Solar Energy, Vol. 18, pp. 467-473. Pergamon Press 1976. Printed in Great Britain

TECHNICAL NOTE

A method for estimating hourly averages of diffuse and direct solar radiation under a layer of scattered clouds*

MARVIN L. WESELY and ROBERT C. LIPSCHUTZ

Radiological and Environmental Research Division, Argonne National Laboratory, Argonne, IL 60439, U.S.A.

(Received 24 October 1975)

1. INTRODUCTION

The amount and type of cloud cover prevailing at a given time and location largely determine the amount and type of solar radiation received at the Earth's surface. It is conventional to consider the total shortwave irradiance (visible plus some near infrared) at a horizontal, rather than inclined, surface and to express the irradiance as consisting of a diffuse component D and a direct component/. Clearly, the total T --- D + I is the solar energy input relevant to considerations of local energy balances at the Earth's surface. However, when the efficiency of radiation receptors that view the sky in a complex or incomplete manner is being investigated, the separate values of I and D should be examined. Such receptors include, for example, many of the solar collectors now being developed for heating and power-production facilities, and also the individual leaves of plant canopies.

The purpose of this paper is to provide a simple procedure for estimating values of D and I that can be used in comparisons of the theoretical performances of solar collectors of different designs. The amount of solar energy collected at the focal point of a Sun-tracking parabolic reflector is, of course, extremely sensitive to the direct-beam attenuation caused by clouds. On the other hand, the collection by flat-plate receptors is less sensitive to such shading because the decrease of the direct component is typically accompanied by increased diffuse radiation. The characteristics of devices such as the compound parabolic concentrator [1] lie in between. These collectors view a fairly large portion of the sky (obviating the need for continuous Sun tracking) so that both the direct and some portion of the diffuse radiation is gathered. Owing to the partial concentration of radiation onto

tWork supported by the U.S. Energy Research and Develop- ment Administration.

collection strips, considerably higher temperatures are obtained than with flat plates.

A first step towards determining the desirability of including some diffuse as well as the direct radiation in solar-energy collection schemes is to consider the variations of I and D to be expected as the amount and type of cloud cover changes. Relationships can then be derived to assist the prediction of the amount of energy that could be collected at a given location. Averaging times of at least I hr are necessary for an adequate statistical sample; longer periods probably should not be considered since it is unlikely that cloud and haze conditions would remain constant. In this paper, the amounts and types of cloud cover are examined with regard to hourly averages of the magnitudes of D and I, when small zenith angles of the Sun and partly cloudy skies prevail. Also investigated are cloud- induced transients in the direct-beam irradiance. Although considerable effort by other workers has already gone into analyses of daily averages of D and I (e.g. see Refs. [2] and [3]), to our knowledge an investigation of hourly data has not previously been reported.

2. MEASUREMENTS

Total radiation was measured with an Epply black-and-white pyranometer, while the diffuse component was measured with the same type of instrument mounted in the device shown in Fig. 1. A solar-tracking occulting disk attached to a clock-driven equatorial mount was arranged so that a small shadow always just covered the protective dome of the sensor. In practice, the value of I was found by taking the difference between the levels of total and diffuse irradiance recorded on strip charts for the two devices.

Nearly continuous measurements of radiation were obtained at Argonne National Laboratory during the summer of 1974. On 28 occasions during June-September, the patterns on the strip chart

Fig. 1. The shielded pyranometer for continous measurement of diffuse radiation.

467

468 Technical Note

appeared uniform for at least 3 hr. During these periods, routine synoptic observations taken by the National Weather Service personnel at Chicago Midway Airport (located 22 km northeast of Argonne) indicated both uniform conditions and the presence of a single cloud layer. For all observations, the sun zenith angle was less than 600 , and the surface in the vicinity of the radiation instruments was mostly grass or tree covered.

