Solar Energy, Vol. ~ , pp. 373-381. Pergamon Press 1979. Printed in Great Britain

PRACTICAL DESIGN CONSIDERATIONS FOR CPC SOLAR COLLECTORS?

A. RAaL Argonne National Laboratory, Argonne, IL 60439, U.S.A.

and

N. B. GOODMAN and R. WINSTON Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, U.S.A.

(Received 12 February 1977; revision accepted 14 August 1978)

Abstract---Several practical problems are addressed which arise in the design of solar collectors with compound parabolic concentrators (CPC's). They deal with the selection of a receiver type, the optimum method for introducing a gap between receiver and reflector to minimize optical and thermal losses, and the effect of a glass envelope around the receiver. This paper also deals with the effect of mirror errors and receiver misalignment, and the effect of the temperature difference between fluid and absorber plate. The merits of a CPC as a second stage concentrator are analyzed.

nVrROI~CTION The merits of a CPC as a second stage In recent papers[I-4] the general optical and thermal concentrator[13, 14] are analyzed in Section 7, and a properties of the compound parabolic concentrator simple criterion is given for deciding whether or not to (CPC) were studied. In the actual design of CPC solar use a second stage concentrator. A CPC is shown to be collectors certain practical problems arise which require preferable to a V-trough even under conditions most further analysis. The purpose of this paper is to discuss favorable to the latter. the most important of these [5].

Section 1 treats the relative merits and drawbacks of L COMr~dUSON OF vwrgnr.s'r crc several geometric configurations, specifically when the For specific applications, one or another of the CPC absorber is flat (Fig. la), a fin (Fig. Ib), an inverted vee types shown in Fig. I may be preferred. We therefore (Fig. lc) and tubular (Fig. ld). All of these have come to summarize their similarities and differences. For all of be referred to as CPC-type collectors although some them the concentration,~; C, and acceptance half-angle, 0, configurations (e.g. Fig. Id and most second stage concentrators) are not parabolic. They are all maximally . [ ' ~-s-0----7 ~ ~ r ---~ concentrating devices designed for each specific absor- \ / , / J

ber shape [6, 7]. ~\ / / I ",\ , //////],/ A gap between absorber and reflector may be required \ , '

to reduce heat leaks and/or to accommodate a glass '\\ // envelope as shown in Fig. 2. We discuss design \x o!o/ \0~ 0, modifications in Section 2 which minimize optical losses ~\] /7 \ '<!//~ / produced by such gaps. These losses are actually smaller \ ~,l/ / tors than those for an equivalent flat array of tubular collec- ! / / i~\ / b.

Reflection losses caused by a glass envelope around \ . / ' \ / the receiver are computed and compared with a flat ~ - -

;= 0-..~ stationary glass plate in Section 3. Section 4 deals with the relationship between the height and cohcentration of a collector. Section 5 concerns the imperfect alignment [7 ~ ~ , ~] -.a--s,~s --71, °f the abs°rber and err°rs in the mirr°r shape" I , , -~l / ", / / ~' ~ i' //'/ ,"

The performance of a CPC solar collector has pre- , , ', i / viously been described relative to the absorber plate ', , ', ' temperature[3, 11, 12]. The effects of the temperature , 0 difference between the fluid and absorber plate are dis- ~"' cussed in Section 6. ', / " '

/ c.

lWork supported by the U.S. Energy Research and Develop- ,. /~-~ / ment Administration.

:~The concentration C is defined as ratio of aperture area over Fig. 1. Four types of CPC. All have the same absorber perimeter a receiver surface area ("geometric concentration ratio"), and acceptance half angle 0.

373

374 A. RABL, N. B. GOODMAN and R. WrNSTDN

are related by the fin, wedge or tube absorber will receive less heat than a flat absorber by an amount

C = 1/sin 0 Aq = (1 - ~/p)~oS = 20-50 W/m 2

if the reflector is untruncated; the effect of truncation[3] is practically the same for all types. The comparison of per unit area of the collector aperture. If a glass tube the different CPC types is made under the assumption of surrounds the absorber an additional difference in optical totally isotropic illumination of the absorber[15]. The losses is incurred because with a fin, wedge or tube most important differences arise from the fact that a fin receiver some of the rays must traverse the glass en- or tube absorber is illuminated on all sides, thereby velope more than once. For example, for a CPC with requiring only half as much absorber material. This concentration 1.8 and a tube receiver of radius 0.74 in. results in lower material costs, smaller conductive losses enclosed by a glass tube of outer radius 1.03 in. and inner to the back, and gains in performance since the transient radius 0.98 in., the average number of traversals through response is improved, the glass envelope is 1.20, making an antireflection coat-

