Solar Energy Vol. 22, pp. 155-164 O0384192XI7910"201-OI55FIg/$02.0010 © Pergamon Press Ltd., 1979. Printed in Great Britain

THE AVERAGE DISTRIBUTION OF SOLAR RADIATION-- CORRELATIONS BETWEEN DIFFUSE AND HEMISPHERICAL

AND BETWEEN DAILY AND HOURLY INSOLATION VALUESt

MANUEL COLLARES-PEREIRA~: and ARI RABL§

Enrico Fermi Institute, University of Chicago, 5630 S. Ellis Ave., Chicago, IL 60637, U.S.A.

(Received 15 June 1978; revision accepted 14 August 1978)

Abstract--The correlations of Hottel, Whillier, Liu and Jordan between diffuse and hemispherical, and between instantaneous (hourly) values and daily totals of solar radiation are recalibrated against pyrheliometer data for five stations in the U.S. The validity of the Liu and Jordan approach is confirmed, and numerical inaccuracies of the original correlations are found to arise from three factors: (i) reliance on uncorrected measurements of diffuse insolation with pyranometer plus shade ring, (ii) use of a single value of extraterrestrial insolation for a whole month, and (iii) neglect of seasonal variations in the diffuse/hemispherical ratio. The new correlation for the daily total ratio of diffuse over hemispherical insolation agrees with results reported for India, Israel and Canada which included the shade ring correction. This suggests that latitude independence is a good approximation. The new correlations imply that the diffuse component is significantly larger than that predicted by the original formulas of Liu and Jordan. Analytical expressions are presented the parameters of which are obtained by a least squares fit to the data. The resulting formulas provide a complete description of the long term average insolation incident on surfaces of arbitrary orientation; the only meteorological input is the long term average daily total of hemispherical insolation on the horizontal surface. Comparison between model and data shows an accuracy better than 3 per cent for the long term average insolation available to solar correctors, both fixed and tracking. The model provides a framework for optimal matching between solar radiation and ideal concentrators (based on the principle of phase space conservation). In addition to computing radiation availability for solar collectors, the model can be used for calculating heating and cooling loads of buildings.

I. INTRODUCTION

The only measurements of solar radiation for which long term records are available from a large number of loca- tions are measurements of hemispherical insolation (also called total or global) on a horizontal surface. However, in order to predict the energy delivery of any solar collector (other than a horizontal flat plate) one needs to know both the beam and the diffuse component of in- solation. Correlations of the type first derived by Liu and Jordan[l] supply that information, but questions[2-5] have been raised about the general validity of these correlations.

All previous studies of this type have been based on measurements of diffuse radiation by means of a pyranometer with shade ring. Since the shade ring blocks not only the solar disk but also a sizeable portion of the diffuse radiation, corrections must be applied which depend on insolation and incidence angle[6]. This prob- lem is avoided if the solar beam radiation is measured with a pyrheliometer. Recently pyrheliometer data have become available from the Aerospace Corporation [7] for Albuquerque, New Mexico; Fort Hood, Texas; Liver- more, California; Raleigh, North Carolina; and Maynard, Massachusetts; with approximately 2 yr at each station. We have analysed these data to reexamine the cor-

;Work supported by the U.S. Department of Energy, Division of Advanced Energy Projects.

:~Supported by the Instituto Nacional de Investigacfio Cientifica and the Centro de Fisica de Mat6ria Condensada, Lisbon, Portugal.

§Also at SERI (1536 Cole Blvd., Golden, CO 80401) and on leave of absence from Argonne National Laboratory.

relations between diffuse and global and between hourly and daily values of insolation.

As for possible sources of error, there are several differences between a pyranometer with shade ring and a pyrheliometer. A shade ring is a simple nontracking device, requiring only tilt adjustments every couple of days, and the relative calibration of diffuse and of hemispherical measurements can easily be established by removing the shade ring. A pyrheliometer (or equivalently a pyranometer with tracking occulting disk) measures the beam (or the diffuse) radiation directly, provided it tracks the sun accurately. The main problems with a pyrheliometer are its susceptibility to errors if the sun gets outside the 2.80 acceptance half angle, and the difficulty of verifying the calibration relative to the pyranometer. We have found slight variations in the behavior of diffuse insolation from one station to another (see Fig. 2d), but the data base is too short to determine whether such effects are real or due to instrumentation. We have, therefore, lumped all data together without distinguishing between different stations. Thus, any cal- ibration errors are likely to be washed out, and the statistics are improved to the point where seasonal trends can be established with confidence.

Since the present investigation concerns correlations for daily totals of radiation, we had to discard any days with incomplete records. We also screened the weather tapes for data which were suspect for the following reasons:

(i) zero pyranometer signal during central hours of the day (more than 2 hr from sunrise and sunset).

(ii) beam and hemispherical data imply negative diffuse component

155

156 M. COLLARES-PEREIRA and A. RABL

(iii) pyrheliometer reading too low when pyranometer shows high insolation, a condition strongly suggestive of misalignment. We rejected data when the hourly beam and hemispherical values lh,, and lh failed to satisfy the condition [8]

0.5 (1) L,., -L , . , cosO

where 0 is the incidence angle of the sun and L., the solar constant.

We discarded about 10 per cent of the data, mostly days with Incomplete or nonexistant records. Ap- plication of criteria (i), (ii) and (iii) does not have any significant effect on our results. Table 1 lists the stations, their latitude, longitude and elevation above sea level, the years of observations, and, as indication of the statistics, the number of days of data used for this study.

