Simple procedure for predicting long term average performance of nonconcentrating and of concentrating solar collectors

Solar Enerey Vol 23. pp. 235 253 Pergamon Press Ctd., 1979. Printed in Great Britain

SIMPLE PROCEDURE FOR PREDICTING LONG TERM AVERAGE PERFORMANCE OF

NONCONCENTRATING AND OF CONCENTRATING SOLAR COLLECTORSt

MANUEL COLLARES-PEREIRA+ + and ARI RABL§

(Received 14 August 1978; revision accepted 6 April 1979)

Abstract - The Liu and Jordan method of calculating long term average energy collection of flat plate collectors is simplified (by about a factor of 4), improved, and generalized to all collectors, concentrating and nonconcentrating. The only meteorological input needed are the long term average daily total hemispherical isolation /7 h on a horizontal surface and, for thermal collectors the average ambient temperature. The collector is characterized by optical efficiency, heat loss (or U-value), heat extraction efficiency, concentration ratio and tracking mode. An average operating temperature is assumed. If the operating temperature is not known explicitly, the model will give adequate results when combined with the ~, f-chart of Klein and Beckman.

A conversion factor is presented which multiplies the daily total horizontal insolation/7 h to yield the long term average useful energy (~ delivered by the collector. This factor depends on a large number of variables such as collector temperature, optical efficiency, tracking mode, concentration, latitude, clearness index, diffuse insolation etc., but it can be broken up into several component factors each of which depends only on two or three variables and can be presented in convenient graphical on analytical form. In general, the seasonal variability of the weather will necessitate a separate calculation for each month of the year ; however, one calculation for the central day of each month will be adequate. The method is simple enough for hand calculation.

Formulas and examples are presented for five collector types: flat plate, compound parabolic concentrator, concentrator with east-west tracking axis, concentrator with polar tracking axis, and concentrator with 2-axis tracking. The examples show that even for relatively low temperature applications and cloudy climates (50°C in New York in February), concentrating collectors can outperform the flat plate.

The method has been validated against hourly weather data (with measurements of hemispherical and beam insolation), and has been found to have an average accuracy better than 3 per cent for the long term average radiation available to solar collectors. For the heat delivery of thermal collectors the average error has been 5 per cent. The excellent suitability of this method for comparison studies is illustrated by comparing in a location independent manner the radiation availability for several collector types or operating conditions : 2-axis tracking versus one axis tracking: pola( tracking axis versus east-west tracking axis; fixed versus tracking flat plate; effect of ground reflectance; and acceptance for diffuse radiation as function of concentration ratio.

I. INTRODUCTION

The performance of solar collectors is usually specified in terms of instantaneous or peak efficiency, based on clear days and normal incidence. For practical applications, however, one needs to know the long term energy delivery averaged over all cloud conditions and incidence angles. To answer this need, many re- searchers have advocated average diurnal efficiency as a collector performance measure. Unfortunately such average efficiency curves may depend strongly on peculiarities of the weather for the tes t day and test

t" Work supported by the Solar Heating and Cooling Research and Development Branch, Conservation and Solar Applications, U.S. Department of Energy.

Enrico Fermi Institute, The University of Chicago, 5630 S. Ellis Avenue, Chicago, IL 60637, U.S.A. Supported by the Instituto Nacional de Investigac~o Cientifica and Centro de Fisica de Mat6ria Condensada, Lisbon, Portugal.

§ Now on leave of absence from Argonne National Labo- ratory to Solar Energy Research Institute, 1536 Cole Bou- levard, Golden, CO 80401, U.S.A.

location, and are therefore limited in their general applicability.

One approach to this problem is to use a computer program with instantaneous efficiency and hourly insolation data as input. The results of such a calculation can be considered valid only if the weather data are representative of long term weather behavior. Various choices have been used, for example, real hourly data for a single year, real hourly data for several years, averaged hourly data, and stochastic data. Use of real data for a specific place and year provides only a performance simulation for that place and year, but its reliability as prediction for the long term average is uncertain - after all, fluctuations in monthly total insolation from one year to the next commonly exceed + 10 per cent, and the .resulting output fluctuations for thermal collectors are even larger.

While some of this uncertainty can be eliminated by generating a typical meteorological year for each locat ion[l] , there are other drawbacks to computer

235

236 MANUEL COLLARES-PEREIRA and ARI RABL

simulation arising from the required time, money and expertise. Even though the computing time is incon- sequential for a few sample simulations, the large number of parameters to be considered will make any meaningful system optimization or comparison study costly and time consuming. A computer can produce unlimited quantities of numbers, but it provides little intuitive understanding of functional relationships.

As an alternative, we propose a model which is based on the utilizability concept pioneered by Hottel, Whillier, Liu and Jordan[-2, 3]. It offers the advantages of automatically averaging over year-to-year weather fluctuations and of being sufficiently simple to permit hand calculation of long term performance. In many important cases it will even permit prediction of system performance without recourse to computer simulation. We have generalized the utilizability method to all collector types and improved its accuracy by calibrating it against the latest insolation data. Simplification by about a factor of four relative to Ref. i-2] has been achieved by defining the utilizability with respect to the day rather than the hour1-4]. The model needs as meteorological input only the long term average daily total hemispherical irradiation//h on a horizontal surface, (and for thermal collectors), the long term average day time ambient temperature T a. This information is readily available for a large number of locations[2, 5] ; Ref.[5], for example, lists 170 stations in the U.S. and Canada. The collector is characterized by optical efficiency, heat loss (or U- value), heat extraction efficiency, concentration ratio and tracking mode. (The user is urged to consult the Appendix for a consistent definition of the collector parameters.) In general the seasonal variability of the weather may necessitate a separate calculation for each month of the year; however one calculation for the central day of each month will be adequate. Since the dependence on individual design variables such as tilt angle, concentration ratio and operating temperature is displayed explicitly, it is easy to study the effect of changing any of these variables. The influence of climate and location can be assessed systematically. This gain in intuitive understanding can be of great help for system optimization and for comparison studies. For illustration we have compared some typical collectors (fiat plate, CPC, collector with east-west tracking axis, collector with polar tracking axis, and collector with 2-axis tracking). Comparison and optimization of collectors by themselves is particularly important for manufacturers who want to sell a single collector for use in a variety of different systems.

The calculationt proceeds in one, two or three main steps depending on the application. The first step yields the long term average insolation /~coll reaching

"1" We must warn the reader not to mix the symbols and formuas of this paper with those of other authors. This applies above all to the R factors and to the ~b curves which differ markedly from those of other investigators[2-6], both in interpretation and in numerical value.

the collector within its acceptance angle. For collectors wfiose efficiency r/ is independent or nearly independent of solar intensity nothing else is required, and the delivered energy per unit aperture area is simply

Q./A = r//~co,. (I-1)

This is the case for photovoltaic collectors. Thermal collectors with significant heat loss, on the

other hand, will be turned on only when the insolation is sufficiently high, hence the relation between //co~l and delivered energy () is nonlinear. This case requires the second step, the calculation of the utilizability. The utilizability ~b is that fraction of the total insolation which is above a specified threshold, q~ is a function of the heat loss, expressed as dimensionless critical energy ratio, and is defined in such a way that the delivered energy () per aperture area A is

Q./A = F floC~Aco H (I-2)

where r~o is the average optical efficiency (which has also been called (T~) product); F accounts for the heat extraction or the heat removal efficiency and depends on the specification of the collector temperature (fluid inlet temperature~ fluid outlet temperature, mean fluid temperature or absorber surface temperature). Eqn. (I- 2) is the final answer in all systems where the average operating temperature of the collector is known. This is likely to be the case for most applications in process heat and shaft power.

The third step is needed only if the average collector temperature is not known. For example, in home heating applications the average storage temperature is not known a priori, but a short hand method, the./: chart, has been developed by S. A. Klein et al.[5] which predicts the fraction of the load supplied by solar, taking into account the interaction with thermal storage. Recently a generalized version, the 4, f- chart[6], has been introduced in order to model closed-loop solar thermal systems with any specified minimum temperature (rather than just 20°C as in the original f-chart). Even though the q~, f-chart was developed only for fiat plate collectors, it can be extended to all collector types because it treats utilizability and storage effects separately. This extension is accomplished by replacing the product q~Hr of Ref. [6] with the product ~b/-/co . of our paper. Hence step three can be summarized by [7]

(~b//~o,,)loureq,.~i.2~ ~ (tp/'/7.j]Klei, of qS, f-chart (1-3)

In order to evaluate the accuracy of this model we have compared its predictions with the results of adding hourly contributions from weather tapes. We have avoided the usual uncertainties associated with estimating the beam component when only hemispherical insolation is known, by relying only on tapes which conl~ain both pyranometer and pyrheliometer measurements. Such data have recently become available through the Aerospace Corporation for the five stations: Albuquerque, NM; Fort Hood, TX; Liver-

Procedure for predicting performance of solar collectors

Table 1. Function R h and Rd for flat plate collector

237

a) tile B = latitude I , azimuth # = 0

b)

c)

{{, ,-ql cos

+ (1 - cos ~ a sin ~c + ~ (sin ~c cos me + u

_ 0_2 (i cos ~) cos ~s [a u c + b sin Uc]l

- - 2 ( i + cos I s in toc + (I + cos l ) cos m toc

tilt B # latitude )., azimuth ~ = 0

_ [ oo.<~-~, <o. o '+ ~= <,.-oo. ,>oo. o_1 r. m - I ' b s i n to_l/ L c o . , s ~ ='I I '; ';II

• - J z. j )

/roo, <,+ oo. ,>/ ,~.o.

