Solar Energy Vol. 23, pp. 169-173 0038--092X/79/0801-0169/$02.0010 © Pergamon Press Ltd., 1979. Printed in Great Britain

CORRELATION OF AVERAGE DIFFUSE AND BEAM RADIATION WITH HOURS OF BRIGHT SUNSHINE

M. IQBAL

Department of Mechanical Engineering, The University of British Columbia, Vancouver, B.C., Canada V6T 1W5

(Received 24 August 1978; revision accepted 5 March 1979)

Abstract--Empirical equations have been developed which correlate the monthly average daily horizontal diffuse and beam radiation with the fraction of maximum possible number of bright sunshine hours. These correlations are based on measured data from three widely spread Canadian stations.

Depending upon whether or not the total horizontal radiation is known, the two correlations for the diffuse radiation are:

H_d = 0.791 - 0.635 (N) H

Ito

The correlation for beam radiation is:

/~b -1.12 , for ti >0. -:- = 0.176 + !,45 Ho

INTRODUCTION

The total shortwave radiant flux reaching the earth's surface is composed of two parts. The part arriving directly along the line of sight from the sun is generally called beam or direct radiation. The second part coming from the remaining portion of the hemisphere is called diffuse or sky radiation. The fraction of diffuse radiation at any instant depends upon the elevation of the place and its latitude, the solar altitude, the sun's declination, the degree of turbidity, the amount of water vapour present in the atmosphere and cloudiness. Variability in the amount and type of cloud cover is the major factor in determining the ratio of diffuse to total radiation.

The total solar irradiation (diffuse plus beam) is by far the most important factor in such applications as clima- tology and agriculture, for instance. Beam radiation is very important in some industrial applications such as solar furnaces and other solar energy concentrating devices. On the other hand, an assessment of diffuse radiation is required for interior illumination of buildings and other energy related problems associated with build- ing research.

In some research problems associated with solar energy, instantaneous fluxes of the diffuse and beam radiation may be needed. On the other hand, engineering design of many solar devices may require only long-term monthly averages of the hourly or daily beam and diffuse radiation.

For clear-sky conditions, the beam radiant fluxes usu- ally can be calculated from purely theoretical con- siderations. The ASHRAE Handbooks[I,2] may be referred for calculation procedures in this instance. For engineering purposes, the clear-sky diffuse radiation estimation procedures also given in these handbooks may be used.

For cloudy or partly cloudy skies, however, theoretical procedures for calculation of the irradiation on the earth's surface are not yet well developed for engineer- ing use. Under these conditions, one is obliged to search for the measured data, or, when the measured-data is not available, estimate the insolation from known values for similar conditions.

The solar radiation measuring networks generally record only the total daily horizontal radiation, H; or the total hourly radiation, /. To compute insolation on in- clined planes [3], one needs the corresponding daily or hourly diffuse components Ha or Ia respectively. Sta- tions measuring diffuse radiation are extremely rare. Therefore, methods are required to estimate the diffuse component of the total horizontal radiation. Liu and Jordan[4], Kalma and Fleming[5] and Ruth and Chant[6] have presented correlations between the daily horizontal total and the daily horizontal diffuse radiation. Bugier[7] and Orgill and Hollands[8] have presented correlations between the hourly horizontal total and the hourly horizontal diffuse radiation.

The correlations between the diffuse and total radia- tion mentioned above apply to particular days or hours. However, most design problems require monthly averages of the above quantities. Liu and Jordan[4], Page[9] and Iqbal[10] have treated the estimation of monthly diffuse radiation,/td, as a fraction of its daily total value, /~. In these studies, the fraction of diffuse radiation /td//'I has been expressed in terms of the cloudiness parameter, /t//~o; where /~o is the monthly average extraterrestrial horizontal radiation.

In a recent paper, Iqbal[ll] on the basis of physical arguments has proposed that the average horizontal diffuse radiation may be expressed in terms of the frac- tion of maximum possible sunshine hours; as,

qF Vc~l "~ Nc~ ~--F |69

170 M. IQBAL

N

In eqn (1), fi is monthly average number of bright sun- shine hours per day and N is monthly average day- length.

The purpose of the present study is to extend (1) by correlating diffuse radiation with bright sunshine hours using measured data. Proceeding in the same way, beam radiation (H -/'/d) would also be correlated.

