Quasi-optimum pseudo-lambertian reflecting concentrators: an analysis

Transcription

1 Quasi-optimum pseudo-lambertian reflecting concentrators: an analysis Antonio Luque n this work we analyze a two-stage concentrator in which the first stage is reflective and the second stage considers the first one as a Lambertian source in order to obtain the highest possible gain. We determine the profile and the position of this first stage, which happens to be a parabola, and we calculate the value of the gain as a function of the acceptance angle and the focal length of the first stage. 1. ntroduction The purpose of this paper is to analyze the first stage of a reflecting nature of a two-stage concentrator, whose second stage looks at the first one as if it were a Lambertian source.' t is assumed that the second stage is designed so that rays emitted by this source that penetrate through its entry aperture must reach the collector and that there exists conservation of etendue inside the second stage. The largest possible concentration of the system is obtained when the collector is illuminated by incident rays with leveling grade angle (900). n this case the etendue of the second stage is E = P, (1) where P is the collector's perimeter. n the case of monofacial and bifacial solar cells 3 submerged in a transparent medium of high index of refraction n the etendue is, respectively, E = nw, () E = 4nW, (3) where W is the width of the cell used. The use of bifacial solar cells submerged in a medium with a high index of refraction leads to the optimum photovoltaic concentration concept. 4 5 Figure 1 shows the relative arrangement of the first and second stages in schematic form. This type of ar- The author is with Universidad Politecnica de Madrid, nstituto de Energia Solar (ETST), Madrid-3, Spain. Received 5 January /80/ $00.50/ Optical Society of America. rangement, in which in the second stage solar cells were surrounded by air, was first suggested by Spectrolab and Arizona State University. 6 n Fig. 1 the second stage is suitable to obtain optimum photovoltaic concentration and therefore uses bifacial cells submerged in a medium with a high index of refraction. However, the nature of the second stage is irrelevant to this work. When the first stage is a reflecting surface that reflects a portion of the sky, considered in this case as a Lambertian source, the first stage is not itself a Lambertian, and therefore etendue E with which the second stage has been designed is higher than etendue E 1 produced by the sky on the first stage. As a consequence, the collector is bigger than this in an optimum system. n this work we try to analyze the gain of the proposed system and compare it with that of the optimum system to determine the shape and position of the first stage that gives a maximum gain. The technological interest of the configuration of Fig. 1 lies mainly in the reduction of the mirror area that can be obtained with respect to optimum concentrators of the compound parabolic concentrator (CPC) type 78 based on the fact that these concentrators intercept the rays at more oblique angles than in the concentrator under study. 11. Optimum Configuration n Fig. segment 00', which we assume for the moment to be of fixed length c, represents the secondstage entry aperture, while that of the first stage is represented by segments AA' with different subindices. The direction of an extreme ray is the vertical, while that of the other extreme ray is deflected an angle 0 clockwise. Etendue E of the second stage can be calculated using the method proposed by Winston and Welford 9 that results in 398 APPLED OPTCS / Vol. 19, No. 14 / 15 July 1980

2 ncident rays from sun where c is the length of 00'. from sun te o ncident rays G = D/c, (8) -j A. Maximum Concentration for a Given To obtain the maximum gain we must make the j! quotient D/E as large as possible, where D is the pro- '! j jection of segment AA' on the horizontal axis, and E is proportional to the cell's width (or to the perimeter of the collector). Let us assume that the extremes of primary mirror AA' are found in positions Al and A. Moving the points on their respective hyperbolae, the etendue of the second stage is not altered, arriving this way to points A and A of intersection with the outline of the allowed region and increasing D to its maximum value. Then, moving point A upward over vertical O'A the value Fig. 1. Quasi-optimum pseudo-lambertian reflecting concentrator of D is maintained, but the value of a is increased [we for bifacial solar cells. must keep in mind from Eq. (4) that the hyperbola's branches to the left of the MM' axis have negative values of semidiameter a], and the value of E is decreased. n this way we arrive at point A 3 of the intersection of said vertical with the vertical axis parabola and focus E = (A'O - A'') - (AO- AO') = a' - a. (4) at O' which crosses A. There is no continuous curve 1al locus f that could cross the vertical at a point above A This equation shows that the geometric 3 because points of equal etendue is constituted by] h locb