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Solar Energy Vol. 24, pp. 491-503 Pergamon Press Ltd., 1980. Printed in Great Britain PREDICTION OF HOURLY DIFFUSE SOLAR RADIATION FROM MEASURED HOURLY GLOBAL RADIATION ON A HORIZONTAL SURFACE M. IQBALt ~ Department of Mechanical Engineering, University of British Columbia, Vancouver B.C., Canada (Received 27 June 1979; accepted 3 December 1979) Abslraet--A statistical procedure has been employed to develop correlations between the hourly global horizontal radiation and its diffuse component. Several years', hourly radiation data from three Can- adian stations and two French stations have been employed for this purpose. The relationships have been developed in dimensionless form which predict IJlo for particular solar altitudes when I/Io is given. Under heavily cloudy conditions or when the sky is completely covered if~Iv < 0.35), diffuse radiation increases linearly with the global radiation. In this region, solar altitude has no bearing on the fraction of diffuse radiation. As I/Io goes beyond 0.35, the effect of solar altitude begins to appear and the region immediately following this may be considered as partly-cloudy-skies conditions. In the beginning of this region, the diffuse component increases briefly with the increase in global radiation and then begins to decrease as the partly cloudy skies become clearer. At particular solar altitudes, a minimum value of the diffuse radiation is reached. The value of I/Io where Id/lo reaches its minimum value varies with solar altitude. The region beyond which a minimum value of Id/lo is reached may be considered as mainly-clear-sky conditions. In this region, Id/lo increases again with I/1o, lower solar altitudes giving a higher percentage of diffuse radiation. Under partly cloudy skies and under clear skies, solar altitudes lower than 30° had a marked effect on the fraction of diffuse radiation. However, solar altitudes greater than 30° had minimal influence on the fraction of diffuse radiation. INTRODUCTION In order to design any solar energy system or study of the potential of solar energy in a region, information on the availability of solar radiation is required. Under clear-sky conditions, solar radiation on the earth's surface can generally be computed in purely theoretical terms. In cloudy or partly cloudy regions, however, long-term averages of the global (direct plus sky-diffuse) irradiation on a horizontal surface can be estimated through climatic parameters, such as the number of hours of bright sunshine [1, 2]. The coeffi- cients in the correlations linking insolation with the number of hours of bright sunshine are themselves based on measured values of global solar radiation. Therefore, it is indispensable to have access to measured values of radiation. In many parts of the world, daily or hourly values of insolation on hori- zontal surfaces are regularly recorded. These measure- ments are generally of global radiation. In order to compute insolation on inclined planes, it is necessary to estimate the diffuse and the direct components of the horizontal radiation. Methods of predicting dif- fuse (and hence direct) components of the measured "t'This study was carried out partly at the Ecote Polytech- nique, Universit6 de Montr6al, while the author was on sabbatical leave. :~Professor. global radiation on a horizontal surface will now be discussed. While there are many stations recording global radiation, those recording diffuse radiation are very few. The basic procedure is to develop correlations between the global radiation and its diffuse com- ponent using measured values of these two quantities, and then to apply such correlations at locations where diffuse radiation data are not available. The quantities which are generally correlated can be divided into the following four groups: 1. Correlations between the daily global radiation H and its diffuse component H d. 2. Correlations between the monthly mean daily global radiation H and its diffuse component Ha. 3. Correlations between the monthly mean hourly global radiation f and its diffuse component id. 4. Correlations between the hourly global radiation I and its diffuse component ld. In order to better explain the purpose of the present study, it is necessary first to review the litera- ture dealing with the above mentioned four groups. 1. Correlations between H and H d Prediction of the daily diffuse radiation when the daily global radiation for a particular day is given was 491

Prediction of hourly diffuse solar radiation from measured hourly global radiation on a horizontal surface

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Solar Energy Vol. 24, pp. 491-503 Pergamon Press Ltd., 1980. Printed in Great Britain

PREDICTION OF HOURLY DIFFUSE SOLAR RADIATION FROM MEASURED HOURLY GLOBAL

RADIATION ON A HORIZONTAL SURFACE

M. IQBALt ~

Department of Mechanical Engineering, University of British Columbia, Vancouver B.C., Canada

(Received 27 June 1979; accepted 3 December 1979)

Abslraet--A statistical procedure has been employed to develop correlations between the hourly global horizontal radiation and its diffuse component. Several years', hourly radiation data from three Can- adian stations and two French stations have been employed for this purpose. The relationships have been developed in dimensionless form which predict IJlo for particular solar altitudes when I/Io is given.

Under heavily cloudy conditions or when the sky is completely covered if~Iv < 0.35), diffuse radiation increases linearly with the global radiation. In this region, solar altitude has no bearing on the fraction of diffuse radiation.

As I/Io goes beyond 0.35, the effect of solar altitude begins to appear and the region immediately following this may be considered as partly-cloudy-skies conditions. In the beginning of this region, the diffuse component increases briefly with the increase in global radiation and then begins to decrease as the partly cloudy skies become clearer. At particular solar altitudes, a minimum value of the diffuse radiation is reached. The value of I/Io where Id/lo reaches its minimum value varies with solar altitude.