2. CORRELATION FOR DAILY VALUES OF DIFFUSE AND HEMIS- PHERICAL INSOLATION

Because of the variability of the weather the dis- tribution of diffuse solar radiation is very irregular, both in time and in space. Nevertheless the average charac- teristics of diffuse radiation are quite regular and can be correlated with an effective transmission coefficient of the atmosphere, as was first recognized by Liu and Jordan. Several different correlations can be obtained, depending on the averaging procedure and on the time interval chosen for the insolation data (e.g. hourly or daily). The present investigation is concerned with the relation between Ha, the daily total of diffuse insolation on a horizontal surface, and Ha, the daily total of global insolation on a horizontal surface. As correlation parameter one uses the ratio

Kh = H, IH,, (2)

of terrestrial over extraterrestrial radiation, Ho being the daily total radiation which would be incident on the same horizontal surface in the absence of any atmosphere.

The extraterrestrial radiation Ho is given by[9]

H,, T Io,[l+O.O33cos{ 2~rn ~1 = ~ ' \365.24/ cos3. cos 6

x (sin w, -w~ cos o)~) (3)

with T = length of day, 24hr; /o., = 1353 W/m 2 = solar constant: n = day of year (starting 1 January): 3, =geog- raphic latitude; tos = arccos [ - tan ;t tan 3] = sunset hour angle: and 8 = solar declination given by

sin 6 = 0.3979 sin 3,

(Note 0.3979 = sin 23.45°).

(4)

In the approximation of a circular orbit of the earth 3, is

21r(n + 284). (5a) 3 '~3,o= 365.24 '

for greater accuracy the expression

y[rad] = yo + 0.007133 sin 3'o + 0.032680 cos 3̀ o - 0.000318 sin 23,o + 0.000145 cos 23,0 (Sb)

is recommended by Ref. [10]. For the present analysis we used 8 and Ho as provided for each day with the weather data; they are based on the more accurate eqn (5b).

Kh has been called cloudiness index[l] but the name "clearness index" is more descriptive. When Ha/Hh data for individual days are plotted vs Kh, as for example in Fig. 3 of Ref.[3], they show a characteristic pattern, data points being clustered around the solid line in our Fig. l(a) but with considerable scatter. This scatter reflects the short term day-to-day fluctuations of the weather. To isolate the long term characteristics one averages over all values of Ha and Hh corresponding to a particular value o[ Kh. If the number of data is finite one must group data with slightly different Kh values together. With large Kh intervals, one reduces the stan- dard deviation of Ha/H, from the true average, but at the price of losing information about the correlation with K,. For the data at hand we found an interval of AKh = 0.05 to be a good compromise. The resulting pattern is shown in Fig. l(a). For each K, interval the error bars show the standard deviation

S.D. o f - ~ = ~/(((Ha/Hh ):) - ((Ha/Hh))2)

where (HdHh) indicates the average over all HdHh

Table 1. Data base for present study

Station Lat. deg. Long. deg.

Elevation above

sea level, m Years Number of Days

Albuquerque, New Mexico Fort Hood. Texas Livermore, California Maynard, Massachusetts Raleigh, North Carolina

35.05N 106.62W 1618 1961--64 1284

31.08N 97.85W 329 1974-76 589

37.7N 121.7W - 160 1974-75 230

42.42N 71.48W 63 1975-76 696

35.87N 78.78W - 130 1975-76 283

1.0

0.8

H---~d 0.6 Hh

0.4

0.2

o ' o'.2 ' o' . . . . .4 0.6 0.8 K h = H h / H 0

Fig. l(a). Plot of ratio HalHh between daily total diffuse and daily total hemispherical irradiation vs clearness index Kh = (HdHo), found in the present investigation and compared with the results published by Ruth and Chant[4] and Liu and

Jordan [l].

The average distribution of solar radiation correlations

- t , - ~ - 4 ~ | ' Fil ' ' '

LIu ond JordonJ''~.~

157

curve was based, were not corrected[Ill, whereas the data in Refs. [2-4] included the correction.

Until now the correlation between HdHa and Ka (as well as the one between /td//~a and /~h, see following section) has been assumed to be independent of time of year. However, since the amount of scattering of radia- tion passing through the atmosphere depends on the path length (or air mass AM = 1/cos 0, 0 being the incidence angle), and since average path length varies with time of year, one might expect a seasonal variation in the cor- relation for diffuse insolation. We have therefore separated the data according to their value of sunset hour angle to, [in rad] ~o, = (21rtJT) where t, = sunset hour and T = 24hr. The HdHh points of Fig. l(b) were obtained by grouping all data into three tos bins

I 0

0 5

[ ] Over All Averoge

W-Winter

S- Summer

F-Foil- Spring

- - PolynomioI fit

\

0 0.5 lO

Kh

Fig. I(b). Same as in Fig. i(a), but with data points separated according to season--Summer, Spring-Fall, Winter.

values in the Ka intervalt. A least square fit with a fourth degree polynominal yields the curve

H.

0.99 for Ka -<0.17

1.188 - 2.272 K, + 9.473 K, 2 - 21.856 Ka 3

+ 14.648 Ka 4 for 0.17 < K, < 0.8

(6)

which is shown by the solid line in Figs. l(a) and (b). It agrees quite well with the correlations for India[2], Israel[3] and Canada[4], but lies significantly above the original curve of Liu and Jordan[l]. The most plausible explanation for the discrepancy between the Hd/Hh curve of Liu and Jordan and all subsequent in- vestigations lies in the shade ring correction. The data of Blue Hill, Massachusetts, on which the Liu and Jordan

winter: t o s < 7 - 0 . 1 5

spring and fall: ~ - - 0.15 -< to, -<~+0.15

summer: oJs ->~-+ 0.15.