% r ' I "I -I

t i l t 6 ~ l a t i t u d e ~,, azCmuch ~ ~ 0

conver t from B and ) (wi th r e s p e c t co h o r i z o n t a l and v e r t i c a l )

Co B o and ~o (wi th r e s p e c t Co equaCocial p lane and po l a r a~d.s)

by means of

s i n ~o " cos B s i n I - s i n S cos I cos

and s in S s i n

t a n ~ o " c o s ~ c o s ~ + s i n B s i n ~ c o s

"~[L oo,~ + ~ c~- oo~)] ~ +-,~. q

+ 2 ( s i n ~c+ cos ~c+ - s i n t0c_ cos ~ c - + ~c+ - ~ c - ) [ i

J cosBosln4)~ [ b 2. 2 ] a (eos to¢+- cos to e ) + 2 (cos t~e+ - cos we_)

- c o s X

, < , _ : o : , > < : : g [ = < o o + _ o _ , o _ . + t a ~ l cos A

L

- L I r c o s B o c o ' , o 1 _ cos ' ] l"d " Zd / l ;< - ~ (1 + ~o,, 13) (s:tn to,:+ - ~:t.. ,,,_) %

_ c o s So s in i o (cos toc+ - cos ~ c _ ) cos

+ t a n 6 / c o s ). 2 (1 + cos B) tam i (mc+ - l c _ )

more, CA; Maynard, MA; and Raleigh, NC; with 1-4 yr at each station.

We have computed the deviation

n c o l l . d i t l - - ncol l .model ~" = /'/¢ou. d. , . ( I -4 )

between data and model; this is documented in Refs. [9] and [' 10]. We found that the average error (over all stations and months) is less than 3 per cent for the

insolation/~'~o, available to the collector, both for flat plate and for concentrating collectors.J9] The model appears to be free from any significant bias with respect to location or time of year at least as far as could be discerned from this rather limited data base. For a particular month of a particular year the discrepancy between model and data can be much larger, on the order of 3 per cent (standard deviation) for a flat plate and on the order of 10 per cent (standard

238 MANUEL COLLARES-PEREIRA a n d AR1 RABL

Table 2. Functions R d and R, for concentrators with fixed aperture, e.g. compound parabolic concentrator (CPC)

a) t i l t 5 " l a t £ t u d e ~, a z l ~ t h ~ - 0

- ~ a s ~ % + ~ (sin% cos%+%)

alad

IE *I .... -oi Rd- ~ ~0 - -~ -~ , i ~ o + c

t:[11: B ~ l a t £ t u d e ~, a z ~ u t h t " 0

/ ' ' c o s (X-B) (.a - b cos ~s ) sin ~c - a cos "s we Rh = d cos

+ -~ (s±n ~c cos ~e + uc)| )

R.-~ L ~ o . ~ • - ~ ,,n~o

+ " COS ;~ COS ~11

deviation) for concentrators. These fluctuations reflect the month- to-month variations in the ratio of diffuse over hemispherical insolation. Even though large, the fluctuations do not matter, provided the long term average is correct. After all, the model is designed to predict only the long term average collector performance, not the performance for a particular month of a particular year.

The computat ion of the utilizability q~ involves further approximations of several per cent, and hence the uncertainty in the energy delivery of a thermal collector will be on the order of 5 per cent (for thermal collectors the model in its present form is conservative, i.e. it tends to underpredict)[10].

In view of the evidence for the general validity of the meteorological correlations underlying the present model, we believe that the model is applicable for all locations and latitudes from - 4 5 ° to +45 ° . We urge, however, that the model be tested again and if necessary recalibrated when long term data with pyranometer and pyrheliometer become available from a larger number of stations. The present paper provides only a users' manual[8]. Derivation and validation of the model are documented in two companion papers, Ref.[9l for the available insolation and Ref. [-10] for the calculation of the conversion factors R d and Rh and of the utilizability function.

II. I N S O L A T I O N / / c o i l R E A C H I N G C O L L E C T O R APERTURE WITHIN ITS ACCEPTANCE ANGLE

The long term average daily total irradiation incident on the collector within its acceptance angle is obtained from the daily hemispherical inso la t ion/ /h on a horizontal surface by means of the formula

//co,, = [Rh -- Rd//d///h]//h. (II-l)

For the energy actually absorbed per aperture area A of the collector one has to multiply this by the long term average optical efficiency~" r~o

(Qabs A) = ~ o / / c o l l (II-2)

t The optical efficiency has also been called (re) product in the flat plate literature. The bar indicates that incidence angle modifiers for all-day average have been taken into account. For a test procedure defining 4, see Section III of the companion paper[10]. For the specification of the insolation base, see the Appendix of the present paper.

Table 3. Functions Rk and Rd for a collector which tracks about east-west axis

~ c d~ 4 ' c o s 2 ' 1 [o (a + b cos ~) ~ + ten 2 6 Rh " d cos

1 f o d~ ~ o s 2 ~ + t a n 2 6 f o r h i g h c o n c e n t r a t i o n

c ~ 10 Rd ~c

~ cos ~ f o dw [ 4 c o s 2 ~ + t a n 2 6 c ° ~ I ( c o s u - co s ~ s ) ]

f o r l o w c o n c e n t r a t i o n C ~ 10

t o e v a l t m t e t h e s e I n t e g r a l s one can u s e S i m p s o n ' s r u l e o r e l l l p c £ c £ n t e g r a l e ,

a s e x p l a i n e d be low.

S i m p e o n ' s r u l e w i t h one e t ~ p ~ t 1 1 be a d e q u a t e i n mos t c a s e s ( e r r o r < 3Z) ;

f o r g r e a t e r a c c u r a c y two s t e p s can b e u s e d .

x + 26x o

f f ( x ) d x ~ - ~ [ f (Xo) + 4 £ (x o + Ax) + f (x o + 2&x)]

X O


Table 4. Functions Rh and R4 for collector which tracks about north-south axis

239

a) tilt of tracking axis B = latitude g (polar mount)

and

b)

+ b s i n w awe C

h, d cos X

~ ~c Rd " -- co-----s l

w c c o s

f o r h i g h c o n c e n t r a t i o n

Sit1 tO c - ~C COS ~S f o r low c o n c e n t r a t i o n C d

t i l t o f t r a c k i n g a x i s B ~ l a t i t u d e X ~¢

1 gh d cos X f o de ( s + b cos e ) g(w)

with g(e) = 4sin2w + [cos(X-B)eos w + tan 6 sin(l-B) l 2

and e c

J Rd " f d - - ' ~ - - ~ f o de g(w) f o r h i g h c o n c e n t r a t i o n

~C S ~ e c -- W C coa ~s

~ e ' ~ ' s ~ f o d~ g(m) - C d

f o r low c o n c e n t r a t i o n .

The integrals can be evaluated by Simpson's rule. See Table 3.

The functions Rh and Rd are given in Tables 1-5, and Ha/H h is the long term average ratio of diffuse over hemispherical i rradiat ion on a horizontal surface. This ratio is correlated[9] with clearness index gh and sunset hour angle co, as

//h - 0.775 + 0.347 to, - (II-3)

2)] x cos(2(g , - 0.9)) for 0.4 ~ gh < 0.75

For flat plate collectors the co, dependence can be neglected by setting co, = n/2 in this equat ion; this curve is shown by the solid line in Fig. 1 (the dashed lines show eqn. (II-3) at to, = n/2 - 0.2 and at to, = n/2 + 0.2). The clearness index g , is the long term average ratio

K, = f l , /Ho (II-4)

of daily hemispherical insolation on a horizontal surface and extraterrestrial insolation Ho. The equation for Ho can be found in any textbook on solar energy ; it is not needed for the procedure in this paper if the data base lists Kh in addi t ion to/~h, as do for example Refs. [2] and [5].

R, and Ra depend on collector type, collector orientation, latitude, and collector turn-on and turn-

off time. We have calculated these functions for the following collector types:

(i) Nonconcentrat ing collector with fixed aperture, e.g. flat plate collector (a) tilt fl = latitude 2, azimuth ~b = 0 (Table l(a)) (b) tilt fl :/: latitude 3., azimuth ~b = 0 (Table l(b)) (c) tilt fl # latitude 3., azimuth ~b :~ 0 (Table l(c))

(ii) Concentrators with fixed aperture, azimuth q~ -- 0 e.g. compound parabolic concentrator (= cpc) (a) tilt fl = latitude 3. (Table 2(a)) (b) tilt fl ¢ latitude 3. (Table 2(b))

(iii) One-axis tracker of concentrat ion C, tracking about east-west horizontal axis (Table 3)

(iv) One-axis tracker of concentrat ion C, tracking about north-south axis of tilt fl (a) tilt fl = latitude 3. ( = polar mount) (Table 4(a)) (b) tilt fl ¢- latitude 3. (Table 4(b).