The flat receiver configuration has a higher optical ing such as etched glass[18] desirable. transmission, T, through the concentrator since its shape The smaller back losses for the fin and tube absorber factor[16] for direct radiation is larger. • can be ap- configurations will more than compensate for their higher proximated by [17] optical losses in most cases. For a flat absorber the back

losses are typically Ub~k = 0.2-0.5 W/m2°C. If the fluid = p(n) temperature is sufficiently higher than ambient the fin or

tube absorber types will have lower losses and will where p = reflectivity of the CPC walls and (n) = average therefore be preferable. number of reflections. For reasonable truncations, the variation in (n) with angle of incidence is less than -+ 10 per cent and can therefore be ignored [3]. 2. GAPS BETWEEN RErLr.CTon Am) A a S O ~

The exact values for (n) given in Fig. 11 of Ref. [3] and The design principles[19] of ideal concentrators Fig. 8 of Ref. [13] can be fitted with an error of less than require that the receiver touch the reflector, as illustrated _+0.1 for any practical truncation and concentration by: in Fig. I. In many solar thermal applications, however, a (n)=0.5+0.07C for flat receiver (Fig. la); (n)= gap between the reflector and the absorber is needed to 1.0+0.07C for fin, wedge and tube receiver (Figs. lb-d), incorporate an evacuated glass envelope[8-10] as shown

For p --0.85 an error of -+0.1 in (n) results in an error in Fig. 2, and/or to reduce conductive heat losses since a in ~- smaller than +0.02. Therefore, all other properties reflector made of aluminum sheet in direct contact with being the same, the optical efficiency ,o for configuration the absorber is an efficient cooling fin[20]. l(a) will be a factor of ~/p higher than for the others. For Since a gap between the reflector and the receiver typical values of reflectivity, p(0.85--0.95), optical causes optical losses, a compromise between optical and efficiency, ,0(0.6--0.7) and insolation, S(600-1000 W/me), thermal performance must be made. If the basic concen-

trator geometry is to remain unchanged, three ap-

i'An alternative approach to the gap problem which was found proaches can be taken:t (i) reduce size of absorber (Fig. subsequent to the completion of this work is discussed in Ref. 2); (ii) truncate edge of reflector near absorber (Fig. 3); [25]. (iii) modify absorber to form a radiation cavity (Fig. 4).

Fig. 2. Four types of (?PC Solar Collectors with diminished absorbers surrounded by glass tubes. Optical losses are L = 2gla for cases (a)-(c) and 2~la for case (d).

Practical design considerations for CPC solar collectors 375

[a) ".. . . . .

,4 /3 B c

Fig. 3. Four types of CPC with truncated reflectors to create gaps each of width g between the reflector and the receiver with perimeter, a. For each case the_.~pticai losses and gap width, g, are: (a) L -- [1 + cos (3~r/4- 02~g/a; ranges from 0.3g/a to 0.Sg/a; where g=BC; Co) L~z/a; where g=BC; (c) L~g/a; where ~=BC; (d)

L = 1/~{~/[2g/r + (g/r) 2] - arc cos (r/(g + r))}; where g + r is the distance from the center of the tube to B.

k 7

Fig. 4. Four types of CPC with receiver modified to form a radiation cavity.

Explicit solutions for the optical loss, L, where for collector types 2(a)-2(c) is

radiation lost in gap L = 2g/a. L =

radiation on absorber + radiation lost in gap

For the tube receiver of Fig. 2(d) the loss is are presented here for a 2-dimensional (trough-like) geometry but the extension to the 3-dimensional (cone- L = 1 - Fdesigncd.reduce d

like) case is straight-forward. The optical loss for solution (i) can be easily under- where Fd.*iSn~-r.duc.d is the shape factor[16] for radia-

stood if the flux distribution on the absorber is tion from the absorber for which the collector was diffuse[15]. If the original absorber has perimeter a and designed to the reduced absorber. For example, if a is truncated to create gaps each of width g then the loss circular tube of radius r' = r - g is placed concentrically

SE Vol. 22, No. ~--F

376 A. RABL, N. B. GOODMAN and R. WINSTON

with the original tube of radius r = a/2~r then L = to form a radiation cavity. Figure 4 indicates this sche- 1 - r'/r = 2¢rg[a. A different solution with lower losses is matically for various concentrator geometries. The described later in this section, cavity can easily be designed to avoid all optical gap