The region beyond which a minimum value of Id/lo is reached may be considered as mainly-clear-sky conditions. In this region, Id/lo increases again with I/1o, lower solar altitudes giving a higher percentage of diffuse radiation.

Under partly cloudy skies and under clear skies, solar altitudes lower than 30 ° had a marked effect on the fraction of diffuse radiation. However, solar altitudes greater than 30 ° had minimal influence on the fraction of diffuse radiation.

INTRODUCTION

In order to design any solar energy system or study of the potential of solar energy in a region, information on the availability of solar radiation is required. Under clear-sky conditions, solar radiation on the earth's surface can generally be computed in purely theoretical terms. In cloudy or partly cloudy regions, however, long-term averages of the global (direct plus sky-diffuse) irradiation on a horizontal surface can be estimated through climatic parameters, such as the number of hours of bright sunshine [1, 2]. The coeffi- cients in the correlations linking insolation with the number of hours of bright sunshine are themselves based on measured values of global solar radiation. Therefore, it is indispensable to have access to measured values of radiation. In many parts of the world, daily or hourly values of insolation on hori- zontal surfaces are regularly recorded. These measure- ments are generally of global radiation. In order to compute insolation on inclined planes, it is necessary to estimate the diffuse and the direct components of the horizontal radiation. Methods of predicting dif- fuse (and hence direct) components of the measured

"t'This study was carried out partly at the Ecote Polytech- nique, Universit6 de Montr6al, while the author was on sabbatical leave.

:~Professor.

global radiation on a horizontal surface will now be discussed.

While there are many stations recording global radiation, those recording diffuse radiation are very few. The basic procedure is to develop correlations between the global radiation and its diffuse com- ponent using measured values of these two quantities, and then to apply such correlations at locations where diffuse radiation data are not available. The quantities which are generally correlated can be divided into the following four groups:

1. Correlations between the daily global radiation H and its diffuse component H d.

2. Correlations between the monthly mean daily global radiation H and its diffuse component Ha.

3. Correlations between the monthly mean hourly global radiation f and its diffuse component id.

4. Correlations between the hourly global radiation I and its diffuse component ld.

In order to better explain the purpose of the present study, it is necessary first to review the litera- ture dealing with the above mentioned four groups.

1. Correlations between H and H d

Prediction of the daily diffuse radiation when the daily global radiation for a particular day is given was

491

492 M. IQBAL

studied by Liu and Jordan [3]. They used data from one station in the U.S.A. and developed a correlation between Ha/H and the cloudiness index Kr (where Kr = H/Ho). Following Liu and Jordan's approach, Choudhury[4-1 and Stanhill[5], using data from single stations in India and Israel respectively, presented correlations between Ha/H and Kr. Estima- tions of [4, 5] give higher values of Ha compared to the one obtained from [3]. Ruth and Chant [6], using data from four stations in Canada (one within the arctic circle), have also concluded that Liu and Jor- dan's correlation predicts conservative values of the diffuse component. A number of reasons have been adduced for the lack of correspondence between Liu and Jordan's correlation and those reported in [4-6]. The main reason for the discrepancy appears to be due to the fact that the U.S.A. data were not corrected for the shade ring effect while those in [4-6] were. Ruth and Chant have indicated that latitude should also have an influence on these correlations. However, Buckius and King [7] have proven that it is not the latitude itself but higher average air-mass and also the higher albedos associated with northern latitudes which result in higher values of the diffuse com- ponent.

In [3-6], functional relationships have been devel- oped between Ha/H and H/Ho. Stanhill has shown that it is also possible to correlate Ha/H with the ratio of actual hours of bright sunshine to the maximum possible hours of bright sunshine.

From the point of view of many users of solar radi- ation data, values for particular days (of diffuse and beam radiation) are of little importance compared to the daily values based on long term averages. In fact many design problems are worked out using only long term averages of radiation, ambient temperature and wind velocity data. Predictions for the mean daily diffuse radiation are now presented.

the statistical approach of [3]. They have concluded that a universal correlation has yet to be developed.

Using the number of hours of bright sunshine, Iqbal [12, 13] has presented simple expressions whereby Ha can be predicted either from H or Ho. The advantage of using these correlations is that records of bright sunshine are widely available throughout the world.

3. Correlations between I and Id

Based on an analysis of multiple reflections of solar radiation between ground and cloudcover, Hay [14] developed a correlation between ia and i. Hay's cor- relation has, unfortunately, two disadvantages. It is sensitive to the regional albedo, a quantity for which one finds in general only a crude approximative value, and furthermore, measured values of i are not easily available. Iqbal [15, 16], in a study which reviewed in detail various correlations, has recommended the use of Liu and Jordan's equation

ia io (1) Ha Ho

in order to estimate la. In eqn (1), Ha in itself has to be predicted through one of the correlations refer- enced under group (2) above. In this way, given the values of H, which are widely available, ia can be easily estimated.

