Solar Energy, Vol. 21, pp. 385-391 0038-092X/78/1~01--0385/$02.00/0 © Pergamon Press Ltd., 1978. Printed in Great Britain

OPTIMAL GEOMETRIES FOR ONE- AND TWO-FACED SYMMETRIC SIDE-WALL BOOSTER MIRRORSI"

K. D. MANNAN Mechanical Engineering Department, Punjab Agricultural University, Ludhiana, India

and

R. B. BANNEROT Department of Mechanical Engineering, University of Houston, Houston, TX 77004, U.S.A.

(Received 21 February; revision accepted 18 July 1978)

Abaraet--Moderate concentration in east-west aligned non-tracking, infinitely long, trough-like solar energy collectors is examined. Two designs are evaluated in detail. They are the one-and two-facet plane side-wall configurations. The maximum performance designs are shown not to be the most practical designs since they tend to require disportionately large reflectors. A significant reduction in reflector area can be made with only a small degradation of performance. At solstice a 9* acceptance angle is necessary for a minimum of eight hours of collection at optimum performance. Under this restriction the practical concentration ratios are limited to about 2 and 2.6 for the one- and two-facet designs, respectively.

INTROI~CTION q, L _ /

Moderate concentration with flat side-mirrors is a tech- ~]~/~" nique to improve collector performance without tracking. While the incorporation of curved mirroi's into the design ~ A =i ~ A and tracking can impr°ve c°llect°r perf°rmance t° a ~ ~ ~ _ _ ~ / ~ ~ T ~ = 2 ~ f greater degree, there are many situations where these sophistications are impractical or unnecessary. For example, fabrication and materials may be a problem in ~, N ~ ~ / developing countries. Some applications requiting a S s particular quality heat may be only marginal with a

a. One-Facet Mirror b. Two-Facet Mirror flat-plat collector, but the modest boost provided by plane mirrors may be sufficient to improve collector Fig. 1. Trough designs. performance enough to make the process viable.

Flat mirror designs associated with stationary collec- mirror case. Additional design possibilities are discussed tars are impractical due to the requirement for a large here. Work on a two-facet conical configuration has acceptance angle and the resulting low concentrating been discussed in one paper [9]. No other work has been effect. However, with periodic tilt adjustments, smaller reported previously for the two-facet mirror. acceptance angles are possible and higher concentration For the analysis the reflectors are assumed to be is achievable. Seasonal adjustments are not unrealistic, perfectly specular and non-absorbing. Optimal designs They have been utilized with conventional flat-plate will be developed based on the following criteria: For the collectors to even out seasonal performance by single facet wall, the geometry will be specified such that diminishing the "cosine effect." This can be important radiation incident within the (acceptance) angle 8 (that is, for a constant load use like heating domestic or process the component of this radiation in the plane of the cross water, section) of the normal to the groove will strike the base

In this paper two trough-like (infinitely long)east-west in one reflection or less. For the two-facet wall, the aligned collectors are discussed. A thermal absorber or incident radiation within 8 of the normal will strike the possibly solar cells are located at the base of the trough, base in two or less reflections. If two reflections are Seasonal tilt adjustments occur about the east-west axis. required, they will occur on each of the two facets on The side walls are symmetric about the plane normal to one wall. Figure 2 illustrates the resulting geometric the base. One collector type utilizes a single plane restraints for the extreme rays. Rays entering the aper- reflector on each side; the other, two plane reflectors, ture at angles greater than 8 may or may not strike the The two configurations are illustrated in cross section in base depending upon the location at which they enter the Fig. 1. groove. This behavior results in a gradual decrease in

Previous papers [1--8] have dealt with the one-facet performance as the acceptance angle is exceeded. This characteristic is discussed further in Ref. [7] (e.g. Fig. 4

tPresented at Flat-Plate Solar Collector Conference, Orlando, in that reference). Florida; 28 February-2 March 1977. For the one-facet case, requiring the extreme ray to

385

386 K. D. MANNAN and R. B. BANNEROT

Fig. 2. Geometric design criteria for the one- and two-facet configurations.

strike in the opposite corner of the groove assures all other rays incident at S or less from the normal will strike the base with one or less reflection. For the two-facet case proper values for a,, a2, D,/B and h/B will assure that if the extreme rays shown strike the edges of the base, that all other rays within 6 of the normal will also. Rays striking the upper facet may reflect directly to the base or may reflect to the lower facet (on the same side) and then to the base.

