Daily efficiency of solar collectors

Applied Energy I0 (1982) 189-202

DAILY EFFICIENCY OF SOLAR COLLECTORS

B. BARTOLI, V. CUOMO, M. FRANCESCA and C. SERIO

lstituto di Fisica della FacolttJ di lngegneria Universitft di Napoli, Naples (Italy)

and

G. BARONE and P. MATTARELLI

SoGesTA, Urbino (Italy)

SUMMARY

It is shown that the daily eJficiency of flat-plate solar collectors working at constant .flow rate can be evaluated with a simple algorithm when their structural features are known. It is also shown that long-term performances can be calculated starting from monthly ralues of global radiation.

INTRODUCTION

In a previous paper ~ it has been shown that it is possible to obtain statistically significant daily values of global radiation from monthly average data. Moreover, it is possible to evaluate 2'3 the long-term performances of solar fiat-plate collectors using only daily values of meteorological data. For these reasons, we expect that daily efficiencies of fiat-plate collectors can be expressed as analytical functions of the parameters which describe their technological properties, their operating conditions and daily solar radiation data.

Let us define {O } as the set of parameters which take into account the operating conditions of the collector, { T} as the set of their technological features, Fex p as the global incident radiation and % as the daily efficiency of the collectors. Then:

E.d r/d - - F e x p , ~

where: E,e = useful energy delivered by the system and d = subscript indicating daily values.

If r/d can be expressed as:

r/d = ~/a({O }, {T}, Fexp) (1) 189

Applied Energy 0306-2619/82/0010-0189/$02-75 © Applied Science Publishers Ltd, England, 1982 Printed in Great Britain

190 B. BARTOLI, V. CUOMO, M. FRANCESCA, C. SERIO

it would be possible to evaluate the monthly efficiency (that indicates the monthly useful energy) simply by using the method presented in reference 1 and integrating on Fex v weighted with the probability distribution of Fex p values:

Euu = I qa({O}, {T}, F~xp)P(Fexv)F, xpdF, xp (2)

where: P(F, xp)= probability distribution of F**p and M = subscript indicating monthly values.

Since flat-plate collectors working at a constant flow rate are generally used, this case will be examined in the present work. In Section I the determination of {O} and { T} for collectors working at constant flow rate will be discussed. In Section II an explicit expression of qe will be proposed and the predictions of this hypothesis will be compared with the results obtained using the Hottel and Whillier formula with hourly data. In Section III the approximation in which such parametrisation holds will be discussed in detail.

SECTION I

Parameters defining the operating conditions of a collector working at constant flow rate are implicitly included in the Hottel and Whillier equation, where the useful power delivered by the collector is expressed as:

W, = F R . A. [ctr~b - UL(T/- Ta) ] (3)

where: F R =collector heat removal factor, A =collector area (m2), ctr =optical absorptance-transmittance collector coefficient, q~ = radiation hitting the collector surface (W/m2), U L = thermal loss coefficient (W/m 2 °C), T,. = inlet temperature of the working fluid (°C) and T, = ambient temperature (°C).

A complete discussion of eqn (3) is presented in reference 4. It must be noted that the collector heat removal factor, F R, depends only on the

collector features and on the thermal flow rate:

GCv 1 - e - (4) Fu= U--~ GCp J

where: G =flow rate of the working fluid per unit of collector area (kg/sm2), C v = thermal capacity of the fluid (J/kg°C) and F'= collector efficiency factor (depending only on the structural features of the collector).

Equation (3) shows that W u is linear in FR, and the operating threshold of the collector, UL(T i -- T,), is independent of the flow rate. As a consequence also, E~e and r/d are linear in F R.

Furthermore, eqn (4) shows that F R is constant when the technological features and the flow rate of the collector have been fixed. It follows that it is sufficient to calculate r/e for F R = 1 : the actual efficiency will be obtained simply by multiplying this value by the actual FR value.

DAILY EFFICIENCY OF SOLAR COLLECTORS 191

In eqn (3) ar and U L take into account the technological features of the collector; F a and UL(T i - T , ) take into account both the technological features and the operating conditions. The ~z coefficient--that is, the percentage of solar radiation really absorbed by the black absorber of the collector---depends on the angle, 0, between the incident radiation and the collector's transparent cover: it is, therefore, a function of time and depends on the zenith, ~, and azimuth, ~b, angles of the collector. When ~ and ~/are known, ar can be evaluated easily as a function of the transparency features of the cover surfaces and of the black plate absorptancefl 4

These considerations show that it is possible to write eqn (1) as:

rld =J [ UL ( Ti - T.), ~rqS, F¢~v] (5)

SECTION II

We have assumed:

] F U L ( T i - T,,) Ernax~ . r/d= 1_ ~ h ~ - i 2 "F~-~p.J = j ( X ) (6)

where: ~r~blh= 12 = solar radiation absorbed by the black plate at mid-day under clear-sky conditions, Eexp=daily global experimental solar radiation and Ema x = maximum global radiation that would hit the collector under clear-sky conditions.