The foregoing references under groups (2) and (3) give the necessary elements that an architect or an engineer might need to estimate the monthly mean values of daily or hourly diffuse radiation. On the other hand, for research purposes and development of simulation methods, etc. diffuse and global radiation for particular hours are needed. This is considered below.

2. Correlations between H and FIa

In this group, radiation values for particular months, based on the averages of many years, would be treated. The terms "monthly mean daily" or simply "mean daily" would be employed to distinguish these averages from the "daily" radiation reviewed in group (I) above.

Liu and Jordan [3], employing a statistical method, developed a functional relationship between Hd/H and the index Kr, (Kr = H/Ho). They used diffuse radiation data from one station and global radiation data from several widely separated stations. Page [8], eoaploying regression analysis approach, presented a linear correlation between Hd/H and K'r. He used dif- fuse and global radiation data from a number of stations lying in the northern as well as the southern hemisphere. Liu and Jordan's correlation does not agree with that of Page. Klein and Duffle, using daily diffuse correlations of [4-5, 10, 11], have recalculated the relationships between Hd/H and Kr employing

4. Estimation of I a from I

Hourly global radiation on horizontal surfaces is now recorded at many stations in the industrialized world. These records are generally available on mag- netic tapes in machine-compatible form. While the hourly global radiation is recorded in many places, stations measuring hourly diffuse radiation are ex- tremely few. It is therefore necessary to develop methods of predicting the diffuse component of the hourly global radiation. Orgill and Hollands[17], Bugler [18], and Bruno [19] have made attempts in this direction. Orgill and Hollands, using data from one station in Canada, presented a correlation between Ia/l and I/Io. They correlated in fact simple averages of all I/Io values within a certain range and the averages of corresponding ld/l values, with the result that their correlation gives an unacceptable estimate of la, except in the cloudy-weather con- ditions. Also, Orgill and Holland's comparison of their correlation of hourly values with the corre-

Diffuse solar radiation 493

lations of 1,3, 6] based on daily values is not valid as pointed out by [ l l ] L

Bugler, employing data from one station in Austra- lia, presented correlations between l d l o and I/Io which are valid for various solar altitudes. He took account of the effect of solar altitude by plotting data separately for each 10 ° change in altitude and then obtaining a best fit curve for the data.

The purpose of the present report is to reinforce some salient points of Bugler's study and to bring out some new features of the correlation. In order to pro- vide a wider applicability of the results, data from three Canadian and two French stations have been employed. The procedure followed in this study and its results are given in the next section.

PROCEDURE AND RESULTS

Hourly radiation data (global and diffuse) from three stations in Canada and two in France (Table 1) were obtained on magnetic tapes in machine- compatible form. The Canadian diffuse radiation data include a correction of 2 per cent uniformly applied in time and space. The French data have varying amounts of correction for each station applied every ten days 1-20].

As a tirst step, data li'om each statmn were con- sidered separately. The combined data from the three Canadian stations were considered. A .similar pro- cedure was followed with the French data.

In the first part of the study, data within + 1 ° range of solar altitudes of 10, 20, 30, 40 and 50 ° were separ- ated. Each value of I and ld was divided by the corre- sponding value of Io. Fractions I/Io were arranged in an ascending order of magnitude. Groups of 1/lo in fractional steps of 0.05 were formed by a procedure similar to that employed by I-3] in correlating Ha and H. Within each group, averages of I/Io and l,dlo were obtained and plotted. By a similar procedure, plots of I J l against I/Io were drawn (Appendix A).

As a second step, the above process was repeated for data within the + 2 ° range of solar altitudes men-

tA reader in this subject might well be confused by the fact that various authors-have used the same nomenclature for daily and hourly, as well as mean daily and mean hourly, values of radiation. This is the reason why in this report, various correlations have been divided into separ- ate groups and particular attention has been paid to the nomenclature.

tioned earlier. There was a slight difference between the two results. The main reason for this difference is that the data containing the + 2 ° range of solar alti- tudes contain a greater number of points than do those with the _ 1 ° range. As such, the averages of various quantities for these two ranges are slightly different. The difference is somewhat accentuated when I/Io > 0.85. Presuming that the data with the + 1 ° range of altitudes contain a sufficient number of points (to be illustrated later), these data have been retained and employed for the desired correlations to be discussed in the following paragraphs.

Figures 1--4 contain data plots of l d I o vs I/Io for Toronto, Montreal, Goose Bay and combined data from all three cities. In these four diagrams, the plots are identical when the hourly clearness index Mr (Mr = 1/lo) is less than a value of about 0.4. This is a cloudy-weather region and solar altitude has no bearing on it. In this region, as the global radiation increases, the diffuse radiation increases correspond- ingly and linearly with it; in fact, global radiation is mainly diffuse radiation. In this cloudy region, present results correspond very well with those of Bugler 1-18].