Based on the geometry of Fig. 2, it can be shown that; For the one-facet case, the geometric concentration

ratio,

CR = A = sin (3o + 8) R sin(a t 8)

D cos(2atS).cosa -= B sin (a t 6)

R _ 2 cos (2a t 6) A- cos(3atS)' (3)

For the two-facet case

CR=A=2cosar.sin(2al-aztS)-sin(2a2tS)_1 B sin(a,tS).sin(a2tS)

D CR+1 B=2tan(2a2tS)

D, cos(2a,+S)*cosa, -_= B sin@,+@

(4)

(5)

4 D LA -=---= B B B

2 cos (Y, . cos (2a, + 8). sin(al - (12). cos ((II + 8) sin (a, + 6). sin (a2 t 8)

(7)

(8)

For the one-facet case the maximum concentration for a given 8 can be found by setting

which implies that a must satisfy the following tran-

scendental equation:

sin (4a* t 28) = 2 sin (2a*). (9)

For the two-facet case the maximum concentration for a given 8 and a, can be found by setting

( > a(CR) aa2 b, = O

which implies that a2 must satisfy the following tran- scendental equation:

sin (a? t 8). sin (2ar - 3at) = sin (af * Sin (2al t S - at).

(10)

The maximum concentration ratio attainable for a given 6 with the two-facet geometry can be found by solving the following two equations simultaneously.

( > a(CR) o

aal a2,8 =

( > a(CR) da2 .,.s=’

which implies a1 and a2 must satisfy the following pair of transcendental equations:

sin (a2+ t 6). sin (2a1+ - 3a2+)

= sin a2+ * sin(2aI+t6-a2’) (11)

sin (a,+ t 6). cos (3al+ - a2+ + 15)

= cos aI+ . sin (al+ - a2+). (12)

The above equations are now used to generate a series of graphs to illustrate the geometric characteristics of the two grooves.

llm7JLnANDDIsu~oN

One-facet wall

Figure 3 illustrates a plot of eqn (1). Concentration ratio is plotted as a function of acceptance angle S and wall angle a. The dashed line passes through the points of maximum concentration ratio for each 8. The optimal values ,for a (eqn 9) as well as the corresponding concentration ratios (eqn 1) and the ratios of the reflector area to base area (eqn 3) are shown in Fig. 4 as a function of the acceptance angle.

Optimal geometries for one- and two-faceted symmetric side-wall booster mirrors 387

3.0 ---,wt= / q, m 3.0 CURVE DESCRIPTION

8 = 0 ° ~ A " ~ # ~ , ~ I MAXIMUM CR ,s 2 0..o LESS MATE.,AL REOU,REO #

¢l ~ i'~/'~ V"'~ Z(I ~ , \ \ FOR MAXIMUM CR AT SAME8 \ \ \ 3 5O-~o LESS

¢ ~ , ~ 4 70% LESS ~"18 /q , ["- B ~ z , \ ~"-.SL" 90% LESS I,~-- A ~-,,I~

o

_ o T ~ - I 5

'~ 5 fO 15 20 25 30 n," 8 (DEGREES)

z 2.0 0 C-- Fig. 5. Effect of reflector area reduction on performance of the '~ one-facet groove collector. tw I.-

Z

"' correspond to moving along the right legs of the curve in U

o z Fig. 3. Figure 5 generally bears out the conclusion that '-' ~ relatively large reflector area reductions only moderately

reduce performance. For example, with an acceptance angle of 10 ° a 50 per cent reduction in reflector material reduces the concentration ratio from 1.94 to 1.70.