o it would be necessary for the curve that represents the uiypterdas mirror to be at some point more horizontal than the from a set of hyperbolas that we will call e( with the foci at points 0 and 0', a' and c bein the parabola, and in this case it would reflect the vertical te ig he, ray that falls on that point into a direction that would largest semidiameters of the hyperbolas th at cross A' leave O to its left. The conclusion is that the vertical and A, respectively. )timum all axis parabola, with focus at O' that crosses A, is the On the other hand, if the system is to be 01 the sunrays comprised between the extrem( )timuo ll optimum outline for a given value of X (see Fig. ). intercept the entry aperture of the second stage. As a consequence, any mirror point X of the first stage must be the vertex of an oo' angle. This angle must be 0. This means that all points in the first-sstage mirror must be in the inside of the circle whose poihits subtend segment 00' with angle 0. AM For the same reason they must be to the left of the vertical that crosses O' so that the second sta ge does not shadow the first. The allowed space to extraames A and A' of the entry aperture of the first stage is indicated in Fig.. Let us suppose for the moment that, angle a (see Fig. ) of OO' with the vertical is given. Fo r given val- respect ues of 00' and 0 the circle and its position vvith to the vertical and consequently the allowedzone for the primary mirror are determined. f the extreme A' of the entry aperture is placed in a horizontal level above A, as is usually the case, the concentrator is not symmetric, and its may:imum gain occurs for the vertical beam of rays. The Nvalue GTH = DP = D/E, (5) where D is the projection of entry aperture AA' on the horizontal plane. According to Eqs. () and (3) the photovoltaic gain for monofacial and bifacial cells is respectively. is simply GPHM = ngth, (6) GPHB = ngth, (7) The gain of the first stage of this structure Fig.. Diagram of the position of the reflector of the first stage (points AA') with respect to the entry aperture 00' of the second stage. Only reflectors inside the shadowed contour can give high gain concentrators. The hyperbolas of equietendue are also represented by dotted lines. The outline of the reflector with the biggest gain is A3A July 1980 / Vol. 19, No. 14 / APPLED OPTCS 399

3 B. Maximum Concentration Outline We will try to analyze if it is better to move point A around the circumference in the direction of O' or in the opposite way. n the latter case a' decreases as A 3 goes down, which decreases a as well. t is impossible to predict if E increases or decreases without a more detailed analysis. Moreover, when A' moves around the circumference, projection D of mirror A A 3 on the horizontal plane changes, also changing the concentrator gain. An analysis of the variation of etendue E of the first-stage gain G 1 and of the thermodynamic gain GTH is accomplished in Appendix A as a function of angle o, which defines the position of point A over the circle that subtends 00' with angle 0. n Fig. 3 the GTH gain appears as a function of angle co of the A position for several values of acceptance angle 0 and of angle a of the second-stage position. t can be observed that the gain presents a maximum for small angles corresponding to parabolas in which point A 3 is located outside the circle subtending 00', and therefore they are unrealizable. When A 3 is outside the circle in Fig. 3, the gain appears as a dotted line. mmediately following there is a region of decreasing gain with co represented by a continuous line. Finally, there appears a minimum and last increasing portion that, in the case of high angular apertures, could give gains apparently higher than the optimum gain of the concentrator. n reality, these high gains cannot be obtained, since for very high angles X the mirror casts a shadow on itself hindering the rays' entrance through the 00' aperture. On the other hand, the concentrators in this region lack technological interest because they have their A point located above 0, which means they are deep concentrators that can be better achieved with the classical theory of the CPCs. The value of X at which A is placed over O' has been marked on each curve of Fig. 3. Beyond this point the concentrators are considered to be too deep. Besides, these concentrators present a very low first-stage gain G 1. As a consequence, the second stage must supply all the gain for the system; this would require a second stage of excessive dimensions. As a result, the only region of technological interest is the one with decreasing gain. t is clear that the value of co that gives the most gain is the smallest possible that locates the A 3 point inside the allowed region; this is in the A 4 position. The gain that can be achieved is the one that represents the transition from the dotted gain curve and the continuous curve, and it is somewhat. lower than the optimum gain. n this case the outline of the first stage is a vertical axis parabola with focus at O' that crosses point A 