Solar Energy, Vol. 16, pp. 89-95. Pergamon Press 1974. Printed in Great Britain

PRINCIPLES OF SOLAR CONCENTRATORS OF A NOVEL DESIGN

R O L A N D W I N S T O N *

Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637, U.S.A.

(Received 7 March 1974; in revised form 14 May 1974)

Abstraet--A new principle for collecting and concentrating solar energy, the ideal cylindrical light collector, has been invented. This development has its origins in detecting Cherenkov radiation in high energy physics experiments. In its present form, the collector is a trough-like reflecting wall light channel of a specific shape which concentrates radiant energy by the maximum amount allowed by phase space conservation. The ideal cylindrical light collector is capable of accepting solar radiation over an average ~8-hr day and concentrating it by a factor of -10 without diurnal tracking of the sun. This is not possible by conventional imaging techniques. The ideal collector is non-imaging and possesses an effective relative aperture (f-number)= 0.5. This collector has a larger acceptance for diffuse light than concentrating collectors using imaging optics. In fact, the etficiency for collecting and concentrating isotropic radiation, in comparison with a fiat plate collector, is just the reciprocal of the concentration factor.

INTRODUCTION cal light collector. The advantages of this new The inherent attractiveness of directly using solar approach are briefly: radiation to meet man's energy needs has motivated (a) There is no need for diurnal tracking; only an intense search for practical solar power seasonal adjustments are required. For concentra- schemes[l]. For most of these, it is necessary to tion factors f o r - 3 , even seasonal adjustments may concentrate the solar radiation by at least an order not be needed. of magnitude in order to achieve high temperatures. (b) The efficiency for accepting diffuse light is This poses no in principle problem because sunlight much larger than for focusing collectors. In fact, is quite parallel (the half-angle 0~ subtended by the the fraction of the total sky light collected (as solar disk is only - I / 4 °) provided one tracks the compared to a flat plate) is precisely the reciprocal sun's location in the sky with an accuracy of the concentration factor. comparable to 0~. Because of the formidable This may represent a break-through in the ability technical problems associated with tracking to this to economically utilize solar energy for central precision, it would clearly be an enormous advan- power as well as the cooling and heating of tage if the required concentration was achievable buildings. by a stationary collector. This possibility was, in fact, explored by Tabor [2] in 1958 who reached the IDEAL CYLINDRICAL LIGHT COLLECTOR disappointing conclusion that the maximum possi- FOR SOLAR CONCENTRATION ble concentration obtainable by a stationary Background of the invention collector[3] was 3. This result has been generally The ideal light collector is a non-imaging accepted to the present time by the solar energy reflecting wall light channel that concentrates a community, divergent beam of light by the maximum amount

It is now possible, however, to concentrate solar allowed by phase space conservation[4]. If the half radiation by a factor of - 1 0 without diurnal angle of maximum beam divergence is 0 .... this tracking using a new design principle for collecting maximum permissible concentration factor (ratio of and concentrating solar energy--the ideal cylindri- entrance pupil diameter dt to exit pupil diameter d2)

may be shown to be

*Supported in part by the Enrico Fermi Institute Greenewalt Fund. d~/d~_ = (nz/nn)(1/sin Ore.x) (1)

89

90 ROLAND WINSTON

d~ ~ employed for particle physics instrumentation for some years.

Ideal cylindrical light concentrator Recently it was determined that a trough-shaped

cylindrical collector, possessing in transverse section the specific profile shape already described,

A ×is of / ~ ~ 3 c ~ is exceptionally well-matched for concentrating solar radiation as received on earth (see Fig. 2)[6].

parabola The acceptance of an ideal cylindrical collector Parabola may be described using optical direction cosines

i / ~ Kx, Ky, K. where, for a constant index of refraction we may take /< as the unit light ray direction (the

~ / general case is treated in the Appendix). Then K~, K,. become true Hamiltonian variables conjugate to x, y when the light ray is parameterized by z. Here z

Focus of is measured along the optic axis of the collector. parabola This geometric formulation is particularly useful

for describing light ray trajectories in non-imaging / /

- - ~ d 2 J~-- systems. We may consider the variables x(z) , y(z), Kx(z), K,.(z) as coordinates in a four dimensional

Fig. 1. Profile curve of ideal collector. The axis of the phase space. The volume in this space occupied by parabola is inclined at angle O,,ox to the optic axis (OA). an ensemble of light ray trajectories at fixed values

where n, and n2 are the indices of refraction at z entrance and exit. In the language of conventional imaging optics, this property would imply a relative aperture (f-number) of 0.5 which is the limiting value set by the Abbe sine condition beyond the practical limit for imaging systems. For clarity, we shall restrict our discussion of solutions in the text ~ , to the case n2 = n ~. These solutions readily general- ize to nz > n~ as shown in the Appendix.