I.o - - ~ Acceptance angle o ~o 2o 30 40 Before continuing to the two-facet analysis, it would

a (DEGREES) be helpful to discuss some of the limits associated with Fig. 3. Concentration ratio as a function of acceptance angle and the acceptance angle. These east-west aligned groove

wall angle for the one-facet groove collector, collectors are designed to be non-tracking; that is, they do not continuously follow the sun. However, to be

a \ practical, occasional tilt adjustments are necessary. The g= =_6 3.0 I " ' frequency of these adjustments depends on the accep- \ ,~ ,~ tance angle for the collector and the relative daily north- -" south movement of the sun. It is the projection of the

~- = I~ - A ~¢(q' ~ sun's direction vector onto the north-south vertical plane ~ / . --~\ \~- / ' ~ ~ that determines whether or not the sunbeam reaches the z t ~

o t O hJ / r-o_ 2.o ,o 9 base of the groove. If the angle which the projected z o * beam makes with the normal to the groove is less than or z~ ~, equal to 8, the beam will reflect to the base. (Walls are 8 ~ assumed to be perfectly reflecting.) As discussed above,

d 1.0 . . . . 5 if the beam's projected angle exceeds & part or none of = ~ o 5 ~o ~5 2'o 2~ 30 the beam reaches the base, depending upon where it

8 (DEGREES) enters the groove (e.g. near the south wall, halfway Fig. 4. Maximum concentration ratio and optimal design across, etc.). For the discussions here, however, all in- parameters as a function of acceptance angle for the one-facet solution outside of 8 is ignored. Tabor[l] has determined

groove collector, that the angle that this north-south projection of the sun makes with the normal can be expressed as (called EWV

The relative "flatness" around the maximum concert- altitude by Tabor) tration ratios in Fig. 3 leads to the important conclusion ( tan O that for a fixed acceptance angle, relatively large changes 3' = tan - t \ ~rt/ (13) in the wall angle have little effect on the concentration cos -~ ratio. Consequently large savings in reflector material can be realized by accepting a slightly off-maximum where 0 is the sun's declination and t is the time in hours design. The designs would correspond to wall angles from solar noon. This angle, 3', is plotted in Fig. 6 taken greater than those required for maximum concentration, from Ref. [3]. It is measured from the equinox position. Decreasing wall angle increases rather than decreases the If the east-west trough is tilted from the zenith by an reflector area. A quantitative evaluation of the savings angle equal to the geographical latitude, the projected possible is given in Fig. 5. First the maximum concen- angle of insolation is 1'. If the trough is then tilted from tration ratio has been plotted as a function of acceptance this position by an angle, //, the projected angle of angle (as in Fig. 4). The amount of reflector material incidence of the solar beam is (R/A in eqn 3) required for the maximum concentration ratio at each acceptance angle was determined with 30 @ = 3 ' -B. (14) per cent (then 50, 70 and 90 per cent) less reflector material. The results are plotted in Fig. 5. These designs The daily north-south "swing" of the sun can now be

388 K.D. MAr~NAN and R. B. BANNEROT

determined for any day; thus setting the requirements for In Fig. 8 are plotted the maximum concentration ratios the collector acceptance angle 8. For instance, at the attainable as a function of at, for several different ac- solstices where the daily swing is the most severe, the ceptance angles, 8. Also indicated are the appropriate 8-hr swing (noon plus or minus 4hr) is 17.5 ° (40.94 °- values of a2. For example, with 8=9 . , the maximum 23.45 °, from Fig. 6). Thus a 9* acceptance angle (17.5/2 = concentration ratio is seen to be approximately 2.7 at 8.7°)wouldassureShroffullconcentrationontheso lstices at = 21 ° and a2 = 8°; a result already determined from if the trough were tilted to bisect the diurnal "swing" of the Fig. 7. The dashed line connecting the local maxima sun. The tilt angle, /3, would be adjusted as necessary represents the solution to eqn (11). throughout the year. In Fig. 9 the maximum concentration ratio attainable

for a given acceptance angle is plotted. Also given are Two- face t wall the optimal groove wall angles and the amount of