One can see the seasonal trend most clearly at large K, where summer has a higher ratio Ha/Ha than winter. At intermediate Ka the trend is reversed and less pronounced.

For a possible explanation of this effect we calculate the average air mass during the day

(AM) = (I/cos 0).

The precise averaging procedure does not matter for the sake of the present argument. The weighting factor used for this averaging should account for the variation of irradiance with incidence angle 0 which is given by

cos 0 = cos (5 cos A (cos co - cos to~) (7)

where co = (2¢rt/T) is the hour angle. It is most con- venient to assume a weight proportional to cos 0; for correct normalization the weighting factor cos 0 must be divided by

fo =' dto cos 0 = cos A cos (5 (sin tos - to, cos tos). (8)

(See also Section 4). The resulting daily average air mass is

0)s (AM) = (l/cos 0) cos ~ cos h (sin to~ - to~ cos ~o~)"

*There is a slight difference between averaging the ratio HdHh and averaging Ha and Hh separately since the same Kh value may be associated with several different values of Hh. We found, however, the numerical difference between these two averaging procedures to be insignificant. Similar conclusions apply to FIa/fI~ and to HdHo.

For the stations in Table 1 the lat. A is approx. 35 ° and (AM) varies from 1.64 at summer solstice to 1.92 at equinox to 2.95 at winter solstice.

In a homogeneous atmosphere with either isotropic or strongly forward (within a few deg.) scattering the ratio of diffuse over hemispherical insolation increases with

158

air mass. Such an effect shows up in the data only at intermediate K, and it is not very pronounced. Several other factors seem to mask this effect. Firstly, scattering by aerosols is in the forward direction with relatively large scattering angles, on the order of 20--60 deg. and hence a significant portion of the diffuse radiation fails to reach the surface of the earth when the incidence angle is large. Secondly, the atmosphere is not homogeneous but contains clouds of finite width and finite thickness. Therefore, an intermittant cloud cover transmits less radiation at large than at small incidence angles, and high values of Kh can be reached only on very clear days. This agrees with the trend observed by Bugler for hourly insolation data, see Fig. 8 of Ref.[12]. It also seems to be consistent with the results of Ref.[13]. Thirdly, there may be seasonal variations in atmospheric turbidity.

M. COLLARES-PEREIRA and A. RABL

I O

0 8

O 6

0 4

0 2

Winter

El Fall and Spring

X Summer

~ . x i ~

0 0 2 0 4 0 6 0 8 I O

Kh

3. CORRELATION FOR MONTHLY AVERAGES OF DIFFUSE AND HEMISPHERICAL INSOLATION

As shown by Liu and Jordan[l], the monthly averages /~d and Hh of diffuse and hemispherical insolation can be correlated with the monthly average clearness index

Fig. 2(a). Plot of the ratio fl, dlYth between "monthly" diffuse and "monthly" hemispherical irradiation vs "monthly" average clearness index /(h, for all stations. The data are grouped into bins according to their sunset hour angle; the solid and the dashed lines are plots of a least squares fit for each of the three

seasons.

g;h = fI,/ no. (9) A least squares fit yields the correlation

The difference between the correlation for HJH, and the one for Hd/Hh arises from the underlying averaging procedure. Whereas the Hd/Hh curve is obtained by averaging Hd and Hh data corresponding to one value of K,, a point of the Hd/Hh plot is the result of averaging Hs, H, and K, over all days of a month. The Hd/H, curve incorporates therefore the distributio,~ of clear and cloudy days during the course of a month.

Of course, the choice of a month as an averaging period is rather arbitrary. If one had data for a very large number of years, one could evaluate Hd/fth and/~, for a single day, say the N' th day of the year, by averaging lid, HE and K, measured on the N'th day of the year over all years for which data are available. The smaller the number of days used for averaging, the larger the resul- ting scatter of the FId/fth points. This is illustrated by the large scatter in Fig. 2(a) where each point represents the average over a 26 day period for a particular station and a particular year. (We chose 26 days as length of the "month", because it divides the year into equal "months", with only a minimal leftover of 1 or 2 days to be discarded). By averaging the monthly data over several years one reduces the scatter and arrives at the universal relationship between ffld/fflh and /~h.

TO test for seasonal variations we grouped the data into bins according to their sunset hour angle to,, as shown by the *, [] and X symbols in Fig. 2(a)

< E _ 0.15 (winter, *) t o s - - 2

E - 0.15 <tos < ~ + 0 . 1 5 (spring and fall, f7]) 2

> E + 0.15 (summer, X). ~os - 2

[ /~'d//~" = l -- [0.505 + 0.261 (to, -- 2) ] COS [2(gh -- 0.9,]

(10)

shown for co, = (z-/2) by the solid line in Fig. 2(a). The dashed lines are eqn (10) evaluated at to, =(~r/2)-0.2 and co, = (7r/2) + 0.2.

The seasonal trend in Fig. 2(a) is obscured by the random scatter of month-to-month variations. For a clearer demonstration of the seasonal pattern we have averaged the monthly points of Fig. 2(a) o v e r / ( , inter- vals of 0.05. The resulting seasonal averages in Fig. 2(b)

1.0

08

0.6

r="

0.4

02

X Summer

o Foll and Spring

i Winter

~ x

. It × ~ ×

×

x

0.2 0.4 0.6 0.8

Fig. 2(b). The same as in Fig. 2(a), but averaging the data points within /~h intervals of 0.05 but separated according to season.