An optimised non-imaging collector was de- ~ / veloped in the course of elementary particle physics research[5]. It arose from the need to j ~ . detect faint Cherenkov light from electrons travel- × ing faster-than-light in a medium. The first ideal End walt collector was a hollow, axially symmetric "cone" with light reflecting walls. The plane profile curve of the "cone" is a parabola with focus at the y opposite edge of the exit aperture, and axis inclined at an angle 0 .... with respect to the optic axis (see Fig. 1). At its termination, the tangent to the parabola is parallel to the optic axis. The resulting length of the cone is just sufficient to transmit direct rays at angle 0 .... It follows from phase-space conservation that the length of the collector

L = (l/2)(d, + d,.) cot 0 ..... (2)

is the minimum possible. Thus, we have found a Fig. 2. Trough-shaped cylindrical collector. Reflective unique solution satisfying both relations (1) and (2). vertical end walls make collector appear optically Conical ideal collectors have been successfully infinitely long.

Principles of solar concentrators of a novel design 91

of z is preserved as the rays propogate from sun's motion in the sky. To an adequate approxima- entrance to exit. Hence tion, the apparent motion of the sun as viewed from

a fixed point on earth, describes the cone depicted

f dxdvdK~dK, , is conserved. (3) in Fig. 4. In this figure the x axis direction is along " north, the y axis direction along west and the z axis

z = constant direction along the vertical. The cone axis in the x, z plane, inclined at angle A, which is the latitude. The

In deriving the acceptance of such trough collectors cone opening angle, c~, is the angle between the earth's axis of rotation and the earth-sun direction.

in the Kx, K~. plane, the light ray trajectories Since the earth's axis is inclined at an angle of

projected on a constant y plane behave as though the collector were two-dimensional and concentrat- approximately 23-5 ° with respect to the normal to

the plane of its orbit (the ecliptic plane), the angle ing meridional rays only. Therefore,

varies between the approximate limits 66.5 ° -< c~ -< K,2/(K, ~ + K: ~-) -< sin ~" 0 ...... (4) 113.5 ° during the course of a year. Except at a time

of equinox, when a = 90 ° and the apparent solar

however, Kx 2 + K~-" + K:" = 1 s ince/~ is the unit light path describes a great circle wherein the sun does ray direction so that, not "r ise" or "fai l" in the vertical, the problem of

collecting solar radiation is non-trivial and becomes most demanding at solstice (c~ = 900- + 23.5°). Col- K,2/sin 2 0 ..... + K~ 2-< 1. (5) lection and concentration of solar radiation by high

Thus the acceptance fills an ellipse of semi-minor factors at the time of solstice for a reasonable fraction of the day, say 6 to 8 hr, may be considered axis equal to sin 0 ...... and semi-major axis equal to 1,

as shown in Fig. 3. The collector concentrates by a the fundamental problem of solar collection. This is factor 1/sin 0 ...... so because at such times the apparent "rise" or

"fall" in the vertical requires following or tracking the solar disk "upwardly" about 12 ° within the

K¥ three or four hours prior to its reaching the noon r position and "downwardly" about another 12 °

~ x within three or four hours after its reaching the noon position. Clearly, a stationary collector which would continuously accept direct solar radiation

V ~ / ~ / ~ throughout the period of the above-mentioned -+ 12 ° -t I K x excursion during the time before and after reaching

U ~ / / ~ the noon position, and which further was capable of high orders of concentration, approaches the ideal

~ , ~ in solar energy collection. The extent to which collectors of the present invention approach this

-I ideal is set forth hereafter. It is easily shown that the apparent motion of the

Fig. 3. Acceptance of an ideal cylindrical collector in the sun in the K~, K~. plane is also an ellipse. A K~, K~ plane.

