Solar Energy Vol. 29, No. 2, pp. 137-140, 1982 003g-092X/821050137-04503.00/0 Printed in Great Britain. © 19~2 Pergamon Press Ltd.

U T I L I Z A T I O N O F L U N D E ' S C O M P U T I N G P R O C E D U R E S

W I T H O U T E X T E N S I V E M E T E O R O L O G I C A L D A T A

RENATO M. LAZZARIN Istituto di Fisica Tecnica, Universit~ di Padova, Padova, Italy

(Received 26 June 1981; revision accepted 13 November 1981)

Abaraet--A group of useful computing procedures was proposed by Lunde to predict the performance of solar heating systems. The computations require an analysis of extensive meteorological data. In this note an estimation procedure is suggested to evaluated the parameters needed in Lunde's methods starting only from monthly average irradiation and hours of sunshine. The procedure was tested against extensive meteorological data in different seasons for the climate of Padova with acceptable results.

l . INTRODUCTION

The prediction of monthly and annual performance is crucial in sizing solar plants. It is often obtained through f-Chart for solar heating systems[I,2]. Only monthly horizontal irradiation and diurnal temperatures are needed. Unfortunately f-Chart computations are suitable for a very diffuse, but particular, plant lay-out and for a minimum useful temperature of 20°C. Frequently the minimum useful temperature is higher; another proce- dure is available and well known: the 4)-f Chart[3]. Nevertheless particular systems such as those ones with annual storage cannot be considered by this method.

Another interesting approach was attempted: this generated a group of very useful computing procedure by which it is possible to predict the performance of solar heating systems at different temperatures and storage capacities as well as of systems equipped with annual storage. The procedures are due to Lunde[4-7] therefore in the next they will be indicated as Lunde's ones.

Briefly Lunde considers daily useful solar heat 4, as the difference of absorbed ~, and lost heat ~L:

4, = 4o - #L (1)

The daily absorbed heat is:

Ch = FR fo'T(m)la dt = Fe(-~a)[a+ tr. (2)

In order to know tT and fo+, it is useful to consider the intensity threshold for solar collection at the average inlet temperature Tl:

ic = L ) (s) (ra)

An analysis of extensive meteorological data is needed to find out for every month in a given locality the number of hours when the radiation is above the threshold and the average intensity in those hours. In fact it is neces- sary to build up the cumulative curves for radiation. As regards ambient temperature it is often enough to con- sider only the monthly average.

This is a disappointing feature of Lunde's procedures because:

(a) It is necessary to process the meteorological data for every locality; as the author suggested, this is not a big obstacle, as it is done only once;

0a) It is necessary to possess extensive meteorological data; this limits the procedures to few localities (today in Italy 2-3 only).

In this note an estimation procedure is suggested for the two parameters tr and fa+ which requires only monthly average irradiation and hours of sunshine [8]. It has been tested with extensive meteorological data measured in Padova by the meterological station of National Research Council.

whereas the daily lost heat is:

fo 'T qL= FRUL (Ti- T~)dt= FRUL(~- T,)tT (3)

la + is the average solar radiation intensity when the collector works:

_ [ ' r 1~ dt i + = ' ° (4)

tr

tr is the daily working period of the collector. The two parameters are requited to predict the per-

formance of a plant over a set of storage capacities.

2. PAIiAMETF._.iIS F.STIMATE

The average monthly utilizability ~ is defined by[9]:

N nd

N/Ja (6)

As usual the plus sign indicates that the summation is extended over all hours in which (la - Ic) is positive and a well controlled plant is therefore working.

The summation o~y be written as:

N nd N nd N nd

X y. (1a- y. X i ; -X 2 i; (7)

137

138 R. M. LAZZAPdN

If the period tr when the system is on were known:

N nd N nd

la + would then be obtainable by (6):

fF=I,++A~t,

It easy to see that the prediction is quite good for/:~+ with the only exception of February for the higher critical intensities: this is due to the fact that the period

(8) of working is very short owing to the unfavourable climate conditions. In so extreme conditions the dis- crepancy between generalized and cumulative curves is very large, but not so important for the computation of

(9) monthly collected heat.

The evaluation is achieved by Liu Jordan generalized cumulative curves[10] (Fig. 1): given the cloudiness in- dex the time fraction fc is determined for which the insolation is below HJ/~B:

H_c = n / c (10)

where n is the number of sunshine hours. The period tr is then:

tr = n(1 -f~). (11)

3. TESTING OF THE PROCEDURE The estimation procedure was tested against extensive

meteorological data for three characteristic months in winter, midseason and summer (February, April, June). Radiation data were recorded for horizontal and 45 ° tilted surfaces in Padova. The comparison was developed for various critical intensities ranging from 750 to 2000kJm-2h - ' , that is from a minimum inlet temperature in a simple liquid flat plate collector of about 35°C in winter and a maximum of 90°C in summer.

