A two-band model for the calculation of clear sky solar irradiance, illuminance, and photosynthetically active radiation at the earth's surface

Solar Energy. Vol. 43,. No. 5, pp. 253-265. 1989 0038-092X/89 $3.00 + .00 Printed in the U.S.A. Copyright © 1989 Pergamon Press plc

A TWO-BAND MODEL FOR THE CALCULATION OF CLEAR SKY SOLAR IRRADIANCE, ILLUMINANCE, AND

PHOTOSYNTHETICALLY ACTIVE RADIATION AT THE EARTH'S SURFACE

Christian Gueymard* Serres Bio-Solaire Inc., Ham-Nord, GOP IA0 Canada

AbstractmA two-band radiation modelling technique is proposed for the clear sky case. The solar spectrum is divided into a UV/Visible band (0.29-0.7 IJ.m) and an infra-red band (0.7-2.7 ~m). In each band, the transmittance of each extinction layer (ozone, water vapor, mixed gases, molecules and aerosols) is pararneterized using preliminary integrations of state-of-the-art spectral transmittance functions. The beam and diffuse radiation components are obtained as functions of these layer transmittances. The photosynthetically active radiation (0.4-0.7 In.m) and illuminance (0.39-0.76 ttm) components may be easily derived from the UV/Visible band irradiances. In the case of typical atmospheric conditions, the model predictions are generally in good agreement with results from three accepted rigorous spectral codes. The effect of changing the solar elevation, the Angstr6m turbidity coefficient and the site altitude on the beam, diffuse and global components (or their ratio) is further discussed.

1. INTRODUCTION

The network of radiation measuring stations is rather scarce throughout the world. For many solar energy applications, it is thus generally necessary to com- pute the radiation input data from available meteorological data. For example, a clear sky radiation model is generally used to check the validity of measured data during the quality control process of data in radiation networks. Reference design curves are also needed in a number of engineering calculations. In particular, clear sky design curves or design values are necessary in applications like cooling load calculations for the design of air-conditioning systems, photovoltaic system design or predictions of temperature swings in passive systems.

Other engineering applications, like energy calculations, necessitate all-weather hourly radiation data. In that case, a clear-sky radiation model must be used in conjunction with an appropriate cloudy-sky model. The present contribution concentrates on the former type of model and presents an accurate method to develop design curves. The cloudy-sky sub-model will be reported subsequently.

Among the models that have been proposed to determine the clear sky radiation components on a horizontal surface, most consider a full-spectrum (or one- band) calculation. Typical models of this kind (e.g., [1]-[3]) are reviewed in [4]-[6]. The most sophisticated models are of the spectral type (e.g., [7]-[9]). These have the greatest flexibility and are particularly adapted to those applications where a receiver

*ISES member

has spectrally varying optical characteristics. How- ever, such spectral models are also of an extreme computational complexity, and often need very specific input data (e.g., spectral optical characteristics of aerosols), that are rarely available in practice. They are rarely used by engineers, except if spectral data are specifically needed.

At the other extreme, very simple one-band radiation models of the ASHRAE type[10] are widely used in engineering applications like thermal simulation and design. The empirical equations of this approach allow for a very fast computation, but at the expense of serious limitations, particularly regarding climatic versatility[11,12].

In order to obtain an accurate, versatile and non- climate specific determination of solar irradiance, the "physical" modelling approach is retained here. Physical refers to the principles used in the derivation of the model, which consists of a set of parameteri- zations based on preliminary integrations of spectral transmittance functions, taking separately into account the most important extinction sources of the atmosphere. This leads to a compromise between the pure theoretical approach used in "rigorous mode l s ' - - based on a solution of the equation of radiative trans- f e r - a n d the empirical approach used in "regression models ' - - s imple but climate specific. The physical approach should give relatively accurate results in any climate while maintaining the computation effort within limits compatible with most engineering projects.

After Lacis and Hansen[13] and Paulin[14], it appears logical to consider separately two wide bands of solar spectrum: the ultraviolet (UV)/visible band (where ozone absorption and molecular scattering are

253

254 C. GUEYMARD

concentrated) and the infra-red (IR) band (where water vapor and mixed gases absorptions are concentrated). This spectrum separation into two bands facilitates the transmittance modelling of beam and diffuse clear sky irradiances because overlaps between scattering and selective absorption are limited. Moreover, such an approach may be of interest in those applications where calculations for a limited region of the spectrum are needed (e.g., daylighting, biomass, PV engineering). For instance, the derivation of illuminance and photosynthetically active irradiance from the UV/visible irradiance will be detailed in Section 5.

The proposed model is called CPCR2 (Code for Physical Computation of Radiation, 2 bands). Though it is restricted to the computation of radiation on horizontal surfaces, it may be easily supplemented with a companion model called CDRS[ 15]. The latter model takes the anisotropic scattering effects of aerosols into account to determine the irradiance components on tilted surfaces. CDRS has been shown to give accurate results for each of the two bands considered here[15].

Examples of design curves obtained with CPCR2 are given in Section 3, along with some selected in- terrelationships between the various radiation components. In order to assess the performance of CPCR2, its predictions will be compared to theoretical if'radiance values obtained with rigorous spectral codes (Section 4).