As the hourly clearness index M r goes beyond a value of 0.4, the effect of solar altitude begins to appear. This region may be referred to as the partly- cloudy region. In this region, as the clearness index increases, the diffuse radiation does not increase with it linearly. After a short increase it begins to decrease until it reaches a certain minimum value. In this partly-cloudy region, the minimum value of M,~ = l,dlo varies with the solar altitude. For a nominal solar altitude of 10 °, Mn reaches its minimum value (except for Goose Bay, Fig. 3) when M r is about 0.5. For higher solar altitudes (studied within this report), a minimum value of Md is obtained when Mr is in the neighbourhood of 0.75.

The region of Mr beyond which Ma begins to in- crease again may be considered as the mainly-clear- weather region. This region is long for a nominal solar altitudes of I0 ° and is shorter for higher solar altitudes. The reason for this is obvious: in clear weather, at lower solar altitudes, the global radiation has a high percentage of diffuse component due to the scattering effects of thicker air mass. In this region, as the global radiation increases, the diffuse radiation increases correspondingly.

The main difference between Bugler's study and the present one is in the region of mainly-clear-weather.

Table 1. Canadian and French stations used in this study with regular hourly measure- ments of diffuse and global solar radiation on a horizontal surface

Latitude Longitude Station Country 0 0 Record used

Toronto Canada 43 48 N 79 33 W Aug. 1967-Dec. 1975 Montreal Canada 45 30 N 73 37 N Oct. 1964-Dec. 1975 Goose Bay Canada 53 18 N 60 27 W May 1962-Dec. 1975 Trappes France 48 46 N 02 01 E Jan. 1967-Dec. 1976 Carpentras France 44 05 N 05 03 E Feb. 1968-Dec. 1976

494

._.o

-o

i i

£

0.8

0.7

0 , 6

0.5

0.4

0.3

0.2

0.1

I I

M. IQBAL

I I I I I

TORONTO ( t . 3 * / . 8 ' N; 79 "33 ' W )

[] SOLAR ALT ITUDE 10-¢1 °

0 SOLAR ALT ITUDE 20-* 1 °

SOLAR A L T I T U D E 3 0 ± 1 °

© SOLAR A L T I T U D E 1,0 + 1 °

O [ ]

o

I O .0 0.1

0 [] " o o ~ ~ ° z~

0

8

I I I I I I 02 0.3 O.t, 0.5 0.6 0.7

M r = I / Io

Fig. 1. Variation of Md with Mr for Toronto.

I I

[ 3

[ ]

0 0

8 ~ o

1 I 0,8 0.9 1.0

Bugler obtained best fit curves from total data within a certain nominal solar altitude. On the other hand, in this study, the plots are based on the weighted values of data points and as such reflect the true be- haviour of the correlations between Md and Mr.

It is now necessary to compare with each other the plots in Figs. 1-4 when M r > 0.4. In this region of

0.4 < MT < 1.0, there are some differences between each diagram. The differences are somewhat pro- nounced when M r > 0.8. The main reasons for these differences appear to be that; (1) for each station the number of data points (Figs. 5--10) at various altitudes and various M r values were different, (2) in clear weather the diffuse radiation is strongly affected by

o

-o

u

0 . 8 ..L

0.7

0 .6

0.5

0.1,

0.3

0.2

0.1

0.0

I I I I I

M O N T R E A L 145"30 ' N; 7 3 e 3 7 ' W )

0 SOLAR ALTITUDE 10 +- 1 °

0 SOLAR ALT ITUDE 20-* 1 °

A SOLAR ALT ITUDE 30 + 1 °

o SOLAR A L T I T U D E 40 z 1 °

O

~ o a []

0

0

I I

0

0

0.1

o o O

2 o

[] 0 o o 4>

o 2 2 0 0

o @

0 . 8 0.2 0.3 0.4 0.5 0.6 07 M. r = I / I,

Fig. 2. Variation of Mj with Mr for Montreal.

LX

0 . 9 1.0

o m

.o n

tl

0.8

0.7

0.G

0.5

O./,

0,3

0.2

0.1

I I

Diffuse solar radiation

I I I I I I

GOOSE BAY [53°18" N: 6 0 o 2 7 ' W }

[]

e

[ ] SOLAR ALTITUDE 10 ± 1 °

0 SOLAR ALTITUDE 20-* 1 °

A SOLAR ALTITUDE 3 0 + 1 ° [ ]

o SOLAR ALTITUDE /,0-* 1 ° [ ] "

[ ] 0

z~ 0

g o g ~ 0 O o

D

I I 0.8 0.9 1.0

I I I I I I I 0.0 0.1 02 0.3 0J, 0.5 0.6 0.7

M r = I / I o

Fig. 3. Variation of M d with Mr for Goose Bay.