For solution (ii), shown in Fig. 3, the gap losses can losses. However, it may pose practical disadvantages for also be evaluated readily employing the calculus of radi- manufacturing, and the heat losses are larger due to the ation shape factors. The procedure consists of identify- increased absorber surface. ing a surface over which the flux distribution is totally For a CPC with a flat receiver there exists a special diffuse[15] and then evaluating the relevant shape factor, solution (iv) as illustrated in Fig. 5. By spreading the Since light rays are reversible, the fraction of radiation mirrors, M~ and M2, and positioning the focus of each reflected onto the absorber is equal to the shape factor mirror, F~ and F2, respectively, at the opposite edge of F,b,**b,r-pp~ for radiation from the absorber to the gaps. the absorber, rather than at the base of the opposite Hence, the gap loss is mirror, the optical losses can be cut in half. For isotropic

incident radiation, each mirror section by itself will L = Fabsorber-gaps- produce uniform illumination over the plane of the exit

aperture. Therefore, for isotropic incident radiation up to We can calculate this for the collector types shown in the cut-off angle, each mirror section will produce uni- Figs. 3(a)-(c) using the method of Hottel's Strings[16]: form illumination between its own base and focus. It

follows that for two mirror sections together but spread apart by 2g, the radiation density in the gap region will be

L = 2. lAB + B C - AC] just one half the value between the foci. By positioning the receiver between the foci, the amount of radiation

1 g2 T)~/2]. lost is L = a [ a + g - ( a 2 + - 2ag cos

L.pread mirrors = g / ( g + a) ~ g/a Since g ,~ a, we can approximate this by

where a is the absorber width the mirrors are designed L ~ (1 + cos T)g/a. for. This loss is intermediate between that of solutions (i)

and (il). In case 3(a), 3' = (3~r/4)- (0/2) so L is between 0.3g/a and A special solution also exists for a collector with tube 0.5g/a. In cases 3(b) and 3(c), 3' = ¢r/2 so L ~- g/a. In case receiver. Rather than altering the physical dimensions of 3(d) one can take advantage of the fact that radiation the absorber or reflector, one can simply move the tube passing the dashed surface AB is isotropic. (AB extends away from the cusp of the reflector and along the axis of tangentially from the absorber to the edge of the trun- the collector. This is a special case of displacing the tube cated reflector.) Let r be the radius of the absorber and g from its designed position, as illustrated in Fig. 7. As the distance from the edge of the truncated reflector to calculated in Section 5, the losses introduced by displac- the absorber. Then the shape factor is ing a tube of radius r = al2w by a distance g is:

Fas-,b,o,b,r = ~k/tan 6, 2 L = 1 - - - arc cos (g/2r).

where 6 = arc cos (r/(g + r)).

It follows that the gap loss is given by The first term of the series expansion gives the simple

L = "~-~ur ( - aS-,b,o*b,r) '\," S.~+ g__7/

= l { v ' [ 2 g / r + (g/r) 2] - arc cos [r/(g + r)]}. \' ' \ // "If \\ t I

\ \ /

Thus L is < I0 per cent for g/r as large as 0.5. The \\ / / specific formulas for each case are included in Fig. 3. \, 0 0

\ // They imply that truncation of the reflector edge (solution Ms ~ r - ~ l (ii)) results in lower optical losses than truncation of the , , / [ absorber (solution (i) by at least a factor of two. Solution ,, , , / / / (ii) has the additional advantage of spreading out the losses overall angles of incidence whereas losses in ~ / i " / \ r l / / / ' ~ ' k " \ ~ 2 / solution (i) are highest near the acceptance angle. For practical configurations [8--10] this loss is in the range of 1-2 per cent. By comparison, gap losses in a nonconcen- trating flat array of tubular collectors are L = glr. We see gfl-~-* -

that the use of reflectors can reduce these losses Fig. 5. A CPC with fiat receiver and mirrors M~ and M2 spread substantially, apart to maintain their loci, F~ and F~ respectively, at the edges

In solution (iii) part or all of the absorber is distorted of the absorber.

Practical design considerations for CPC solar collectors 377

, tory analysis of the fraction of fight incident on a collec-

2.s , , l , , ' /~ 1 I tor at an angle • from its optic axis which reaches gaps 20 .I between the absorber and the reflector. This has been

I done for collector types shown in Figs. 2(a) and 5 (CPC ~.s I i with truncated flat receiver with fixed and spread mir-

L i rors) for the gaps occupying up to a total of 32 per cent ,o "" of the exit aperture.

