So/ar E~rgy, Vol. 22, pp. 225-228. Petlltmoe Press 1979. Prieted in Great Britain

AN ATMOSPHERIC MODEL FOR COMPUTING DIRECT AND DIFFUSE SOLAR RADIATION

S.BARBARO, S. COPPOLINO, C. LEONE and E. SINAORR Istituto di Fisica della Facolta di Ingegneria, Universit~ di Palermo, Palermo, Italy

(Received 23 May 1978; revision accepted 18 July 1978)

Abstract--This paper proposes an atmospheric model, which extends the computation of the direct radiation given by Cole's model to the cloudy sky and shows a method to calculate the diffuse radiation.

Therefore the monthly average values of the global radiation incident on a horizontal surface at Palermo are computed by this method. These values are compared with the experimental data provided by the lstituto di Idraulica Agraria, Univeralti di Palermo and generally exhibit a mean deviation not more than 10 per cent. The deviations become lower than 8 per cent taking into account the effect of the underlying surface albedo.

In order to better verify the validity of the proposed method, it should be suitable to extend its application to other locations provided with actinometric stations. This should allow to use this method with more realibility to predict the radiation incident on the locations lacking in actinometric data.

INTIIOmJCrlON

Several attemps have been made in order to give a theoretical basis to the various formulas used in the computation of the solar radiation. Hence, different at- mospheric models have been proposed, such as Moon's, Cole's and Hottel's[l--4]. The aim of this paper is to extend Cole's atmospheric model, which interprets in a simple way the effect of the attenuation processes (ab- sorption and scattering) of the solar radiation in the atmosphere, but deals only with the direct radiation in a cloudless sky. The here proposed model shows a method for the computation of the direct and diffuse radiation in a cloudy sky. It is based upon the following, generally accepted, statement:

The variation of the spectral composition of the ex- traterrestrial radiation depends mainly upon (a) the selective absorption of the water vapor in the near infrared, (b) the absorption and scattering processes by aerosols, (c) the Rayleigh scattering.

I. ~ AND DII~U~ SOLMI ~TION

The attenuation law of the instantaneous direct solar radiation can be expressed in a simple form similar to the one used for a monocromatic radiation [3]:

Iz = Ae -e" (I)

where m is the air mass, A is the apparent solar ir- radiance or the apparent solar constant at air mass zero and B the atmospheric extinction for a "fictitious" monocromatic radiation.

The values of A and B could he computed using Moon's fundamental relations[l] for atmospheres with different values of atmospheric parameters. In order to avoid the difficulties of applicability of Moon's relations, Cole has performed a fitting for different values of air mass m. Particularly considering the air mass range m = 1-5, Cole has determined the attenuation effect of the precipitable water depths w for an assumed dust

content d (d = 400 particles/cm3). Then he has derived the following general formula as a function of the parameters m, w, d

Iz = ~ exp [al + bl w- as(d - 400)]

× exp [- (a~ + b2w + bs(d - 400))m} (2)

where I, is the extra-terrestrial solar irradiance in the N-th day of the year, expressed by:

/~ = Io[ 1 + 0.035 cos (21r(N - 4)/366)] (3)

and the coefficients a, b have the following values:

ai=-0.13491; b,=-0.00428; a2=0.13708;

b~=0.00261; as=O.368x 10-4; b3 = 1.131x 10 -4.

The computation of w requires the knowledge of the vertical profile of water vapor content B(h), generally given by the exponential law[5]:

8(h ) = 8oexp ( - h/H - Bh 2) (4)

where 80 is the water vapor content in g/m s at sea level and h is the height in Km of the considered point. It is generally accepted [6] that H = 2 Km, B = 0 and then, by integration of eqn (4), we obtain.

w = 8(y) dy = 28(h) (5)

At sea level, eqn (5) becomes:

w = 28o. (6)

We note that 8o can be computed by the relation 80 = m*~, where # is the relative humidity in per cent and m* is the mass density of saturated water vapor.

225

226 S. BAt~ARO et. al.

By integrating eqn (2) we derive the daily direct radiation

f t 2 lc = I~ cos z dt (7) i °

where the limits of integration t, and t2 are respectively the sunrise and sunset, and z is the zenith distance. The direct solar radiation I¢ decreases in the presence of clouds by a factor K depending on a certain parameter c characterizing the sky cloudiness. Therefore the direct radiation I, is given by:

In = K I~. (8)

Because the scattering functions involved in the computation of diffuse radiation are extremely compli- cated, it is convenient to use empirical formulas. Then the instantaneous diffuse radiation D: incoming in a clear sky on a horizontal plane can be computed by the following relationship:

D~=K~(L~-I,) (11)

where K: is an empirical coefficient depending on solar height. For albedo near 0.25, K~ can be expressed by [11]:

K, = 0.5 cos ~13z. (12)

Generally K can be expressed by the relative insolation s/S, which is the ratio of the real sunshine durations s to the possible value S. Such an approximation is necessary because the information on cloudiness is unsatisfactory.