2.2 Direct normal irradiance It is assumed that the direct rays entering the at-

mosphere encounter extinction processes which are limited to: ozone absorption, molecular scattering, uniformly mixed gases absorption, water vapor absorption, aerosol scattering and aerosol absorption. Separated extinction layers are considered, so that each band atmospheric transmittance for beam radiation may be obtained as a simple product of layer transmittances. Thus, for each of the two bands, the beam irradiance at normal incidence is given by

1,,, = ToiTmTGiTwiTAi io~i (1)

where i = 1 for band Ba and i = 2 for band B2. Sub- scripts O, R, G, W, and A refer to ozone absorption, Rayleigh (molecular) scattering, mixed gases absorption, water vapor absorption and aerosol extinction. The total beam irradiance at ground level is simply: L=L, +in2.

The spectral-average transmittance of extinction process j in band Bi is calculated from

Tj, = fB T~xdX/Jo~, t

(2)

where h is the wavelength. For instance, the ozone transmittance may be cal-

culated from eqn (2), using

2. MODEL DESCRIPTION Ton = exp(-mouoko~) (3)

2.1 Solar spectrum

The preliminary integrations which are necessary to obtain average transmittances for each atmospheric extinction effect are based on the solar spectrum proposed by the World Radiation Centre (WRC), as adopted by the World Meteorological Organization (WMO). The corresponding solar constant is 1367 W / m z, or 1% above the commonly used Thekaekara spectrum[16] of 1353 W / m 2. Tabulations of the WRC spectrum appearing in [6] have been used throughout.

The shortwave limits of the model have been chosen as 0.29 and 2.7 I~m. These values approximately correspond to the average sensitivity limits of two widely used pyranometers, the Eppley PSP (0.285- 2.8 Ixm) and the Kipp & Zonen CM6 (0.3-2.5 p,m). The cut-off wavelength between the UV/visible band, Bt, and the IR band, B2, has been set to 0.7 I.tm. This is also the upper limit adopted for the photosynthetically active radiation, and the lower cut-off wavelength of the frequently used RG695 red filter. Ac- cording to [6], the energy contained in bands B~ and B2 is respectively 46.04% and 50.57% of the solar constant. Therefore, coefficients fH = 0.4604 and f2 = 0.5057 are to be applied to the total extraterrestrial irradiance in order to obtain the band extraterrestrial irradiances io~z and io~2, corresponding respectively to B, and B2.

where koa is the spectral absorption coefficient, interpolated from data in [6], Uo is the ozone amount in a vertical column, and mo is the ozone mass. The ozone amount may be predicted from latitude, lon- gitude and day of the year[17]. From the graphical data in [8], mo has been fitted by

mo = 13.5(181.25 sin2h + 1) -0.5 (4)

where h is the solar elevation. Equations (2)-(4) have been solved numerically

for 0.25 < moUo < 10. The results have been fitted with

T m = 1 - exp[ -2 .5686 + 0.6706 ln(moUo)] (5)

The ozone absorption is negligible in band B2, so that To2 = 1.

The transmittance corresponding to Rayleigh scattering is calculated from eqn (2), using

T~ = exp(-m~rRx) (6)

where mR is the absolute optical air mass and ~rRX the Rayleigh spectral optical thickness. The absolute air mass is given by

mR = m ( p / p o ) (7)

Calculation of clear sky solar radiation

where p and Po are the station pressure and the sea- level standard pressure (101.325 kPa), respectively; m is the relative air mass, given by Kasten[18] as

m = [sin h + a(h + b)~] -1 (8)

where a = 0.15, b = 3.885 °, and c = -1 .253. When station pressure is unknown, P/Po may be approxi- mated by (1-0.1 Z) where Z is the station altitude in km. Different expressions for trRx have been given in the literature. For instance, Bird[19[ proposed

trRx = k - 4 / ( l 1 5 . 6 4 0 6 - 1 . 3 3 5 / h ~) (9)

More recently, Davies[20[ used the following general equation:

trRx = o'mx=l) k -(q°+q'x+qz/x) (10)

Coefficients qi in eqn (10) were shown[20[ to be dependent on the depolarization factor value, d. The value of trR~x=,) was not given in [20], but three of its possible values have been determined by the present author from Rayleigh scattering theory to be 8.7347 x 10 -3 for d = 0.035 (the depolarization factor value used by Elterman[21], among others), 8.6296 x 10 -3 for the corrected value d = 0.0279 proposed by Young[22[, and 8.7485 x 10 -3 from eqn (9). From the preliminary tests performed for band B~ with eqns (2), (9), and (10); eqn (10) with d = 0.035 has been selected because the corresponding determinations of the band-average optical thickness, trR~, were close to the average of the two other determinations, for any air mass between 0.8 and 30. These results have been fitted very precisely so as to obtain trR~ as

trm = 1 - exp [-0.24675 + 0.0639 In

(1 + mR) - 0.00436/n2(1 + mR)[ (11)

As Rayleigh scattering is only residual in band B2, the TR2 values should be close to 1. More exactly, T~ has been obtained from eqn (2) and parameterized as