Z

A characteristic property of ideal light collectors . . II /~a as revealed by detailed ray tracing is the small ~ t number of reflections when averaged over the solid ~ , . . 1 angle of angular acceptance. This is plausible since ~ ' x " ~ ' ~ l

R

rays in the transverse (x, z) plane incident at large x

angleSsolar concentrationO <~ 0 ..... have at most one reflection. 1 / ~ v Y ~ ~ , v

Inasmuch as it is desired to concentrate solar radiation with ground-based collectors, it is conve- Fig. 4. Apparent motion of the sun. The x axis points nient for the purposes of discussing solar tracking North, the y axis West, and z axis vertical. The angle ~ is problems to adopt a "Ptolemaic" description of the the latitude.

92 ROLAND WINSTON

convenien t way to visualize this is to reconsider Figure 6 shows the ellipse described by the sun on a Fig. 4 and take as the z ' direction the noon posit ion solstice, the most difficult period for collection. On keeping the y ' direction west as before (see Fig. 5). the same figure has been superimposed the

z ' acceptance of a sin 0m~x=0.l collector which z concentrates the direct sunlight by a factor of 10. d Clearly, such a collector accepts most of the useful

day (7 to 8 hr) at solstice. More rigorously, one f / must choose the z ' axis to place the origin in the x '

" ' ~ " ~ K ' , K'y plane at the center of the collector ellipse. ¢ ~ . , ~ , ~ One then finds that for the solar ellipse

x

~ f ~ ,,,.ff -sin(ZT +O~,,O~K'<-sinO .... -cos T <- K;,<-cos T. (12)

Hence, Fig. 5. Apparent motion of the sun in rotated coordinates.

The z' axis points to the noon position of the sun. a = 1/2[sin (2 T + 0o1,,0 + sin 0m,,d (1 3)

b = cos T. Clearly, the project ion of the cone on the x ' , y '

plane is an ellipse, and Introducing a phase angle ~b for the solar ellipse,

where d' = 2w (=360 °) corresponds to the 24-hr - s i n 2 a - < K ' - < 0 (6) day, we find that at the intersection of the two

_ t - K v - s i n a <-Ky<~sma, I-K~, (7) ellipses

or, in terms of T = w/2-a, cos ~h = [a2-2a sin O,,,,x+(b sin Om,,O2]/[a 2

- s i n 2 T ~ < K ' ~ 0 (8) - ( b sin 0m,x)2]. (14)

-- COS T -- K ~ - cos T. (9) Therefore, the accepted number of daylight hours is given by

Hence, the semi-minor axis equals Number of hours = 2(qS/2w)(24) = (4#~-)(24).

a = 1/2 sin 2a = 1/2 sin 2T (10) (15)

and the semi-major axis equals Where the factor 2 results from the fact that cos d,

is even in ~h. Table 1 shows the number of accepted b = s i n a = c o s T . (11) hours per day for a collector with 0 .... = 6 °

K y, (concentrat ion factor, 9"6) throughout the year in the approximation of a point-like sun. Averaged I

~...P',, ~ r.o over the year this gives approximately 8-hr of 1 collected sunlight. For 0 .... = 7 ° (concentrat ion

~'~ ~l factor of 8.2) one may obtain an average day of ~i~-~,:.~ ~ approximately 9-hr of sunlight. Table 2 illustrates !~i ['i~!~ V~/ concentra t ion factors for collectors having 0 .....

K x' Table 1. Collector operating hours (0 .... = 6 °)

ii!,] ~;-z,~ ,~z Collected hours ~] i']% ~ Season 2T cos 05 05 per day

] ~i~'iil ,,¢J (Full solar I.,' ~_J -I.0 Equinox 0 ellipse accepted) (Full daylight)

15 ° 0.37 68 ° 9 30 ° 0.57 55 ° 7.4

Fig. 6. Solar ellipse at solstice. The acceptance of a xl0 Solstice 47 ° 0.61 52.4 ° 7.0 collector is superimposed.

Principles of solar concentrators of a novel design 93

Table 2. Collector concentration factors

Concentration Hours 0 ...... 2T cos ¢h ,¢, factor collected

7 ° 47 ° 0.567 55.5 8.2 7.4 6 ° 47 ° 0.610 52-4 9.6 7.0 5 ° 47 ° 0-657 48-9 11-4 6-5 4 ° 47 ° 0-709 44.8 14.3 6.0 3 ° 47 ° 0-769 39.8 19.1 5.3 2 ° 47 ° 0.835 33.4 28.6 4.4

within the range of 2°-7 ° a long with hours of sunl ight col lected at sols t ice ( 2 T = 4 7 °) in the app rox ima t ion of a point- l ike sun.