~ - ~ _~.._ / The results are plotted in Figs. 6(a) and (b) and can be Qs ~ seen to match well to the derived L = 2gla for fixed

mirrors (solution i) and L = g/a for spread mirrors o , i I I i i i (solution iv). It should be noted that the calculated -Is- -,2- -~. -4- o- 4- 8- ,2- ,s. values of L are averaged over all [6[ - 0 and that without to) Fi~d mirrors spreading the mirrors a large fraction of the total losses

are within 1-2 ° of the acceptance angle. So the losses 2.s , , , I ' , , within a reduced acceptance angle are less than 2gla.

With the computer simulation it was possible to obtain 2 0

an average loss within the full range in ~ and it indeed was L = 2gla.

h5

g / o • " . , . . 3. G L A S S T U B E S I . q ~ O U N D I N G

,.o / ~ . - f f " - ' ~ " " - - A glass tube surrounding the receiver of any type of 1 / / ~ ~ ' ~ " " - concentrator creates optical losses. The absorption by

as / / y ~ the glass amounts to only a few percent ff low iron glass . . . . is used. A larger fraction of the light is reflected off the

0 ~ • • i i I I I -e- -,2. -e. -4. o. 4. e- ,2- ~- surfaces of the tube, the exact amount depending on the tu} ~ mirrors angle of incidence of light on the tube and the index of

Fig. 6. Computer simulation (---) and experimentally deter- refraction of the glass. The average angle of incidence mined ( ~ ) ratio L/(g/a) where L, g and a are defined in the varies with the time of day for a non-tracking collector text for (a) CPC with truncated receiver; gia = 0.03, 0.05, 0.07, 0.09. (b) C?C with truncated flat receiver and spre~ mirrors; so average reflection losses for various times are given in

gla = 0.06, 0.09, 0.13, 0.16. Table I. The numbers in Table 1 were calculated assuming a

refractive index of n = 1.5 with the use of computer raytracing specifically for the collector type shown in Fig. 2(a) with acceptance half-angles between 5 ° and 35 °, but should be approximately correct for all collector types shown in Fig. 2. These losses are comparable to those in a stationary fiat glass plate. In both cases the use

~ . of antireflection coating or etched glass[18] may be a d - ~ '~\~ visable.4. COLLEC'TOR H E I G H T F O R DESIRED C O N C E N T R A T I O N

Another practical consideration is the height of a collector for a given concentration ratio and absorber perimeter, o. For any untruncated CPC there are two so-called shadow lines, each forming an angle 0 with the optic axis of the collector. Irrespective of concentrator design, there is a constant distance, h, = al(2 tan 0 sin 0),

Fig. 7. A CPC with tube receiver with radius r which has been displaced from its designed position (dashed circle) to another Table 1. Amount of light reflected at various times of (solid circle) by a distance g. Losses caused by this mi~li~mcnt day by a glass tube surrounding the receiver of a CPC

are L = 1 - 211r arc cos (g/2r) = gl3r. and by a stationary flat glass plate

Percentage of fight reflected approximation

Hours away Glass tube Stationary flat from noon around receiver glass plate

L =-g- for g ¢{r; ~rr 0 (9.8 + 0.1)% (7.7 ± 0.1)%

! (9.9±0.1)% (7.8±0.1)% the relative error is below 5 per cent, even if g/r is as 2 (10.4±0.1)% (8.1±0.2)% large as 1.0. 3 ( 1 2 . 2 ± 0 . 1 ) % (10.0±0.5)%

We have performed computer simulated and labora- 4 (18.9±0.1)% (17.2± 1.0)%

378 A. RABL, N. B. GOODMAN and R. WrNSTON

from the point where they cross, P, to the top of the Table 2. Total collector light for various CPC's with collector, acceptance angle and absorber perimeter a

The distance, h2, from P to the level of the bottom of the collector depends on the receiver shape. The ab- Receiver type Total height of collector (h)

sorber is constrained to lie entirely between the shadow a [ 1 + 1 ] lines and to have a total perimeter, a. If the absorber Flat h =2 Lsin 0 tan 0 ~-O does not extend from one shadow line to the other, there a r l 1 must be a reflecting involute[6] of it which does. Table 2 Fin h =2 L ~ + I J gives expressions for the total height (h = h, + h2)of a f 1 1 each collector type shown in Fig. 1. The differences Wedge h=~- L ~ + c ° s 0 J among them are relatively small. The shortest type of a [" 1 +1+ I 1 concentrator is that for which the absorber is tangent to Tube h =2 [ ~ 2 ~---~-0j both shadow lines at all points (Fig. lc). The effect of truncation on the Height/Concentration ratio varies only slightly from one collector type to another. \ /