The results are summarized for the three months in Fig. 2 for horizontal and in Fig. 3 for 45 ° tilted surface as regards tr and in Figs. 4 and 5 as regards [~+. The experimental points are also indicated.

2.8 [ I I I I I ~ ~ L I

2.4

H ==r H

2.0

1.6

1,2

O,8

0.4

O 0 0.2 0.4 0,6 0,8 1.O F r i c t i o n a l t i m e f d u r i n g w h i c h r a d . ~ H

Fig. 1. L iu J'ordan generalized radiation distr ibution curves for a horizontal surface.

tT 12 (h) 11 HORIZONTAL SURFACE

10t " ~ ,.,..

'~ b,,,, O I 7 - -

= 6 5 1 P . .

2 1.~.

o 750 1000 1250 1500 1750 2000

[c (kJ rn'2 h'1)

Fig. 2. Comparison between estimated and measured tT parameter for a horizontal surface in the three different months

as a function of critical intensity.

11 tx g t t45° TILTE0 SURFACE (h) 10 ~,~ . . . .

7' '_ 6 , , . . - ' . , , , ; . .

I

750 1000 1250 1500 1750 2000 Ic (kJm'Zh "~}

Fig. 3. Comparison between estimated and measured t T

parameter for a 45 ° tilted surface in the three different months as a function of critical intensity.

4O00 I HORIZONTAL ISUNFAC[

-- 3000 ~ .'

. . , ~ ~.'Ip r IL , .-

1ooo/~ ~''~i~: -

0 1750 201~0 750 1000 1250 1500 lc (kJrn'zh '')

Fig. 4. Comparison between estimated and measured ~+ parameter for a horizontal surface in the three different months

as a function of crit ical intensity.

Utilization of Lunde's computing procedures 139

4000 [ &5' TILTED SURFACE I

-- 3000 --

2ooo ! . . . q ~ , " . r ~ p ~ t '

,ooo I i 750 I000 } 2000

I c (kJm'Zh "I)

Fig. 5. Comparison between estimated and measured f~+ parameter for a 450 tilted surface in the three different months as

a function of critical intensity.

The evaluation of tr is less precise. The estimate is almost always too large by 10-20 per cent. The best evaluations are for horizontal surface; the computations for tilted surfaces are affected by the introduced ap- proximations owing to further hypothesis used, such as the utilization of Liu Jordan distribution curves for horizontal surface on tilted surfaces.

It may be concluded that the errors in the procedure with estimated parameters are within 10-20 per cent, acceptable for many applications, considered that only monthly data were requested.

However it is useful to look at a full application of one of Lunde's procedures such as ambient heating at a minimum temperature of 50°C during February and April (45 ° tilted surface) and process heating at 60% during February, April, June (horizontal surface) for a specific monthly load of 0.2 x 109 J m -2 month -m. The solar col- lectors are simple single glazed flat plate Fe(~a), = 0.85, FeUL = 7.5 W m -2 K- ' , specific storage 100 kg m -2. The results are reported in Table 1; for the meaning of ~t and

see [7]. The solar fraction prediction is very good with an

absolute error less than 0.09 for 45 ° tilted surfaces and 0.05 for horizontal ones; this error leads to bad predic- tions for low solar fractions (February horizontal, even if the estimates of ~ and .~ are particularly good). In some cases the computation is very precise.

monthly meteorological values and allows the estimation of working period tr and average insolation in the meanwhile as a function of critical intensity. The two parameters are necessary for the utilization of Lunde's procedures.

The estimates were tested against extensive meteorological data for the climate of Padova with ac- ceptable results. This suggests to extend the tests to other different climates in order to utilize a series of very suitable computing procedures not only in localities with continuous data recording, but also in the many other ones with monthly averages.

Example Estimate tT and/~+ for a horizontal surface in April in

Padova for a critical intensity Ic = 1000 kJ m -2 h -m (From analysis of extensive meteorological data one obtains tT =7.7h and 1=8 + = 1881 kJ m-2 h-t).

One knows: Monthly average daily isolation H = 16343 kJ m -2 day -m Cloudness index /~h = 0.495 Sunset angle a,s = 99.7 ° Number of hours of sunshine n = 10.3 h.

Ratio of the diffuse radiation at noon to the daily diffuse radiation:

• " 1 - cos (as = 0.120. rd., = 24 . ¢r

s in ¢as - l - ~ a)s COS ~os

Ratio of the radiation at noon to the daily total radiation:

rh., = (1.07 + 0.025 sin (w, - 60))ra., = 0.130.

