The dependence on solar elevation of the correlation between monthly average hourly diffuse and global radiation

Solar Energy Vol. 41. No 4. pp. 335-340. 1988 0038-092X/SS $3.00 + ,00 Pnnted in the U.S.A. Copyright ~ 1988 Pergamon Press plc

THE DEPENDENCE ON SOLAR ELEVATION OF THE CORRELATION BETWEEN MONTHLY AVERAGE HOURLY

DIFFUSE AND GLOBAL RADIATION

A L F O N S O S O L E R *

Departamento de Fisica, Escuela Trcnica Superior de Arquitectura, Avenida Juan de Herrera s/n, 28040 Madrid, Spain

Abstract-- ln the presentwork, the dependence on 5t of the correlation between /~ = [u/fo and /~, = [//o is studied./, Iv, and I0 respectively being the monthly average hourly values of the global, diffuse. and extraterrestrial radiation, all of them on a horizontal surface, and "y the solar elevation at midhour. The dependence is studied for Uccle for the following sky conditions. Condition A: clear skies (fraction of possible sunshine = 1) and the maximum values of direct radiation measured during the period considered (each of the hours before or after the solar noon for which radiation is received); Condition B: corresponding to all the values of radiation measured when the sunshine fraction is 1 during the period considered; Condition C: corresponding to all the data collected, independently of the state of the sky; Condition D: corresponding to overcast skies (/-= fa).

From the available values of f and [o (monthly average hourly direct radiation on a horizontal surface), values of/~d and/~, for 5 ° <-- g/-< 45* and A~/= 5 ° are calculated using Newton's divided difference interpolation formula. The interpolated Ka--K, values occupy three clearly different regions in the /~a-- /~, plot, one for each of the conditions A.B, and C. For Condition A and each value of ~ best linear fits with high correlation coefficients are obtained for the/~a-/~, correlation. The influence of the Linke turbidity factor on the correlation for Condition A is studied for 5 ° -< ",/ -< 35 °, A'Tt = 5 °.

1. INTRODUCTION

The purpose of this work is to study the dependence on solar elevation of the correlation between /~d = fa//0 and/( , = l i fo where In , / ' and 10 are the monthly average hourly diffuse, global, and extraterrestrial radiation, respectively, all of them on a horizontal surface. This study is of interest because apparently there are no published works where the dependence is studied for sets of data obtained for the following sky conditions. Condition A: Clear skies and minnimum turbidity,

as given by the Linke turbidity factor, corresponding to the maximum values of direct radiation measured when the sunshine fraction is 1 during the period considered (each of the hours before or after the solar noon for which radiation is received). Condition B; Clear skies and average turbidity, cor-

responding to all the values of radiation measured when the sunshine fraction is 1 during the period considered. Condition C: Corresponding to all the data collected

during the period considered, independently of the state of the sky. Condition D: Overcast skies, corresponding t o / a =

£

2. DATA USED

Values of !, foN (monthly average hourly direct normal radiation), and f~ for Uccle (50 ° 49 'N) have been published by Dogniaux[ 1,2], and correspond to the solar radiation measurements performed in the period 1951 to 1980. The absolute accuracy was al-

*ISES Spain member.

ways better than 5%. This was made possible from the early measurements due to a detailed study and important modifications of the available devices and the generalized use of electronic potentiometers. A detailed description has been given in [3]. The following are given in [1,2]. For Condition A: land/DN; for Condition B: I , / a , and IDN; for Condition C: I, In, and foN; for Condition D, f = /a. Values of Io, the monthly average hourly direct radiation on a horizontal surface have been obtained from [o = I~N sin ~t, "Y being the solar elevation angle at midhour. Val- ues of ~/ are given in [1]. For the purpose of comparison between results obtained for different conditions, values of i'd used in this work have been obtained from ld = I -- Io. All the values of [ and ft~N used in this work correspond to all the hours before the solar noon for which data are available. Sim- ilar results are obtained for the hours after the solar noon. Values of ,~ have been calculated using the expression given in [4] with the characteristic declinations given in [I].

3. RESULTS FOR THE/~t--/~t CORRELATION

Using Newton ' s divided difference interpolation formula, values o f /~a and /{t have been calculated for the hours centered at 5 ° ~ 3, -< 45* and A'y = 5 °. The following results are obtained:

3.1 Condition D For overcast skies/~d = /~, and 0.15 --</~, ---< 0.19.

3.2 Condition C Results obtained for this condition are given in

Table 1. ff best linear fits are calculated for each value

335

336

0

0

0

0 tFl

C

!