3. TYPICAL RESULTS

Partly cloudy conditions result in many types of radiation records. When clouds are uniformly broken or scattered, the patterns obtained on strip charts appear similar to those illustrated

Io TOTAL-~, a) N4=Ij.=I = 0

DIFFUSE- N DO . . . . . . . . . .

cu

I° ~ = I

'-,'--:-.4 c) N = O S Co

~ = 0.95 .~ Io ~. = 0.85 o Z

,~ Do d) a = 0.5 Ci

ac I o a : 0.6 & = l

Do ,. , ..,,~ . . - - ~ e ) N = 0 . 5 Ci

[o ~ ' ~ F " ~ f / ~ ~ = 0.6 = 0.85

i

Do ,~ I hr ~,

Fig. 2. Typical traces of diffuse and total radiation. Part (a) is for a clear clean sky, parts (b) and (c) are for scattered fair-weather cumuli, and parts (d) and (e) are for scattered cirrus clouds. The

ordinate values are in terms of Io.

in Fig. 2. (The cloud parameters will be explained in the analysis to be given later.) During days with scattered cumulus clouds and no haze, the amplitude X of the fluctuations of the direct irradiance closely approximates the value Io of the direct beam that would have existed if a clear, clean sky had been present. This suggests that clouds of this type are nearly opaque. This case is illustrated in Fig. 2(b). In the presence of more translucent, usually cirriform, clouds, the magnitude of X is considerably less than Io; as a result the strip-chart traces are similar to the example shown in Fig. 2(d). Hence, although the size, speed of translation, and number of clouds determine the duration and frequency of transients in direct-beam irradiance, the amplitudes of these fluctuations primarily depend upon the extent to which the clouds transmit the direct beam.

With clouds overhead, the value of the diffuse component D is considerably greater than the value Do defined as that observed under a clear clean sky. When both haze and clouds are present, the increase in D is more than would be expected from cloud effects alone (see Figs. 2c and 2e). It should be noted that the presence of haze in the subcloud layer reduces the values of X, but even the relatively large reductions of 15 per cent present in Figs. 2(c) and (2(e) are hardly noticeable in the illustration. Also, light reflected from the sides of cumulus clouds momentarily adds to the total radiation received at the surface and results in small peaks in the traces, typically just before and just after the Sun is obscured. This minor effect is ignored in the simple model discussed below.

Figure 3 shows 28 values of mean hourly diffuse and direct irradiances used in this study. Each value of D and I has been normalized by dividing by Io, the direct radiation available above the cloud layer. After determination of the solar zenith angle 0, the value of Io was found using the formula[4]

/o = (1050/m) exp (-0.1 m) (I)

where m = sec 0 represents the optical air mass and & is in W per m 2. The large scatter of data points evident in Fig. 3 is caused by variations in the optical properties of the clouds, as well as by changes in the amount of haze present. Obviously, for a meaningful description of the partitioning of solar radiation into diffuse and direct components, the sources of the scatter in Fig. 3 need to be taken into account.

4. A SIMPLE DESCRIFHON OF THE EFFECTS OF CLOUDS

With the radiation measurements available for this analysis, we can deal only with quantities that are related to the net effects of

o

0 .6 -

0.4

0.2

I ' ~ ~ I I I i

7.50 x 'v x9 6 07

7e 706.5 e 5

7"5x×5 e 4 ° o4 5 " ~ 6o o 4 •

~..4o#3 3oxx 4 ~ 2 . I, 5e205~ 4 3 2 e 1.5e~O~

CUMULUS STRATOCUMULUS CIRRUS

O i I I 1 i I ~ I I O 0.2 0.4 0.6 0 .8 I.O

I / I o

Fig. 3. Observations of diffuse and direct irradiances. The number near each plotted point designates cloud cover in tenths. "BINOVC" refers to "breaks in overcast clouds", and the solid line is a 45 ° line.

