of output aperture, a = 1.05 (see Section ). The intensity ap is the projection on the x/y plane of the absorbed irradiation (in W/ ). Soe aps of the absorbed flux are shon in Fig. 3 for the all reflectivity R = 0.9, a typical value for realistic solar concentrators. The angular aperture of the ertian bea has been varied fro 10 to 90. Fro Fig. 3 e can see that the flux density on the all progressively oves fro the exit to the input aperture, and, starting fro 60, the region adjacent to the exit opening is copletely devoid of flux. It is interesting to study in detail the average distribution of the flux along the z coordinate (the optical axis). To do this, e take the radial profiles of the aps of Fig. 3, plotted as function of the z coordinate, and apply to the the correction factor cosα, to reove the projection operation ade by the progra, here α is the angle that the tangent to the CPC profile akes ith the optical axis, given by []: π dz π α = arctg = +... (34) dx e... arctg d (16 f sin θacc ) b 4(4 f x+ c) sinθ acc here x is the coordinate on the axis perpendicular to the optical axis and passing through the center of the exit opening, hile z is the coordinate on the optical axis, easured fro the center of the exit opening []. a) b) c)

19 100 Antonio Parretta et al.: Optics of Solar Concentrators. Part III: Models of Light Collection of 3D-CPCs under Direct and Labertian Irradiation d) e) f) g) h) i) Figure 3. Maps of the flux density on the internal all, projected on the x/y plane orthogonal to the optical axis z, siulated for different values of the angular aperture of the ertian bea at input: θ = 10 (a); 0 (b); 30 (c); 40 (d); 50 (e); 60 (f); 70 (g); 80 (h) ; 90 (i). Wall reflectivity: R = 0.9 Absorbed irradiance (W/ ) 3,5x10 9 R 3,0x10 9 Exit opening =0.9,5x10 9,0x10 9 1,5x10 9 1,0x10 9 5,0x10 8 θ=0 θ=15 θ=10 θ=8 θ=6 θ=5 θ=4 θ= Absorbed irradiance (W/ ) 6x x x x10 10 x x10 10 θ=80 θ=70 θ=60 θ=50 θ=40 θ=30 θ=5 θ=0 θ=15 θ=10 θ=8 θ=6 θ=5 θ=4 θ= Input opening R =0.9 0, Z coordinate () Z coordinate () a) b) Figure 33. Distribution of the absorption irradiance along the optical axis, fro z = 0 (the exit opening) to z = 150 (the entrance opening), siulated for angular divergence θ fro to 0 (a) and fro to 80 (b). Wall reflectivity: R = 0.9 The final result is the profile of the absorbed irradiance, as reported in Fig. 33a for incidence angles in the 0-0 interval and in Fig.33b for incidence angles in the 0-80 interval. The plot of the irradiance profiles ends at about 137, due to the liited resolution of the siulations. The profiles of Fig. 33a sho that, at lo θ values, the flux is restricted in a thin zone near the exit opening, ith the peak of irradiance increasing and oving toards higher z values at increasing θ. For θ 10 a large band appears in proxiity of the input opening and increases at increasing θ, reaining centered at about z = 115 and leaving a hollo in the center of the CPC. A siilar result as observed ith parallel beas at input increasing the incidence angle fro θ in = 0 to θ in = 0 []. For θ values higher than 30, this band becoes doinant and