Critical ratio Xc:

L X~ = = 0.470. rh.,/-~

Coefficients A, B, C ([9] Table 1):

4. CONCLUSIONS

An estimation procedure which allows to use Lunde's computing methods without extensive meteorological data has been proposed. The procedure requires only

A = 2.943 -9.271/~h +4.031 /~h z = --0.658

B = - 4.345 + 8.853/~h - 3.602/~h 2 = -- 0.845

C = - 0.170- 0.306/~h + 2.936/~h 2 = 0.398.

Table l. Comparison between predictions obtained with estimated and measured parameters. Ambient heatin8 in February and April at a minimum temperature of 50°C (45 ° tilted collectors); process heating in February, April, June at a minimum temperature of 60°C (horizontal collectors). The load is uniformly at 0.2 x 109 J m -2 month-m;

= 0.061 h -m

A M B I E N T H E A T I N G P R O C E S S H E A T I N G

e s t i m a t e d m e a s u r e d e s t i m a t e d m e a s u r e d

39 5£ f J~ S# f ,99 5£ f ,99 5£ f

FEBRUARY

APRIL

JUNE

1.533 O.90B 0.406 1.225 0.733 0.410 0.56g 0.486 O.07g 0.507 0.473 0.033

2.060 1.201 0.631 1.726 0.944 0.501 1. 786 1.182 0.476 1.655 1.10(J 0.43g

. . . . . 2.782 1.348 0.866 2.645 1L266 0.850

140

Average daily collector utilizability:

= exp ((A + B(RdR))(Xc + CX~2)) = 0.432.

Rn is the ratio of radiation on a tilted surface to that on a horizontal one at noon and R is the ratio of the monthly average daily radiation on a tilted surface to that on a horizontal one. In this example R~ and R are, of course, both equal to one. Ratio of daily critical radiation to daily insolation:

/-L n/c . . . . 0.629. t~ n

n is the number of sunshine hours. From Fig. 1 one obtains fc = 0.240; from (11):

tT = n(1 - f c ) =7.83 h

and from (9):

I~+ = Ic +-~-~ = 1000+0.432 × 1634317.83

= 1903 El m-2 h -I.

A,B,C F~

/

f~ H

rt, lc

gh n

nd

NOMENCLATURE

coefficients collector overall heat removal efficiency factor proportion of monthly heating demand met by solar

energy in a particular month fractional time corresponding to critical intensity ratio monthly average daily total radiation on a horizontal

surface monthly average daily total radiation on the collector

surface critical level solar radiation intensity on the collector surface average solar radiation intensity on the collector sur-

face during h- cloudness index number of hours of sunshine number of hours from sunrise to sunset

R. M. LAZZARIN

N rd, n

rh, n

Rn

r, T, tT

x~ ( 2

tb

number of days in a month ratio of the diffuse radiation at noon to the daily diffuse

radiation ratio of the radiation at noon to the daily total radiation ratio of radiation on a tilted surface to that on a

horizontal surface at noon ratio of monthly average daily total radiation on a tilted

surface to that on a horizontal surface ambient temperature collector fluid inlet temperature total time of collector operation collector overall energy loss coefficient dimensionless critical level transmittance absorptance product average daily collector utilizability sunset hour angle correlation coefficients

REIrERENO~ 1. S. A. Klein, W. A. Beckman and J. A. Duffie, A design

procedure for solar heating systems. Solar Energy IS, i13- 127 0976).

2. S. A. Klein, W. A. Beckman and J. A. Dute, A design procedure for solar air heating systems. Solar Energy 19, 509--512 (1977).

3. S. A. Klein and W. A. Beckman, A general design method for closed-loop solar energy systems. Solar Energy 22, 269-282 (1979).

4. P. J. Lunde, Prediction of average collector efficiency from climatic data. Solar Energy 19, 685-689 (1977).

5. P. J. Lunde, Prediction of the monthly and annual per- formance of solar heating systems, Solar Energy 20, 283-287 (1977).

6. P. J. Lunde, Prediction of the performance of solar heating systems utilizing annual storage. Solar Energy 22, 69-75 (1979).

7. P. J. Lunde, Prediction of the performance of solar heating systems over a range of storage capacities. Solar Energy 23, 115-121 (1979).

8. R. M. Lazzarin, Sistemi solari attivi: manuale di calcolo (Active solar systems: computing handbook). Chap. il. Muzzio, Padova (1981).

9. S. A. Klein, Calculation of fiat-plate collector utilizability. Solar Energy 21,393-402 (1978).

10. B. Y. Liu and R. C. Jordan, A rational procedure fur predict- ing the long-term average performance of flat-plate solar energy collectors. Solar Energy 7, 53-74 (1963).