CORRELATION OF DWFUSE AND BEAM RADIATION Measured diffuse and total solar radiation data of three

stations, Toronto, Montreal and Goose Bay (Table 1), were obtained from Atmospheric Environment Canada. Monthly average values of these data are reported in[10].

Monthly average bright sunshine records of Canadian stations have been compiled by Yorke and Kendall[12]. These records are for the period 1940-70 which is not identical to the corresponding period for radiation records. However, both periods are long enough that their data may be employed simultaneously (Table 2).

Mathematical development of the expressions dealing with a number of empirical approaches follows.

tA word of caution is necessary. The coefficients as, a6 and 07 are obtained directly from the regression analysis and not by the algebraic manipulation of the coefficients in (2) and (3). The same remarks apply to the coefficients in eqn (7) to follow.

MATHEMATICAL FORMULATION

Equation (1) can in fact be written as

--=- = a, + a= , (2) H

where it was assumed that al = 1 d2 = - 1. Values of at can be obtained by correlation with actual data. Equation (2) may be considered as a correlation procedure where

has to be known, experimentally or by estimate. When H is not known experimentally, it can be esti-

mated through the well-known empirical expression,

11o

Equation (3) was first suggested by ]~ngstr6m[13] and has since been widely used[14] to estimate total in- solation at a location on the surface of the earth. Schulze[15] has critically reviewed the estimation methods which employ eqn (3).

Multiplying (2) with (3), the following quadratic expression is obtained,

_--- = a~ + a6 + a7 (4)t Ho

Equation (4) is interesting. It shows that the diffuse

Table 1. Canadian stations used in this study with regular hourly measurements of diffuse as well as total solar radiation on a horizontal surface

Lat. I Long. Station (N) (W) Record Used

o , o v

79 33 Aug. 1967 - Dec. 1975 Toronto, Met. Res. Stn. 43 48

Montreal, Jean Bre~beuf 45 30

Goose Bay

73 37 Oct. 1964 - Dec. 1975

53 18 60 27 May 1962 - Dec. 1975

Month

January

February

March

April

May

June

July

AuguSt

September

October

November

December

Table 2.

Station

I . . . . Montreal Goose Bay Toronto

87 96

ii0 122

145 171

179 188

221 241

256 248

281 261

257 234

197 196

153 150

82 75

77 79

88

119

141

147

179

194

212

205

143

102

71

73

Correlation of average diffuse and beam radiation with hours of bright sunshine

radiation can be estimated directly from the local extra- terrestrial radiation.

Equations (2) and (4) may be considered as two different approaches to determine H,,. Before discussing the regression analysis to determine the coefficients a~ in (2) and (4), the empirical procedure for determination of beam radiation will be outlined.

The monthly average daily beam radiation on a horizontal surface can be written as,

Hh : H - He. (5)

Dividing the right hand side of (5) by/'to we obtain,

(6) /-70

Inserting (3) and (4) in the right hand side of (6), we obtain,

--=-- = as + a9 +alo Ho

(7)

It should be noted that a purely algebraic manipulation of (6) would result in a,o being negative. However, a posi- tive sign is placed in front of this coefficient as the actual sign is obtained through the regression analysis.

Equation (7), just like eqn (,it), is independent of the monthly average total radiation, H.

A procedure for the determination of a~ and discussion of the results is presented in the next section.

171

DISCUSSION OF RF~JLTS

A curve-fitting program[16] that employs the Gaus- sian-Newton least-squares technique was used to obtain the regression coefficients a~ in eqns (2), (4) and (7). Each of the three stations provided twelve data points. This meant that for each of the three equations, combinations of 12, 24 or 36 data points could be employed to evaluate a~. Use of only 12 points from a single station would, of course, provide a local correlation. On the other hand, combined data from all three stations would result in a regional correlation. In this study a number of com- binations were tried.

Tables 3-5 list the coefficients a~ and standard error of estimate for eqns (2), (4) and (7) respectively. Each one of these tables also lists the names of the stations from which the data is used. From an examination of the values of standard errors of estimates it is evident that very good correlations have been obtained. By and large the data of single stations give lower values of the standard error of estimate compared to that obtained from the combined data from two or three stations.