IO

:E ~ 0 ,5

The average distribution of solar radiation correlations 159

In principle one could work with rb = [b/fib or rb., = [b.dflb., instead of either ra or rh, but in practice use of r~ and r, seems to be simpler (We did prepare plots of rb.. from our data, but found the pattern less clear than

Page for rd and rh). As shown by Hottel and Whillier[14] and ~ i by Liu and Jordan[l] the factors ra and rh depend to a

ptes~rtt

0.5 I 0

Fig. 2(c). The same as in Fig. 2(a) but comparing the least squares fit of the present investigation with the fits of Liu and Jordan[l]

and Page [16].

show a definite separation of summer (higher Ha/Hh) and winter (lower Hd/H,) from spring-fall, at least for all intermediate/~h in the range from 0.4 to 0.65. A possible reversal of fall-spring and winter for /~h <0.4 is not significant because of poor statistics. The pattern is less definite for very clear months/~h >0.65; most of these data come from a single station, Albuquerque, with high altitude 1600 m, and we do not know how typical they are of other locations; also the scatter for/~h > 0.65 in Fig. 2(a) is very large.

We have computed X_ 2, the sum of squares of the difference between the Ha/Hh points of Fig. 2(b) and the analytical fit (10). Without including seasonal dependence, i.e. using (I0) with to, = (¢d2) for all data, the value of X 2 is 3.11, whereas X 2 for the correlation with the to, depen- dence is only 1.55.

A comparison between the Ha/Hh fits of Liu and Jordan[I], of Page[16], and of the present investigation is shown in Fig. 2(c).

good approximation on only two variables: the time of day t, expressed as hour angle to = 2rrt/T, and sunset time t,, expressed as sunset hour angle to, = 2~rtJT.

The functions ra(to, tos) and rh(to, to,) are found empirically by plotting for each hour the average ratio of hourly f and dai ly /~ insolation on a plot of r = I-//4 vs to,. To reduce the scatter one averages over all days within an tos interval, for example, a month. We com- bined, as was done in Ref. [1] a.m. and p.m. data, because for most of the applications of this model slight morning- afternoon asymmetries do not matter. We did find slight evidence for possible systematic variations of ra with/~, and with location, but these variations are small com- pared to the scatter in the data, and the statistics of our data base is too limited for any definite statement about deviations from the simple relations established in Refs.[1] and [14]. Liu and Jordan have shown that rs is represented very well by the expression

7r cos t o - cos to, (13) ra( to , to,) = ~ sin to, - to, cos to,

with T = length of day = 24 hr. This is actually the ratio of extra-terrestrial irradiance on a horizontal surface, L cos 0, over extra-terrestrial daily irradiation Ho on the same horizontal surface. The reason for the remarkably close agreement between L cos O/Ho and [a/ffIa lies in the fact that the average transmission coefficient for diffuse irradiance ~d = [,dip cos O is quite insensitive to cloud conditions and incidence angle. This feature is also reflected by the near constancy of /~a in Fig. 2(d).

0 4

4. RELATION BETWEEN INSTANTANEOUS AND DAILY TOTAL IN-

SOLA] ' ]ON 0 3

For many applications one needs to know the in- stantaneous insolation, also called irradiance, Ib (beam), z ° la (diffuse) and Ih (hemispherical), rather than the dally ,2 o2 totals Ha, H, and lib = H , - Ha. Given the long term average daily total irradiation on a horizontal surface Hh, one can find the long term average irradiances [a and [ , o, and [b = [h - [a by first calculating/~a from eqn (10) and then multiplying Ha and Hh by the conversion factors

r a = _--.-- H a

O 0

and

+ FT Hood

[3 Livermore

X Rayleigh

• Maynard

Albuquerque

© Total average

h /'h ~ 7 .

0 2 0 4 0 6 0 8

(11) K.

Fig. 2(d). Plot of the ratio /-)J//o between the "monthly" average total diffuse and the extraterrestrial irradiation vs the "monthly" average clearness index /~h, for the five locations considered in this study averaging data points within/<h intervals

(12) of 0.09; the dashed line is a linear least squares fit to the total a v e r a g e .

160 M. COLLARES-PEREIRA and A. RABL

(Future investigations concerned with refinements of this model for ra might start with an investigation of the dependence of rd on Kh and 0). The solid lines in Fig. 3(a) show ra of eqn (13) for t = 0.5; 1.5; 2.5; 3.5; 4.5; 5.5; 6.5 hr from solar noon. The crosses are the correspond- ing averages for all data of the stations and years in Table 1. The close agreement for ra between Liu and Jordan and the present investigation comes from the fact that the ratio of hourly diffuse to daily diffuse radiation is insensitive to the shade ring correction.

For the hemispherical insolation rh differs markedly from eqn (13) because atmospheric attenuation of beam radiation increases with incidence angle. In Refs.[1] and [14] curves for rh were presented, without analytical expressions. As our new data agree so closely with the r, curves of Ref.[I] and as there is no reason to expect any discrepancy, we decided to combine the new data with corresponding points from the r, curves of Ref.[I], each data base weighted according to the number of years of observation. The resulting averages are shown by the crosses in Fig. 3(b). A least squares fit to these data in the form

71" COS to - - COS tos rh(to, w,) = ~ (a + b cos to)

sin 1 tos O) s COS (]4)

yields the coefficients

a = 0.409 + 0.5016 sin (to, - 1.047)

b = 0.6609 - 0.4767 sin (to, - 1.047).