(v) Two-axis tracker ofconcentrat ion C (Table (5)) Table 2 applies also to high concentration systems

with fixed reflector and tracking receiver such as the hemispherical reflector[l l , 12] and the segmented cylindrical reflector[12] (developed by General Ato- mic), provided their incidence angle modifiers are


Table 5. Functions R h and R d for collector with 2-axis tracking

a{a + b sin u c c Rh d COS X COS 6

and

R d = I Wc c o s k c o s d

~c c o s X c o s d

for high concentrat ion

B~n ~C -- ~C COS ~S

Cd for low concen~ratlon

included in the long term average optical efficiency ~,. Tables 4 and 5 hold for both reflective (mirror) and

refractive (lens) concentrators if the aperture moves as a single unit ; included is almost any reasonable solar concentrator with trough or dish reflector or with Fresnel lens. This is in contrast to Fresnel reflector systems, e.g. the power tower[12] whose aperture consists of reflector segments which follow the sun

individually; for this latter case use of Tables 3-5 is not quite correct. If more accurate formulas are needed for Fresnel reflectors, they can be derived by the method described in Ref. [,10]. For linear Fresnel reflectors with east-west axis, the results obtained from Tables 2 and 3 provide lower and upper bounds.

We find it convenient to express all times t in dimensionless form as hour angle co from solar noon

2nt c o = - - with T = l e n g t h o f d a y = 2 4 h r ;

T (11-5)

note that throughout this paper all angles are in

radians, except for a few cases where degrees are explicitly indicated. The sunset hour angle

2~ts cos = T

corresponding to sunset hour t, is given by

cos co, = - tan 2 tan 6 (1I-6)

where 2 = geographic latitude and 6 = solar declination = arcsin [-0.3979 sin (2n(n + 284)/365.24)] for the nth day of the year starting 1 January. The quantities a, b and d in these tables are functions of co,

a = 0.409 + 0.5016 sin(co s - 1.047) (II-7a)

b = 0.6609 - 0.4767 sin(co, - 1.047) (lI-7b)

and

d 0- sin co, - co, cos co, (II-7c)

(Note that 1.047 radians = 60°). In the equations for flat plate and C P C co', is given by

cos co', = - tan (2 - / ~ ) t a n 6. (I1-8)

The reflectance p of the ground in front of a flat plate collector is also needed for Table 1 ; recommended[-11] values are p = 0.7 with, and p = 0.2 without snow, in the absence of better information.

One further variable remains to be explained, the collector cut-of f time to, or equivalently, the cut-off angle

2ntc co¢ - (II-9)

T

If the collector is placed due south, i.e. with zero azimuth, and if its time constant is short, it will operate symmetrically around solar noon, being turned on at

turn-on time to_ = - t¢ (ll-10a)

and turned off at

turn-off time to+ = t~. (II-10b)

This has been assumed for all collectors with zero azimuth. For the flat plate with nonzero azimuth the asymmetric turn-on and turn-off angles are explicitly shown, with the sign convention that co,_ (morning) is negative and coc+ (afternoon) is positive, as in eqn. (II- 10). If collectors with zero azimuth are operated with co~_ ~ - co,+ to account for transient effects or for asymmetric shading problems, the formulas of Tables 2-5 can still be used in the combinat ion

R~..¢,~,,. = ~[,R(-co~_) + R(w¢+)]. (II-lla)

1.0

"~- 0.5 ,:Z

I 0 0.5 1.0

Kh

Fig. 1. /74//4h versus /~h (eqn. (II-3); the solid line corresponds to co, = ~o/2 and the dashed lines correspond to w,

= n/2 - 0.2 (bottom) and (o, = n/2 + 0.2 (top).

Procedure for predicting performance of solar collectors 241

When -co c_ and o~+ do not differ very much, the simple approximation

( - m c - + ~ + t ( I I - l lb) Reflectiv e ~ R co c = 2

is acceptable. The model has been written for explicit input of cut-

off time t¢ in order to permit greater flexibility and applicability in situations with any shading configuration. The cut-off time is limited by optical constraints, and may be further reduced by thermal considerations for thermal collectors. The procedure of finding t¢ for thermal collectors is described in Section III.

The highest possible value of ~o, is the sunset hour angle to, for a completely unshaded collector. For fixed collectors ~o, also has to be less than oJ', of eqn. (II-8) except in the unlikely case of a collector which can operate on diffuse radiation alone. In collector arrays some shading between adjacent rows will usually occur close to sunrise and sunset, and ¢o~ has to be calculated from the trigonometry of the collector array. This is straightforward for an array with continuous collector rows, for example with long horizontal parabolic troughs. For arrays with rows of separate collector units, for example parabolic dishes, the analysis of shading is more complicated. In either case a good albeit slightly optimistic approximation is obtained by setting t, equal to the time at which half of the collector aperture is shaded.

For nontracking concentrators of the CPC type[13-17] the optical cutoff time depends on the acceptance half angle 0 of the collector. If a trough-like CPC with east-west axis is mounted at tilt fl = latitude 2, ~c is given by

tan[6l (II-12) cos ~Oc - tan 0

For CPCs with concentration C > 2 the tilt will generally differ from the latitude, with tilt adjustments during the year, and w~ is given by

tan 6 cos ~oc = tan{2 - fl + 0,~/I,~ll tli-131

Note that for a CPC with tilt adjustments one should always verify that the sun at noon is within the acceptance angle, and that ~o~ ~< rr/2.

IlL HEAT LOSS, UTILIZABILITY, AND C U T O F F TIME

If all days and hours were identical, Q could be obtained by simply subtracting the total daily heat loss

((~io,JA) = 2t~U(T¢on - ' T a m b ) ( l l l - la )

from the absorbed solar energy OilY/co,. The operating temperature Too n of the collector can be taken as absorber surface or fluid temperature, depending on choice of temperature base (see Appendix). Since the heat loss from transport lines between collector and

storage or point of use occurs at the same time as the loss from the collector, i.e. only when the circulating pump is turned on, the equation for Qio~s should include the loss qti,e from the transport lines

O.io~, = 2tc[AU(Tco, - 7",rob) + qiine] ; (III-lb)

q,~., depends, of course, on the system. Due to the variability of the weather, the true energy

gain can be significantly higher. This feature can be illustrated by the following two artificial climates. Climate 1 has identical days, all uniformly overcast, while climate 2 has clear days half of the time and no sunshine for the rest; both climates have the same long term average insolation/-/. If the heat loss of a collector equals the peak insolation of climate 1, no useful energy can be collected. Under climate 2, however, the same collector can collect some useful energy on the clear days.

It is convenient to calculate this effect once and for all for any concentrator type and for any climate, and to summarize the result in terms of the utilizability function ~. ~b depends on the critical energy ratio

((~,o,,/A) X = qoB¢on (III-2)

and is defined in such a way that the long term average collected energy (~ per aperture area A is

O./A = FdPFloHco, 0-2)

where F is the heat extraction or heat removal efficiency factor[ l l , 12]. F depends on the type of operating temperature which has been specified, as discussed in the Appendix, and is given by

I for average receiver surface temperature 7",

I F' of eqn. (A-14) for mean fluid tem-

F = perature Ty (III-3) FR of eqn. (A-16) for fluid inlet temperature Tin Fg/[l - FsUA/(ffaCp) ] of eqn. (A-17) for fluid outlet temperature Tout

The calculation up to and including 4, is the same regardless of which temperature base (T~., Tout, TI or 7",) is used to specify the instantaneous efficiency. Only at the last step the temperature base is accounted for by inserting the appropriate factor F in eqn. (I-2) for Q.

In principle, q~ is a complicated function of many variables, but fortunately the dependence on most of these is rather weak. For the climatic variation, Liu and Jordan[2] have shown that consideration of a single factor, the clearness index gh (see eqn. II-4) is adequate. From a large number of numerical simulations[10], we conclude that ~b can be approximated within a few percent by a function of only three variables, the clearness index gh, the ratio

R = --Rd (III-4) Rh

2 4 2 MANUEL COLLARES-PEREIRA a n d ARI RABL

1.0

0.8

0.6

0.4

02

0

I I I I

Kh= 0.7

I I I I 0.2 0.4 0.6 0.8 1.0

X

1.0

0,8

0.6

q'o.4

o.2

I I I I I

I I I I I 0.2 0.4 0.6 ().8 LO

X 1.2

Fig. 2. Utilizability 0 versus the critical ratio X for Kh = 0.7 Fig. 4. Utilizability 0 versus the critical ratio X for/(h = 0.5 and nine values of R from 0 to 0.8. and nine values of R from 0 to 0.8.

and the critical energy ratio X of eqn. (III-2). For nontracking collectors, ~b is given by the para- metric expressions

th = e x p [ - X + (0.337 - 1.76K h + 0.55R)X 2] (III-Sa)

for 0 . 3 < / ~ h ~ 0 . 5 a n d 0 < X < 1.2,

and

tk = 1 - X + (0.50 - 0.67gh + 0.25R)X 2 (III-Sb)

for 0 .5< /~h <0 .75 a n d 0 < X ~< 1.2

For tracking collectors ofhigh concentration (C ~ 10), " the R dependence can be neglected and the fit

q~ = 1 - (0.049 + 1.44/~h)X + 0.341KhX 2 (III-Sc)

can be used for all values ofgh < 0.75 and for 0 < X < 1.2. For exceptionally clear climates, i.e., with Kh > 0.75, the simple expression

= 1 - X for gh > 0.75 (III-5d)

should be used for all collector types. These ~b curves are shown in Figs. 2-7.