Acceptance of diffuse light Fig. 7. Trough-shaped concentrator wilh energy receiver. To u n d e r s t a n d the proper t i es of the ideal

cyl indrical co l lec tor when i l luminated by diffuse light, it is useful to r e fo rmula t e its angular a c c e p t a n c e in more geomet r i c terms. The angular accep t ance , as v iewed f rom any point on the en t r ance ape r tu re fills an elliptic cone where the axis of the cone is parallel to the opt ic axis of the co l lec tor t rough, the semi -minor axis of the cone is in a d i rec t ion t r a n s v e r s e to the t rough and sub t ends an angle 0 ...... and the semi -ma jo r axis of the cone sub t ends an angle app r oach i ng 90 ° as the t rough b e c o m e s very long or, equiva len t ly , if the t rough is t e rmina ted at the ends by vert ical reflecting walls. Al te rna t ive ly , we may r ep re sen t the field of view f rom any point on the en t r ance ape r tu re by an infinite r ec tangu la r str ip w h o s e width sub tends an angle -+0 ...... It fo l lows that the solid angle for Fig. 8. Concentrating flat plate configuration. diffuse light is jus t 40 ...... s teradians . More useful ly , the f rac t ion of i so t ropic light accep ted as c o m p a r e d the scale of the individual col lec tor until one with a flat ab so rb ing sur face is precise ly the a p p r o a c h e s the wave- leng th of visible light. (b) rec iprocal of the c o n c e n t r a t i o n fac tor . In essence , Fabr i ca t ion is faci l i ta ted s ince the optical e l emen t the ideal co l lec tor maps the full phase space is s i m p l y a r e p e a t i n g " q u a s i - t r i a n g u l a r " b e a m . (c) A a c c e p t a n c e of the ene rgy rece iv ing cavi ty into the por t ion of the energy re rad ia ted f rom the rece iv ing precise band of the sky t r ave r sed by the diurnal cavi ty would be select ively ref lected back by an t r a j ec to ry of the sun. inf ra- red reflecting glass cover and re -accepted by

the col lector . Collector configurations Such modules combine the s implici ty of the

We are aware of at least two conf igura t ions of we l l -known flat-plate conf igura t ion with the abili ty solar c o n c e n t r a t o r s wh ich are of in te res t to explore , to a t ta in high t e m p e r a t u r e s th rough concen t r a t ion . One of these is the single co l lec tor fo l lowed by an We bel ieve that this flat plate conf igura t ion may ene rgy rece ive r as, for example , in Fig. 7. A n o t h e r have advan tages both for centra l power and conf igura t ion which appear s to us to be most hea t ing and cool ing of bui ldings. p romis ing is the c o n c e n t r a t i n g fiat plate configura- t ion depic ted in Fig. 8. Such rows of ideal cyl indrical co l lec tors in flat-plate modules offer Acknowledgements--I am grateful to R. G. Sachs for

motivating me to conduct the present inquiry, to R. cer ta in advan tages : (a) by reduc ing the scale of the Levi-Setti for valuable insights in the applications of ideal individual col lec tor , the p rob lem of co l lec tor dep th light collectors and to J. A. Simpson for support and is a l leviated. In principle the re is no lower limit to encouragement. I am indebted to R. A. Swanson for

SE Vol. 16, No. 2 - -C

94 ROLAND WINSTON

ass is tance with the description of the apparent solar motion and to M. F. Borun for clarifying portions of the exposition.

Note added in proof: I recently learned of related papers in 3 - the Soviet literature by V. K. Varanov. The close parallel d'_ __ _ _ _ be tween the Soviet papers and the early work of the author sugges ts that this is an idea whose time has arrived. I am grateful to Dr. Charles Miller of Jet Propulsion Labora tory for bringing the Soviet papers to my attention.

NOMENCLATURE 0~ Half angle subtended by solar disk, 0s = 1/4 °

0max M a x i m u m divergence (half angle) of a light Fig. 9. Const ruct ion of an ideal light collector for the case beam. Also the angular acceptance (half n~. > n,. A typical light ray t raversing the entire sys tem is angle) of a collector shown. In this example , n~./n, = 1.5 and 0m,. = 35 ° at the

d, Diameter of ent rance pupil ent rance aperture. d.. Diameter of exit pupil n Index of refraction