To verify the hypothesis contained in eqn (6), values for several collector types have been calculated for the periods 1964-1969 in Macerata and 1964-1972 in Genoa for which daily radiation data are available. 6'7

Possible combinations of the following design and operating conditions have been analysed:

(a) Collectors covered by one or two normal glass panes (3mm thick). (b) Thermal loss coefficient U L = 4, 6, 8 (W/m 2 °C). (c) Absorptance of the plate ~ =0.9. (d) Tilt angle of the collector ® = 0 °

In order to investigate a possible seasonal dependence, the calculations have been performed for each month for both localities. We have divided the range of variability of X into 100 intervals of equal length. For each we present the average value of% and its standard deviation computed using the many values of X falling in each interval: only intervals with three or more points have been considered. The results show, in particular, that all points lie on the same curve independently of month and station--see Figs 1 to 24 in which daily values ofr/d(givenin per cent) are presented. In the same Figures we also show the general best fit:

rl d = a X + b + [(aX + b) 2 + K] U2 (7)

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196 a. BARTOLI, V. CUOMO, M. FRANCESCA, C. SER|O

with the following values of the parameters:

a = - 62.79

b = 31.84

K = 407.36

The Z 2 test, at the 95 per cent confidence level, confirms that the points o f Genoa and Macerata are well fitted by eqn (7).

The root mean square deviation o f the points to the fit is:

A =0 .336 per cent

In Fig. 25 we show the r/d efficiency values averaged over months and stations with their s tandard deviations for each interval o f X, and the general best fit. The result is completely satisfactory: as shown by Fig. 25, the fit introduces an error which is negligible in all significant cases.

120-

110 i

1°° i

8O

7O

N 6O

5O

30

20-

-0.40 0.00 0.40 0.80 1.20 1.60 2.00 X

Fig. 25. Daily efficiency

Er, tax

for all months and both stations. The curve is the general best fit t/a = aX + b + [ (aX + b) 2 + K.]1/2.


SECTION nl

The result we have obtained seems to be a consequence of the scaling properties of solar radiation during the year.

For sunny days, the time behaviour of solar radiation intensity can be described to a good approximation by the following expression:

[ctz(7)]sqSs(7) = g(j)Fo(7)

where: [Tr(7)]s~bs(~,,)= radiation absorbed by the black absorber in the j th day at time, 7; 7 = t/I, with t = current time and / = length of the day and g(j) = generic function of the jth day in the year.

The shape of F o is essentially independent of the season, while g(j) and / are smooth functions of the day of the year.

Also, the hourly behaviour of the threshold intensity UL(T i - Ta) varies during the year so that its shape remains essentially unchanged. These two hypotheses mean that the radiation and threshold scale during the year are as shown in Figs 26(a) and 26(b). In this approximation the efficiency (i.e. the ratio between the dotted and dashed areas in Figs 26(a) and (b)) results, for the similarity principle, are a function of the ratio between the peak value of the radiation and the corresponding threshold value only. As a consequence:

Unfortunately, the scaling law for radiation is only a first approximation to the actual behaviour of radiation, as shown also in Figs 27 and 28. Nevertheless, this approximation is quite good so that eqn (8) can be considered still valid.

Actually, during cloudy days the radiation behaviour is quite different, as shown in Fig. 29. However, if the daily time distribution of clouds does not vary too much during the year, it is possible to suppose that the efficiency varies as a function only of the ratio between the daily clear-sky radiation, Ema ~ and the experimental global radiation, Fexp: i.e.

rle=J(Ua(Ti- T:)~.g(E':x3 (9)

~ h = l : / \Fexp / /

We have shown that, in our case, eqn (6) holds. Such a result is a satisfactory improvement on the utilisability curves obtained for the hourly and daily basis by Liu and Jordan :5 in fact, utilisability curves depend on the average value of the rate F~p/H o where H o is extra-atmospherical radiation, while our efficiency curves, which can be obtained with a little complication in the algorithm, seem to be independent of season and station.


• a~ ~(wlm2)

1000

800

600

40C

20C

] 1 1 I I I I I ~

4 6 8 10 12 14 16 18 20 TIME (heurs)

0

• a f# (w/ , , ,= )

1000

800

600

4OO

200

3 - L #,

4 6 8 10 12 14 16 18 20 TIME (heurs)

b Fig. 26(a) and (b). Consequences of scaling properties. If the solar radiation and threshold scaled exactly during the year then r/d would be a function only of the ratio of mid-day threshold to peak

absorbed radiation ctrtklh- t 2.

800 / / ~ ÷

V % , // \

~. // / \ ~: 4 0 0 { , _ , . ~,

m ~

/ ' z+ / , , / ,,\

0 4 8 12 16 20

Time {h)

Fig. 27(a). Global clear-sky solar radiation on a horizontal surface at the Spring equinox ( + ) , the Summer solstice (X) and the Winter solstice (or) in Macerata. `9 = 0 °.