The fits were derived with emphasis on accuracy at reasonably large values of ~b because collectors with low utilizability will not collect enough energy to be economical. The above expressions for q~ are reliable

t" One could extend the utilizability curves beyond X = 1.2 by drawing tangents at X = 1.2. However, for reasons explained in Ref.[10], accuracy cannot be guaranteed.

only when ~b is larger than approximately 0.4. At smaller values of~b, the above fits are not recommended (nor is it likely that a collector will be practical if its heat loss is so large as to imply ~b < 0.4). Since the above fits may increase with X at very large X, they must not be used outside the specified range of X-values.t

The values of R will range from about - 0.1 to 0.8 for nontracking collectors and from 0.95 to 1.05 for collectors with high concentration. For tracking collectors with significant acceptance for diffuse radiation (i.e. (C < 10), R may fall between 0.8 and 1.0. For such a configuration, we recommend linear interpolation in R between the R = 0.8 value of eqns. (III-Sa) or (Sb) and eqn. (III-5c) assuming that the latter equation corresponds to R = 1.0. (This is not very accurate because the variation of ~b with R in this range is not uniform for all gh; but in any case, tracking thermal collectors of very low concentration appear to have little practical interest.)

The cutoff time tc for thermal collectors can be determined by the following simple iteration procedure (which is justified in Ref.[10]):

(i) start with tc = t~l = maximum permitted by optics, as discussed at the end of the previous section; for example, tel -- ts for flat plate or for tracking collectors if there is no shading. For the CPC, t¢1 is given by eqns. (II-12) or (II-13).

(ii) calculate corresponding output Q. (iii) decrease t, by Ate to get new t~z = tot - Ate (Ate

= 0.5 hr will give sufficient accuracy in most cases). (iv) calculate output Q2 for t¢2 and repeat procedure

until maximal Q is found.

1 . 0 I I I l

0.8 " ~ ~'h: 0"6

0.4

0.2

I I I I 0.2 0.4 0.6 0.8 1.0

X

Fig. 3. Utilizability ~b versus the critical ratio X for/(h = 0.6 and nine values of R from 0 to 0.8.

I.C

0.8

0.6

*0.4

0.2

O'

I I I I I

I I I I I 0.2 0,4 0.6 0.8 f.0 L2

X

Fig. 5. Ut i l izabi l i ty q~ versus the crit ical rat io X for gh = 0.4 and nine values of R from 0 to 0.8.


i .0 I I I I I

0.8 ~ ~h'°3

0.4

0.2

I I I I I 0 0.2 0.4 0.6 0.8 1.0 1.2 X

Fig. 6. Ut i l izabi l i ty ¢ versus the critical ratio X for K h = 0.3 and nine values of R from 0 to 0.8.

[.(] I I

0.8 ~ _ . . . i

o.6 0.4

0.2

, I 0 o12 0'.4 016 o'8 ,o ,2

X

Fig. 7. Utilizability ~b versus the critical ratio X for collectors with high values of concentration and five g, , from 0.3 to 0.7.

The smaller the heat loss, the closer the optimal tc will be to t d. This is illustrated by the sample calculations in Tables 6 and 7 which were carried out with a rather small decrement At c = 0.1 hr. This small value was chosen only for the sake of illustration. For example in Table 7(a) a value of Ate = 0.5 hr would have yielded (~ = 3.714 MJ/m 2 on the second in- teration, 1 per cent less than the value of 3.743 MJ/m 2 obtained with At, = 0.1 hr on the 14th iteration.

The maximum is broad and quite insensitive to uncertainties in t~.

IV. SAMPLE CALCULATION

To provide an example, wc calculate the energy delivery of several collector types in New York, NY, on February 15. The latitude is 3. = 40.5 ° and the sunset time t, = 5.24 hr. The relevant values of insolation/4h,

clearness index gh and daytime ambient temperature T a are obtained from Ref. [5] as

/~, = 8.33 x l 0 6 J /m 2 per day

K, = 0.41

T. = 1.0°C

The correlation for the diffuse/hemispherical ratio, cqn. (II-3) or Fig. 1, yields a diffuse component o f / t a = 3.78 x 106 J /m 2 per day for these conditions. For the ground reflectance we assume p = 0.2. The comparison is based on collector performance only, without regard to system properties such as losses q,i,, between collector and storage. With economically justifiable quantities of insulation, line losses can be quite large. In particular for large arrays of parabolic dish collectors, they may degrade the energy delivery down to the level of parabolic trough collectors.

Table 6. Col lector parameters and energy collected 15 February in N e w York at 50°C

Parabol ic trough Parabol i c trough Flat Plata C2C evacuated E.W. tracking axis polar tracking Two axis

a x i s tracker

n0 .75 V[wlm2Cl 4

C 1

F' .9

t c (hrs from noon) 3.934

R h 1 .626

R d .783

R-%/R h .481

Hcoll[M J/m 2] Eq. II-I 10.616

X(Eq. llI-i and III-2) .697

.6

.8

1.5

.99

4.651

1.723

1.089

.632

10.278

.213

.65

.7

20

.95

5.234

1.874

1.932

1.031

8 .385

.237

.65

.7

20

.95

5 .234

2 .377

2 .549

1 .072

10 .274

.194

.65

.2

500

.9

5.234

2.441

2.617

1.072

10.550

.054

¢(~, R, X) .470

(Q/A) - n o ¢ Hcoll[MJlm2]

Tr=50"C 3.743

(Q/A) I = F' n o @ Hcoll[ttIlm2]

I Tav. fluid = 50"C 3.369

.807

4.976

4.926

.856

4.667

4.434

.881

5.886

5.592

.966

6.624

5.962


Table 7(a). Some results of the iteration procedure to determine the cutoff time corresponding to the max imum energy (max Qo,t) collected. Flat plate, tilt = latitude, at T, = 50 ° in New York in February. Iterations start with t¢ = t, = 5.234 sunset time (hours from noon); decrement At c = 0.l hr, for the sake of

illustration Ate has been chosen much smaller than necessary

I t e r a t i o n # ( c u t - o f f time) - X - t ~ Rd Hcoll Q o u t

c

Ist

10th

14th (Hax.Qou t)

15th

5.234

4.334

3.934

3.834

1.801

1.700

1.626

1.604

.943

.839

.783

.768

11.482

11.027

10.616

10.491

• 858

.739

.697

• 688

3.405

3.714

D 3.738

We consider the following five collectors: (i) Flat plate collector with optical efficiency ~o =

0.75, U-value U = 4.0 W/m 2 °C, and heat extraction efficiency factor F' --- 0.90, typical of double glazing and selective coating.

(ii) Fixed CPC collector with evacuated receiver, having ~]o = 0.60, U = 0.8W/m 2 °C, F' = 0.99, concentration C = 1.5 and acceptance angle 2 0 = 68 °. These values are typical of the present generation of CPC collectors, for example the 1.SX CPC described in Refs. [14] or [16]. The heat extraction efficiency F' is excellent because of the combination of vacuum and selective coatings in the receiver (even if air is used as heat transfer fluid).

(iii) One-axis tracking concentrator with east-west axis (horizontal) and collector parameters ~o = 0.65, U = 0.7 W/m 2 °C, F' = 0.95 and C = 20. These values are typical of good collectors with parabolic trough reflectors and single glazed nonevacuated selective receivers of current technology[17].

(iv) Same collector as (iii) but with polar mount, i.e. tracking axis in north-south direction with tilt equal latitude.

(v) 2-axis tracker of high concentration, for which we arbitrarily assume ~o = 0.65, U = 0.2 W/m 2 °C, C = 500 and F' = 0.9 because no reliable test data are available for such a collector at the present time.

The above collector parameters represent currently

Table 7(b). Some results of the iteration procedure to determine the cutoff time corresponding to the max imum energy (max Qo,t) collected. Fixed tilt = latitude CPC (1.5X), E.W. tracking axis parabolic trough, polar mount tracking parabolic trough, 2-axis tracker, for New York, and 1', = 100°C in February. Due to small heat loss tc is close to t,, and omission of iterations (i.e. setting t, = t,) would affect result for Q by less

than l per cent

CPC (1.5X)

Parabolic trough E.W. Tracking axia

Parabolic trough Polar t racking axis

2 axis t racker

I t e r a t i on #

1

3

(Hax. ~out)

1

5

(~x. ~out )

1

(Hax.qout) 2

• 1

(Hax. ~out) 2

t c

(cut-off time~

4.651

4.451

5.234

4,834

5.234

5.134

5.234

5.134

%

1.723

1.695

1.874

1.827

2.377

2.344

2.441

2.407

R d

1.089

1.062

1.932

1.865

2.549

2.500

2.617

2.567

Hco~l [Kl/m']

10.728

10.153

8.385

8.247

10.274

10.182

10.550

10.455

.430

.417

.472

.450

• 381

.387

.109

.108

Qout [~/m 2]

3.985

3.955

3.~1

5.119

6.339


available technology. All collectors stand to gain from improvements in reflector materials, low cost antireflection coatings and selective surfaces. The magnitude of the gains to be expected varies from collector to collector. Of the collectors we have considered, the evacuated CPC has the lowest optical efficiency, because of double glazing, losses in the reflector, and low absorptivity of currently available selective coatings on glass[18, 19] ; it therefore offers the greatest potential for improvement. Optical efficiencies above 0.70 have recently been demon- strated for evacuated CPC collectors[16]. The collector parameters are listed in the first four rows of Table 6.