495

the atmospheric constituents which are different at the three stations because of their varying proximity to different types of industry. For instance, the histo- grams for Toronto show that at the nominal altitudes of 30 and 40 ° (Figs. 5 and 6), there were no data for Mr > 0.9. The same is true for Montreal (Figs. 7 and 8). Therefore under these specific conditions, Fig. 4 is

based on-only the Goose Bay data (Figs. 9 and 10). Even for latitudes of 10 and 20 °, there were not enough data points when Mr > 0.9. Therefore, in Figs. 1-4, the validation of the correlations may be considered as limited to M r < 0.9. Figure 4 is based on combined data of all three Canadian stations. It is believed that each plot in this diagram contains sufti-

- 9 ,%

0.8

0.7

0.6

I I I I | | | I |

COMBINED DATA OF TORONTO. MONTREAL, AND GOOSE BAY

[] 0.5 ~ ~SOLAR ALTITUDE 10 - + 1 °

~ - - -4 )SOLAR ALTITUDE 20-+ 1 ° b - - - -~SOLAR ALTITUDE 30-+1 ° / ~

OJ. ~ - - - -~SOLAR ALTITUDE /,0~-1 ° ~ /

0.3 ~ 0

/ - j - ~.,~,X~_._~.~. ~ . ~ .

0.1

o.o o!, 0!2 o!3 o!, o'6 o!, o!8 o!, Mt = I / I0

Fig. 4. Var iat ion of M~ with MT for the combined clara of the three Canadian stations.

t []

O

A

o

1.0

496 M. IQBAL

> -

z I,LI :D C l I ,LI

LL

900 -

800

700

600 -

500

400

300

200

100

0 0.0

I I I I ! I I f I I I I I I 1

TORONTO (/ ,3"48' N. 79 ' 33 ' W)

SOLAR ALTITUDE 10d:1'

I I I

SOLAR ALTITUDE 20 2 1'

I I I I I I I ~" 0.2 0.4 0.6 0.8

K T =

I I I I I I I I i 0.2 0.4 0.6 0.8

I / I o

1.0

Fig. 5. Histogram for Toronto for solar altitudes of 10 ± 1 ° and 20 + 1 °.

cient data points (except for M r > 0.9) to represent acceptable relationships between Md and M r for the wide geographical area bounded by these three stations. Contrary to Figs. 1-3, Fig. 4 has curves drawn through the points to present it as a workable diagram.

The above mentioned process has been repeated for

the two French stations. Figures 11-13 represent cor- relations between Ma and Mr based on data from Trappes and Carpentras and the combined data from these two stations, respectively. Figures 14-17 contain the corresponding histograms for the two stations.

Comments based on the Canadian data can almost be repeated here. Comparing Fig. 11 with Fig. 12, at

I I I

900

8 0 0 - SOLAR

?00

600

500 -

4 0 0 m

3 0 0 -

2 0 0 . ~ - - i

100

0 I I 0 . 0 0 . 2

I I I I I I l I I I I I I I I

TORONTO 143048 . N. 7 9 * J 3 ' W )

ALTITUDE 30*- 1" SOLAR ALTITUDE 40 + 1'

I J I I I i ' - 1 1 0 . 4 0 . 6 0.8

K . r =

_p_7 I I I I I

0 . 2 0 . 6 0.8 I / I 0

I I

0 . 4 1.0

Fig. 6. Histogram for Toronto for solar altitudes of 30 + 1 ° and 40 + 1 °.

Diffuse solar radiation 497

>.- t ~ Z Lfl

O I.IJ ew LL

9 0 0

800

700

600

500

400

300

200

100 ~, .

0 0 .0

I I I I , I w , , I I , I , , I t I

MONTREAL ( / , 5 ° 3 0 ' N 7 3 o 3 7 . W)

S O L A R A L T I T U D E 10~10 SOLAR ALT ITUDE 20 -+ 1'

m

L I I I I I I I I I I I I I I I I " - - !

0 . 2 0 . / , 0 . 6 0 . 8 0 . 2 O . / , 0 . 6 0 . 8

K T = I / I o

Fig. 7. Histogram for Montreal for solar altitudes of 10 + 1 ° and 20 _+ 1 °.

M r < 0.3, the correspondence of results between Trappes and Carpentras is exact. At MT > 0.3, the variation of Md with M r for different solar altitudes is similar for the two stations. However, in the range 0.3 < M r < 0.6 (partly-cloudy-weather), the two stations exhibit some differences in the magnitudes of Md. In this region, Trappes shows relatively less

dependence of the diffuse radiation on solar altitude. This can be probably ascribed to the fact that Car- pentras has more hours of bright sunshine than Trappes [21].

Again, in the range 0.8 < Mr < 0.9, the two stations exhibit differences in the magnitudes of M~ mainly at the nominal solar altitudes of 10 and 20 °.

>-

Z W

0 W

IJ.

9OO

8 O 0

700 -

600 -

5 0 0 -

& 0 0

300 J'- -1

r ' ° ° - . . . . . . .

K T =

_ MONTREAL ( 4 5 o 3 0 . N 7 3 ° 3 7 ' W )

_ SOLAR A L T I T U D E 3 0 t 10 SOLAR ALT ITUDE 4 0 Z l °

- ' - t _ . 8 O .2 0 . / . 0 . 6

I / I o

Fig. 8. Histogram for Montreal for solar altitudes of 30 _+ 1 ° and 40 +_ 1 °.