Let us now consider the diffuse radiation which in- comes on the earth's surface as a result of simple or multiple scattering. Assuming that absorption occurs before scattering, the scattered radiation can be computed by the difference between the radiation depleted only by absorption and the direct one when absorption and scattering effects are included.

Since the prevalent contribution to absorption is given by water vapor, we propose the following relation for the computation of I~, that is the instantaneous value of the radiation transmitted in absence of scattering:

l~,=l.-d~ (9)

The daily diffuse radiation De is obtained by eqn (11) multiplying by cos z and integrating through the whole solar day:

D¢=ft"D, coszdt. (13)

In the presence of clouds the diffuse radiation Dn can be computed by:

D. =D¢(I - n ) + k * n ( I ¢ + D c )

= D¢slS+ k* (1 - slS)(l¢ + D¢) (14)

where k* is the empirical transmission coefficient, whose values shown in Table 1 have been obtained by Berland for different latitudes L[12].

where 4' is the solar radiation absorbed by water vapor in the cloudless atmosphere.

MacDonald has analyzed Fowle's experimental data on solar radiation absorption in the atmosphere and has derived the energy absorbed by water vapor in the different bands[7]. The obtained values are reproduced with a good approximation using the formula here we propose:

I n = / r [0.938 exp ( - 0.0154 row)] + 0.004 (row) "l

-1.1086 X lO-S(mw) 3

+ (121.948(1 + mw))/(l + 10(row) 2) (10)

where--row, representing the real thickness of the pre- cipitable water layer, is expressed in mm H20, I,,z and I. in cal.cm-2.min-L The range of validity of eqn (10) is larger than other simple formulas generally used[8-10].

2. A ~ C MOD~ APPLICATION

The atmospheric model shown in this paper has been applied to the computation of the direct and diffuse radiation incident on horizontal surface in Palermo.

Here are used the experimental data of temperature and relative humidity measured during the period 1970--74 by the Institute of Idraulica Agraria and sunshine records of the Astronomic Observatory during the above period. The monthly average values of relative sunshine s/S, temperature 0, relative humidity ~, and precipitable water depth w are reported in Table 2. Assuming K* -- 0.33 and d = 200 particles/cm 3 for the clean atmosphere in Palermo, the monthly average values of the following quantities are computed and listed in Table 3:

(a) direct radiation fc and diffuse radiation /5c in a cloudless sky,

(b) direct radiation .T and diffuse radiation/Sn in a cloudy sky.

Table 1. The coefficient k* values for different latitudes L

L ° 75 70 65 60 55 50 45 40 /

K x 0 .55 0 .50 0.45 0 .40 0 .38 0 .36 0 .34 0 .33

L ° 35 30 25 20 15 10 5 0

K ~ 0 .32 0 .32 0 .32 0 .33 0 .33 0 .34 0 .34 0 .35

An atmospheric model for computting

Table 2. Monthly average values of the relative sunshine siS, the temperature 0, the relative humidity ~ and the precipitable water

depth w in Palermo (1970-74)

JAN

FEB

MAR

APR

MAY

3UN

JUL

AUG

SEP

OCT

NOV

DEC

s/S 8(oC) ,.p(%] w(mm)

0 . 3 8 3L'14.67 7 3 . 6 18 .46

0 . 3 9 1 5 . 1 2 6 9 . 8 16.O1

0 . 4 4 1 5 . 8 4 6 7 . 0 16.05

O.51 19 .57 6 2 . 8 2 1 . 2 6

0 . 6 4 2 2 , 7 2 6 8 . 3 2 7 . 6 8

0,70 25.78 68.4 32.87

O,80 2 6 . 4 3 6 6 . 5 38 .02

0,77 2 9 . 7 5 6 9 . 3 3 7 . 9 9

0 .61 2 7 , 4 6 72 .6 38 .25

0 . 5 3 2 3 . 5 7 71.9 30.60

0,56 19.20 70.2 23.19

O.38 15 .71 74.7 19 .97

direct and diffuse solar radiation 227

January and December (see Table 3). The computed values are always lower than the experimental ones probably because the underlying surface albedo a is not taken into consideration. Hence, it is necessary introduce a corrective factor in the global radiation G and for this purpose we have used the relation proposed by Averkiev [ 13]:

G* = G/(I - ay) (15)

where y = 0.2 + 0.5 (1 - siS). The monthly average values G* for a = 0.2 are listed

in Table 4 and compared with the experimental data: the deviations do not exceed 8 per cent,

It should be suitable to extend the application of the proposed method ot other locations provided with actinometric stations in order to better verify its validity.