For the operating temperature Tcon we consider two possibilities, either average receiver surface temperature T, = 50°C or average fluid temperature TI = 50°; this temperature is in the range appropriate for some space heating applications. The calculation up to and including q~ is the same regardless of which temperature base (Tin, Tout, Ty or T,) is used to specify the instantaneous efficiency. Only at the last step the temperature base is accounted for by inserting the corresponding factor F in the equation

O = AFqo cb Bco. (I-2)

The values for F are given in eqn. (III-3) or in the Appendix.

The entries in Table 6 are obtained by iteration over t, with Ate = 0.1 hr; details of the intermediate iterations are provided in Table 7. Rows 5-8 list the cutofftime to the function Rh and Rn of Tables 1-5, and the ratio R = Rd/Rh. The insolation/r/co . incident on the aperture during operating hours is obtained from eqn. (II-1) /-/coil = [Rh -- RdHd/Hh]Hh and listed in row 9. /too . reflects several competing influences: on one hand the gain of diffuse radiation for the flat plate, and on the other hand the longer collection time due to reduced heat losses for concentrating collectors.

Row 10 lists the critical energy ratio X = O.joss/(AflcFIcou) of eqn. (III-2), and the corresponding value of the utilizability q) of eqn. (III-5)is entered in row 11. The final result for the delivered energy Q./A at specified receiver surface temperature T, = 50°C is given in row 12. If a different temperature base was specified, the result in row 12 is simply multiplied by the appropriate factor for heat extraction or removal efficiency. Row 13 gives the energy delivered Q/A if the mean fluid temperature is "F/= 50°C ; it is the product of Q/A of row 12 and the heat extraction efficiency F' in row 4.

The large cutoff time t c for the tracking collectors results from the assumed absence of any shading. In practical installations tc is likely to be reduced by about 1/2 hr to 1 hr, and the energy output will be on the order of 5 10 per cent lower. However, even with severe shading and 20 per cent reduction of (~ the tracking collectors would still be as good as (for east- west axis) or much better than the flat plate. The cutoff times for the CPC and the flat plate are realistic and

unaffected by shading in any reasonable installation. The good performance of the 1.5X CPC compared to the parabolic trough with east-west tracking axis is due to its ability to collect most of the diffuse insolation and due to the low heat loss of its evacuated receivers. Collectors with polar tracking axis or with two-axis tracking can deliver significantly more heat per aperture area than fixed collectors or collectors with east- west axis. On the other hand, considerations of field layout, plumbing connections and line losses may make concentrators with horizontal tracking axis more attractive for large installations.

To evaluate the effect of different operating temperatures on long term average performance, we list in Table 8 the energy output of the above mentioned collectors for operation at several temperatures, including ambient. The location is Lemont, Illinois, a suburb of Chicago[20].

The examples show that even in relatively cloudy climates and even for rather low temperatures (e.g. some space heating applications with heat exchanger) concentrating collectors can significantly outperform the fiat plate. This conclusion stands in strong contrast to the conventional wisdom that concentrating collectors are impractical for such applications. It is consistent with findings of recent studies[21 24] of insolation availability for various collector types and climatic regions. It is important to underscore the performance potential of concentrating collectors at a time when such collectors are beginning to enter the market at cost nearly competitive with flat plate collectors[25]. Since the concentrator industry is still undeveloped compared to the fiat plate industry, and since the materials requirements can be smaller, greater cost reductions can be expected than for flat plates. Therefore, the conventional wisdom is likely to be wrong, and concentrating collectors may replace flat plates for many applications.

V. C O M P A R I S O N S T U D I E S

In this section we demonstrate the use of our method for comparison studies by evaluating and comparing in a location independent manner the energy availability for various collector types and operating conditions: (a} 2-axis tracking versus l-axis tracking; (b) polar tracking axis versus east-west tracking axis; (c) fixed versus tracking flat plate; (dl effect of ground reflectance; and (el acceptance for diffuse radiation as a function of concentration ratio. For cases (b) through (d) we consider only equinox since this section is to be just an illustration, not an exhaustive study. At equinox co, = n/2, the declination is zero and the coefficients a, b and d of Tables 1 5 are:

a = 0.6598

b = 0.4226 (V- 1

d = 1.0

while the ratio (II-3) of diffuse over hemispherical insolation becomes

SE Vol. 23, No. 3--E

246 MANUEL COLLAP-J~S-I~R~RA a n d ARI RABL

8 8

£

0

8

0

0

0 °~

0

I

Z , d ~ 4

" ! . .

O

I

I

O

,q .

II

,q-

II

t"...-

tl

e ~

I.

o

O

II

° .

II r...: ,,~ ~ ~ eq

t -

II

"q" "q" oO ~ ..-*

o6 ,.6 ~ ,.4 ~

- - , o ° ~ t e 3 c ~ t ~ f , * 4

~ q q ~ q

t~

06 t-...: ,,4 ~ ,-4 II

~O r ~ 'q" ---~ ,.--*

II

e~

,6 ~ ,n: ~ e,i

Q

f q t ' ~ - -* t ~ l

~ t .~-

II

tl

I

Q

II

O

II

[.-,


/~d ,o,=,~/2 0.775 0.505 COS(2(gh - 0.9)) Hh (V-2)

(a) One-axis (Polar) tracking versus 2-axis tracking

By tracking with two degrees of freedom one keeps the collector normal to the sun at all times and has more radiation available than with one axis tracking. As for total radiation available during the day the difference between a polar mounted 1-axis tracker and a 2-axis tracker can easily be evaluated from the equations for R~ and Rh in Tables 4 and 5. For simplicity we assume high concentration to neglect the 1/C contribution of diffuse radiation. Since both R a and Rh differ only by a factor cos5 between these collectors, the available radiation differs also by this factor:

Hco,. l-axis polar COS 3 (V-3) /'~coll, 2-axis

where 3 is the solar declination. The cosine of the declination varies only from 0.92 to 1.0 during the year with a year-round average of 0.96. In practice this difference may be enhanced (perhaps doubled) by end effects if the polar axis tracker is short and has no end reflectors; this depends of course on the optical design. In any case the energy collection ratio is so close to one, that one could hardly justify the preference for 2- axis tracking over polar axis tracking on account of enhanced radiation availability.

(b) Polar versus east-west tracking axis

In the following subsections we consider only equinox; then the sun moves in a plane and the formulas for nontracking (Table 2) and for tracking (Table 3) concentrators with east-west axis become identical. For high concentrations the functions R d and Rh become in this case

( 2 ) Ew.x,, - sint°c (V-4) R4 t~s = COS ~.

and

R*(og' = 2 ) ~w,,, . - cosl 2 {0.6598 sin o9c

+ 0.2113(costocsinto ~ +toc) } (V-5)

The irradiation available to a concentrator with east- west tracking axis is therefore

I-

//¢o,t.Ew = / 0.6598 sin o9, + 0.2113

x (cos to~ sin 60, + ta¢)

/-/~ ] f/h -//~ sin toc, cos 2 ' (V-6)

with H.,/Hh given by eqn. (V-2). For a concentrator with polar tracking axis the available irradiation is

l-

] '~co l l , po lar = / o . 6 5 9 8 ~ , L

/~d coc ] /-/h + 0.4226 sin toc - /_/, Jcos2 (V-7)

which is the same as for the 2-axis tracker, the time of year being equinox. To compare the collectors and display the dependence on cutoff angle co, we have plotted in Fig. 8 the ratio (lower curves)

//~o,,. ~w (co. = 2. co~ )

\

and the ratio (upper curves)

( : = _~) versus toc" The sensitivity /~coll.polar O)s = 2 'm,

of these quantities to the diffuse/hemispherical ratio is relatively small, as shown by the fact that the curves for /(h = 0.4 are close to those for g', = 0.7. These ratios are independent of geographic latitude.

Clearly, the polar tracking axis exceeds the east-west tracking axis in potential for energy collection. But, despite a difference of 20-30 per cent (and even more for thermal collectors) one will usually choose a horizontal tracking axis for large concentrator arrays in order to minimize expenses and heat losses due to heat transfer lines. On the other hand, in small installations with a single row of concentrator mo- dules, the polar axis mount is likely to be preferred. In a home heating or cooling system, for example, lightweight polar mounted reflector troughs could be placed under a glass roof, an arrangement which also minimizes problems with windloading and dirt on the reflector.