1.0

498 M. IQBAL

>,-

Z LU

0 LU r,- LL

900

800

700

600

500

400

300

200

100

0

GOOSE BAY 153 =18'N, 60 = 2 7 ' w ) n

_ SOLAR ALTITUDE 10 + 10 SOLAR ALTITUDE 20 ~ 1 o

1

-L

.0 0.2 0.4 0 .6 0.8

K T =

LLLr _ k

0.2 0 t. 0.6 0.8 1.0

I / I 0

Fig. 9. Histogram for Goose Bay for solar altitudes of 10 + 1 ° and 20 + 1 °.

This is partly due to the fact that in this range, there is a disparity in the number of data points from the two stations (Figs. 14-17).

The combined data of Trappes and Carpentras are plotted in Fig. 13. This figure may be considered as a regional representation for locations between Trappes

and Carpentras. Consequently, lines have been drawn through the data plots so that the graph may be used for calculation purposes.

The correlations based on the data of the three Canadian stations (Fig. 4) can now be compared with the corresponding correlations based on the two

900

800

700

600

500

400

300

200

100

_ GOOSE BAY

- SOLAR ALTITUDE 3 0 ± 1'

L o ~ 0.0 0.2 0.4 0 , 6 0 .8

K T =

(53 = 18' N, 60 = 27' W ) i

SOLAR ALTITUDE 40 ± 1 ° -

0.2 0.4 0.6 0.8 I / I o

1.0

Fig. 10. Histogram for Goose Bay for solar altitudes of 30 + 1 ° and 40 + 1 °.

_ o

1=

0 . 8

0 .7

0 . 6 - -

0 . 5 -

0 . 4 -

0 . 3 -

0 . 2

0 . 1

0 . 0 3

I

g

I

0.1

[ ] S O L A R

O SOLAR

SOLAR

o S O L A R

Diffuse solar radiation

1 ' ' I I I I

T R A P P E S { / ' 8 % 6 ' N; 02"01 ' E }

A L T I T U D E 10± 1 °

A L T I T U D E 20"- 1 °

ALT ITUDE 3 0 " - 1 °

A L T I T U D E / '0 - * 1 °

[]

D [] CI l~ 0 v

A

8

I I I I I I 0 . 2 0 .3 0 . 4 0 . 5 0 . 6 0 . 7

M T = I I I o

Fig. 11. Variation of M a with Mr for Trappes.

O O

A O Z~

D

D

0

O O

I I 0 . 8 0 . 9 1.0

499

French stations (Fig. 13). In the range 0 < M r < 0.35, the correspondence between the two figures is almost total. In the range 0.35 < M r < 0.6, the correspon- dence is close for all solar altitudes, although the Canadian data result in slightly higher values of M~. For all values of Mr, at solar altitudes of 30 and 40 °,

the correspondence between the two figures is very close. However, at these altitudes, the French stations did not have enough data points for M r > 0.8.

The main difference between the two plots (Figs. 4 and 13) lies in the clear-weather range at solar alti- tudes of 10 and 20 °. The reasons behind these differ-

o

%

u

0 . 8

0 . 7

I

0 . 6 -

0 . 5 -

0 . / , -

0 . 3 -

0 . 2 -

0 .1

0 . 0

O

I

0 .1

I ! I I

C A R P E N T R A S

I I I

( 4 4 ° 0 5 ' N : 0 5 ° 0 3 ' E )

SOLAR ALT ITUDE 10-+ 1 °

SOLAR A L T I T U D E 20 +-1 °

S O L A R A L T I T U D E 30 + 1 °

SOLAR A L T I T U D E / '0 + - 1 °

O O

D [] ~) D D 0 o

D <>

o

D

! !

0

[ ]

[3 0 0

I 0 . 8

0 Q zs

0

.1 I I I , I I 0 . 2 0 . 3 0 .& 0 .5 0 . 6 0 .7

M; = I / I o

Fig. 12. Va r i a t i on o f Md w i th M r for Carpentras.

I 0 . 9 1.0

500 M . IQBAL

o B

=;

0 . 8

0.7

I i I I I I I I

COMBINED DATA OF T R A P P E S AND C A R P E N T R A $

0 . 6 i []

/

0.5 - o nSOLAR ALTITUDE 10"- 1 /

e-----eSOLAR ALTITUDE 2 0 " 1 E] / ~ ~'

a,----,-~SOLAR ALTITUDE 30 +- 1

0 . 4 - e , - . - .eSOLAR ALTITUDE /,0 ~ 1

0.1

O.C , 0.1 0 . 2 0 .3 O.& 0.5 0 .6 0 . 7 0 . 8

Mr = I / to

Fig. 13. Variation of Ma for the combined data of the two French stations.