Technical Note 469

radiation exchanges between the clouds and the ground. It is assumed that there is no substantial interaction between the clouds, so that the radiative effects of each cloud add or subtract linearly. This allows the use of variables, for individual clouds, that are independent of the total amount of cloud cover. One of these variables is the net fraction a of the direct beam that is attenuated within each cloud. As illustrated in Fig. 4, the downward direct irradiance beneath each cloud can then be written as (1 - a)Io. Since the optical thicknesses of clouds are not estimated in standard surface weather observations, cloud type alone is used in an attempt to uniquely identify values of a. Empirical values have been determined from analysis of irradiance measurements.

We choose s to represent the fractional transmission through "dry" aerosols. It is the net fraction of the direct beam that remains after some portion has been scattered and absorbed by suspended particles beneath the cloud layer. As illustrated in Fig. 4, the direct irradiance in a cloud shadow can be expressed as s (1 - a )Io; in the sunlight between clouds it is slo. The amplitude of the changes of total irradiance at the surface is therefore

X = aslo. (2)

It is convenient to determine s with use of estimates of visibility V, using the classical formulation for optical attenuation:

s = e x p ( - I / V ) (3)

where the magnitude of I depends on several atmospheric factors, including the height of the planetary boundary layer within which most of the attenuating particles are found. Table 160 of the Smithsonian Meteorological Tables[5] suggests I = 3-5 km, but since the present application deals with attenuation in the vertical rather than along the horizontal, considerably different values might be more appropriate.

In synoptic weather observations, the fraction N of the sky covered by clouds is usually estimated to the nearest tenth. If all clouds were flat and had well-defined edges, then the averaged direct irradiance at the surface would be

I = s(1 - aN)Io , (4)

as shown schematically in Fig. 4. On the other hand, if the tops of clouds mask the direct beam of the sun at large zenith angles, then it might be appropriate to replace N with, for example, N(1 + dw -~ tan 0). Here, d represents the mean vertical depth of the clouds, w is the mean cloud width and 0 is the sun zenith angle. For our purposes, we will accept eqn (4) unaltered, with N representing an "effective" cloudiness. It will be shown that this cloudiness agrees with that reported by a surface observer, even though the cloud tops should be ignored during surface observations of cloudness.

The amount of diffuse radiation received at the Earth's surface for a given sun angle is governed by three factors. First, the "clean sky" diffuse radiation (i.e. that part which is attributable to clear-sky effects above the existing clouds) is altered by the factor 1 - N, where again the effects of vertical cloud extent are neglected. Our measurements have shown that the diffuse radiation Do for clear clean skies at Argonne is about 0.09/o when 0 = 25 °, and about 0.16Io when 0 = 65 °. The first value is an appropriate estimate of Do for the small values of 0 assumed here. Second, the bottoms of the clouds emit diffuse light, and here we shall assume that the net, time-averaged amount of this light detected by a pyranometer is YN, where Y is expressed as per unit of cloud area. Third, some of the direct beam attenuation caused by aerosols is recovered as an increase in diffuse radiation. It is shown in radiation measurements [6] following the eruption of Mount Agung, Bali, that the recovery is about 75 per cent, whereas a recovery of 60 per cent has been found in measurements taken at Argonne[4] for presumably tropospheric aerosols. Here, the representative value of 70 per cent will be used for aerosol scattering beneath a cloud layer. Summing these three contributions to diffuse radiation gives (see Fig. 4)

D = Do(l - N ) + Y N + 0 .7( I / s ) ( l - s). (5)

In order to take into account the effects of various sun angles, we can normalize eqns (2)-(5) by division by Io, to obtain

X I I o = as (6)

I / Io = s(1 - a N ) (7)

DIIo = 0.09(1 - N ) + t N + 0.7(I/Ios)(1 - s) (8)

where t = Y/ lo is, in effect, the means fraction of direct radiation transformed into diffuse radiation within the clouds.