A plot of eqn (4) for a 9° acceptance angle is presented reflector material required. This figure is the two-facet in Fig. 7. Concentration ratio is plotted as a function of counterpart to Fig. 4. the two wall angles. The maximim concentration ratio is As in the one-facet designs, considerable reduction in 2.68 at at = 21 °, and a2 = 8 °. In the figure it can be seen reflector material is possible with only a small degrada- that for each at there is an a2 which maximizes the tion in performance. Figure 10 illustrates this effect for concentration ratio. The dashed line is the locus of the the two-facet wall and a 9* acceptance angle. The maxi- local maxima and represents the solution to eqn (10). For mum concentration ratio is 2.68, requiring a reflector at less than 21 °, the optimal a2 is less than 8*; for at surface 3.05 times as large as the aperture of the groove, greater than 21 °, the optimal a2 is greater than 8 °. Each (R/A = 3.05). If 10 per cent less reflector is used than constant at line terminates at the right at at = a2. The required for the maximum concentration (R/A = 2.75), locus of these points, indicated by the second dashed then at each at the resulting concentration ratio is plot- line, represents the performance curve for the one-facet ted in Fig. 10 just below the maximum attainable at the wall. In this case the maximum one-facet concentration a,. At approximately at = 250 the maximum concen- ratio is 2.0 occurring at at = a2 = a = 15 °. These results tration ratio is 2.65, and this can be attained with 10 per can be verified with Fig. 3. Figure 7 illustrates that for cent less material (R/A = 2.75) than is required for the a 2 < a t the two-facet wall will outperform the cot- maximum concentration ratio of 2.68. This fact is responding one-facet wall with a = at. A design with represented in the figure by the coincidence of the a2 > a, should not be considered. Similar curves result maximum concentration ratio line and 10 per cent less for other acceptance angles, material at at = 25 °. Similar curves for 20, 30, 40, 50 and

60 per cent less reflector material are also given. The respective ranges of a2 are also given and are seen to

ao -+s vary only slightly for a wide range of at. With 20 per g ~ - , s _ 60 -,1:4 Hours FROM cent less reflector material the maximum concentration g --'t'3 SOLAR NOON ratio at a, = 290 is 2.55, only about 5 per cent less than 0. 4 0 - . 2 x ~ -+o the.maximum for this acceptance angle. ° 2 2 ° Each of the curves in Fig. 10 represents a given :D~:: w=°° o , A amount of material per unit of aperture. Each curve is

-~-zo ~ concave down resulting in a maximum concentration at ~- one value of at. These maxima are indicated by the ,9, -40 ~ dashed line. This line, then, represents the maximum

~ -6o concentration for a given amount of material. = Limi ts to a or a, . One possible trough design would be

-80 ,: . . . . . . . . . . . . . . . . . to have a thermal or photon (solar cell) absorber at the 0 40 so izo ~60 200 z ~ 2ao 320 360 base of the groove extending from the base of one wall

DAYS FROM EQUINOX to the base of the opposite wall. Such an absorber would

Fig. 6. Projection of sun's position into the north-south plane, probably have a glazing over it. An air-to-glass interface

= 3.0 < . ~ ~ _ _ ~ M A X I M U M CR FOR EACH =~ ~ / q t

6 A :L/ '""

" . . / / . . M A X I M U M CR ,FOR Zs'-.'X~ ~'= O N E - FACET GROOVE ] ~ . , , . . ~

~ 3 9 " 1.0 . . . . . . . . . i . . . . t . . . . i . . . . , : , , i i . . . . i , , , , i

0 5 10 15 2 0 2 5 3 0 35 4 0

= z ( D E G R E E S )

Fig. 7. Concentration ratio as a function of each wall angle for the two-facet groove collector with a 9 ° acceptance angle.