The angles are specified in rad, and 1.047rad = 60 °. This fit is shown by the solid lines in Fig. 3(b). This fit satisfies the normalization condition

~ ' dt[h=Fl. t~

to within I per cent for all to, under consideration.

5. APPLICABILITY OF INSOLATION CORRELATIONS The correlations discussed in the present paper pro-

vide a complete model of average instantaneous insolation [15], both beam and diffuse, on a horizontal surface. Together with the additional assumption that the diffuse sky radiation is isotropic[16], this model uniquely determines the average insolation at any instant and on any surface. The only meteorological input needed is H,, the long term average daily total hemispherical insolation on a horizontal surface. The seasonal variation of /-1, will usually necessitate separate calculations for different times of the year, typically one calculation per month. Monthly averages of H, can be found in many pub- lications, for example, Ref.[l] lists 80 and Ref.[17] lists 200 stations in the U.S. and Canada. The present model is quite different from traditional insolation models which are based on data of atmospheric transmission coefficient, cloud cover and sunshine hours (10, 21, 23). Those models are appropriate for locations where only the latter type of data are available but the radiation has

Sunset Hour Angle, Ws, in Rodions from Solor Noon 1.04... 1.57... 2.09...

Hours from Sunrise to Suns'et 8 9 ~0 12 13 14 15 16

0.2 1

~ H o T~me of Doy urs from Solor Noon)

o.oi / . , I

6'0 7'5 9'o ,65 Sunsel Hour Angle ~s, in Degrees from Solor Noon

Fig. 3(a). Plot of rd (conversion from daily total to instantaneous for diffuse component) vs sunset hour angle for different times of

the day: eqn (13) is shown to the solid lines.

0.2

i l i g

0.0

Sunset Hour Angle, w s in Rodions from Solor NOO~ 1.04 L57 2.09

I I I Hours from Sunrise to Sunset

8 9 I0 II 12 13 (4 t5 16

' Time o'f Doy' ' s from %lor Noon)

3 1 1 2 x x ~ x ~ l '

, S .

6b 7'5 9'0 Sunset Hour Angle, =s, in Degrees from Solor Noo~

Fig. 3(b). Plot of rh (conversion from daily total to instantaneous for hemispherical component) vs the sunset hour angle for different times of the day; the solid lines are the least squares fit

to these data, eqn (14).

not been measured. In those models the insolation H, is calculated. The model presented in this paper assumes the measured Hh as a starting point, which is more directly relevant and therefore provides a more reliable and accurate basis for predicting radiation intensities. Furthermore, even for locations where Hh has not been measured or where the data are questionable (among the dubious stations quoted in Ref.[17] the most infamous is Inyokern with unrealistically high values o f /£ , greater than 0.8) the model can still serve as unifying reference frame for comparison studies of solar collectors [18].

The average distribution of

The model can be used to calculate the solar con- tribution to heating and cooling loads of buildings as well as the energy incident on solar collectors of various kinds. The method is applicable not only to f/ate plate collectors and collectors that employ optical focussing but also to non-imaging collectors that concentrate the solar radiation as much as permitted by the second law of thermodynamics[24, 25] (phase space conservation). For photovoltaic collectors the delivered energy is pro- portional to incident energy. For thermal collectors, on the other hand, the relationship between incident and delivered energy is nonlinear because the collector is turned on only when heat gain exceeds heat loss. This effect can be accounted for by means of a utilizability function which is based on the fractional time dis- tribution of clear and cloudy days. We have treated this problem in another publication where explicit formulas for the energy delivery of solar collectors are derived [ 18].

For an example, we calculate the long term average insolation on an east facing vertical wall in Rome on 15 March at 8:00 in the morning. The latitude is ~. = 41.8 °, the sunset hour angle is to, = 1.57 rad and the hour angle is to = -7r/3. The irradiance on the wall is

COS Owall . . . . 1 = + ~ [h

with incidence angles given by eqn (7) for 0 and

cos 0wa, = -- COS 8 sin to

for 0wa,. From Ref.[22] the long term average hemispherical

insolation H, is Hh = l l.96J/m 2. Intermediate steps of the calculation yield 14o = 27.94 M Jim 2, /(h = 0.43, Ha = 5.72 MJ/mL rd = 0.064, rh = 0.056, [h = 186 W/m s, /'~ = 102 W/m 2. Assuming the ground reflectance to be p = 0.2 the result is

L~. = 284 W/m 2.

The remarkable simplicity of this insolation model derives from the observation of Liu and Jordan that the average effects of clouds on transmission of solar radia- tion can be characterized by a single parameter K, ("clouds are clouds", so to speak). This appears to be quite a good model of nature.

6. ACCURACY OF INSOLATION MODEL

In order to evaluate the accuracy of the present model, one should compare the model predictions with insolation data for tilted and for tracking surfaces. There are almost no long term insolation data for tilted or tracking surfaces[19], and even hose that are available are problematic unless the reflectivity of the ground is known and the pyranometer has been carefully cali- brated for tilt dependence of sensitivity[26, 27]. Instead we have used the data base in Table 1, calculating/~, for each month and comparing the model predictions/4=o~e~ based on this value of/4h with values/qdata obtained by

solar radiation correlations 161

summing hourly insolation data (with the usual isotropy assumption for the diffuse component). This procedure may appear to be circular in the sense that this very data base was used for deriving the correlations. This is unavoidable, however, because one must use all avail- able reliable data in order to develop such a model to its greatest accuracy, and then one has no choice but to reuse the data to test the model predictions. This test is genuine insofar as it yields a measure of the average and of the maximum error of the predictions.