5. EVALUATION OF PARAMETERS With a proper choice of I in eqn (3), s can be determined and

used in eqn (4), from which a calculated effective cloudiness N' can then he found for comparison with the observed value of N. When the horizontal visibilities reported at Chicago Midway Airport are used in such calculations, the best agreement between N and N' usually obtains with I = 1.6 kin. We suspect this single value might be appropriate for times from the middle of the morning to late afternoon when a well-mixed atmospheric layer of about the same depth is present. Since as shown in Fig. 5 there is good agreement between N and N' , it appears justified to continue with the analysis and to use eqns (6)-(8) to determine a and t for each case. When this is done, no systematic variation of a or t with either N or N' is found. Figure 6 shows that the value of a for fair-weather cumulus appears to be about 0.95, regardless of t, whose average value is about 0.5. When a = 0.95 and t = 0.5 are assumed for the 17 cases of cumulus, the computed values of

DIRECT I ; 0 - : 8 >/i'o,('cosol I / " I~o Io

HOURLY ~l(l-aN)Io AVERAGE:

D I F F U S E

' . . : t "'? t " t '

,i, ;Y ' 0ZI 100 / / / / / 1 1 / I / / i / / / / /

• YN D o ( I - N ) 0.7 (I/~)(I-~) or or

¢I o N O.091o(l-N)

Fig. 4. Schematic diagram showing the components of solar irradiance used in the simplified model.

470 Technical Note

-z

0.8

0.6

0.4

0.2

0 0

i I i I

• X

• o

0.2 0.4 0.6 N

] I ~ ~ i . i I /

BINOVC x

Q

I I 0 8 1.0

Fig. 5. Comparison between the estimated cloudiness N ' computed from radiation measurements and visibilities and the cloudiness N reported in standard surface observations.

1.0

0.8

0.6

0.4

0.2

i I F I ; X I

o x x

• CUMULUS x STRATOCUMULUS o CIRRUS

I I I I t I i I l 0 0.2 0.4 0.6 0.8 1.0

t

Fig. 6. Measured absorption of the direct beam by each type of cloud, plotted as a function of the estimated cloud transmittance.

T, I and D are within 25 per cent of the measured values, as illustrated in Fig. 7. For cirrus, however, the value of a appears to increase rapidly with increasing t, so calculation of the radiation components by this simple method is not possible. Typical values of cirrus cases are 0.6 for a and 0.5 for t. The behavior with stratocumulus clouds present appears to be quite similar to that for cumulus clouds, although in the former case uniform coverage for N < 1 may be rare; the values t =0.6 and a =0.95 are appropriate.

Figure 2(c) can be used to illustrate this simple model. These

records were chosen to illustrate the calculations that follow, and are not meant to be a test of the formulations. We assume that fair-weather cumuli that attenuate the direct beam by 95 per cent cover 30 per cent of the sky, and that the reported visibility is about 6 kin.

1. It is assumed that 1 = 1.6 km and a = 0.95 are appropriate. Then with a typical noon value of 840 W m -~ for Io, eqn (6) indicates that X = 0.8081o = 768 W m-L

2. The averaged direct irradiance determined by eqns (4) or (7) is 0.608•0 = 511 W m 2.

Technical Note 471

I000

800

n I I I u I n I / _

x_ xi>

6 0 0 •

o • y , ~ x ×

m- × ×

..J

....1

4 0 0 (.,)

200 / o q~° x TOTAL, T .o~ o • D,RECT', I

o o DIFFUSE, D

0 I I I I J I i I i 0 2 0 0 4 0 0 6 0 0 8 0 0 I000

MEASURED (Wm -2)

Fig. 7. Comparison between the measured irradiances T, I and D and values calculated on the basis of zenith angles of the Sun and observations of cloudiness and visibility.

3. The diffuse radiation caused by molecular scattering from the portions of the sky that are clear is 0.027/o = 23 W m -2. With the assumed value of 0.5 for t in the second term on the right side of eqn (8), the diffuse radiation received from the base of the clouds is found to be 0.15/o = 126 W m -2. The third term on the right side of eqn (8) indicates that the amount added by particulate scattering is 0.12/o= 63 W m -2. The value of the total diffuse irradiance is then about 212 Wm -=, which is about 25 per cent of Io, as shown in Fig. 2(c).