4. Point A3 constitutes the other extreme of the concentrator which is the best that can be obtained for a given orientation a of the second stage Quasi-Optimum Concentrations Orientation a of the second stage determines the position of the circle in Fig.. Since its radius is determined by c and 0, a also determines the focal distance O'A 4 of the parabolic concentrator of maximum gain that corresponds to angle 0 and the aperture horizontal projection D of said concentrator. The goal of this section is to determine the relationship between the maximum gain achievable and the relationship of focal distance to D (f/no.) of the concentrator. We also give the necessary parameters to build the concentrator. To obtain this relationship we will consider all the dimensions referred to the parabola's focal distance, which is taken as unity. The details of the calculations are given in Appendix B. The etendue of the second stage can be written as E = DS(D) sino - (D /4)(1 - coso), (9) where S(D) is a polynomial defined in Appendix B [Eq. (B)] in which S(O) = 1. The etendue of the first stage 9 is the difference of the projection of line A 4 A' 3 on a line inclined 0 with the vertical and on a vertical line. A simple calculation shows that this difference is E = D sino - (D /4)(1 - coso). () GTH E (D/4) tano 77 = -= (11) GTH E S(D)-(D/4) tano represents to which extent the two-stage concentrator differs from an ideal one (in which E 1 = E ). n the above expression the ideal thermodynamic gain is As we said previously, the gain of the concentrator is obtained from E by means of Eqs. (5)-(7). The quotient E 1 sino coso(1 - (D/4) tano) 1-1., - 11,,.. a hi U0 (1) Fig. 3. Gain of the concentrator as a function of position w of point A (see Fig. ). The parameters are the orientation a of the first stage and the acceptance angle APPLED OPTCS / Vol. 19, No. 14 / 15 July 1980

4 t can be observed that, when aperture D of the concentrator is very small, it approaches the ideal. The departure from ideality 1 is larger when acceptance semiangle 0 is high. For very high values of 0 it casts shadows over the rest of the concentrator so that Eqs. (9)-(1) become invalidated. The gain of the first stage is given by Eq. (8). Using expression c that appears in Eq. (B6) of Appendix B and also using the expression in Eq. (B3) in the same Appendix we have G, = D/c = -+-D + - D 4 1/ sino. (13) 3 56 / Curves with values of 1 and G 1 sino appear in Fig. 4 as a function of D. Even though 7j depends on acceptance angle 0, this dependence is very small for 0 <. The values of i7 shown in Fig. 3 have a margin of error smaller than ±1% for 0 <. Once these values are known, it is possible to design the first stage which will always be a vertical axis parabolic arc. ts horizontal projection D, normalized to the focal distance of the parabola, will be considered as an independent variable. The length of the aperture of the second stage will be obtained from G 1. The entry aperture of this second stage will contact its edges on the circle formed by the focus and the two edges of the parabolic arc, making sure that one of them coincides with the focus. Figure 5 shows values of the total gain of the concentrator as a function of D for several values of the semiangle of acceptance 0. V. Discussion The specific interest of this type of two-stage concentrator lies in the small dimension of the first stage, practically normal to the sunrays, as compared with the CPC in cases where high concentration and large acceptance angles are desired. The conditions for obtaining maximum gain with a second-stage concentrator have been discussed in detail. This gain increases when the F/No. of the first stage increases. The outline of the concentrator is, in this case, a vertical axis parabolic arc with focus at one of the second-stage entry aperture's edges. Though theoretically optimum this outline may not be the most desirable from a practical viewpoint. The reason for this is that any technological imperfection in the making of the mirror throws the extreme vertical rays that fall over the whole mirror, outside the second stage, resulting in a considerable reduction of the acceptance angle of the concentrator for rays near the extreme vertical ones. For the extreme rays falling with an inclination of 0 the problem is smaller since only those reflected near points A 4 and A 3 miss the second stage. The intermediate rays, even the ones outside the theoretical acceptance angle in fact, go into the second stage, producing a gradual fall of the effective concentration that extends outside the mirror's acceptance angle. n certain cases, it would be convenient to make the gain curve less asymmetric. This can be accomplished using curve arc A, A 4, located between points A in the circle in Fig., nearer O' than A3. Naturally, this system will have less gain and a larger acceptance angle 1.0. D ~ ~ * * Fig. 4. Approximate value to within +1% for semiacceptance angles smaller than of that of the structure's output GTH/GTHOPT and exact value of the gain of the first stage (multiplied by the sine of the semiacceptance angle) as a function of the first stage. 