TR2 = 0.999523 -- 0.010274 m R

+ 7.375 X 10 -5 m~ (12)

The absorption effect due to the uniformly mixed gases (mainly Oz and COs) is concentrated in band B2, so that T~, = 1. The spectral values of the gaseous absorption coefficients tabulated in [6] and [23] have been interpolated where necessary and inserted in the equation first proposed by Leckner[24] and also used in [6] and [23]:

T~x = exp [ -1 .41 k~xmR/

(1 + 118.93 k~xmR) °'aS[ (13)

The following parameterization has been obtained after combining eqns (2) and (13) for 0.8 < mR < 30:

To2 = 0.9776 - 0.0094 In mR

- 0.0019 In 2 m s

255

(14)

The water vapor selective absorption is essentially concentrated in band B.,, so that Tw~ = 1. From Leck- ner[24], the water vapor spectral transmittance is given by

Twx = exp [-0.2385 kwxuw/

(1 + 20.07 k~uw) °45] (15)

where mw is the water vapor optical mass, w the precipitable water, uw = mww and k ~ the water vapor spectral absorption coefficient. The formula used here for mw is identical to eqn (8), but its coefficients are changed to a = 0.0548, b = 2.65 °, and c = -1.4521181. The precipitable water is generally obtained from measurements or from empirical equations using surface meteorological data like dew-point temperature or vapor pressure (see the review in [6]). Monthly average values of w are also available for a number of stations (e.g., [6]). The tabulated data of kw~ from [6] and [191 were used here and interpolated where necessary. Equation (15) has been combined with eqn (2) and the numerical results for 0.1 < Uw < 100 cm have been parameterized so as to give

T ~ = 0.8221 - 0.0519 In uw

- 0.0033 In 2 uw (16)

The average aerosol transmittance for each band is obtained from eqn (2), using

Tan = exp (-ma13 k-~) (17)

where mA is the aerosol optical mass, and et and 13 are the Angstr6m turbidity parameters. The aerosols are concentrated in the first two or three "kilometers above sea level, like water vapor. Thus, it is assumed that the aerosol optical mass is equal to the water vapor optical mass, mw.

It is worth discussing here the physical meaning of et and 13, as well as their possible values, because they have a very significant effect on TAx, thus on the transmitted irradiance.

[3 is a function of the aerosol loading of the atmosphere. Its minimum value is 0 for an ideally dust- free atmosphere, while values greater than 1 have been estimated in extremely turbid climates. Mean monthly values of 13 for various stations in the world are tabulated in various references (e.g., [6]). These values may be used to choose a design curve appropriate to the local aerosol climatology. For hourly calculations, a relationship between 13 and visibility may be used (see, for example, [6]). Visibility observations are available at a number of airports and meteorological stations.

ct is a function of the aerosol size: low ct values correspond to large particles, and vice-versa. In most studies, the frequently quoted value et = 1.3 is im- plicitly assumed spectrally independent and season-

256 C. GUEYMARD

ally constant. In reality, this is only a climatological average, so that these assumptions are not appropriate at all times for all types of aerosols and atmospheric conditions. Some studies have been devoted to the day-to-day--as well as the seasonal--variations of a. Moreover, some recent investigations (e.g., [25,26]) indicate that a may vary significantly with wavelength. Therefore, the band-average values of a, which are noted at and a2, may not be equal, and the following inequality is expected in most cases: a= < a < a2. If irradiance computations corresponding to model atmospheres (for which the spectral optical characteristics of aerosols are fixed) are needed, it is possible to define values of at and a2 by spectrally averaging these characteristics. However, the aerosol characteristics are generally not known at any in- stant. Thus, because of present knowledge limitations, it is generally necessary to assume that a= = a , = a, in computations dealing with real atmospheres.

The results of the numerical integrations of eqn (2), combined with eqn (17), for 0.05 < mAI3 < 8 and 0.5 < a < 2.5, have been parameterized using a formulation similar to eqn (17):

TAi = exp (-mal3h~/'~') (18)

where hei is the effective wavelength for band B~. The values of h,~ depend on two parameters, ma~ and ai. The computed minimum and maximum values of he= and k~2 are 0.50 and 0.66 ~m, and 1.09 and 1.47 p.m, respectively. The following parameterization, obtained with a least-square polynomial method, allows to evaluate h,~ from the above integrations:

where

kei = aio + a i l UA + ai2 UA2 (19)

ua = In (1 + ma 13) (20)

(21a) at0 = 0.510941 - 0.028607 at

+ 0.006835 a~

(21b) art = -0.026895 + 0.054857 a=

+ 0.006872 a~

(21c) al2 = 0.009649 + 0.005536 at

- 0.009349 a~

am = 1.128036 - 0.0642 a2 + 0.005276 a~ (21d)

(21e) a2= = -0.032851 + 0.036112 a2

+ 0.005066 a~

(21f) a22 = 0.027787 + 0.064655 a2

- 0.021385 a22

2.3 Diffuse irradiance The diffuse irradiance at ground level is modeled

as a combination of three individual components corresponding to the two scattering layers (molecules and aerosols), and to a backscattering process between ground and sky. It is assumed that respective frac- tions BR = 0.5 and Ba of the Rayleigh and aerosol scattered fluxes are directed downwards. Assuming that Ba is independent of wavelength, its expression has been fitted from experimental data[27]:

B a = l - e x p ( - 0 . 6 9 3 1 - 1.8326 sin h) (22)

Assuming that the spectral model proposed in [28] is also applicable to band average calculations, the diffuse components for band B~ may be written as follows:

[dm : BR Toi rGi Twi TAai (1 -- TRi ) [oi (23)

l ~ i = Ba Toi TGi Twi ZAai TRi (1 -- Tasi) loi (24)

Idol = PsPsi(l.i sin h + l,~i + lain)l(1 -- pgp~i) (25)

lai = 1,~i + l,~i +Idoi (26)

where loi = Io,,~ sin h, TAa~ and Ta,,- are the aerosol transmittances due to absorption and scattering, respectively, lag is the spatial average regional albedo, and Psi is the sky albedo. The aerosol transmittances are related by

TAi = TAa i TAM (27)

with

In TAxi =toAi In TAi (28)

where toA~ is the aerosol single-scattering albedo, a function of the aerosol optical characteristics. It may thus vary with the origin of the air masses and pol- lution level, among other factors. Typical values of toAi have been obtained after integration of spectral data[29], corresponding to three different standard aerosol characteristics. Two other values have been added so as to represent intermediate cases. The proposed set of values appears in Table I.

The back-scattered diffuse component is a rather strong function of pg. Typical values of the latter are

Table 1. Band-average values of the single-scattering albedo as affected by the type of aerosol

Aerosol type

Marit ime/ Rural Rural/ Urban Urban Band rural clear average urban average pol luted

0.29 - 0.70 I~m 0.965 0.931 0.865 0.800 0.667 0.70 - 2170 iJ.m 0.913 0.632 0.754 0.676 0.516 0.29 - 2.70 p.m 0.940 0..883 0,81 t 0,740 0.595

Calculation of clear sky solar radiation

0 .10 -0 .25 for snow-free areas, but extreme monthly average values of 0 .6 -0 .8 may be encountered at high latitudes. The reflection process is considered purely isotropic, and p, is assumed to be wavelength soo- independent.

The sky albedo, p,~, results from the reflectance of the diffuse flux (emanating from below) on the scattering layers. Following [28], P,i may be ex- pressed as:

p , , = [ ( 1 - B , ~ ) ( ] - T;,,) + (1 - B~)(1 - T'A,,) T'e~] 7b, T~,, T'A,,, (29)

The primes indicate that the transmission functions are calculated for the effective value (1.66) of the relative optical masses.

3 . R E S U L T S

Some examples of typical radiation cu rves - -o r design curves - -a re given hereafter. These curves correspond to average rural aerosols (type 2 of Table 1), with the following fixed set of parameters: p =

Po = 101.325 kPa, Uo = 0.34 cm, w = 1 cm, ct~ = et., = 1.3, 9x = 0.2 and]o , = 1367 W / m 2.

The functional dependence of in, ld and J (= in sin h + la) on solar elevation and turbidity is shown in Figs. 1-3 , for the chosen standard conditions. It is noticeable that 13 has a great effect on both ], and ld, but a rather limited effect on J. This is caused by the large forward scattering of aerosols, and their rela-

I I 0 0 "

I 0 0 0 -

900"

800-

700- N

E 600-

. - " s o o -

4 0 0 -

3oo-

~oo-

I 0 0 -

0 ZO

/3 °

0.1

O.Z

0.3

0.4

40 60 80

h ( * )

"0.8

" 0 . 7

-0.8

- 0 . 5

o"

- 0 , 4

"0,3

" 0 . 2

.n!

- O

Fig. 1. Direct normal irradiance, ]n, predicted by CPCR2 for standard conditions (see text), as a function of solar elevation, h, and turbidity coefficient, 13. The apparent at-

mospheric transmittance, Kb = 1~/i~, is also indicated.

257

4 0 0 -

E

3 0 0 -

v

2 0 0 -

/3

0 . 4

0 .3

0 . 2

0.1

0 . 0 5

1 0 0 -

0

i i ~ -- i i i i I 0 ~0 4 0 6 0 80

h ( ' 1

Fig. 2. Same as Fig. 1, but for the diffuse irradiance, Jd.

tively high single-scattering albedo. Consequently, a rough estimate of 13 is a sufficient input if only the global radiation component is to be evaluated.

In Fig. 4, it is apparent that i , t is highly dependent on h and 13, like in, and that J,l is generally significantly less than (f~ in) because of the high optical

I I 0 0 "

I 000 "

9 0 0 "

8 0 0 "

~" roo- E \

6 0 0 "

5 0 0 -

4 0 0 -

3 0 0 -

l O O -

/ 3

0

0 .2

0 . 4

o io ~ ' 6'o ,~ '

h ( 0 )

Fig. 3. Same as Fig. 1, but for the global irradiance, 1.

258 C. GUEYMARD

i

o 0.5

0.4

'• 0.3"

0.2.