[]

[]

0

0

[]

m

o

I 0 . 9 1 , 0

ences are not apparent and need to be explored; nevertheless the energy associated with low solar alti-

tAs a personal opinion, this writer favours the French procedure where the amount of shade ring correction varies with time and space. On the other hand, the Cana- dian measuring systems have an international reputation for being very reliable.

tudes is not critical. The different methods of shade ring correction followed by the two countries may have some bearing on this matter.t

Finally, Bugler's [18] study may now be compared with the present results in Figs. 4 and 13. In the range 0 < M r < 0.3, the correspondence between the two studies is perfect. In the range 0.3 < Mr < 0.65, the

9 0 O

80O

700

60O o z

o~ 500

u. ~00

3OO

20O

IO0

0

I I I I I I I I I I I I I I I | I !

T R A P P E S 1480 4 6 " N . 0 2 0 0 1 . E )

-- SOLAR ALTITUDE 10.. 1" SOLAR ALTITUDE 20 4" 1 °

|

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 K T =

'Fig. 14. Histogram for Trappes for solar altitudes of 10 4- 1 ° and 20 4- 1

q I I I - - '

0 . 2 0 . 4 0 . 6 0 . 8 I / i o

Diffuse solar radiation 501

>- (J z LU

o w r,- u.

900

800

700

600

500

400

300

200

100

0 O.

TRAPPES 1480 46' N 02* 01' E l

S O L A R A L T I T U D E 30 +- 1' SOLAR ALT ITUDE 40 *- 1" -

0 0.2 0./. 0.6 0.8 K T =

m

0.2 0.4 0 . 6 0 .8 1.0 I / I o

Fig. 15. Histogram for Trappes for solar altitudes of 30 + 10 and 40 + 1 °.

correspondence is better for solar altitudes of 30 and 40 ° and is poor at lower solar altitudes due to the reasons given earlier.

In conclusion, it may be said that correlations between the hourly diffuse radiation and the hourly global radiation have been developed for five different stations. Each of these correlations may safely be used

in the neighbourhood of the stations studied. Two regional correlations have been developed, one for Canada and another for France. For M r > 0.7 (mainly-clear-weather), the present correlations should be considered as tentative. Further work is advised after another decade, when more data in this range become available. Meanwhile all histograms are

>- (J Z

0

900

800

700

600

500

400

300

200

I00

0

- S O L A R A L T I T U D E 10 +- 1" SOLAR ALT ITUDE 20 " 1" - -

- 7 _

.0 0 .2 0 .4 0 . 6 0 .8

Kr

f m

I I

0 .2 0.4 0 .6 0 . 8 1.0 I / I 0

Fig. 16. Histogram for Carpentras for solar altitudes of 10 _+ 1 ° and 20 _+ 1 °.

SE Vol. 24, No. 5--F

502 M. IQBAL

) -

o Z i i i

0 kfl ¢w i t

g O 0

800

700 -

6 0 0 -

5 0 0 -

/ . 0 0 -

3 0 0

2 0 0

1 0 0

0 0.0

I I I

- SOLAR

: / i i l I I I

0.2 0.~ 0.6

I I ~ I I I I I I I I I I ! I I

CARPENTRAS (4/.* 05' N, 050 0T E)

II ALTITUDE 30"- I* SOLAR ALTITUDE t.O t I* -

I 0.

| I 0.8

K T =

-4-

-4-

I I I 0.2

I/I o

I I I 0./. 0.6

Fig. 17. Histogram for Carpentras for solar altitudes of 30 + 1 ° and 40 + 1.

10

included in this report as archival record on which

further work could be based. In the present study, in view of the above remarks, no mathematical ex- pressions have been developed between Ma, M r and the solar altitude.

Acknowledgements--The financial support of the National Research Council of Canada is gratefully acknowledged. Computation work and the preparation of diagrams was done by Cecilia Cameron. Thanks are also due to the Uni- versity of British Columbia for providing sabbatical leave during which period this report was written.

NOMENCLATURE

H Ha,re.y), global solar radiation received on a horizon- tal surface on a particular day, i.e. jth day, mth month and yth yeari', MJm- 2 day-

H Him), monthly mean daily global solar radiation received on a horizontal surface during ruth month, averaged over several years, MJm- 2 day- 1

Ha Hdtj.,..y~ sky diffuse solar radiation received on a horizontal surface (from a solid angle of 2n with the exception of the solid angle subtended by the suns disc) on a particular day, i.e. jth day, ruth month and yth year, MJm -2 day-

Ha Harm), monthly mean daily sky diffuse solar radi- ation received on a horizontal surface during mth month, averaged over several years, MJm -2 day-

Ho Hoo.m ), extraterrestrial solar radiation received on a horizontal surface on a particular day, i.e. jth day and ruth month, MJm -2 day-

Ho Ho(,,), monthly mean daily extraterrestrial solar radiation received on a horizontal surface during the ruth month, MJm -2 day-

I I(~j.,,.h~ global solar radiation received on a hori- zontal surface during a particular hour, i.e. ith hour, jth day, ruth month and yth year, kJm -z h-

tThe year is mentioned only to identify the data.

i 1,,~,), monthly mean hourly global solar radiation received on a horizontal surface during ith hr of ruth month, averaged over several years, kJm- 2 h-