Optimal geometries for one- and two-faceted symmetric side-wall booster mirrors 389

, . M A X I M U M CR ._~/~O;_

4 . 0 = l

~ - - --p2"

$ IS* ~O2. O T" ~14"

Z 15

5 IO 15 2 0 2 5 :50 :55 4 0

= l ( D E G R E E S )

Fig. 8. Maximum concentration ratios attainable as a function of the lower wall angle and the acceptance angle for the two-facet groove collector.

d

fl::~:

p< I =, 4 0 IO

QC

O A ~ \ \ o

< :5.0 5

=, / B

0 5 IO 15 2 0

z o c~ 8 ( D E G R E E S )

Fig. 9. Maximum concentration ratio and optimal design parameters as a function of acceptance angle for the two-face{ groove collector.

j =

A 8 = 9 ° CURVE NO DESCRIPTION

~ " MAXIMUM CR AS A FUNCTION OF ~1 G ~ / (AT (11,22 l, C R , 2.68, RIO • 3.O5]

2 J 0% LESS MATERIAL : { R/O • 275), 9" -<CI2(* 9.5" a 3 2 0 % , IO.5"-~ a = s rl °

3 0 4 3 o % t z . s ' : a z s l a " I [ O " s 40% ; t5 °s a z s Is .s" P - B ~ j

6 ~ % ; 18'*-* ~2 "~ I S S " 7 60% 22"s ~l ~225" _ . . . . . . ~ M A X I M U M CR DESGN

' z AND REDUCED MATERIAL DESIGN MERGE

~ 2 . O Z i OPTIMAL CR FOR A GIVEN

OL> ' i i , i i , ,5 2o 2'5 30 ,o

¢== ( D E G R E E S )

Fig. 10. Effect of reflector area reduction on performance of the two-facet groove collector for a 9 ° acceptance angle.

has the characteristic of high solar transmittance for important to limit the incident angle on the base to less incident angles within about 65 ° of normal. However, at than about 65 °. From Fig. 2 this implies that larger angles the transmittance quickly drops to zero as all energy is reflected back into the air. If glass is used for the internal cover plate on the absorber, it would be 2a + 8 --- 65 °

SE Vol. 21, No. 5- -C

390 K.D. MANNAN and R. B. BANNEROT

for the one-facet case and (3) The wall curvature of the CPC is quite larger near the base of the groove resulting in grazing incidence at

2a,+8_<65 ° the base for many rays. As previously discussed, depending on the absorber design, this could result in

for the two-facet case. reflection rather than absorption at the base. As can be seen from Figs. 3 and 4, this requirement is

not restrictive at all for the one-facet trough for ac- ceptance angles up to about 150 . However, for larger CONCLUSIONS acceptance angles and in the two-facet design this The maximum performance designs for grooved restriction could be important. From Fig. 8, for example, collectors are not practical designs. In general, only small it is seen for the larger acceptance angles the maximum degradation in performance results from significantly concentration ratios occur for values of at near the limit, off-optimal configurations. For example, if 8 = 90, The design of a non-tracking groove collector is a

strong function of the required acceptance angle. The -65°-9° o choice of the acceptance angle is, in turn, a function of

a l < ---28. 2 collected heat requirements. Smaller design acceptance

angles increase attainable concentration ratios but The maximum concentration ratio occurs at a l = 21 °, require more reflector area and increased maintenance

but from Fig. 10 it is clear that the optimal use of by increasing the frequency of tilt adjustments. material dictates a value of a l possibly as high as 30*. For the 9 ° acceptance angle, corresponding to maxi-

mum apparent daily north-south solar movement, the Comparison of performance one-facet has a practical concentration approaching 2.0.