We have computed the deviation

] 'I coll. model -- /~ coil,data 6 = n coil, data

between data and model, both for a point focus concen- trator (2 axis tracker) and for a flat plate collector. (For the point focus concentrator/4co, is the daily total beam radiation at normal incidence). We found that the average error (over all stations and months) is less than 3 per cent for the insolation Hco, available to the collector, both for flat plate and for concentrating collectors. The model appears to be free from any significant bias with respect to location or time of year at least as far as could be discerned from this rather limited data base. For a particular month of a particular year the discrepancy between model and data can be much larger, on the order of 3 per cent (standard deviation) for a flat plate and on the order of 10 per cent (standard deviation) for concentrators. Results for other collector types fall between those for flat plate and for point focus concentrating collector.

The error ~ for the point focus concentrating collector is plotted vs tos in Fig. 4(a) for each month at Ft. Hood, Livermore, Maynard and Raleigh, as indicated by the letters F, L, M and R. In view of the year to year fluctuations the available statistics is too limited to decide whether a particular station deviates from the general pattern. (Livermore with the largest apparent deviation also has the poorest statistics). Even a station with almost constant sunshine such as Albuquerque shows strong differences between different years. This is demonstrated in Fig. 4(b) where the individual years 1961--64 for Albuquerque are labeled separately; there noticeable underprediction (-0.017) for 1962 and over- prediction for 1964 (+ 0.027).

In order to isolate the origin of these large fluctuations in the prediction of the daily total beam irradiation, we have rerun this test, replacing Ha//4h from eqn (10) by the value actually measured for each month. The resul- ting average error remains nearly unchanged (from 0.005 for all data to 0.014), but the standard deviation is reduced significantly (from 0.09 to 0.03). Therefore the fluctuations are mostly due to the ratio Hd/H, and not due to rd of r,. The point focus concentrator represents, of course, the extreme in its sensitivity to the diffuse/hemispherical ratio ISIa/ffI,, whereas the flat plate is much less affected by it.

In any case, even though large, the fluctuations do not matter, provided the long term average is correct. After all, the model is designed to predict only the long term average insolation not the insolation for a particular month.

162 M, COLLARES-PEREIRA and A. RABL

0 . 3

o

0 . 2

%

~ o ,

£

i..r ~u

1

- 0 5

F • F

1 ' 2 - " ~ " ' ~" h I 6 ~ I 7 18 19' 2 0

" ~ 2 " ( in r o d i a n s )

Fig. 4(a). Plot of the error e = ( l " ~ ' c o l l . m o d e l - Ucoll.dalJ/'lcoll.data) VS W, = the sunset hour angle, for each month at Fort Hood (F), Livermore (L), Maynard (M), Raleigh (R).

° t =" O I

o 0

E

I ~ - O I

~E

-oz

. O 3 [

o B [ !

0 2 1 i

l

I I 1'2 ~ , ~ I 9 2 1 0

z ~ oj s

( i n r o d i e n s )

Fig. 4(b). The same as in Fig. 4(a) but for Albuquerque with the years 1961--64 labelled separately by the Indexes I--4.

It is instructive to show why the Liu and Jordan method should predict the long term average correctly by its very construction.

As stated above, with the isotropy assumption for the diffuse component, the long term average insolation on any aperture is uniquely determined by [h and [d. These in turn can be written in a form which shows explicitly the long term average as a summation over a period of N days

h(o,) = 1 ~ I,,(to). (15)

Mter multiplying and dividing by

1 / ~ = ~ H h (16)

The analogous statement for fd is

1 fd(oJ) = ~ Y ~ Id(~o)

_ ~Id(co) ~.Hd I ~ H h Y.Hd ~Hh

= r.(~o)FI.. (18)

Therefore the long term average insolation must be cor- rect if rh, rd and /'~d//4h are correct. The validity of the present model is guaranteed to the extent to which these correlations can be described by location independent analytic fits.

this can be rearranged as

Z Is(t°) l Z = rh(to)flh

i . ( , , , ) = Y~ 1-I.

using the definition of rh, eqn (12)

(17)

Acknowledgements--We are grateful to Dr. S. A. Klein and Dr. M. L. Weseley for valuable comments, and to Prof. R. Winston for support and encouragement during the course of this work.

NOME2qCLATURE In an attempt to be systematic and precise we were forced to

depart slightly from the notation established by Liu and Jordan, tin particular with regard to subscripts. We use the symbols I for

The average distribution of solar radiation correlations 163

irradiance (or instantaneous insolation, in Wire 2) and H for irradiation (or daily total insolation, in j/m2), together with subscripts b for beam (also called direct), d for diffuse and h for hemispherical (also called global or total). All radiation values refer to a horizontal surface, unless otherwise specified. Beam radiation on a surface normal to the sun is indicated by an additional subscript n as in Ib.n and Hb.n. Bars indicate long term average. Note that beam is defined with respect to the 2.8 ° acceptance half angle of the pyrheliometer, and not with respect to the solar disc; thus it includes the circumsolar component[20].