(c) Flat plate:fixed versus tracking

The enhanced radiation availability for tracking surfaces has led some investigators to consider the possibility of tracking a flat plate collector. We do not wish to take sides in a debate whether such an arrangement could ever be practical, and present this paragraph only as illustration.

lc(hours) 2 4

1.0 ' I ' I '

Hcoll polQr {we) Kh ~ 0 . 4 ~ , / / _ ' ~ = 0.7

.

r~

0 I 7;'/4 rr/2

w c (radions)

Fig.& Comparison of radiation availability for east-west and polar tracking axis at equinox, as function of cutoff time.


Strictly speaking Tables 3-5 with C = 1 do not quite correspond to a tracking flat plate because of differences in brightness of ground and of sky. How- ever, as shown in the following subsection, neglect of this difference results only in errors of the order of 1-4 per cent, for collectors with tilt less than 40 ° . Further- more, a comparison between fixed and tracking flat plates is hardly effected at all by this difference, if both are based on formulas which consistently equate the brightness of ground and of sky. Working at equinox we can represent a fixed flat plate with tilt = latitude by Table 3 and a 2-axis tracking (or a polar axis tracking) flat plate by Table 5, provided the concentration ratio C is set equal to 1.0. The resulting radiation availabilities are

+ 0,2113(cos ¢oc sin oJ¢ + ~o¢)

-sinto~(l- c o s 2 ) ~ l / t h /-/h ICOS X

(v-8)

(d) Effect of ground reflectance In the preceding subsection we compared radiation

availability to fixed and to tracking flat plate collectors by neglecting the difference in brightness between ground and sky. The effect of the ground reflectance on flat plate collector performance can readily be evaluated by means of Table 1. For equinox and tilt = latitude Table 1 yields

= ~ - ~(1 + cos 2)] sin mc

and

(1 - cos 2)] x [0.6598 sin to c P + 3

+ 0.2113(cos to c sin to, + toe) ]

(V-IO)

(V-ll)

and

F

= [0.6598 m, + 0.4226 sin o9~

• . , ~ , l / L - (m~ - - sm to~ cos JtA , - - ~ | - -

n h J cos 2 (V-9)

The ratio of these two quantities is listed in Table 9 for different values of latitude and clearness index. The enhancement in available radiation increases with latitude and clearness index, from about 13 to 33 per cent.

where p is the reflectance of the ground in front of the collector. If the brightness of ground and sky were equal, the ½(1 + cos2) term in eqn. (V-IO) could be replaced by one and the ( 1 - cos A) term in eqn. (V-11 ) replaced by zero; the resulting/-?co, would be just eqn. (V-8).

To evaluate the effect of the ground reflectance p we list in Table 10 the ratio

/-Icoll,flatplatc.sround= p £0 a ---~ ~ , ¢"0c

Table 9. /lco,.natp~.t..t,,cal,e//Tco,.n.tp~.t,.tl,d. ratio of radiation availability for tracking and for fixed fiat plate, at equinox

.3

O* 1 0 °

1.13

20*

1.14

30*

i. 14

40*

1.15

50 °

1.17 1.13

.4 1.18 1.18 1.19 1.20 1.21 1.22

.5 1.23 1.23 1.23 1.24 1,25 1.27

.6 1.27 1.27 1.27 1.28 1.29 1.30

.7 1.29 1.29 1.30 1.31 1.32 1.33


for different values of latitude and clearness index. For latitudes beyond 30 ° the sensitivity to p increases, and we have included the p = 0,7 case (for snow) in addition to p = 0.2. The ratio of radiation available with and without snow can readily be obtained from Table 10. For example, for latitude 40 ° and clearness index 0.5 this ratio is

] ' Icol l . flat plate.p = 0.7 1 .02 = 1.05

/~coll,flatplat¢.t,=0.2 = 0.97

One learns from this table that the error due to neglect of difference between sky and ground reflectance is negligible for tilt below 20 ° and small (1-4 per cent) even for tilt of 400.

(e) Collection of diffuse radiation as function of concentration

The Rd and R, functions for concentrators of low concentration C contain a term proportional to 1/C to account for their ability to accept part of the diffuse radiation. By decreasing C, one can collect more of the diffuse component, but at the price of in- creasing the receiver surface and the heat losses. A

complete optimization of the concentration ratio should take into account several additional factors, in particular mirror and tracking errors and circumsolar radiation - an undertaking beyond the scope of the present paper. But, to provide at least some evaluation of the importance of the diffuse component, we have calculated the insolation available to a concentrator with east-west axis at equinox (from Table 2 or 3 ) as a function of concentration C

o.,,.,w ...... io.6=9=,,o..

+ 0.2113(sin toc cos to, + me)

a (l 1 a, (V-12)

~C 1.0 I0.0 5D 2,0 1.0

..~.

~ 0.5

b o eL, 0.2 o'.5 ,.o

Fig. 9(a). Radiation availability for concentrator with east- west axis at equinox, as function of concentration ratio C, for

different values of clearness index gh.

Fig. 9(a) shows the ratio

21 /~colI.EW,conc=C ~ s = 2,¢d)c =

o . . ...... ,=2t 0.9917 H* L

(v-13) B. 0.9917 - ~-~,[1 - cos2]

for different values of the clearness index and latitude k = 35 °. The analogous ratio for concentrators with polar tracking axis is

Hcoll.polar.conc= (' r~ s = ~ , ( D c =

/-Icoll.polarconc = 1 (Os ~ --

(; (v-14)

1.459 - ~ x - cos 2

This is shown in Fig. 9(b) also evaluated at k = 35 °. The polar axis concentrator is seen to be less sensitive

< £

o

C,

4 8

1,0 i0,0 5.0

- C

2.0 1,0

0,5

0

"-.0,6

0,1 0,2 0,5 1.0

I/C -

Fig. 9(b). Radiation availability for concentrator with polar axis at equinox,as a function of concentration ratio C, for different values of clearness index Kh.


Table 10. Effect of ground reflectance, evaluated by means of the ratio of radiation availability ifground = p and if brightness of ground and sky are equal:

O"

0=0.2

20*

p=0.2

.3 1.00 1.01

.4 1.00 1.01

.5 1.00 1.01

,6 i.oo 1.o0

.7 1.00 1.00

30*

0=0.71 0=0.2

0.99 1.03

0.99 1.02

0.98 1.01

0.98 1.01

0.98 1.01

O=.?

40"

P ' . 2

0.99 1.04

0,90 1.03

0.97 1.02

0.97 1.02

0,96 1.01

50*

0=0.7 0=0.2

0.98 1.06

0.97 1.04

0,96 1.03

0.96 1.02

0.95 1.01

to loss of diffuse radia t ion than the concent ra tor with the east-west axis; the reason lies in the reduced beam incidence angle for polar axis tracking. For very clear climates, gh = 0.7, a high concen t ra t ion collector with east-west axis receives only 73 per cent of the radia t ion available to a fixed flat plate, whereas a high concent ra t ion collector with polar axis receives 80 per cent of the radia t ion avai lable to a polar axis t racking fiat plate. For c loudy climates, K, = 0.3, the corresponding rat ios d rop to 46 per cent for the east-west axis and to 52 per cent for the polar axis.

Table 10 and Figs. 9(a) and (b) provide general rules of t h u m b abou t the effects of t racking and concentrat ion. For example for gh = 0.5, typical of average cloudiness, Table 10 indicates tha t t racking enhances the energy availabili ty for flat plates by abou t 25 per cent, while Fig. 9(b) shows that high concen t ra t ion reduces the energy availabil i ty by abou t 30 per cent. Therefore the improvement in average incidence angle b rough t abou t by polar axis t racking nearly compensates for the loss of diffuse radia t ion associated with high concentra t ion.

Acknowledgements - Throughout the course of this work we have benefited from discussions with many of our colleagues. In particular we should like to thank Dr. S. A. Klein for valuable comments, and Mr. H. W. Gaul and Dr. K. H. Macomber for checking some of the results.

NOMENCLATURE

We use the symbols I for irradiance (or instantaneous insolation, in W/m 2) and I t for irradiation (or daily total insolation, in J/m 2), together with subscripts b for beam (also called direct), d for diffuse and h for hemispherical (also called global or total). To minimize subscripts in the present paper we refer irradiation H (except for A~on) to horizontal surface, but irradiance I to normal incidence. Bars indicate I o ~ term average. Note that beam is defined with respect to the 2.8 ° acceptance half angle of the pyheliometer, and not with respect to the solar disc: thus it includes the circumsolar component.

A net aperture area of collector C geometric (or area) concentration = ratio of aper-

ture area over receiver surface area ; for example a parabolic trough of aperture width D and receiver tube diameter d has C = D/(nd).