To obtain a visual appreciation of the regression analysis, the estimated monthly values of the diffuse and beam radiation were compared graphically to the experimental data. A large number of plots were pre- pared. The estimated values corresponded very well with the actual data. The differences for individual months depended upon the correlation and the location (or loca- tions) from which the data were obtained.

In the following examples, only the correlations based on data from all three stations will be presented.

In Fig. 1, the monthly variation of diffuse radiation Ha

Table 3. The coefficients al in ffd117I : az + a 2 ( t i / / Q ) . . . . . (2)

Source of Data Coefficient

a I a 2

0.7294

0.7860

0.8281

0.7650

0.7910

Standard Error of Estimate

-0.5337 0.0228

-0.6142 0.0394

-0.7044 0.0223

-0.5900 0.0277

-0.6350 0.0317

Montreal

Toronto

Goose Bay

Montreal plus Toronto

Montreal, Toronto and Goose Bay combined

Number

( 2 a )

( 2 b )

( 2 c )

(2d)

(2e)

Table 4. The coefficients al in/'f,d/to = a5 + ar(ti//~') + a T ( t i / J Q ) 2 • • • (4)

Source of Data

Montreal

Toronto

Goose Bay

Montreal plus Toronto

Montreal, Toronto and Goose Bay combined

Coefficients Standard Error of Number

a 5 a 6 a 7 Estimate

0.0945 0.6336 -0.7398 0.0163 (4a)

0.2427 0.0439 -0.1161 0.0187 (db)

-0.1017 2.0408 -2.7905 0.0273 (4c)

0.2020 0.1775 -0.2421" 0.0196 (4d)

0.1633 0.4778 -0.6555 0.0267 (4e)

172 M. IQBAL

Table 5. The coefficients al in/tb//to= as+adNIV)+ato(~lV)2...(7)

Source of Data

Montreal

Toronto

Goose Bay

Montreal plus Toronto

Montreal, Toronto and Goose Bay combined

Coefficients

a 8 a 9 alO

-0.2536 1.8690 -1.6494

-0.1844 1.4578 -I.0868

-1.0576 6 .2283 - - 7 . 4 9 8 8

-0.1791 1.4561 -i. I169

- 0 . 1 7 6 3 1.4497 -1.1193

S t a n d a r d E r r o r o f E s t l m a t e

N~Iber

0.0163 (7a)

o.o311 (To)

0.0173 (7c)

0.0253 (7d)

0.0250 (7e)

1 4 | - - ] i I i i I i 1 I I 1

t MONTREAL

~ 8

B 6

g. ~> MEASURED ' ~ _h

2 . . . . EQ. (2) ; (8)

. . . . EQ (4) ; (9) 0 I I I t I I I I I I

F M A M J J A S 0 N

MONTH

Fig. 1. Monthly variation of the average horizontal daily diffuse radiation.

estimated from eqns (2) and (4) is compared with the actual data for Montreal. Comparing the results of eqn (2) with that of (4), it is evident that the correspondence is very close. Now, comparing the estimated values with the experimental data, it can be said that except for some summer months, the correspondence is good. Equation (2) corresponds better with the actual data as compared to eqn (4). A similar comparison carried out for Toronto and Goose Bay has shown that while the estimates from eqns (2) and (4) correspond well with the actual data, the performance of eqn (2) is somewhat superior.

The correlation of the monthly average beam radia- tion, eqn (7), is shown in Fig. 2. The estimated and the measured values of (/~ - / t a ) for Montreal are plotted in this figure. The estimated values are again based on the correlation using data from all three stations. This figure indicates that correspondence between the measured and the estimated values is very close. A study of the other two locations, not presented here, shows that estimation of the average beam radiation through its correlation with the bright sunshine hours is a valid one.

It can be concluded that for the region bounded by the

t Just before this article went to press, author's attention was drawn[17] to two studies[18, 19] where correlations very close to (8) have been developed for France.