0.1,

g : o s

0.2

0,3

0.4

' ; o ' , 'o ' ~o ' ~'o

h(*) Fig. 4. Same as Fig. 1, but for the visible/fidl spectrum

ratio of the beam irradiance,/,t/i,.

thickness of Rayleigh scattering at the shorter wave- lengths• Figure 4 may be used also in a reverse way to obtain 13 if i , i is known, for example from pyr- heliometric measurement with a RG695 filter.

Figure 5 shows the apparent sky diffuse transmittance Kd = JJJo as a function of 13 and of the apparent direct transmittance, Kb = J,/Jon. It is note- worthy that Ka is almost linearly related to Kb for very low turbidities.

A comparable transmittance plot relating Kd and Kr = J/Jo is shown in Fig. 6. The variation at constant 13 is somewhat steeper than in Fig. 5, but the linearity is conserved at very low turbidities. The effect of solar elevation is also shown, so that a comparison can be made with experimental results, like those reported in [30].

Figure 7 shows a plot of K = ld/] as a function of 13 and Kr, for 2.5 --< h --< 90 °. The important effect of turbidity on K is clearly identified, particularly for Kr > 0.5. The effect of solar elevation is relatively smaller. For comparison purposes, the Org i l l -Hol - lands formula[31] is superimposed, though it is not restricted to clear sky conditions. This empirical correlation fits rather well the CPCR2 results if 13 is limited to the range 0 . 0 5 - 0 . 2 , which are typical values during winter and summer, respectively, for the climate of Toronto, Canada- - the station which data have been used to derive the Orgill-Hollands formula. While this finding should not be considered as a validation of one or the other model, it suggests that a model of the type K -- f (Kr, 13) may be more accurate than a classical model of the type K = f (Kr ) .

The ratio C = id/i, is plotted in Fig. 8 for different values of h, 13 and w. It appears that C is almost constant for given values of 13 and w, if h > 20 °. However , large variations of C occur at low solar elevations if 13 > 0.15. These results support the claim that C is generally not a conserved quantity throughout the day, except at very low turbidities[32]. On the other hand, it is evident from Fig. 8 that the range of values of C (0.057 to 0.136) that have beem em- pirically derived for use in the A S H R A E model[10] only correspond to conditions of very low turbidity (13 < 0.07). In other words, the A S H R A E model may be accurate enough for design purposes in very clear sky conditions, but will not predict a correct balance between the diffuse and direct components in hazy conditions, when 13 largely exceeds 0.07 (see also the discussion in [12]).

Finally, Fig. 9 shows the variation o f / , as a function of w and 13, for h = 10, 30 and 60 °, and for two altitudes, Z = 0 (or p = P0) and Z = 3 km (or p = 0.7 Po)- It appears that w has a strong effect on in only when values of w lower than 1 cm are considered, i.e. in the case of a rather dry atmosphere. This could be expected because of the saturation effect of

o.,1 /3

o,A • 0 . 3

2 0.2

0 .2

0 . I

0, I

0

I I I 1 I ~ I I

0 0.1 0.2 0.3 0 .4 0.5 0.6 0.7 0.8

Kb

Fig. 5. The dependence ofKd = ]~/]o on Kb = ]./io. and 13, for 2.5 ~ h --- 90 ° and the standard conditions defined in the text.

Calculation of clear sky solar radiation 259

0.4

0 .3 -

v

0.2-

0.1 o

h f

13 30" 45" \ zo.-. ". .. 6o..-~9. ~ 0 . 4

f - . • -., 0.3

5*- ~ 0 " " . : . .'- 0.2

i i ! i i

o12 a4 0'.6 ola

K-r

Fig. 6. The dependence of K~ = ]d//o on Kr = //Jo, h and 13, for the standard conditions defined in the text.

0.9 °

0.8"

0.7-

0.6-

0.5-

0.4-

0.3-

0.2-

0 .1

1.0

\

/3

0.4

0.3

bO. 2

%

i ! | | ! s i i i

0 0.2 O. 4 0.6 O.e 1.0

K- r

Fig. 7. The dependence of K = /d/i on Kr and 13, for 2.5 h ~ 90* for the standard conditions defined in the text.

The Orgill-Hollands empirical function is indicated by a dashed line.

the water vapor absorption. The small difference between the curves for P/Po = 1 and P/Po = 0.7 should be carefully interpreted. It is stressed that the indicated values of w and 13 correspond to the atmospheric loading of water vapor and aerosol above the station. As these const i tuents are concentrated in the first three kilometers or so above sea level, it is con- cluded that, for otherwise constant condit ions, the apparent values of w and 13 would considerably decrease with an altitude increase. From data in [6] and [33], it appears that w decreases exponent ia l ly with altitude and that it is about 4 t imes lower at 3 km altitude than at sea level. The decrease of 13 with al- t i tude may be obtained f rom tabulat ions of the alti- tudinal variat ion of the aerosol optical thickness[21 ]. These data show that the ratio 13(z = 0)/13(z = 3) is 3.85 for the typical aerosol profile considered. Therefore, it will be assumed that both w and 13 are reduced to about a fourth at Z = 3 km. Thus, if w = 4 cm and 13 = 0 .2 at sea level, the respective values at Z = 3 km would be about 1 cm and 0.05. For these condi t ions, the effects of altitude and atmospheric loading combined are now particularly evident , as for example ]n increases f rom 164 W / m 2 at sea level to 506 W / m 2 at Z = 3 km, if h = 10 °.