I d la,,i,,.,h~, sky diffuse solar radiation received on a horizontal surface during a particular hour, i.e. ith hr, jth day, ruth month and yth year, kJm -2 h-

id [~(i.,.), monthly mean hourly sky diffuse solar radi- ation received on a horizontal surface during ith hour of ruth month, averaged over several years, kJm -2 h-1

To Iooj,,. ), extraterrestrial solar radiation received on a horizontal surface during a particular hour, i.e. ith hr, jth day and ruth month, k J m-2 h -

io l~.m ~ monthly mean extraterrestrial solar radiation received on a horizontal surface during ith hr of ruth month, kJm- 2 h -

I~c solar constant 4871, kJm 2 h- K H~/H

KT H/Ho K Ha/H

Kr H/Ho M Id/l

Ma ld/I o MT I/Io

REFERENCES

1. A. K. Angstr6m, Solar and atmospheric radiation. Q.J.R.M.S. 20, 121-126 (1924).

2. G. O. G. Liar, J. A. Duffle and C. O. Smith, World distribution of solar radiation. Report No. 21, Engin- eering Experiment Station, Madison (1966).

3. B. Y. H. Liu and R. C. Jordan, The interrelationship and characteristic distribution of direct, diffuse, and total solar radiation. Solar Energy 4(3), 1-19 (1960).

4. N. K. O. Choudhury, Solar radiation at New Delhi. Solar Energy 7(2), 44-52 (1963).

5. G. Stanhill, Diffuse sky and cloud radiation in Israel: Solar Energy 10(2), 96-101 (1966).

Diffuse solar radiation 503

6. D. W. Ruth and R. E. Chant, The relationship of dif- fuse radiation to total radiation in Canada. Solar Energy 15(2), 153-154 (1976).

7. R. O. Buckius and R. King, Diffuse solar radiation on a horizontal surface for a clear sky. Solar Energy 21(6), 503-509 (1978).

8. J. K. Page, The estimation of monthly mean values of daily total short-wave radiation on vertical and inclined surfaces from sunshine records for latitudes 40°N-40°S. Proc. UN Conf. on New Sources of Energy, Paper No. S/98 (1961).

9. S. A. Klein and J. A. Duttie, Estimation of monthly average diffuse radiation. Presented at the National Solar Energy Conf. (1978).

10. S. E. Tuller, The relationship between diffuse, total and extraterrestrial solar radiation. Solar Energy 18(3). 259-263 (1976).

11. M. Collares-Pereira and A. Rabl, The average distribu- tion of solar radiation correlations between diffuse and hemispherical and between daily and hourly insolation values. Solar Energy 22(2), 155--164 (1979).

12. M. Iqbal, Estimation of the monthly average of the diffuse component of total insolation on a horizontal surface. Solar Energy 20(1), 101-105 (1978).

13. M. Iqbal, Correlation of average diffuse and beam radiation with hours of bright sunshine. Solar Energy 23(2), 169-173 (1979).

~4. J. E. Hay, A revised method for determining the direct and diffuse components of the total short-wave radi- ation. Atmosphere 14(4), 278-287 (1976).

15. M. Iqbal, A study of Canadian diffuse and total solar radiation data. Part I. Monthly average daily horizon- tal radiation. Solar Energy 22(1), 81-86 (1979).

16. M. Iqbal, A study of Canadian diffuse and total solar radiation data. Part II. Monthly average hourly hori- zontal radiation. Solar Energy 22(1), 87-90 (1979).

17. J. F. Orgill and K. G. T. Hollands, Correlation equa- tion for hourly diffuse radiation on a horizontal sur- face. Solar Energy 19(4), 357-359 (1977).

18. J. M. Bugler, The determination of hourly insolation on an inclined plane using a diffuse irradiance model based on hourly measured global horizontal insola- tion. Solar Energy 19(5), 47%491 (1977).

19. R. Bruno, A correction procedure for separating direct and diffuse insolation on a horizontal surface. Solar Energy 20(2), 97-100 (1978).

20. R. Mittner, Private communication, dated 14 December 1978.

21. M. Gamier, Dur6e et fraction d'insolation en France. Monographie No. 105 de la M~tdorologie Nationale (July 1978).

APPENDIX A

A few peripheral points are added here. In the beginning of this section, a procedure to calculate

Mr has been described. Another procedure would be to obtain separate averages of I in a certain range, divide by the averages of the corresponding values of Io, and then obtain M r and in a similar way Md. This latter procedure was tested for the Canadian stations but the results remained the same. This is being added to emphasize the fact that the estimated value of I d will remain the same whichever of the two averaging pro- cedures is followed.

The second point concerns the plotting of data as I,,/I versus 1/11o. The data for the Canadian stations were plotted in this manner as well. From these plots it appeared that the influence of solar altitude was not brought out strongly. For this reason, the plots of only Md vs Mr are presented in this study.

Because of the latitudes of the five stations considered, no useful data could be obtained for solar altitudes higher than 40 ° . It is recommended that the data of stations nearer to the equator be studied in order to investigate the range of solar altitudes higher than 40 ° .