F factor to account for heat extraction or removal efficiency

' Ho extraterrestrial irradiation on horizontal surface (daily total)

//co. irradiation incident on collector aperture (daily total)

A~ diffuse irradiation on horizontal surface (daily total) / - /h hemispherical irradiation on horizontal surface

(daily total) I b beam irradiance at normal incidence

(pyi'heliometer) la diffuse irradiance on surface normal to sun I, hemispherical irradiance on surface normal to sun

(pyranometer)


Kk /tdHo - long term average clearness index (called /~r in Refs. [ 2 - 6 ] )

qo~ instantaneous collector output (W) ql AU(Tco, - T,= 0 = instantaneous collector heat

loss (W) Q long term average energy (J) delivered by collector

(daily total) ( ~ long term average heat loss of collector (J) (daily

total) Ra} { conversion factors from horizontal irradiation to Rh irradiation on collector aperture R RJR~ t time of day from solar noon (p.m. is positive) t, collector cut off time (if - t c_ = t~+) t,_ collector turn on time (hours before noon) t~+ collector turn off time (hours after noon) t, sunset time T length of day = 24 h = 86,400 sec T o ambient temperature T~o . operating temperature of collectors, for which any of

the following four temperature bases may be used if the corresponding F-factor is inserted into efficiency equation

Tf (Tin + T,mt)/2 = mean fluid temperature T~ inlet fluid temperature To~, outlet fluid temperature T, average receiver surface temperature U U-value [W/m2°C) l/ collector tilt from horizontal surface (positive to-

wards equator) solar declination

2 geographical latitude r/o optical efficiency (also called (ta) product in the flat

plate literature) = fraction of insolation absorbed by absorber = efficiency if receiver at ambient, i.e. no heat loss.

~o long term average optical efficiency 0 acceptance half angle of CPC 0 incidence angle of sun on horizontal surface ~b collector azimuth from due south, relative to horiz-

ontal plane (west is positive, east negative) 4~ utilizability to 2nt/T = hr angle to~ 2m~/T = sunset hr angle

= arcos ( - tan 6 tan ~.) to', arcos ( - tan 3 tan (2 - fl)

Note that throughout this paper all angles are in radians, except for a few cases where degrees are explicitly indicated.

REFERENCES

1. I. J. Hall, R. R. Prairie, H. E. Anderson and E. C. Boes, Generation of Typical Meteorological Years for 26 SOLMET Stations, Environment and Resource Assess- ment Branch, Division of Solar Technology, DOE.

2. B. Y. H. Liu and R. C. Jordan, A rational procedure for predicting the long term average performan~ of flat- plate solar-energy collectors. Solar Energy 7, 53 (1963).

3. H. C. Hottel and A. Whillier, Evaluation of flat plate solar collector performance, Transaction~ of the Con- ference on the use of Solar Energy: The Scientific Basis, Vol. II. Part 1, Section A. pp. 74-104 (1955).

4. For the flat-plate collector this simplification has also been found by S. A. Klein, Calculation of flat-plate collector utilizability, Solar Energy 21, 393 (1978). For the example of the flat plate in our Table 7. Kiein's method agrees with ours to 1 per cent if the same/~//]h is used.

5. W. A. Beckman, S. A. Klein and J. A. Duffle, Solar Heating Design by the f-chart Method, Wiley, New York (1977).

6. S. A. Klein and W. A. Beckman, A General Design Method for Closed-Loop Solar Energy Systems, Solar

Energy, 22, 269 (1979). 7. The validity of this procedure has been verified for

systems with CPC collector by comparison with hour- by-hour simulation. S. A. Klein, personal communication.

8. A preliminary version of this model was presented Sept. 1977 : M. Collares-Pereira and A. Rabl, Simple procedure for predicting long term average performance of nontracking and of tracking solar collectors, proceedings of, Deutsches Sonnenforum, Hamburg Vol. I, p. 275 (26-28 September 1977).

9. M. Collares-Pereira and A. Rabl, The average distri- bution of solar radiation correlations between diffuse and hemispherical and between daily and hourly insolation values. Solar Energy 22, 155 (1979).

10. M. Collares-Pereira and A. Rabl, Derivation of Method for Predicting Long Term Average Energy Delivery of Solar Collectors, Solar Energy, companion paper in this issue. The functions Ra and Rh are integrals over time of day of the conversion factors ra and rh between daily and instantaneous insolation values of Ref.[9], convoluted over incidence angle factors.

11. J. A. Duffle and W. A. Beckman, Solar Energy Thermal Processes, Wiley, New York (1974).

12. F. Kreith and J. F. Kreider. Principles of Solar Engineer- ing, McGraw-Hill, New York {1978).

13. A. Rahl, Comparison of solar concentrators. Solar En- ergy lg, 93 (1976).

14. A. Rabl, Concentrating Collectors. In Solar Energy Handbook, W. C. Dickinson and P. N. Cheremisinoff (Eds) Marcel Dekker, New York, Basel (1979),

15. R. Winston, Solar concentrators of a novel design. Solar Energy 16, 89 (1974).

16. A. Rabl, Optical and Thermal Analysis of Concentrators, in Proceedings Solar Thermal Concentrating Collector Technology Symposium, Denver, CO (14 and 15 June 1978) SERI/TP-34-048.

17. J. Leonard, Linear concentrating solar collectors current technology and applications, Solar Thermal Concentrating Collector Technology Symposium, Denver, CO (14 and 15 June 1978). V. E. Dudley and R. M. Workhoven, Summary Report: Concentrating Solar Collector Test Results Collector Module Test Facility, Sandia report SAND 78-0815 (1978).

18. G. R. Mather, Jr. and D C. Beekley, Analysis and Experimental Tests of High Performance Tubular Solar Collectors, ISES Conference in Los Angeles, CA (July 1975).

19. R. Bingham, Final Report for Contract 31-109-38-3805, General Electric Space Division (1977).

20. The long term average performance of a CPC has also been calculated by R. L. Cole, Long term average performance predictions for compound parabolic concentrator, Proc. 1977 Meeting U.S. Section Int. Solar Energy Sot., Orlando, Florida, p. 36-6 (June 1977).

21. E.C. Boes, H. E. Anderson, I. J. Hall, R. R. Prairie and R. T. Stromberg, Availability of Direct, Total, and Diffuse Solar Radiation to Fixed and Tracking Collectors in the USA, Sandia Laboratories Report SAND-77-0885 (August 1977).

22. D. L. Evans, Simplified Solar Irradiation Data Based on the Aerospace Insolation Data Base, Arizona State University Report ERC-R-77077, for SAND[A Labora- tories Contract No. 02-7850 (1977).

23. W. C. Dickinson, Annual available radiation for fixed and tracking collectors. Solar Energy 21, 249 (1978).

24. G. Treadwell, Low Temperature Performance Com- parison of Parabolic Trough and Flat-Plate Collectors Based on Typical Meteorological Year Data, Sandia Laboratories Report SAND 78-0965 (February 1979).

25. K. C. Brown, Solar Energy Research Institute, private communication.

26. J. E. Hill and E. R. Streed, A method of testing for rating solar collectors based on thermal performance. Solar


Energy 18, 421 (1976). F. F. Simon, Flat plate solar collector performance evaluation with a solar simulator as a basis for collector selection and performance prediction. Solar Energy 18, 451 (1976).

27. T. Tabor, Testing of solar collectors. Solar Energy 20, 293 (1978).

28. F. de Winter, Heat exchanger penalties in double-loop solar water heating systems. Solar Energy 17, 335 (19761.

APPENDIX

The instantaneous collector efficiency[26, 27] serves as basis of the calculation and must be specified in a clear and unambiguous manner. We briefly review the most important characteristics.

1. Specification of insolation

Traditionally the efficiency of flat plate collectors has been referred to hemispherical (also called global or total) irradiance In, and that of collectors with high concentration to beam (also called direct) irradiance It, ; this has been assumed as the basis of the present paper. For the intermediate case of concentrators with low concentration no clear consensus has yet emerged. To a certain extent it is a matter of bookkeeping, and several choices are acceptable, provided one is self- consistent. Within the framework of this paper, it has been most convenient to base the efficiency of such collectors on radiation within the acceptance angle. If the efficiency data have not been presented in this form, correction factors must be applied. Fortunately the conversion from one insolation base to another is straightforward and involves only a multiplicative factor. To find this factor let us add subscripts to the efficiency. If qout is the collector output [in W] relative to net collector aperture area A, then the efficiency with respect to hemispherical irradiance It, (pyranometer) is

qou! tit, - (A- 1 )

Aim

while the efficiency with respect to beam It, (pyrebeliometer) is

qout t i t ' - Aft," (A-2)

The conversion from one to the other is therefore

Ih tib = t ih~

where I d = It, - It, is the diffuse component. Since efficiency measurements should always be done under clear sky, the ratio Id/I b of diffuse over beam is about 0.1 to 0.15. This means that the efficiency curve of a collector is at least 10 per cent higher when stated in terms of beam rather than in terms of hemispherical radiation. For collectors with low concentration 1 < C < 10, e.g. CPC and V-trough which accepts a significant fraction, I/C of the diffuse component[13, 14], the efficiency relative to the irradiance

1 I< = It, + ~ I, (A-61

within the acceptance angle is

quut ti< = - - ; (h-7) AI~

the conversion factor from tim to ti< is given by

Id l + - -

Ib ti, = t i t , - (A-8)

I + - - CI b

and the conversion from t#< to tim is

1 tic = t i s - - (A-9)

ld l + - -

Clb

Test data for the CPC should therefore include a statement about the l,t/lb ratio.