4. COMPARISON WITH RIGOROUS SPECTRAL CODES

4.1 SOLTRAN model SOLTRAN[5] is a direct normal incidence model ,

der ived f rom L O W T R A N [ 2 9 ] , a rigorous spectral t ransmit tance code which numerica l ly solves the radiative t ransfer equation. S O L T R A N uses the transmit tance values of L O W T R A N and the N A S A extraterrestrial spectrum to generate spectral values of the beam irradiance only. Broadband (0 .25-3 .125 0.m) integrated results of SOLTRAN[5] are compared here to those of CPCR2 ( 0 . 2 9 - 2 . 7 ixm) for two synthetic

260 C. GUEYMARD

1.2

il ~ - - - - - W II 0 W a |

i I - - - - - - W , 4 1.0 i I

i ~ II ,!\',

i,",. . . . . . . . . . . . . . . . . . o. 0.8 /~ ,,•

_ . . . . . . . . .

I

0.4 . . . . . . . . . . . . . 0.2

"~"" . . . . " '" -" - ' - " - " 0.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.2 ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.!

o

i ! I i i i i I

0 20 40 60 80 h(° l

Fig. 8. The dependence of C = id/i~ on h, [3 and precipitable water, w (in cm). The hatched area indicates the range of values of C used in the ASHRAE model.

atmospheres. The Mid-Latitude Standard Atmo- sphere (MLSA) contains 2.93 cm precipitable water and 0.31 cm ozone• The Sub-Arctic Winter Atmo- sphere (SAWA) contains 0.42 cm water and 0.45 cm ozone. Both atmospheres are used in conjunction with two rural aerosol models, corresponding to 5 krn and 23 km visibility, respectively. From the available aerosol optical thickness data[19], calculations with CPCR2 have been performed using o~at = 0.94 and c0a2 = 0.87. The relevant turbidities are 13 = 0.47 and 13 ~- 0.133 for 5 km and 23 km visibility, respectively.

Comparisons between CPCR2 and S O L T R A N ap- pear in Table 2 for these different atmospheric conditions. The results show that the CPCR2 predictions are always close to those of the reference model, even at low solar elevations. This is not the case for some

existing broadband models, such as those proposed by Atwater and Ball[I] and Watt[33]. Their predictions have been tabulated in [5] and are displayed in Table 2 for comparative purposes. Results in Table 2 show that broadband radiation models may predict largely different i rradiances--part icularly at low solar e l eva t ions - - even if they take explicitly into account the effect o f the two main sources of extinction in a clear atmosphere: water vapor and aerosols.

4.2 BRITE model BRITE is a rigorous code that solves the radiative

transfer equation problem by Monte-Carlo tech- niques. Tabulated spectral results are given in [35] for various atmospheric conditions. According to [19], the aerosol optical characteristics are not fully compatible with a constant wavelength exponent. In or-


o o ° . o o

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i # I / # s # s so o ## a

v ' ) 0 - - N , t

o o o c~ d

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( ~ u J / M ) Ul

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262 C . G U E Y M A R D

Table 2. Direct normal irradiance, I, (W/m-'), as predicted by different models for two synthetic

atmospheres (MLS: Mid-Latitude Summer and SAW: Sub-Arctic Winter) and two visibilities (5 km and 23 km)

MLSA SAWA I1

CPCR2 AKB" Watt SOLTRAN CPCR2 A&B" Watt SOLTRAN

V • 5km

15 100 64 175 97 115 72 201 1~2 30 271 254 373 270 301 275 404 298 50 426 432 519 - 463 461 54g 461 90 529 548 608 531 571 581 638 569

V = 23km

105 49 144 103 125 65 221 120 1 261 218 336 259 297 254 415 289 15 387 367 477 386 430 413 549 423 30 615 630 704 618 667 684 761 663 50 753 782 832 757 808 836 879 804 90 829 865 899 834 885 917 942 882

" A&B : A~ater and Ball

der to keep 13 constant at its 1 i.tm standard value, the following average band values of ~ have been estimated from data in [19]: at = 1.05 and et: = 1.2. Other estimated parameters are to, u = 0.94, toA2 = 0.87, and 13 = 0.046 and 0.123 for the 150 km and 25 km visibility atmospheres, respectively. Table 3 gives the comparative predictions of CPCR2 and BRITE for the normal beam and global irradiances in band B~. A close agreement may be observed on the global irradiance, while an even closer agreement is obtained on the beam irradiance, at any solar elevation.