6. G E N E R A L I Z A T I O N S

With assumed values of a and t, idealized curves of D/Io vs I/Io can be constructed on the basis of eqns (6)-(8). Figure 8 shows the

results. The effects of cloudiness on the partitioning of radiation are much more evident than in Fig. 3 where the effects of variations in neither the amount of haze nor the optical properties of clouds are evident.

Figures 9 and 10 show the reductions in both direct and total radiations for cumulus and cirrus coverage, respectively. It appears that for cumulus clouds, the reduction in total irradiance is about half the reduction in the direct beam. For cirrus clouds with the optical properties assumed, the reduction in total irradiance is only about one third the direct beam loss.

Since the curves in Figs. 9 and 10 are nearly straight lines, averaging over several hours should yield almost identical curves. For comparison, one example of the many empirical relationships

0.8

0 .6

o 0.4

0.2

0 0

i I ~ I ' I ' I i

a,. " ~ deg

x

J • . - - o - . C,RRUS. CLEAN "~"~\ r - - - X - - CIRRUS, SOME HAZE ]

. C,R S, H A Z Y SOME -

I I I J I i I i 0.2 0.4 0.6 0.8 1.0

I/Io

Fig. 8. Idealized relationships between diffuse and direct radiation, with a = 0.95 and t = 0.5 for cumulus, and a = 0.6 and t = 0.5 for cirrus. Horizontal visibility is assumed to be 100 km in a clean atmosphere, 16 km when the sky is somewhat hazy and 8 km when haze is highly noticeable. Assumed values of cloudiness are 1.0, 0.8, 0.6, 0.4, 0.2 and 0

from left to right at the points, on each curve.

472 Technical Note

0 ~ . . . ~ I i I i I i I I "" - . . . . .

0 .2 x , x. x •

N ~ , ` - N. - • X. \

0 ~' `- \ 0.4 x \ x \ \ ..

C.)

N N. %. " ' . ,,°, 0 . 6 , ,%.,, ` - \ ;% / rr

_ _ _ (I 0 _ i)/I 0 x x \ %. ..., xx, \ .% (T o T ) / I 0 - , , ~ ....

0 . 8 "xxxx, ~ -..

L o ~ i I i I i I I I i 0 .2 0 . 4 0 . 6 0 . 8 LO

N

Fig. 9. Fraction of direct and total radiation incident at a horizontal surface under a single layer of cumulus clouds. The three lines for each component correspond to the haze categories in Fig. 8. The dotted line is from Berliand's formula [7] for daily averages of total

radiation, with the assumption that Do = 0.09/o.

0 .2

o

~. 0 .4 o I-- o El ' " 0 . 6 tw

0.8

~ l I I I I I I /

-..-....-....

_ _ _ ( I o - IVI o ( T 0 - T ) / I 0

I c i I i I J 1 i I i 0 2 0 4 0 6 0 . 8 LO

N

Fig. 10. Fraction of direct and total radiation available on a horizontal surface under a single layer of moderately dense cirrus

(t = 0.6).

between daily averages of T and N is that given by Berliand[7] and shown here as a dotted line in Fig. 9. The agreement between the long- and short-term estimates of N is quite good for N < 0.5; the comparisons may be invalid otherwise, because multiple cloud layers may predominate when N > 0.5.

7. DISCUSSION AND CONCLUSIONS

It has been shown that it is possible to determine an average attenuation of direct-beam radiation for a single cloud, as well as a net transformation of direct to diffuse radiation per unit of cloud area. With mean values of these two quantities, it is possible to reconstruct total, direct, and diffuse irradiance values using standard surface observations of cloudiness and visibility for a given time and location. Estimates of cloud density might be used to provide insight into the variation of the direct beam attenuation a and the corresponding recovery t by increased diffuse radiation for certain cloud types[8], but suitable measurements are ordinarily not available. One can interpret the value of (1 - t) as

being the net shortwave albedo of the cloud, neglecting the few percent of radiation absorbed by cloud droplets. From this viewpoint, the present data indicate that the albedo of cumuliform clouds is about 0.5, which is in good agreement with theoretical estimates of 0.5-0.6 offered by Busygin et aL [9]. Their model gives an albedo approaching 1.0 for stratiform clouds of great optical thickness, but more realistic estimates may be given by Paltridge[10], whose largest values, obtained from aircraft measurements, approach 0.6. Many other workers have provided estimates of cloud transmission and reflection; cloud thickness, cloud shape and ground albedo play important roles. The crude model presented here lacks such refinements, but nevertheless should provide the solar-energy engineer with a method for obtaining good working estimates from standard meteorological observations.