1- a 3 Go S ~ ~ ~ r 0 a 1 Fig. 5. Gain GTH as a function of entry aperture D (normalized to the focal length) with the semiangle of acceptance as the parameter. than the so-called optimum in this work. The etendue of the second stage of this concentrator increases not only because A' moves toward O' but because A 3 also moves toward A 4. The concentrator's outline is not fully defined in this case, giving additional freedom for the design that could take into account the possible errors in the fabrication of this stage. As the acceptance angle increases, the gain of the first stage decreases, which means that the second stage has a large entry aperture. This makes a very big second stage. The significance of the second-stage design is considerable, but the interest decreases as 0 takes higher values. Probably this interest is confined to acceptance semiangles lower than. want to thank Roland Winston for his helpful suggestions in planning this work, which was stimulated by the Ramon Areces Foundation. 15 July 1980 / Vol. 19, No. 14 / APPLED OPTCS 401

5 Appendix A The equation in polar coordinates with origin 0' and axis O'p (see Fig. ) of the equietendue hyperbolas is given by C- a a + c cosco (Al) When using this equation we adopted the formalism of not allowing negative values of p, considering outside the variability field of w all angles that do not meet this condition. This assignment to each value of a positive or negative sign, corresponds to only one branch of the hyperbola. The equations of the circle subtending 00' with an angle 0 and of the parabola of vertical axis and focus at O' are, respectively, c p = - sin(w + 0), (A) sin0 r 1 + cos(w-a) (A3) where r is the radial coordinate of point A 3 (see Fig. ). Given an arbitrary value of w corresponding to the A position, its radial coordinate p is given by Eq. (Al); r is obtained introducing the value p(w) in Eq. (A3): r = - sin(w + 0)[1 + cos(w - a)]. (A4) sin0 To calculate the value of a that corresponds to the equietendue hyperbola that contains point A 3 (r,a), we substitute r and a in Eq. (Al) and get (r + 4c - 4rc cosa) 1 - r (A5) Using an analogous procedure we can obtain the equietendue parabola corresponding to point A (P,w) getting (p + 4e - 4pc cos) 11 / - p (A6) This allows us to find the etendue of the second stage [see Eq. (4)]. The maximum gain of the concentrator appears when the rays fall vertically. n this case the entry aperture is D = p sin(w-a). (A7) The calculation of GTH is now performed by using Eq. (5) and represented vs w for several values of 0 and a in Fig. 3. n case the value of r, which is also calculated, is higher than its maximum allowable value rm = - sin(a + 0), (A8) sin0 curve GTH(w) is represented as a dotted line. Appendix B Setting the distance O'A 4 of Fig. to unity, the equation in rectangular coordinates of parabola A 4 A 3 and of the circle subtending 00' with angle 0, 00'- A3'A4 are given by, respectively, X + y - xxo-y = 0, x = 4y, (B1) (B) where x0 is the abscissa of the center of the circle. As both curves cross point A3 of abscissa D, it is possible to obtain the relation XO = -D + -D 3. (B3) 8 3 To calculate etendue E of the second stage it is necessary to calculate the length of segments A 4 0', A0', A 4 0 and A30. The first two are calculated using the expressions A 4 0' = 1 A 3 0' = 1 + (D /4). (B4) (B5) (This last equation establishes that the distance to the focus of a parabola's point is equal to the distance to the directrix.) For the calculation of A 4 0 and A30 it is necessary to solve triangles A 4 00' and A 3 00', which requires the previous calculation of 00'. This calculation is also needed for the gain of the first stage 00' = (1 + 4X) sin 0 = c. (B6) From the resolution of the first triangle it can be deduced that A 4 0' = cos0 - xo sin0. (B7) From the resolution of the second triangle it can be deduced that AO= (1 + D /4) cos0 + [(1 + D /4) + (1 + 4xo)11/ sin0. The etendue of the second stage, (B8) E = (OA3- 'A3) + ('A 4 -OAA (B9) using Eqs. (B3)-(B9), can be written in the form that is shown in Eq. (9). The function S(D) used there takes the value S(D) = 3 +D D + D 4 1/ \4 161 \ / (B) References 1. A. Rabl and R. Winston, Appl. Opt. 15, 880 (1976); R. Winston, U.S. Patent 3,957,031 (1976).. W. T. Welford and R. Winston, Optics of Nonimaging Concentrators (Academic, New York, 1978), pp A. Luque et al., in Proceedings, Photovoltaic Solar Energy Conference (D. Reidel, Dordrecht, Holland, 1978), pp A. Luque, Spanish Patent 453,575 (1976); U.S. Patent 4,169,738 (1979). 5. A. Luque et al., in Proceedings, COMPLES Conference, Milan, Cassa di Risparmio Calabria e Lucania, Comples talian Session, Roma, 1980; in press. 6. Ref., p R. Winston and H. Hinterberger, Sol. Energy 17, 55 (1975). 8. A. Luque et al., in Proceedings, SES Silver Jubilee Congress, K. W. B6er and B. H. Glenn, Eds. (Pergamon, New York, 1979), Vol. 3, pp , 9. R. Winston and W. T. Welford, J. Opt. Soc. Am. 68, 89 (1978). 40 APPLED OPTCS / Vol. 19, No. 14 / 15 July 1980