. %

%

%

0

0 111

%

v

0 t'q

0

,.-'4

~b ,.=4

0 1.0

l l l ~ ~ l I I I , , ' ' , I

I I I ~ ~ I I I , , ' , ' '

I I , , , , ' , "

I ~ ~ ~

' d d d d d d d d d d '

d d d d d d d d d d d d

0 0 0 ~ 0 0 0 ~ 0 ~ 0

o ~ Q e e e e e * o e

0 ~ 0 0 0 0 0 0 ~ 0 ~ 0

I I I ~ I I I , 0 ,

I I I ~ I I I , = d d

' ' d d d d

-,I" ~ I I • • I I

~2 l-a

t'N u-3 ~.D

• • |

o d c ~

~=~ ~" ~',~ I

I 1 ~ ~ ¢ ~ 1

I ~ ~ ~ 1

' d d d d d d d d d

J # e e o e e e e ~ o

I O Q O t U Q I Q g e O

e o o o e o * e o . e ,

# o o o o Q o •

~ Q O O O O O O ~ O O Q

ALFONSO SOLER

of "y, positive slopes are obtained for 5 ° -< ~ -< 45 °. For -~ = 5 ° the slope is 0 .55 with a correlation coef- ficient r = 0.85. The s lopes diminish when "y increases, and remain < 0 . 1 for -¢ >- 35 °. However, values of r are very low for high values o f ~. Values of g , are in the range 0 .20 --< K, --< 0 .46 Unpublished results for Madrid (1979 -1983) show that the slopes diminish with increasing values of ~/as for Uccle, but become negative for ~/ >- 30 °.

3.3 Condition B Results for this condition are not usually available

for locations where solar radiation measurements are performed because only the daily (not the hourly) sunshine fraction is reported.

Results obtained for this condition are given in Table 2. If best linear fits are calculated, for "y = 5 °, the slope is 0.71 with r = 0 .91, and the slopes diminish when ~ increases. For ~ = 45 °, a zero slope is obtained. However, as for Condition C, values of r are very low for high values of ~/. Values of K, lie in the range 0.41 -< g , -< 0.70.

3.4 Condition A

For this condition, corresponding to max imum monthly average hourly values of direct radiation for clear skies, each of the values of [ and #'ON given in [1] are obtained as an average of 30 values, one for each of the 30 years studied. Thus, taking into account the small number of values available to obtain the averages, it is not clear that the values of ~ to be used to obtain the correlations are the ones corresponding to the characteristic declinations. Values of K, and l~a obtained using values of #'0 corresponding to the characteristic declinations, and the best linear fits for 5 ° -< ~/--< 45 ° are plotted in Fig. 1. Values of K, are in the range 0 .48 -</~, -< 0.84. In Table 3, values o f the empirical constants and the correlation coefficients are given for the linear correlations:

ga = a + bK, (1)

Positive values o f b are obtained for each value of ~, and the values of b are in the range 0.81 -< b - 0 .50.

In Fig. 2, values of Ka and/~t for Conditions A ,B , and C and 5 ° - .T/ _< 45 o have been plotted for the purpose of comparison. The Ka-K, values occupy three clearly different regions in the figure, one for each condition.

The influence o f the monthly average hourly Linke turbidity factor i# on the correlations for this condition has been studied. Values of ~F for Uccle are given in[2]. From the given values of 7 ~, values of 7 ~ were obtained for the hours centered at 5 ° -< ~/ -< 45 °, z~T/ = 5 ° using Newton's divided difference interpolation formula. Figure 3 shows the Kd"-Kt colTe- lations for ~ = 5 ° and 10 °. Values o f ~F have been grouped in different ranges, from 3.1 --< T - 3.3 to 2.2 -< 7 ~ -< 2 .4 . Best linear fits are giyen for the g a - g , values corresponding to each range of values of

.£

e- o

[-

0

0

~- - 0

,,~%

m

g

t '4

I " 0 I,t'l

,-=1

0 I.r'l

I I 1 ~ I I 1 .