w ~f lecTor 5. MIRROR ERRORS AND ABSORBER MISALIGNMENT

As pointed out previously[13], the effect of slope e / o I ~ ~ errors in the mirror surface is the same for all CPC ~ ~ - - J ~ t types. Since all rays incident at or near the cut-off angle ~ j ) r(ece,~er ' undergo exactly one reflection before striking the ab- sorber, an average absolute slope error A in the reflector Fig. 8. Illustration of the temperature drop across a planar decreases the acceptance half-angle by 2A. For rays receiver with a fluid rigged tube attached. incident not near the acceptance angle, the slope errors will redistribute the radiation on the absorber but will not concentrating flat plate collector is the fin efficiency of cause any losses, the absorber because the flux distribution at the absorber

For a fin or tube receiver additional errors can arise varies with time of day. However, for many CPC collec- from incorrect alignment of absorber and reflector, un- tors this fin efficiency can be taken to be one since the less the cavity design of Fig. 4 is chosen. As before, absorber is narrow and the temperature drop across its radiation shape factors permit a simple evaluation of the width is neglible. effect on the average performance and this is illustrated The magnitude of the temperature drop across an here for the case of a tube receiver. Suppose a tube of absorber can be estimated by the following simple radius r (surface T, - T2) is displaced by a small distance method. Since the solar flux distribution at the absorber g to the surface T ; - T~ as shown in Fig. 7. Since the varies with time, it is instructive to consider the worst average flux distribution at surface T , - T2 is isotropic, possible case which occurs when the sun is incident on the fraction f of radiation that reaches tube T~- T~ is the collector at the acceptance angle. Then for a flat, fin found by means of radiation shape factors. By consider- or wedge absorber all the flux is concentrated on the ing the portions of the two surfaces on each side of the edge, E, of the absorber from where it must be conduc- dotted line separately; one obtains: ted into the fluid (see Fig. 8). Let D be a typical point on

the inner tube perimeter located a distance e away from T~ + T2 FT2--T~'. E. (The precise location of D and subsequent value of e

f = T, ÷ T2 FT,-T,, T~ + T2 do not matter in this approximation.) ff the collector has concentration C and optical efficiency "qo and the in-

The individual shape factors are FT,-T1 = T;/T~ and solation is S (Wperm 2 of collector aperture), then the FT:-T~ = 1 and since T~ = T2, power per unit length incident on the absorber is

= _ q - w~oCS f = 2 T J ( T , + T2) 2 arc cos (g/2r) -~ - "rr

= 1 - g/(crr) for g a r. where W is the width of the absorber. If this amount of

For example, if g/r = 1/3, then f = 0.89 so 11 per cent of heat is to be eonducted from E to D through an absorber the incident radiation misses the tube T'. plate of thickness t and conductivity k, then the maxi-

mum temperature drop Tp . . . . between E and D must

6. FIN EFFICIENCY AND FLOW FACTOR FOR A CPC satisfy:

The efficiency of a CPC has previously[3, 11, 12] been q = tkATpm~,/e. described in terms of an average absorber plate l temperature. When referring to the fluid temperature the well-known heat removal factor, FR, must be applied as a Combining the last two equations, one finds correction [21]. For the calculation of FR, the only factor differing from the corresponding analysis of a non- ATpm~ = Cs~oew/(tk).

Practical design considerations for CPC solar collectors 379

As an example, for a copper absorber plate of width and losses we consider the energy balance of the ab- w = 2 cm and thickness t -- I mm with a tube at its center sorber in a system with and without second stage (implying e-- I cm) and typical values of C = 5, S = concentrator. If one defines the optical efficiency 7o as 600W/m 2 and rio=0.60, one obtains ATp=,~=0.9°K the fraction of the direct solar flux S incident on the which is small enough to be neghble, collector aperture which reaches the absorber and is

In an evacuated receiver one may want to use absorbed there, then the energy extracted per unit ab- concentric tubes to avoid the stress problem which arises sorber area if both ends of a glass tube are sealed. The thermal shorting associated with such an arrangement has been qou,--~oC,S- q,o, analyzed by Thodos[22] and by Beekley and Mather[23] with the conclusion that the flow rates can be chosen where C,--concentration of first stage and S =ins- high enough to avoid a serious shorting problem. The use olation (W/m 2) and qto, = heat loss. The collection of a heat pipe can eliminate this thermal short circuit and efficiency ~ is the ratio of qo,, over am = C,S is attractive for other reasons as well.