2. Reference temperature

Several collector temperatures can serve as reference for stating the efficiency, the most useful being

T, average collector receiver surface temperature Ti. fluid inlet temperature Tout fluid outlet temperature T s = (Tin+ Tou0/2 = mean fluid temperature

To a very good approximation only the difference between the collector temperature and To = ambient temperature matters. The heat loss coefficient or U-value U[in W/m2°C] is defined relative to collector aperture area A as

ql U = (A-101

A ( T , - T~)

where qj is the heat loss [in W]. Strictly speaking U is not constant; but its dependence on temperature, wind and other environmental factors is fairly weak, and good approximation is obtained by using an average U-value corresponding to the anticipated operating temperature. For a better approximation we recommend Tabor's parameter- ization[27]

'qtA = U o ( T , - T°) p (A-11)

where p is a collector dependent coefficient, typically in the range 1.1 to 1.3 for nonevacuated collectors and somewhat larger for evacuated collectors. Alternatively one could add a second order term to U. We now omit the subscripts b, c, and h for efficiency and insolation because the form of the following equations is independent of the specification of the insolation. In terms of U the collector efficiency reads

ti = ti, - U(T, - T.)/I (A-12)

if the average receiver surface temperature T, is given, ti, is the optical efficiency or efficiency at zero heat loss; it has also been called (r~tl product in the flat plate literature.

Usually it is more practical to measure the fluid temperature than the receiver surface temperature. In terms of the mean fluid temperature T t the efficiency equals

t /= F'[tio - U(T s - To)/I] (A-13)

where F' is the heat extraction factor (called collector efficiency factor in Ref. [11]) given by the ratio

F ' - Ufa (A-141 U

of the thermal conductance Uso from fluid to ambient over the thermal conductance from receiver surface to ambient (in this equation both U values must refer to aperture area). If the fluid inlet temperature T~. is specified, the efficiency is

ti = FR[ti. -- U(T~ - T.)/t] (A-15)

with the heat removal factor[11, 12]

thc, r I e x p ( - U A F ' ~ ] (A-16) FR = UA I_ - \ t h c ~ / d

th is the mass flow rate [kg/s] through the collector and c r is the fluid heat capacitance [J/kg °C] at constant pressure. Finally, the dependence of efficiency on fluid outlet temperature Tout is given by a modification[28] of eqn. (A-15)

Fa '1 LFI - F'UAq[n*-I U ( T o u t - T,,)II]. (A-1.7)

• tiicl, -I Any of the four expressions for efficiency, (A-12), (A-13), (A- 15) or (A- 17) can be used as starting points for the calculation of long term average performance.


R6sume - La m6thode de Liu et Jordan pour le calcul de la moyenne d long terme de l'6nergie capt~e par des collecteurs plats est simplifi6e (par un facteur de 4) ameliorde et generalisde fi tousles genres de capteurs, concentrateurs ou pas. La moyenne d long terme de la radiation hdmisph6rique journali&e Hh sur une surface horizontale ainsi que la temp6rature ambiante moyenne dans le cas des applications thermiques sont les seuls inputs mdt6r6oiogiques n6cessaires. Le capteur est caracterisd par son efficacitd optique, ses pertes de chaleur (U-value), son efficacit6 darts l'extraction de la chaleur, sa concentration et la fa~on dont il suit le mouvement du soleil. Une tempdrature moyenne de fonctionnement est suppos6e. La nouvelle methodeper- met d'dtudier l'interaction avec I'accumulateur d'energie quand utilisde ensemble avec la mdthode '3C-chart '' de Beckman, Klein et Duffle.

Un facteur multiplicatif est presente qui transforme la radiation hemispherique journali~re dans la moyenne ;i long terme Q qui repr6sente I'dnergie que le capteur est en mesure de fournir. Ce facteur depend d'un grand nombre de variables dont la temp6rature du capteur, I'index "de nuage", la radiation diffuse sont des exemples, mais peut 6tre d6composd en diff6rents facteurs qui ddpendent chacun seulement de deux ou trois variables et peut etre prdsent6 sous forme graphique ou analytique. En general les variations climatiques saisoni~res exigent un calcul sdpar6 pour chaque mois; cependant un seul calcul pour la journ6e centrale du mois est suffisant. La simplicitd de la m6thode permet que ce calcul soit fait fi la main.

Des formules et des exemples sont pr6sentds pour cinq types diffdrents de capteurs: plat, Compound Parabolic Concentrator, concentrateur d axe fixe orient6 E.-O., concentrateur a axe flxe orient6 N.-S. et concentrateur fi mouvement bi-axial. Les exemples montrent que m~me pour des applications dites ii temperatures basses et des climats nuageux (50°C, N. York en fevrier) les concentrateurs ont une performance superieure :i celle du collecteur plat.

La m6thode a 6t~ valid6e en utilisant des donndes exp6rimentales valeurs horaires pour la radiation diffuse et h6mispherique et te resultat final a une pr6cision plus grand que 3 per cent pour la moyenne ~, long terme de la radiation disponible aux capteurs. L'erreur r61ative du calcul de l'6nergie fournie par les collecteurs thermiques est inf6rieure a 5 per cent. L'utilit6 de la mdthode dans les dtudes comparatives est illustree en comparant, independemment de la localit6, la radiation disponible d des capteurs soit diffdrents (concentrateurs fi mouvement bi-axial compar6s d ceux orient6s E.-O., concentrateurs fixes a ceux fi mouvement bi-axial) soit operant dans des conditions diff6rentes (l'effet de la reflectivit6 du terrain, la capacitd de capter la radiation diffuse en fonction de la concentration).

Zusammenfassung - Die Methode von Liu und Jordan zur Berechnung der langzeitlichen Durchschnittslei- stung von Flachkollektoren wird vereinfacht (urn etwa einen Faktor vier), verbessert und verallgemeinert fuer alle Kollektortypen, sowohl nichtkonzentderend als auch konzentrierend. Als einziger meteoroligischer input wird der langzeitliche Mittelwert/7 h der taeglichen horizontalen hemisphaerischen Sonnenstrahlung benoetigt (sowie die durschnittliche Umwelttemperatur). Der Kollektor wird gekennzeichnet durch optischen Wirkungsgrad, Waermeverlust, Wirkungsgrad der Waermeuebertragung, Konzentrationsver- haelmis und Nachfuehrweise. Einc durchschnittliche Arbeitstemperatur wird vorausgesetzt. Wechselwir- kung mit einem Waermespeicher kann beruecksichtigt werden, indem man diese Methode mit der f-chart Methode yon Beckman, Klein und Duffle kombiniert.

Ein Faktor wird gegeben, mit dem die taegliche Strahlung Oh multipliziert wird, um den taeglichen Durchschnittswert der neutzlichen Energielieferung () des Kollektors zu bestimmen. Der Faktor haengt von vielen Variablen ab, z. B. Arbeitstemperatur, Kollektoreigenschaften, Breitengrad und Sonnenstrahlung, aber er kann in mehrere Komponenten zerlegt werden, die nur yon zwei oder drei Variablen abhaengen und daher in geeigneter graphischer oder analytischer Form angegeben werden koennen. Im allgemeinen wird die jahreszeitliche Veraenderlichkeit des Wetters mehrere Rcchnungen fuer das ganze Jahr erfordern. Fuer die meisten Faelle wird eine Berechnung fuer den mittleren Tag jedes Monats ausreichen. Die Methode ist einfach genug fuer Handrechnungen.

Formeln und Beispiele werden gegeben fuer fuenf Kollektortypen : Flachkollektor, compound parabolic concentrator (CPC), Konzentrator mit Ost-West Axe, Konzentrator mit Polar-Axe, und Konzentrator mit zwei-Axen Nachfuehrung. Die Beispiele beweisen, dass sogar bei verhaeltnismaessig niedrigen Arbeitstempe- raturen und wolkenreichem Klima (50°C in New York im Februar) konzentrierende Kollektoren mehr Energie liefern koennen als Flachkollektoren.

Die Methode ist nachgeprueft worden durch Vergleich mit stuendlichen Wetterdaten (mit Messungen yon hemisphaerischer und direkter Sonnenstrahlung). Die durchschnittliche Genauigkeit erwies sich als 3~o fuer den langzeitlichen Mittelwert der Sonn.cnstrahlung, die Solarkollektoren zur Verfuegung steht, fuer die Energielieferung yon Waermekollektoren liegt die durchschnittliche Genauigkeit bei 5%. Die Methode eignet sich hervorragend zu Vergleichsstudien. Dies wird erlaeutert, indem die verfuegbare Sonnenstrahlung fuer mehrere Kollektortypen in ortsunabhaengiger Weise verglichen wird : zwei-Axen Nachfuehrung/Nach- fuehrung mit einer Axe; Polar-Axe/Ost-West Axe; fester Flachkollektor/nachgefuehrter Flachkollek.tor. Ausscrdem wird die Auswirkung des Reflektionsvermoegens des Erdbodens auf den Flachkollektor untersucht, sowie der Beitrag der diffusen Strahlung als Funktion des Konzentrationsverhaeltnisses.

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Documents

Simple procedure for predicting long term average performance of nonconcentrating and of concentrating solar collectors