4.3 Braslau and Dave models Braslau and Dave (B&D) proposed an iterative

spherical harmonics technique to solve the radiative transfer problem[9]. Their model is generally considered as one of the most sophisticated ever proposed, necessitating considerable computer time. CPCR2 and B&D results are compared here for each of the six different atmospheric conditions considered in [9]. As the tabulated results of B&D only display transmittance values, some data reduction is necessary in order to compare irradiances. B&D used an older solar spectrum[36], giving an extraterrestrial irradiance of 1396.4 W / m 2, or 1.02% above the WRC value. Between the spectral limits considered by B&D (0.285-2.5 p.m), the extraterrestrial irradiance they used is 1342. I W/m2[36]. Thus, B&D transmittance results have been multiplied by 1342.1 to obtain the

Table 3. Comparison of direct normal irradiance and global irradiance results (W/m 2) generated by CPCR2

and BRITE for band B, (0.29-0.7 ~m and 0.3-0.7 Ixm, respectively)

~=0 .046 ~=0 .123

h DIRECT GLOBAL DIRECT GLOBAL

CPCR2 B R a E CPCR2 BRITE CPCR2 BRITE CPCR2ERITE

10 147 144 64 58 61 63 56 48 15 220 222 109 101 121 124 97 88 20 274 278 156 148 173 176 143 - 30 345 345 250 240 251 254 237 221 41.8 394 395 353 342 310 317 341 321 53 422 422 436 424 345 347 425 403 90 454 454 563 551 386 388 554 539

corresponding irradiance values, and CPCR2 has been run with a solar constant of 1396.4 W/m: . Atmo- spheric conditions and aerosol optical characteristics used for this test with CPCR2 are detailed in Table 4. The turbidity parameters and the aerosol single- scattering albedo have been estimated from data in [9]. It appears that a is particularly wavelength de° pendent, and varies greatly between 0.29 and 0.75 I.tm. Thus a representative value of et is difficult to estimate for the effective wavelength of band B~. However, the results of CPCR2, which are compared to those of B&D in Table 5, demonstrate that the different components as predicted by CPCR2 compare well to the data of B&D. A part of the difference between these results may be attributed to the dif- ferences in their respective spectra: their extraterrestrial spectral irradiance and integration limits do not match exactly. As pointed out in [6], a 1% difference in solar constant from two slightly different spectra may lead to a 2% difference in the sea level broadband beam irradiance.

S. APPLICATIONS

Two particular applications for which the visible band irradiances may be necessary are presented in what follows.

5.1 Calculation of the photosynthetically active radiation

The photosynthetically active radiation (PAR) is defined as the energy flux contained in the spectrum between 0.4 and 0.7 ~m. It is a necessary input in applications dealing with plant physiology, biomass production and natural illumination in greenhouses. Unfortunately, a worldwide routine network for the measurement of PAR is not yet established. In order to palliate this problem, the PAR is often calculated as a constant ratio of the broadband solar irradiance. For instance, a range of 0.43-0.51 has been empir- ically determined for the global ratio for clear sky conditions (e.g., [37, 38]). Similar ratios for the beam and diffuse components cannot be considered constant, while typical values are still to be proposed. Many parameters are likely to affect these ratios, e.g. station pressure, solar elevation, turbidity, and precipitable water. Moreover, this simplistic approach supposes that solar radiation measurements are available, which is not always the case.

A possible alternative is to generate visible irradiances (band B0 with CPCR2, and then apply the necessary reduction factor to adjust the wavelength limits. Such reduction factors for the beam and global components have been obtained after integrating the spectral data generated by BRITE[35] for the same solar elevations and turbidity conditions that were considered in Section 4.2. A close relationship may be expected between the irradiance in the 0.29-0.7 Ixm and 0.4-0.7 p.m bands. Calculations show that this relationship is a slight function of the air mass and is only marginally affected by turbidity. The fol-


Table 4. Atmospheric conditions applicable to the model ~f Braslau and Dave and the corresponding input parameters of CPCR2

Sub-model Gaseous AEROSOL

absorption Scattering Absorption Concentration ct 1 ct 2 I1

CPCR2 inputs (es~mated values)

(OA 1 ¢0A2

A B X

C X X 0.034 0.3 0.7 0.08 1 1

D X X 0.144 0.3 0.7 0.34 1 1

C1 X X X 0.034 0.3 0.7 0.08 0.90 0.95

O1 X X X 0.144 0.3 0.7 0.34 0.90 0.95

lowing fit from the BRITE data give the ratio between the PAR beam irradiance, Pb, and the visible irradiance, Ibt, for m < 6:

Pb/]bl = 0.87375 + 0.04031 m -- 0.00358 m 2 (30)

with a standard error of 0.0047. Similarly for the global component:

P/]~ = 0.86225 + 0.02084 m - 0.00234 m 2 (31)

with a standard error of 0.0027.