Diffuse and global radiation

~ I I I

d d d d l l I

t"3 I I I ~ I I I •

I I * . -

' d d d d

• • I ! d d ~

• o . . I 1

d d d d d I

d d d d d '

p . . . t . ~ t ~ , c ~ k O t , , c ' l t . ~ t . r l t , ~ I ¢"q C'4 t'~. P,.I t '~ t",4 P.4 C',4 C'q ¢',4 I

I d d d d d d d d d d I




I I 1 ~ ~ 1 1 1 1 I I I d d ~ ~ d I I I I

I I I ~ ~ I

I I ~ ~ I I I . . ' ' ' * ' I

' ' d d d d d d d d ' '

I d d d d d d d d d ' ~

I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ I

' d d d d d d d d d d '

* e e e l e e q e e e e

l e o l l o e o d d = = = d d = = = = =


337

/~. For each value of 7, the smaller the values of 7 ~, the lower lie the best fits in Fig. 3. Figure 4 shows the/~a--/~t correlation for ~/ = 15 °, 20 °, 25 °, 35 °, and 45 °. Values of 7 ~ have been grouped in different ranges from 4.0 - 7 ~ -< 4.2 to 2.5 -.<- 7 ~ -< 2.7. As for 5 ° and 10 °, for each value of ~t best linear fits are given for the Ka--K, values corresponding to each range of values of 7 ~. For each value o f ~/, the smaller the values of T, the lower lie the fits in Fig. 4. Values for the empirical constants and corresponding correlation coefficients for 5 ° -< 3' -< 25 ° are given in Table 4.

4 . C O M M E N T S O N T H E K c . K , C O R R E L A T I O N

In [5], three regions are distinguished by Iqbal in Kf.-Kt corelations after paying attention to their shape. The fol lowing paragraph fol lows Iqbal's description for 10 ° -< 3' -< 40 ° , but it applies to 10 ° -< 3' < 60 ° as similar shapes are found for 40 ° -< 7 - < 60°[6-7] .

1. For overcast skies, Ka increases linearly with Kt. 2. The region immediately fol lowing is considered

as corresponding to "partly cloudy skies" conditions. In the beginning, there is an increase o f Ka with the increase in K,, and then Kd begins to decrease as the skies become clearer. At particular solar altitudes, a min imum of the diffuse component is reached. 3. The region after the min imum is considered as

corresponding to "mainly clear sky" conditions; in this region there is an increase o f Ka with K,, lower elevations giving a higher percentage of diffuse radiation.

In [6 -7 ] , a dependence of K,f-Kt correlations on solar elevation has also been noted. This dependence is very clear for the third region mentioned in [5]. The dependence with solar elevation in this region is similar to the dependence shown for the Kf-K, correlation and Condition A. As a consequence, the present results for the Kf-Kt correlation suggest that values for the K,f.-K, correlation in the "mainly clear sky region" mentioned in[5] may well correspond to "very clear sky" conditions (Condition A).

For Uccle , /~f - /~ , correlations for Conditions B and C show little overlap (Fig. 2). For Condition B (clear skies, average turbidity), comparison of the / f f - /~ , correlation with the "partly cloudy sky" region mentioned in [5] suggests that for this location, values for the Kf-Kt correlation in the mentioned region may wel l correspond to clear skies.

From the preceding remarks, i f a qualitative un- derstanding of Ka-Kt correlations as a function of cloudiness is o f interest, an improvement of the description given in[5] is necessary. This implies the knowledge of, for example, the hourly sunshine fraction v. A study of the Ka-K, correlation for v = 1 and different values o f ~/could possibly be performed for different values of T or the widely used Angstr6m's turbidity factor 13, from min imum to max imum turbidity conditions. Correlations f o r 0 -< v -<- 0.1;0.1 --- v --< 0.2 etc. could also be performed.

Unfortunately, when 13 is used, other factors such as the atmospheric precipitable water w must be taken

338 ALFONSO SOLER

05

O.&

R d 03

02

01

O0

• 5"

o 10"

• 15"

o 20"

• 25" & 30 "

+ 35" x z,O'

~ r ~ 0 ° i r

.p/:

CONDITION A

UCCLE (1951 -1980 )

I l I I i I I I __

01 02 03 04 05 06 07 08 09 Rt

Fig. 1. The/(,,-/(, correlation for Condition A.

Table 3. Constants and correlation coefficients for correlations performed for Condition A.

a b r

5 ° - 0 . 0 7 7 O. 8 1 3 O. 9 7 6

10 ° - 0 . 2 3 0 O. 8 5 9 O. 9 5 0 15 ° - 0 . 2 7 7 0 . 7 9 8 O. 9 3 1

20 ° - 0 . 183 0 . 5 9 9 0.938 2 5 ° - 0 . 1 9 4 0 . 5 6 7 O. 8 9 3 50 ° - 0 . 2 2 8 O. 5 7 6 O. 9 2 2

35 ° - 0 . 261 0.595 O. g18 40 ° - 0 . 3 2 4 O. 6 6 4 O. 9 2 1 45 ° -O. 210 O. 500 O. 863

into account to perform a detailed study of the Ka- K, correlation. In [6,7], the Ka-Ks correlation has been studied for given seasonal values of 13 and w, but the dependence with cloudiness has not been taken into account. A more detailed study in the sense just mentioned could be useful even if performed for just a few locations.