K,, Ky, K~ Direction cosines of light rays. These are y2= I k y project ions of the ray directions along (n2/nl)2k 2+ k ~ ],.c~ the x, y, z coordinate axes × i.o

k Lat i tude angle a Angle be tween ear th ' s axis of rotation and (n2/nl lZ(k× 2+ ky 2)=r

the ear th-sun direction T Angle complementa ry to a

Phase angle for the diurnal solar motion ~ o . e , ~ y / A - O - 8 ' measured f rom the noon position, q5 = 0 at noon and changes 15°/hr -i.o ~ . ~ O o \ 2 4 / , ~ -J-o k~

REFERENCES

1. See, for example, A. B. Meinel and M. P. Meinel, Phys ics looks at solar energy. Physics Today 25(2), 44 ] i~ (1972). I

2. H. Tabor, Stat ionary mirror sys tems for solar collec- tors. Solar Energy 2(3-4), 27 (1958). Fig. 10. The shaded disc of radius n,/n2 shows the region

3. Tabor (op. cit.) finds that by adding a second stage of of phase space occupied by rays refracted f rom a medium concentrat ion, the concentra t ing power can be in- of index n, into a collector filled with a medium of index creased to about 4. n2. The ellipse of semi-minor axis n~/n2 and semi-major

4. R. Winston, Light collection within the f ramework of axis = 1 is the phase space acceptance of the collector. geometrical optics. J. opt. Soc. Am. 60, 245 (1970). Note that rays inside the disc are automatical ly included

5. H. Hinterberger and R. Winston, Efficient light coupler within the ellipse and hence accepted. for threshold Cerenkov counters . Rev. scient. Instrum. 37, 1094 (1966). of index n,. The second is filled with a medium of index n:

6. This deve lopment was in conjunct ion with a proposal and designed for of R. Levi-Sett i for employing ideal conical collectors to concentra te solar energy. 0,~,,~ - arc sin (n,/n~). (A2)

The proof for the two examples cited is similar; we give it APPENDIX specifically for the latter. At its entrance, the second

Distinct indices of refraction collector is in optical contact with a medium of index n,. In case the index of refraction (n2) at the receiver Due to refraction at the ent rance interface boundary , the

exceeds the index (nO at the entrance, Eq. (1) shows that light rays inside this collector have their direction cosines an increase in concentra t ion factor by the ratio (n2/n,) is confined to a disk in the Kx, K, plane achievable. This is readily accompl ished by a straightfor- ward application of the methods already described. For Kx2+ K~2<~ (n,/n~_) 2. (A3) example , by an ideal collector which is filled with a medium of index n2 and designed for This disk is included in the collector 's phase space

acceptance ellipse which has semi-minor axis = n,/n2 and 0",,~ = a r c sin [(n,/n2)sin 0,,.~J, (Al l semi-major a x i s - 1 (see Fig. 10). We conclude that a

cylindrical collector filled with a dielectric of index n, and where 0m,, is the angular divergence of the incident light designed for 0m,~ followed by a second collector filled beam. A more useful solution, for some purposes , is to with a dielectric of index n2 and designed for 0",x = employ two ideal collectors in tandem in the manner arc sin (ndn2) concentra tes light by a factor of (n2/n,) shown in Fig. 9. The first collector is filled with a medium 1/sin Om.~).

Principles of solar concentra tors of a novel design 95

R ~ s u m ~ - - U n principe nouveau de collection et de concentra t ion d'6nergie solaire, le collecteur de lumi6re cylindrique id6al, a 6t6 invent6. L 'or ingine de ce d6veloppement r6side dans la d6tection de radiation Teherenkov dans les exp6riences de physique de haute 6nergie. Dans sa forme pr6sente, le collecteur est un canal de lumi6re en forme de bac avec parois r6fl6chissantes de forme sp6cifique qui concentre l '6nergie radiante au m a x i m u m par conservat ion d 'espace de phase. Le collecteur de lumi6re cylindrique id6al est capable d 'accepter la radiation solaire au cours d 'un jour moyen de 8 heures et de la concentrer par un facteur de 10 sans surveil lance diurne du soleil. Ceci n 'es t pas possible par les techniques conventr ionel les d ' images. Le collecteur id6al est non- imageant et poss6de une ouver ture relative effective (num6ro de f ) = 0,5+ Ce collecteur est de plus grande acceptance pour la lumi6re diffuse que des collecteurs de concentra t ion utilisant l 'optique d ' images. En fait, l'efficacit6 pour la collection et la concentra t ion de radiation isotrope compar6 a un collecteur plat, est exac tement I ' inverse du facteur de concentrat ion.