8ool . ~ / S ~

% I x / ",

"~ 400 / , ',,

4 8 12 16 2o Time ( h )

Fig. 27(b). Global clear-sky solar radiation on a horizontal surface through a single normal glass pane (3 mm thick) at the Spring equinox ( + ), the Summer solstice ( x ) and the Winter solstice (~-) in Macerata.

`9 = 0 ° ; 1 cover glass.

8 0 0

\ J "X

4 0 0 x "" * X

/ ,,

0 - - - " ' ~ " ~ " ~ 4 8 lz 16 zo

Time(h }

Fig. 27(c). Global clear-sky solar radiation on a horizontal surface through two normal glass panes (3mm thick) at the Spring equinox ( + ) , the S ammer solstice ( x ) and the Winter solstice (~-) in

Macerata. ,9 = 0 °; 2 cover glasses.

Boo! J .... - ,

/ • ',,,

400- /'>' ¢ :~",, ,\

/

0 4 8 12 16 20

Time ( h )

Fig. 28(a). Global clear sky solar radiation on a 45°S tilted surface at the Spring equinox (+) , the Summer solstice ( x ) and the Winter solstice ( ~ ) in Macerata. 9 =45 °.

800

% /, 400 /, '

/, ',,, , ~ , , / ~ ,~

4, 8 12 ' 16 2'o Time ( h )

Fig. 28(b). Global clear-sky solar radiation on a 45 °S tilted surface through a single normal glass pane (3 mm thick) at the Spring equinox (+) , the Summer solstice ( x ) and the Winter solstice (-~) in Macerata.

9 = 45 ° ; 1 cover glass.

800-

~E400 , ,~

0 ~ " " i " , - ~ _ ~ 4 8 12 16 20

Time ( h

Fig. 28(c). Global clear-sky solar radiation on a 45 °S tilted surface through two normal glass panes (3mm thick) on the Spring equinox (+) , the Summer solstice ( x ) and the Winter solstice ( ~ ) in

Macerata. 0 = 45 °; 2 cover glasses.

S O L A R R A D I A T I O N F U * X 7 1 ~ - -

/ ' \

\\

/

/

A R B I T R A R Y U N I T S T I M E

Fig. 29. Typical behaviour of radiation on cloudy days. When a cloud covers the sun radiation decreases to only diffuse component.

120 J

110- 4

100~

904 1 80 g 70

~- 6o

5O

4O

30

20

10-

ol -0 5 0 5 1 5 2 5 3.5 45 5.5 6.5

X Fig. 30. Daily efficiencies

( UL(T'-T"}'~m"~=,z } r/d versus X = - ~r-~] h F,,p

on a 45 °S tilted surface, for all months and both stations. The curve is the general best fit obtained for horizontal surfaces. The curve fits well values for qa > 30 per cent. The analytical form of qa depends

on the collector tilt.


As shown in Figs 27 and 28, radiation shape depends on the collector inclination. For this reason we expect that the explicit form o feqn (6) is different for the various tilts.

In Fig. 30 we show the points of Genoa and Macerata corresponding to a tilt angle of 45 °S, compared with the fit (eqn (7)) obtained for horizontal surfaces. This result shows that the analytical form o feqn (6) varies with the collector inclination.

CONCLUSIONS

The method suggested in this paper shows that a simple algorithm can be used to evaluate the daily efficiencies of solar flat-plate collector systems working at a constant flow rate, when their structural features, mode of operat ion and daily global radiation are known.

By means o f this result it is possible to calculate statistically long-term performances o f solar collectors starting from the monthly values o f the global radiation as shown by eqn (2).

REFERENCES

1. B. BARTOLI, S. CATALANOTTI, V. CUOMO, M. FRANCESCA, C. SER10, V. SILVESTRINI and G. TROISE, Statistical correlation between daily and monthly averages of solar radiation data, Nuovo Cimento, 2C (1979), p. 222.

2. G. AMBROSONE, A. ANDRETTA, F. BLOISI, S. CATALANOTTI, V. CUOMO, V. SILVESTRINI and L. VICARI, Long term performances of solar collectors, Applied Energy, 7 (1-3) (1980), pp. 119-28.

3. A. ANDRETTA, G. BARONE, P. BRUNINI, V. CUOMO, M. FRANCESCA, P. MATTARELLI and C. SERIO, Check of a computer program to calculate long-term performance of solar flat plate collectors, Applied Energy, 7 (1980), pp. 93-108.

4. J. A. DUFFLE and W. A. BECKMAN, Solar Energy Thermal Processes, John Wiley & Sons, 1974. 5. B.Y.H. LIU and R. C. JORDAN, A rational procedure for predicting long-term average performance of

flat-plate solar energy collectors, Solar Energy, 7 (1963), p. 53. 6. Meteorological data for Macerata measured by Osservatorio Geofisico di Macerata, Macerata, Italy. 7. Meteorological data for Genoa measured by Istituto Geofisico e Geodetico dell Universit~ di Genova,

Genova, Via Balbi 30, Italy.

Documents

Daily efficiency of solar collectors