qloss 7. SECOND STAGE CONCENTRATORS ~ l = ' ~ o - C I S '

The radiation converging at the focal zone of a con- ventional concentrator such as a Fresnel lens or a If now a second stage of concentration C2 is placed in parabolic mirror has an angular spread which is less than front of the absorber, the absorber size and with it the ±90 °. It can, therefore, be further concentrated by a heat loss can be scaled down by a factor C2 (to a good matching second stage concentrator, as sketched in Fig. approximation). The optical losses in the second stage 9. Second stage concentrators can be employed in solar can be described by a transmission factor ~2 which can thermal as well as in photovoltaic systems, in conjunc- be approximated by the formula tion with line focus (e.g. parabolic trough) or point focus collectors (e.g. central receiver). Such an arrangement 72 = p<"> offers several advantages:

(i) the flux distribution at the absorber can be made where p = refiectivity of second stage and (n)= average more uniform, thus reducing hot spots or avoiding them number of reflections. altogether, In Ref. [3] (n) has been calculated for various types of

(ii) the concentration is increased, by typically a factor CPC under the assumption that the aperture is uniformly of 2 in line focus and a factor of 4-10 in point focus illuminated. However, for the present application, the systems. (Conversely, if a specified concentration value profile shown in Fig. 10 will be more typical of the is to be reached, the tolerances for mirror and tracking radiation reaching the focal zone of the first stage, accuracy can be relaxed significantly.) because of random mirror errors. Therefore only a frac-

The optical design of second stage concentrators has tion of the total radiation will actually hit the reflector of been described elsewhere[13, 14]. Here we discuss their the second stage, and (n) may be as small as 0.3. For advantages and disadvantages and compare a V-trough reflector configuration such as Fig. l(b)-(d) it may be as with a CPC for this purpose. We find second stage large as 1.0. concentrators of the CPC type to be very cost effective An additional attenuation factor should be included in in line focus systems such as linear Fresnel mirrors, or ~'2 if a glass cover is desired next to the absorber. This the fixed reflector plus moving receiver of General factor is typically 0.95 for ordinary glass relative to Atomic[24]. For point focus systems, the situation is less losses in a system without a second stage concentrator clear at the moment, since the materials requirements are and accounts for increased reflective losses at the glass much more stringent (reflective surfaces which remain since the angular spread of radiation is necessarily stable under high solar flux), enlarged by the second stage concentrator. This effect

can be minimized by antireflection coatings (e.g. ADVANTAGES AND DISADVANTAGES OF SECOND STAGE etching[18] of glass) which should certainly be cost

A second stage concentrator will reduce heat losses effective in view of the small surface area involved. because the absorber area is made smaller. On the other There is, however, an effect by which a second stage hand, optical losses are increased. To weigh the gains can actually increase the optical efficiency. Due to shad-

ing by the receiver, the collector aperture is reduced by / S e c o n d ltOge

, ( ~ , ~ ¢ome~mor at least a factor l - 1/C,, which in practice, is more likely

/ •

/ ~ f i rst ~• {_. ", , " t Oge ~ Is ~.."

Fig. 10. Typical flux distribution on the focal plane of a first Fig. 9. Schematic drawing of a second stage concentrator, stage concentrator.

380 A.R.~L, N. B. GOODUT~ and R. Winston

to be around 1 - 2/CI--0.9 to 0.97 because of the need close to the performance of a compound parabolic for thermal insulation at the sides of the receiver. But if concentrator. However, even under conditions most a second stage is used, the receiver will be small enough favorable to V-trough, the CPC appears to be more cost so the insulation will not protrude beyond the front effective as can be seen from the following example. aperture of the second stage; in other words, the shading Suppose a line focus system has a rim angle 0~ = 30 °, and factor can be kept to 1 - 1/CI. Thus the optical efficiency the angle of incidence on a glass covered absorber are to may be enhanced by 1-5 per cent depending on C,. be restricted to values below 02 = 60 ° in order to avoid

The efficiency of v/12 of the system with second stage excessive reflection losses. Then the thermodynamic is therefore limit for the second stage concentrator is[14]

'1712 = ~ O T 2 - - q l o s s ~ -- sin 02 _ CIC2S ,~2~,~ - sin 01 - 1.73.

and the overall gain in efficiency is The V-trough in Fig. 11 has a concentration of 1.6 and a height/entrance aperture ratio of 1.0. For uniform illu-