Ho extraterrestrial irradiation (daily total) Hb beam irradiation on horizontal surface (daily total)

Hb.~ beam irradiation at normal incidence (daily total) /-/con irradiation incident on collector aperture (daily total)

Hd diffuse irradiation (daily total) /'l"d long term average diffuse irradiation Ha hemispherical irradiation (daily total) /~h long term average hemispherical irradiation Io.,, solar constant = 1353 W/m 2

Ib beam irradiance on horizontal surface lb.n beam irradiance at normal incidence

Ia diffuse irradiance I~ hemispherical irradiance

Kd H.,/Ho Kh HdHo = clearness index Kh HdHo = long term average clearness index rd l,dH,~ = conversion from daily total to instantaneous for

diffuse component rh lh/Hh = conversion from daily total to instantaneous for

hemispherical component T length of day = 24 hr = 86,400 sec t time of day from solar noon (p.m. is positive)

ts sunset time 8 solar declination a geographic latitude 0 incidence angle to 2~/T = hour angle

oJ~ 2~dT = sunset hour angle

REFERENCES

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

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

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

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

5. J. F. Orgill and K. G. T. Hollands, Correlation equation for hourly diffuse radiation on a horizontal surface. Solar Energy 19, 357 0977). The correlations for hourly irradiation[5, 13] are different from the correlations for daily irradiation (Refs. [I--4] and present investigation). The daily correlation is the result of averaging hourly irradiation values over all hours, clear and cloudy, occuring during the course of the day. The difference is analogous to the difference between Hd/Hh and /~d/~h. It may therefore, be misleading to present hourly and daily correlations on the same plot as was done in Refs. [5, 13].

6. A. J. Drummond, On the measurement of sky radiation. Arch. Met. Geoph., Series B, L 413 (1956). A brief summary of Drummond's shade ring corrections is given in Ref.[3]. The correction factor for diffuse radiation is significant, for example, at Gilat, Israel, it ranges from 1.12 to 1.34.

7. Aerospace Report No. ATR-76 (7523-I 1)-9, The Aerospace Corporation, P.O. Box 92957, Los Angeles, CA 90009. We should like to thank Dr. C. Randall of the Aerospace Corporation for helpful comments about the data.

8. M. L. Weseley, Meteorology Division, Argonne National Laboratory, Argonne, IL 60439, private communication.

9. J. A. Duffle and W. A. Beckman, Solar Energy Thermal Processes, Wiley, New York (1974).

10. R. L. Hulstrom and M. Imamura, Definition study for pho- tovoltalc residential prototype systems. Final Report, Ap- pendix. Martin Marietta MCR-76-394 0976).

11. B. Y. H. Liu, Dept. of Mechanical Engineering, Univ. of Minnesota, private communication.

12. J. W. Bugler, The determination of hourly insolation on an inclined plane using a diffuse irradiation model based on hourly measured global horizontal insolation. Solar Energy 19, 477 (1977).

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

14. H. C. Hottel and A. Whillier, Evaluation of flat plate solar collector performance. Trans. Con/. on the Use of Solar Energy: The Scientific Basis, Vol. II, Part l, Section A, pp. 74-104 (1955).

15. If hourly radiation values are needed for a detailed system simulation, correlations between Ib and l~ must be used. These have been developed by the Aerospace Corporation and by Sandia Laboratories, see e.g.E. Boes, Estimating the direct component of solar radiation. SANDIA Rep. SAND 75-0565. C. M. Randall, M. E. Whitson, Jr., Final report hourly insolation and meteorological data bases including improved direct insolation estimates. Aerospace Rep. No. ATR-78 (7592)-1.

16. 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. 35/5/98 (1961). Under clear conditions there appears to be a brightening of the sky close to the sun. See Ref.[21].

17. W. A. Beckman, S. A. Klein and J. A. Duflie, Solar/-/eating Design by the f-Chaff Method. Wiley, New York (1977).

18. M. Collares-Pereira and A. Rabl, Simple Procedure for pre- dicting long term average energy delivery of nonconcentrat- ing and of concentrating solar collectors. Presented at Systems Simulation and Economic Analysis for Solar Heat- ing and Cooling, San Diego, 27-29 June 1978, and submitted to Solar Energy.

19. S. A. Klein, Calculation of monthly average insolation on tilted surfaces. Solar Energy 19, 325 (1977).

20. D. Grether, J. E. Nelson and M. Wahling, Measurements of Circumsolar Radiation. Lawrence Berkeley Lab. Report NSFlRANN/SEIAB-5361PRI7414 (1975).

21. On the Nature and Distribution of Solar Radiation. Prepared for U.S. Department of Energy by Watt Engineering Ltd., HCP/T2552-01 UC-59, 62, 63A (1978).

22. S. Barbara, S. Coppolina, C. Leone and E. Sinagra, Global solar radiation in Italy. Solar Energy 20, 431-435 (1978).

23. K. Ya. Kondratyev, Radiation regime of inclined surfaces. WMO--No. 467 (1977).

24. R. Winston, Light collection within the framework of geometrical optics. J. Opt. SOc. Am. 60, 245 (1970).

25. A. Rabl, comparison of Solar Concentrators. Solar Energy 18, 93 (1976).

26. D. J. Norris, Calibration of pyranometers in inclined and inverted positions. Solar Energy 16, 53 (1974).

27. K. A. Reed, Inclination dependence of pyranometer sensi- tivity. Proc. Syrup. Photo-Optical Instrumentation Engrs. Vol. 161 (1978).

Notes added in proof (i) Since completion of this work we received two interesting papers by Dr. John E. Hay, which we would like to

acknowledge: The first: A revised method for determining the direct and diffuse components of the total short-wave radiation. Atmosphere 14, 278 (1976), investigates the effect of multiple reflections between earth and sky on the correlations between diffuse and hemispherical insolation. When the contribution from multiple reflections is

164 M. COLLARES-PEREIRA and A. RABL

subtracted from the diffuse and hemispherical radiation data, the seasonal variation of the correlation is significantly reduced, for the locations considered (Toronto, Montreal, Goose Bay and Resolute in Canada).