A crude comparison among three designs is made in The two-facet design has a practical concentration Fig. l l for a 9 ° acceptance angle. The designs are the around 2.6. Additional facets could increase this. The one-facet wall, the two-facet wall and the compound theoretical maximum possible concentration ratio is parabolic concentrator (CPC) (10, 11]. This comparison about six for a 9 ° acceptance angle. should be applied with caution since:

(1) It is based Oil maximum concentration ratios rather NOMENCLATURE than "optimal use of material" designs. A collector aperture (see Fig. 1)

B collector base width (see Fig. 1) (2) The "complete" CPC design for a 9 ° acceptance angle has a concentration ratio of 6.4 and requires a CR geometric concentration ratio

D collector height normal to base (see Fig. 1) reflector area of about seven times the aperture. Trun- t time in hours from solar noon cation of the reflectors from the aperture end can greatly a wall angle reduce the size of the reflectors while the concentration /3 collector tilt ratio is initially only modestly reduced. However, it is ~, projection of insolation into north-south plane (see eqn

13) substantially reduced with severe truncation (as shown in 8 acceptance angle of collector Fig. 11), Truncation also climates the characteristic sharp 4, solar incidence angle, 3,-/3 (see eqn 14) acceptance or cut-off angle of the complete CPC and 0 solar declination also allows the capture of some insolation from outside the original acceptance angle. In other words, the per- Subscripts formance of the truncated CPC and that of the single- 1 lower wall (two-facet groove) and double-facet trough begin to merge as the CPC 2 upper wall (two-facet groove) truncation becomes severe.

3,0

I" A o ~ ' \ I $ld

i.---A - I /~-= R, *R, y ~ , . "

°,.o ) b

* i i t ' i i 1.0 2 0 3.O

CONCENTRATION RATIO, A/8

Fig. i 1. Pedormance comparison for a 9* acceptance angle.

Optimal geometries for one- and two-faceted symmetric side-wall booster mirrors 391

Acknowledgement--This work is part of a project sponsored by moderately concentrating solar energy collectors. Radiative the Solar Heating and Cooling Research and Development Transfer and Thermal Control, Progress in Astronautics and Branch, Conservation and Solar Applications, Department of Aeronautics, Vol. 49, AIAA, pp. 277-289 (1976). Energy: Contract No. E(40-1)-5100. 7. R. B. Bannerot and J. R. Howell, The effect of non-direct

insolation on the radiative performance of trapezoidal ~ C E S grooves used as solar energy collectors. Solar Energy 19.

1. H. Tabor, Stationary mirror systems for solar collectors. 549-553 (1977). Solar Energy 2, 27-33 (1958). 8. R. B. Bannerot and J. R. Howell, Predicted daily and yearly

2. H. Tabor, Mirror boosters for solar collectors. Solar Energy average radiative performance of optimal trapezoidal groove 10, 111-118(1966). solar energy collectors. Procs. Sharing the Sun! Con[.,

3. K. G. T. Hollands, A concentrator for thin-film solar cells. Winnipeg, Manitoba, pp. 111-125 (1976). Solar Energy 13, 149-163 (1971). 9. K. D. Mannan, Analysis of oven-type conical concentrators.

4. E. L. Ralph, A commercial solar cell array design. Solar 7th Annual Con[. o[ All India Solar Energy Working Group. Energy 14, 279-286 (1973). Ludhiana (1976).

5. R. B. Bannerot and J. R. Howell, Moderately concentrating 10. A. Rabl, Solar concentrators with maximum concentration flat-plate solar energy collectors. ASME Paper 75-HT-54. for cylindrical absorbers. Appl. Opt. 15, 1871-3 (1976). Presented at the Nat. Heat Trans. Con/., San Francisco, 11. R. Winston, Solar concentrators of a novel design. Solar California (1975). Energy 16, 89-95 (1974),

6. J. R. Howell and R. B. Bannerot, Trapezoidal grooves as