= ~ 12- ~l = qiO~C~S~r'_- 1)_ 7o(1-~'2). mination of the entrance aperture, the average number of A

reflections is 0.55, the fraction of radiation reaching the

Clearly one would use a second stage only if the heat absorber at angles beyond 82 = 60 ~ is 16 per cent, and 0.4 losses are sufficiently large to make A T positive, in order per cent of the radiation is rejected. A CPC on the other words if hand has a concentration of 1.73 with a height/entrance

aperture ratio of 1.4. Even if the height is truncated to be qJo .> ~oC2 1 - ~2 the same as for the V-trough, the concentration is still C~$ ~72-'1" 1.68, i.e. 5 per cent larger than for the V-trough. No

radiation is rejected and all rays reach the absorber at Typical values of Ch C2, "~o, p2, (n) and ~2 for line angles less than 60°; the average number of reflections is

focus systems are listed in Table 3. For an illustration essentially the same as for the V-trough. consider a line focus system which is to operate at In order to decide whether such a seemingly small gain To = 300°C. At this temperature the radiative losses from in concentration can justify the use of a CPC over a a black absorber are about 5 kW/m 2. Hence the total V-trough, we note that with V/o = 70 per cent and operat- losses (including convection and conduction on one hand ing efficiency ~/= 50 per cent, the heat loss decreases the and possibly selective coatings, honeycombs, or vacuum efficiency by 20 per cent. If the concentration is in- on the other) can be expected to be in the range of creased by 5 per cent by choosing a CPC over a V- 1.5-6 kW/m 2. A single stage system with concentration trough, the heat loss contribution is decreased from 20 to C, = 40 at an insolation S = 600 W/m 2 will then have a 19 per cent and the actual operating efficiency is in- ratio creased to 51 per cent. Compared to the original energy

output of 50 per cent, this implies a relative increase in q~o, = 0.06-0.25. power output of 2 per cent. Actually, the gain of the CPC C,S

is even greater, because 16 per cent of the radiation

This is generally larger than the critical values of 7/o(1 - leaving the V-trough is at angles of incidence beyond 60 ° for which the absorption and reflection losses are "r2)/(1 - 1]C2) listed in Table 3 and hence a second stage

is desirable, significantly higher. Therefore, the extra expense in- volved in manufacturing curved reflectors rather than

V-~OUCH VS COMPOUND p~axeouc CONCgSal~ATOR straight ones, is certainly justified if it adds less than 2

A V-trough concentrator is attractive because of its per cent to the total systems cost. simplicity, and for large acceptance angles it comes fairly An additional benefit of the CPC is that it can be

modified to accommodate any absorber shape or maxi- Table 3. Parameters for line focus systems mum angle of incidence on the receiver. The relative

merits of various absorber types have been discussed in parameters Values Comments Section 1.

c~ 20--60 with perfect parabolas C~m~ is 70 for cylindrical and 105 - for fiat absorber.

~o 0.65--0.75 ~o=p~Ta (ifp1=0.85,7=0.9, - . . . . . . . . ~ - - - - ? . . . . . ¢

/ / (n) 0.3--1.0 0.3~0.5 for Fig. l(a) 0.8 ~ 1.0 for Fig. l(b)-(d).

• 2 0.9-0.98 Io {b} / 1 - ~2 \ q,o,

~0C2~c--~_ 1. } 0.03-0.2 Use second stage if ~ S is Fig. 11. (a) a V-trough with concentration C = 1.60; (b) a trun- larger than this. cated CPC with concentration C = 1.68; dotted line shows un*

truncated CPC with C = 1.73.

Practical design considerations for CPC solar collectors 381

Note added in proof. In a recent note[25] a new solution to the 11. A. Rabl, V. J. Sevcik and R. Winston, Argonne National Lab. problem of introducing a gap between reflector and tubular Report ANL-75-42 (1975). absorber was presented by imposing the requirement of maximiz- 12. R. Giugler, A. Rabl, K. Reed, V. Sevcik and R. Winston, ing the flux on the absorber. In this solution the design principles for Compound parabolic concentrators for solar thermal power. the reflector shape remain unchanged but the starting point of the Prog. Report. Argonne National Laboratory Report ANL-75- left and right reflector portions is different. For the new solution the 52 (Jan.-June 1975). left and right reflector curves intersect at the end of the gap rather 13. A. Rabl, Comparison of solar concentrators. Solar Energy than at the absorber surface. The optical losses in the gap can be 18, 93 (1976). found by the method radiation shape factors. The result, kindly 14. A. Rabi and R. Winston, Ideal concentrators for finite calculated by M. Collares-Pereira, is sources and restricted exit angles. Appl. Opt. 15, 2880 (1976).