Further development of the present model includes extension to situations in which multiple cloud layers exist and the cover is not uniform. For such conditions, we expect large hour-to-hour discrepancies between measured and predicted values of D and I, but averaging over several days should greatly reduce in- accuracies. Also, the effects of clouds on D and I when the Sun is within 30 ° of the horizon are not adequately understood, and in this case it may not be possible to use the observed cloudiness without estimates of cloud dimensions. Instead, it may be preferable to rely on the estimates of "sunshine" that are commonly recorded (this alternative is not acceptable when the Sun is at large elevation angles at which the presence of cirrus frequently has no effect on the reported sunshine values). Another consideration is that in the mornings and evenings changes occur in the height and structure of the planetary boundary layer that drastically alter the vertical distribution of scattering particles beneath the cloud layer; thus the adoption of a constant value for l can be misleading.

It appears that solar collectors that capture some diffuse radiation offer considerable advantage (in addition to the obvious one of continuing to function to some extent on completely overcast days). The decrease in direct radiation is substantially compensated by a concomitant increase in diffuse radiation. For scattered cumulus clouds, about 50 per cent of the loss in I is regained in D; for fairly heavy cirrus (e.g. cirrostratus), the fraction regained may be as high as 70 per cent.

The results presented here apply to radiation received at horizontal, flat surfaces. For inclined surfaces that view a particular section of the sky, the behavior of the incident diffuse and direct irradiations can be quite different from that described here. This can be taken into account with simple geometric transformations, provided the angular distribution of diffuse radiation is known.

Acknowledgements--The apparatus for measuring diffuse radia- tion was constructed by R. F. Selman, with the advice of P. Frenzen and B. B. Hicks, both of whom made valuable suggestions regarding the methods of data analysis. The radiation data were collected with the assistance of F. Kulhanek. Also, the manuscript benefited from readings by A. Rabl and R. M. Graven of the Solar Energy Group at Argonne.

~ R E N C E S

1. R. Winston, Principles of solar concentrators of a novel design. Solar Energy 16, 89 (1974).

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

3. G. Stanhill, Diffuse sky and cloud radiation in Israel. Solar Energy 10, 96 (1966).

4. M. L. Wesely and R, C. Lipschutz, An experimental study of the effects of aerosols on diffuse and direct solar radiation received during the summer near Chicago. Atmospheric Environment (accepted for publication in 1976).

5. R. J. List, Smithsonian Meteorological Tables. Smithsonian Institution Press, Washington, D.C. (1949).

6. A. J. Dyer and B. B. Hicks, Stratospheric transport of

Technical Note 473

volcanic dust inferred from solar radiation measurements. Nature 208, 131 (1965).

7. Work by T. G. Berliand as reported by J. C. K. Huang and J. M. Park, Effective cloudiness derived from ocean buoy data. J. Appl. Meteoml. 14, 240 (1975).

8. B. Haurwitz, Insolation in relation to cloudiness and cloud density. J. Meteorol. 2, 154 (1954).

9. V. P. Busygin, N. A. Yevstratov and Ye. M. Feygerson, Optical properties of cumulus clouds, and radiant fluxes for cumulus cloud cover. Izvestiya Atmos. Oceanic Phys. 9, 648 (1973).

10. G. W. Paltridge, Infrared emissivity, short-wave albedo, and the microphysics of stratiform water clouds. J. Geophys. Res. 79, 4053 (1974).