Table 3. Monthly average values of solar radiation in Palermo: fc, dail]~ direct radiation in a cloudless sky. cal cm -2 day-';/~¢, d~y diffuse radiation in a cloudless sky, cal cm -z day -j, I,, daily direct radiation in a cloudy sky, cal cm -z day°'; 19,, daily diffuse radiation in a cloudy sky, cal cm -2 day-'; G, daily global radiation computed, cal cm -2

day-'; F, daily global radiation observed, cal cm -2 day-'; p, per cent deviation between G and F

JAN

FEB

MAR

APR

MAY

JUN

JUL

RUG

SEP

OCT

NOV

DEC

Ic bc ~n i n 0 ~

206 49 78 72 150 t 70 - 1 1 . 8

293 62 114 95 209 223 - 6 . 3

410 80 180 129 309 328 - 5.8

508 I00 289 148 407 416 - 2.2

552 121 353 158 511 514 - 0 .6

558 134 391 163 554 572 - 3.1

522 137 4i8 155 573 574 - 0.2

466 124 359 143 502 532 - 5.6

375 105 229 126 385 393 - 9 . 7

294 77 156 i00 256 270 - 5.2

221 55 124 72 196 216 - 9.3

160 46 68 85 133 !55 -14.2

Table 4. Monthly average values of solar radiation in Palermo, corrected by albedo. G*, daily global radiation computed, ca] cm -2 day-'; F, daily global radiation observed, cal cm -2 day-t; p,

per cent deviation between G* and F

G x F 0

FEB

MAR

APR

HAY

JUN

JUL

AUG

SEP

OCT

NOV

DEC

167 170 - 1 . 8

232 223 +3.9

342 328 +4.3

447 416 +7.5

552 514 + 7 . 4

595 572 +4.O

609 574 +6.1

536 532 +0 .8

385 393 - 2 . O

280 270 +3.7

214 216 -O.9

148 155 - 4 . 5

By compa[i_'ng the global radiation G obtained as a sum of I, and D, with the monthly average experimental value F provided by the Institute of Idraulica Agraria we note the deviations do not exceed 10 per cent except for

It shall allow the model may be used with good reliability to predict the radiation incident on locations lacking in actinometric data. Still another improvement of the results would require direct measures of dust as parti- cles/cm 3 and the knowledge of the vertical humidity profile.

i

Wu~U~NCES 1. P. Moon,Proposed standard radiation curves for engineering

use. J. Franklin Inst. 230, (5) 583-617 (1940). 2. g. J. Cole, Direct solar radiation data as input into mathema-

tical models describing the thermal performance of buildings, I. A review of existing relationships which predict the direct component of solar radiation. Building and Environment, Vol. II, pp. 173-179, Pergamon Press, Oxford (1976).

3. R. J. Cole, Direct solar radiation data as input into mathema- tical models describing the thermal performance of buildings, II. Development of relationships. Building and Environment, Vol. II, pp. 181-186, Pergamon Press, Oxford (1976).

4. H. C. Hottel, A simple model for estimating the trans- mittance of direct solar radiation through clear atmospheres. Solar Energy, 18, 129-134 0976).

5. L. T. Matveev, Physics of the Atmosphere. 669, Israel Pro- gramm for Scientific Translations, Jerusalem (1967).

6. A. W. Stratton, Progress in Radio Science. Vol. Ill, p. 279, Elsevier, Amsterdam (1956).

7. J. E. MacDonald, Direct absorption of solar radiation by atmospheric water vapor. J. MeteoroL 17(3), (1960).

228 S. B A I ~ o et al.

8. F. Mfliler, Strahlung in der unteren Atmosphire. Springer, Berlin (1957).

9. F. Kasten et al. On the heat balance of the troposphere. Final Ref. Contrat A F 61 (052), 18 Mainz. (1959).

10. A. Angstr6m, Absorption of solar radiation by atmospheric water vapor. Arkit~Geofyslk 3(23), (1961).

It. N. Robinson, Solar Radiation. Elvesier, New York (1966). 12. T. G. Berland and V. Y. Danilchenko, The continental dis-

tribution of solar radiation. Oidromaeoizdat, Leningrad (1%1).

13. M. S. Averkiev, Precise method for computation of global radiation. Bull. Moscow Univ. 1 (1961).