The second paper, Measurement and modeling of shortwave radiation on inclined surfaces. 3rd Conf. Atmos. Rad., pp 28-30 June 1978, Davis, California, published by American Meteorological Society, Boston, Massachusetts, presents an analysis of radiation data on inclined surfaces, with due regard for such effects as the tilt dependence of pyranometer sensitivity and the radiation reflected by the ground. Under clear sky conditions the isotropic model for diffuse sky radiation seems to result in underprediction, on the order of 3 per cent, of the hemispherical radiation available to tilted south facing surfaces, if the data in this paper (August, 1977, in Vancouver, Canada) axe typical.

(ii) After a closer look at the data in Ref. [22] we are puzzled by the low insolation values reported for Italy. The associated Kh values are markedly smaller than those for seemingly similar climates in the U.S. (see Ref. [17]) and Portugal (Servico Metereoltgico Nacional) thus raising the question of possible calibration errors.

R~sum~---Les corrtlations 6tablies par Hottel, Whillier, Liu et Jordan entre les radiations diffuse et htmisphtrique (globale) et entre leurs valeurs instantantes (horaires) et journalitres sont reajusttes en les comparant avec les donntes exp~rimentales pour la radiation directe provenant de cinq Iocalitts diftrentes aux Etats Unis. La validit6 de la m~thode utiliste par Liu et Jordan est confirmte et les imprecisions numtriques des corrtlations ~tablies aupaxavant sont corrigtes. Trois facteurs sont ~ la source des probltmes associts avec ces corrtlations: (i) des mesures non corrigtes de la radiation diffuse obtenues avec un pyranomttre muni d'un anneau ombrageant; (ii) une seule valeur de la radiation extraterrestre pour tout un mois; (iii) variations saisonnitres du rapport entre les radiations diffuse et htmisphtrique ignortes. Les nouvelles corrtlations sont compatibles avec les rtsultats obtenus pour le Canada, I'Inde et Israel en utilizant la correction de I'anneau ombrageant. Donc l'approximation de rindtpendence par rapport ~ la latitude est excellente. Les nouvelles corrtlations impliquent une valeur pour la composante diffuse significativement plus grande que celle predite par les formules de Liu et Jordan. Les paramttres des expressions anlytiques postultes sont obtenues par la mtthode des moindres carrts. Les formules qui en rtsultent fournissent une description compitte de la moyenne ~ tong terme de la radiation incidente ~ des surfaces d'orientation quelconque; le seul input mtttrtolngique est la moyenne ~ long terme du total journalier de la radiation htmisphtrique sur une surface horizontale. Les comparaisons entre le modtle et les donntes exptrimentales montrent une prtcision sup~rieure ~ 3% pour la moyenne h long terme de la radiation disponible aux capteurs d'tnergie solaire, soit fixes soit mobiles. Le modtle permet de calculer la radiation disponible h tousles types de capteurs.

Zusammeaf~aag--Die Korrelationen yon Hottel, Whillier, Liu und Jordan zwischen diffuser und hemisphaerischer Sonnenstrahlung und zwischen stuendlichen und taeglichen Strahlungswerten werden geprueft und verbessert. Als Gundlage kienen fuenf Stationen in den U.S.A. fuer welche stuendliche Messwerte sowohJ fuer hemisphaerische als auch fuer direkte (Pyrheliometer) Sonnenstrahlung vorliegen. Die prinzipielle Gueltigkeit der Liu und Jordan Methode wird bestaetigt, und numerische Ungenauigkeiten der urspruenglichen Korrelationen werden korrigiert. Wesentliche Maengel der urspruenglichen Korrelationen lassen sich auf die folgenden drei Aspekte der Arbeit yon Liu und Jordan zurueckfuehren: (i) sie verliessen sich auf Messungen der diffusen Strahlung mittels Pyranometer mit Schattenring, welche nicht fuer den Schattenringeffekt korrigiert waxen; (ii) fuer die taegiiche ausserirdische Strahlung benutzten sie einen einzigen Wert ueber den ganzen Monat; (iii) sie bcruecksichtigten nicht die Moegiichkeit jahreszeitlicher Schwankungen im Verhaeltnis yon diffuser zu hemisphaerischer Strahlung. Die neuen Korrelationen fuer diffuse Strahlung stimmen ueberein mit Ergebnissen aus Indien, Israel und Canada; der Schluss liegt daher nahe, dass sic zu guter Naeherung unabhaengig yore Breitengrad sind. Der Anteil der diffusen Strahlung ist wesentlich hoeher als im urspruenglichen Modell yon Liu und Jordan. Alle Korrelationen in dieser Arbeit werden als analytische Ausdruecke gegeben, deren Parameter auf der Methode der kleinsten Fehlerquadrate bernhen. Die Formein bieten eine vollstaendige Beschriebung des langzeitlichen Mittelwertes der Sonnenstrahlung auf Flaechen beliebiger Orientierung. AIs einziger meteorologischer input wkd der langzeitliche Durchschnitt der horizontalen hemisphaerischen Sonnen- strahlung benc~tigt. Vergieich zwischen Modell und Daten zeigt Genauigkeit zu 3% fuer die Straldung, die festen oder nachgefuehrten Solaxkollektoren zur Verfuegung steht. Das Modell kann auch zur Berechnung des solaxen Anteils der Heiz- oder kuehlbelastung yon Gebaeuden dienen.