A t4

i r'r v

I0 OE

g w 6

"c

~2,

i i I i i i i i i i

MONTREAL

\\

/ \, '____ \

I l l l l l l l l l F M A M J J A S O N

M ~ H

Fig. 2. Monthly variation of the average horizontal daily beam radiation.

latitudes in Table 1, the following correlations may be used:

To obtain monthly average horizontal diffuse radiation when the total horizontal radiation is known, use

Hd=0.791-0.635(N). t (8) H

To obtain/~d when/~ is not known, use

/ ' Id=0.163+0.478(N)-0 .655(N ). /to

(9)

To obtain monthly average horizontal beam radiation, u s e

q b = - 0 . 1 7 6 + l . 4 5 ( N ) - l . 1 2 ( N ) 2, fo r~>0 . (10) Ho

Equations (9) and (10) also can be simultaneously used to estimate the monthly average total horizontal radiation where this is not known.

Equations (8)--(10) may be considered as complements to the correlations of Liu and Jordan[4], Page[9], Kalma and Fleming[5] and Iqbal[10, 11].

Correlation of average diffuse and beam radiation with hours of bright sunshine 173

Acknowledgements--Financial support of the National Research Council of Canada is gratefully acknowledged. Numerical work was carried out by Mr. Y. K. Lau.

NOMENCLATURE

ai coefficients in the empirical correlations /~ monthly average daily total (diffuse plus beam) radiation on

a horizontal surface, MJm -e day -~ Ha monthly average daily beam radiation on a horizontal sur-

face, MJm -2 day -~ /'ld monthly average daily diffuse radiation on a horizontal

surface, MJm -2 day -~ /~o monthly average daily extraterrestrial radiation on a

horizontal surface I~c solar constant 1353 Wm -2

ti monthly average hours of bright sunshine per day, hr ~r monthly average day-length, hr

REFERENCES 1. ASHRAE Handbook and Product Directory, Chap. 59.

ASHRAE, New York (1974). 2. ASHRAE Handbook of Fundamentals, Chap. 26. ASHRAE,

New York (1977). 3. M. Iqbal, Hourly versus daily method of computing in-

solation on inclined planes. Solar Energy 21(6), 485 (1978). 4. B. Y. H. Liu and R. C. Jordan, The interrelationship and

characteristic distribution of direct, diffuse and total radia- tion. Solar Energy 4(3), 1-19 (1960).

5. J. D. Kalma and P. M. Fleming, A note on estimating the direct and diffuse components of global radiation. Arch. Met. Geoph. Biokl. Ser. B, 20, 191-205 (1972).

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

7. J. M. Bugler, The determination of hourly insolation on an

inclined plane using a diffuse irradiance model based on hourly measured global horizontal insolation. Solar Energy 19(5), 477--491 (1977).

8. J. F. Orgill and K. G. T. Hollands, Correlation equation for hourly diffuse radiation on a horizontal surface. Solar Energy 19(4), 357-359 (1977).

9. J. K. Page, The estimation of monthly mean values of daily total shortwave radiation on vertical and inclined surfaces from sunshine records for latitudes ~0°N-40°S. Proc. UN Conf. in New Sources of Energy, Vol. 4, Paper S/98, pp. 378-387 (1964).

10. M. lqbal, A study of Canadian diffuse and total solar radia- tion data. Part I. Monthly average daily horizontal radiation. Solar Energy 22(1), 81 (1979).

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

12. B. J. Yorke and G. R. Kendall, Daily bright sunshine 1941- 1970, Atmospheric Environment Service, Department of the Environment, Canada, Rep. No. CLI-6-72 (1972).

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

14. G. O. G. LOf, J. A. Duffle and C. O. Smith, World distribution of solar radiation. Rep. No. 21, Engineering Experiment Station, Madison (1966).

15. R. E. Schulze, A physically based method of estimating solar radiation from suncards. Agr. Meteor. 16, 85-101 (1976).

16. J. Street and C. M. Lee, U. B. C. curve, curve-fitting tech- niques. Computing Centre, University of British Columbia, Vancouver, B.C. (1976).

17. C. Gueymard, private communication. 18. C. Perrin de Brichambaut, Estimation des ressources 6ner-

g~tiques solaires en France. Suppl~ment au Cahiers A.F.E.D.E.S. No. I (1975).

19. J. F. Tricaud, Contribution A l'estimation des ressources 6nerg6tiques solaires, pour des plans diversement orientOs et inclinOs. Unpublished Rep. C.N.R.S. OdeiUo (1977).