5. CONCLUSIONS

The dependence of K,r-K, correlations on solar elevation has been studied for Uccle for different sky conditions and 5 o <: ~ - 45 °, A~/ = 5 °.

The /(a---/(, values occupy three clearly different

3.6 7%

A'A 7% A

]ONDITION B - • 7% '~

05 C - o ,,

z~ A

7% 7% 04 ,~ "" "

zx Kd •

7% • zkO A

• 7% 7x A

Q2 o °° 0 7 % 7 % ~

0.1 0'2 03 0~, 0'5 0'6 0'7 0'8 I~ t

Fig. 2. The/£a-/(, correlation for Conditions A,B, and C.

09

Diffuse and global radiation 339

0.7

0.6

0.5

Rd

CONDITION A 0 / •

• 2s27 / /

o / °

/ o i i

05 O@ 07 S@ 09 Rt

Fig. 3. The dependence on T of the /(a--K, correlation for ~ = 5 °, 10 °. and Condition A.

04

Table 4. Constants and correlation coefficients for correlations performed for Condition A and different

ranges of values of 7 ~

5 ° 2.5-2.? -0.i90 0.982 0.983

2.8-3.0 -0.242 1.048 0.992

3.1-3.3 -0.22I i. I25 1.000

I0 o 2 . 5 - 2 . 7 -0.531 1.250 1.000

2.8-3.0 -0.298 0.952 0.974

3.1-3.3 -0.260 0,931 0.999

15 ° 2 . 8 - 3 . 0 - 0 . 4 2 1 0.990 0.991

3.1-3.3 -0.270 0.803 0.990

20 ° 2.8-3.0 -0.620 1.166 0.971

3. i-3.3 -0.272 0,716 0.999

3.4-3.6 -0.350 0.849 1.000

25 ° 2.8-3.0 -0.540 1.000 1.000

3.1-3.3 -0.427 0.879 0.988

3.4-3.6 -0.260 0.667 0,918

0.3

Rd

02

CONDITION A o ~ A 40 - 42

• 37 - 39 /

/ T o 3, - 36 • 31 - 33 o 28 - 30

• 25-27 , ' / / x ~

/ : / / / , . • / • so .m o / o

O• ° /

, , i

0~ 07 08

Rt

Fig. 4. The dependence on 7 ~ of the Ka.--K, correlation for 15 ° -< ~/ - 45 ° and Condition A.

340 ALFONSO SOLER

regions in the Ka-K, plot, one for each of Condi t ions A, B, and C. For Condi t ion A and each value of ~/, best l inear fits with high correlat ion coefficients are obtained. The influence of the monthly average hourly Linke turbidity factor 7 = on the correlat ion for these condi t ions has been studied. For each value of ~/, best linear fits with high correlation coefficients have been obtained for different ranges of values of 7 = , and for each value of ~/, the smaller the values of 7= in a range, the lower lie the best fits in the Ka-K, plot.

The present results quest ion an avai lable interpre- tation of the different regions observed in Ka-K, correlat ions.

REFERENCES

11 R. Dogniaux, Eclairement energetique solaire direct, diffus et al., global des surfaces orient~es et inclin~es,

I pattie. Miscellanea Sdrie B n ° 59. lnstitut Royal MdtEorologique de Belgique. Bruxelles (1984).

2. R. Dogniaux, De l'influence de l'estimation du facteur total de trouble atmosph~rique sur !'evaluation du ray- onment solaire direct par ciel cl~iir. Miscellanea SSrie C. n ° 20. lnstitut Royal MEt6rologique de Belgique. Bruxelles (1984).

3. R. Dogniaux and R. Pastiels, Techniques modernes de mesure de l'6clairement 6nergetique solaire. Institut Royal Mdt6orologique de Belgique. Publications serie B, n ° 16, Bruxelles (1955).

4. M. Iqbal, An Introduction to Solar Radiation. Academic Press, New York (1983).

5. M. lqbal. Prediction of hourly diffuse solar radiation from measured hourly global radiation on a horizontal surface. Solar Energy 24, 491-503 (1980).

6. J.D. Garrison. A study of solar irradiation for six cities. Solar Energy 32, 237-250 (1984).

7. J.D. Garrison, A study of the division of global irradiance into direct and diffuse irradiance at thirty three U.S. sites. Solar Energy 35, 341-351 (1985)

Documents

The dependence on solar elevation of the correlation between monthly average hourly diffuse and global radiation