15. If the radiation incident on a collector uniformly fills its tan ~ - ~ acceptance angle 20 the flux distribution at the absorber is

Lpp ~r + tan ~ - ~ totally diffuse. This is the case not only if the collector is used as a second stage collector but also to a sufficiently

where 4~ is arc cos (rig + r). good approximation for single nontracking concentrators if Besides maximizing flux on the absorber this solution actually the motion of the sun is averaged over time.

yields somewhat smaller losses than the equation in the caption of Strictly speaking this statement is true only for un- Fig. 3d. In its optical properties the new solution to the gap problem truncated concentrators with perfect reflectivity. In practice is therefore to be preferred over the solution presented in the the reflectors will almost always be truncated for economic present paper, at least for tubular absorbers. For the flat receivers reasons. However, the flux which reaches the absorber from in Fig. l(a)--(c) the requirement of the flux maximization does not the top portion of the reflector is rather uniform, and hence produce a new solution; rather it leads to the solution (I), the flux incident on the absorber is very nearly diffuse even "truncation of the absorber", see Fig. 2(a)--(c), which is generally in severely truncated collectors. The reflectivity of materials less desirable than solution (II), "truncation of the reflector", used in CPC collectors is high enough (;~0.85) not to render

the isotropy assumption invalid. A solar collector should be designed for good performance

REFERENCES both on the average and at any specific time. It is, therefore, 1. H. Hinterberger and R. Winston, Rev. Sci. Instr. 37, 1 0 9 4 advisable to check the behavior of the CPC under the worst

(1966). Also V. K. Baranov and G. K. Melnlkov, Soy. J. Opt. possible conditions. Tech. 33, 408 0966). 16. R. Siegel and J. R. Howell, Thermal Radiation Heat Trans-

2. R. Winston, Solar concentrators of a novel design. Solar fer. McGraw-Hill, New York 0972). Energy 16, 89 (1974). 17. A. Rabl, Radiation transfer through specular passages. Int. J.

3. A. Rabl, Optical and thermal properties of compound Heat andMass Trans. 20, 323(1977). parabolic concentrators. Argonne Solar Group Report SOL 18. R. E. Peterson and J. W. Ramsey, Thin film coatings in 75-01, published in Solar Energy 18, 497 (1976). solar-thermal power systems. J. Vac. Sci. Tech. 12, 471

4. J. F. Kreider, Performance study of compound parabolic (1975). concentrators solar collector. Environmental Consulting 19. W. T. Weiford and R. Winston, The Optics of Nonimaging Services, Inc., Report 1974 (Phase D and 1975 (Phase ID. Concentrators. Academic Press, New York (1978).

5. An abbreviated version of this paper was presented at the 20. Argonne Solar Energy Group Prog. Rep., July-Dec. 1975 and ISES Conf. in Winnipeg, Manitoba in August 1976. Jan.-June 1976.

6. R. Winston and H. l-lmterberger, Solar Energy 17, 255 (1975). 21. J. A. Duffie and W. A. Beckman, Solar Energy Thermal 7. A. Rabl, Appl. Optics. 15, 1971 (1976). Processes. Wiley, New York 0974). 8. U. Ortabasi and W. M. Buehl, Analysis and performance of 22. G. Thodos, Predicted heat-transfer performance of an

an evacuated tubular collector. Presented at ISES Conf. in evacuated glass-jacketed CPC receiver: countercurrent flow Los Angeles, California (July 1975). design. Argonne National Lab. Rep. ANL-76-67.

9. D. C. Beekley and G. R. Mather, Jr., Analysis and experi- 23. D. C. Beeldey and G. R. Matber, Analysis and Experimental mental tests of high performance tubular solar collectors. Tests of a High-Performance Evacuated Tubular Collector. Presented at ISES Conf. in Los Angeles, California (July Owens-Illinois, Inc., Toledo, OH 43666. 1975). 24. J. R. Russell, Central station solar power. Power Engng

10. E. Kauer, R. Kersten and F. Mahdjuri, Photothermal Con- (1974). version. Publication No. 32/75, Philips Forschungslabora- 25. R. Winston, Ideal flux concentrators with reflector gaps. torium Aachen GmbH (October 1975). Appl. Opt. 17, 1668 (1978).