5.2 Calculation of the horizontal illuminance Illuminance calculations are necessary in appli-

cations like daylighting and biometeorology. The network for the measurement of the il luminance is roughly as scarce as the PAR network in the world, though some fixed and mobile stations have been in- stalled recently particularly in the U.S.A.[391. As the illuminance corresponds to the standard eye response curve, with wavelength limits of 0.39 and 0.76 I~m, a close correspondence between the il luminance and the visible irradiance may be expected. Using the same spectral results from BRITE as above, the illumi-

nance, L, for a given solar elevation and atmospheric condition, has been calculated as

0.76

L = KL ~ k~J~dh J0.39

(32)

where k~ is the coefficient of the standard CIE eye response curve, KL is the maximum luminous effi- cacy (683 lumen /W) and J~ is the spectral irradiance generated by BRITE. The ratio i l luminance/irradiance derived from BRITE results for beam radiation has been fitted for m < 6 with

Lb/Jbl = 0.22249 + 0 .01614m - 0.00160 m 2 (33)

with a standard error of 0.0028. Similarly for global radiation:

L/ii = 0.21815 + 0.00581 m - 0.00080 m 2 (34)

with a standard error of 0.0008. As the illuminance and the PAR wavelength limits

correspond closely, it may be expected that their relationship is at least as significant as the i l luminance/

Table 5. Comparison of the irradiance results (W/m 2) generated by CPCR2 and B&D (Braslau and Dave) for the six typical atmospheres of Table 4

Atmosphere A B C C1 D D1

h CPCR2 B&D C P C R 2 B&D C P C R 2 B&D CPCR2 B&D CPCR2 B&D CPCR2 B&D

DIRECT NORMAL

10 960 917 685 688 424 420 424 422 97 95 97 96 30 1146 1119 921 940 776 788 776 789 447 453 447 455 60 1215 1194 1019 1038 922 937 922 938 667 678 667 680 90 1230 1210 1041 1060 955 970 955 971 721 732 721 734

DIFFUSE

10 36 36 31 32 60 62 56 58 97 92 80 81 30 58 51 54 55 113 117 107 111 248 245 223 218 60 71 78 68 70 146 150 139 141 350 348 319 312 90 75 83 72 74 155 159 147 150 377 376 345 338

GLOBAL

10 203 195 150 152 134 135 130 132 114 108 97 97 30 631 621 514 525 501 511 495 505 471 471 448 445 60 1123 1112 950 969 944 962 937 953 928 935 897 901 90 1305 1293 1113 1134 1109 1129 1102 1120 1098 1109 1066 1072

264 C. GUEYMARD

visible irradiance relationship. The correspondence between illuminance and PAR begins to be studied experimentally[39], so that a physically derived fit may be of interest for reference and quality control purposes. Using the same BRITE spectral data as above, the ratios i l luminance /PAR have been fitted for m < 6 with:

Lb/Po = 0.25518 + 0.00591 m

-- 0.00071 m" (35)

L I P = 0.25301 + 0.00058 m - 0.00023 m" (36)

with standard errors of 0.002 and 0.0004, respectively. Equations (30)-(31) and (33)-(34) may be used

to deve lop PAR and illuminance design curves similar to Figs. 1-9 . Such clear sky design curves are necessary in a variety of applications, particularly if no radiation measurement is available. As above, these sets of equations may also be used for quality-control purposes to check the experimental errors and drift of measurement set-ups.

The ratios in eqns (30), (31) and (33)-(36) are essentially independent of turbidity, station pressure and atmospheric ozone thickness, because these parameters affect both terms of the ratios almost equally. Thus, if it is assumed that BRITE results are accurate and spectrally balanced, it is expected that the proposed eqns (30)-(31) and (33)-(36) may be used in any climatic condition.

6. CONCLUSIONS

A physically based irradiance model has been presented. For each of the two wide bands considered (0 .29-0 .7 p,m and 0 . 7 - 2 . 7 i~m), the transmittance of each extinction layer is parameterized using preliminary integrations of state-of-the-art spectral transmittance functions. The two-band approach allows the difficulties caused by the spectral overlapping between some extinction processes to be mostly over- come. Moreover , this permits the derivation of the components of the photosynthetically active radiation and of the horizontal illuminance from the visible band irradiances.

The model may be used to develop ~design curves" for different applications. Examples of such curves are presented for typical conditions, showing that the beam and diffuse components are strong functions of both turbidity and solar elevation. Turbidity has only a second order effect on global radiation. The direct effect of site elevation (on Rayleigh scattering) com- bines with its indirect effects (the precipitable water and turbidity rapidly decreasing with altitude), so that a considerable increase of the beam irradiance occurs with altitude, especially at low solar elevations.

The ratio d i f fuse/global is a strong function of the clearness index, and also a function of turbidity and solar elevation. Though the resulting curves for this ratio are limited to clear sky conditions, they corn-

pare well to the widely used, all-sky, Org i l l -Hol - lands correlation.

As a first step into the validity process of the model, it has been compared to three well-accepted rigorous spectral codes for a variety of atmospheric conditions. These results show a generally good agreement for the beam, diffuse and global irradiances. A further validation step based on comparisons of model predictions with very accurate measurements at different altitudes is underway.

As the m. odel is flexible enough to generate predictions of the horizontal i l luminance and of the photosynthetically active radiation, it may prove useful in various applications from design of solar heating and cooling systems to studies of daylighting or plant growth under typical clear sky conditions.

Acknowledgment--The suggestions provided by Mr. John Inder during the revision of the manuscript are particularly appreciated.

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solar radiation 265

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Documents

A two-band model for the calculation of clear sky solar irradiance, illuminance, and photosynthetically active radiation at the earth's surface