So~r Exoly, Vol. ~, pp. 297-301 oo~&o92xr79/OJOl,.0297ilozoo/o ~) Pergamon Press Ltd., 1979. Printed in Great Britain

TECHNICAL NOTE

Direct solar t ransmit tance for a clear sky

R. K1NGt and R. O. BUCKIUS~ Department of Mechanical and Industrial Engineering, University of IHinois at Urbana-Cbampaign, Urbana,

IL 61801, U.S.A.

(Received 20 April 1978; revision accepted 14 August 1978)

INTRODUCTION The knowledge of the available solar irradiance on the earth's surface is essential to many solar conversion systems in terms of their design, site selection, and performance efficiency as well as in building heating load calculations. The direct radiation is utilized in concentrating devices such as solar furnaces while total radiant energy is needed in solar heating applications. Therefore, for these purposes, it is desirable to have a model of the direct and sky radiation in terms of fundamental measurable quantities including optical depth or air mass, water vapor concentration, and aerosol content in the atmosphere. Such models must include the absorption and scattering of the various constituents in the atmosphere that deplete the incident solar radiation. Various models for the clear sky direct solar trans- mittance have been studied using different approaches. Spectral models have been presented[I,2] including scattering and ab- sorption contributions. A similar model[3] presented the solar spectrum for typical clear sky days under specified climate conditions and is then limited to interpolation procedures for other climate conditions. These models require spectral in- tegrations over all wavelengths for total transmittance cal- culations. A complete analysis of the atmospheric transmittance of solar radiation was presented by McClatchey et aL [4}, through the use of fundamental concepts and expressions. Hottel[5] has

• presented a total transmittance correlation for the specific cli- mates considered by McCiatchey et aL A similar method has been given by ASHRAE[6]. For accurate predictions at specific locations, an expression in terms of fundamental parameters is needed.

The purpose of the present study is to develop a general, universal yet simple, model of the direct solar transmittance in terms of fundamental measurable quantities. The spectral transmittance of the direct solar beam is formulated as a function of measurable quantities. Spectral integrations are performed with various combinations of parameters, m, p, U, a, and ~. Approximate analytical forms are developed, and unknown constants are determined from the numerical spectral integration and expressed in terms of these fundamental parameters. A comparison with Hottel's model shows that the present model is accurate and a summary of the application of the present model employing meteorological data is given.

ANALYSIS The monochromatic direct transmission of radiation along a

path in the atmosphere is

¢oA = exp ( - KL.~m). (I) The units of all variables are indicated in the Nomenclature. For large zenith angle # (> g0°), the curvature of the solar rays due to the refraction in increasingly more dense atmospheric layers must he taken into account. For evaluation of such an air mass, the refractive index as a function of wavelength is required for

fGraduate Research Assistant. ~Assistant Professor of Mechanical Engineering.

spectral integration[7}. The optical thickness is expressed as

~LA : f ; ((r + a) dh (2)

where a subscript A will indicate the aerosol contributions and g the gas contributions. Almost all particles in the atmosphere, including air molecules and particles, contribute to the scattering while the dominant absorbing species are H20, CO2, and 03. The total direct transmittance for all wavelengths is

• .={fo'i--exp[-fo'("+a)dh]d,~}//(fo'i--d~ ) (3)

The value of j'~' i,u, dA, known as the solar constant, is given by Thekaekara[S] with the spectral distribution shown in Fig. I.

Equation (3) is rearranged to give (a = aA + a 8)

"° : { fo i" 'xP [- Io (° + a~) dh ] d' - Io ''"°

x " , [ f o ' ~ * o . ) d ' ]

x[ l -exp(- f ;a , dh)]dA}/i.~ (4,

where i,, = J~ i,u, dA. The gaseous absorption is expressed in terms of the total band absorptance, A, which is defined as

A = L A~dv=L,~[l-exp(-fo 'a. dh)]d~. (S)

The solar spectrum and the aerosol properties are continuous in wavelength while gaseous absorption exists only in the H20, CO2, and O3 banded regions. Thus, the solar spectrum and aerosol transmittance are removed from the last integral in eqn

2100 ~ ~ i i T T ~ 0 T~ (rap

1800 J A ~ ~ TA(m~:O02 ,Q: I) 09 V "~' ~ r~(mp:5) O8

i -

05~-

, 0 0 I/ \

~)0 os 06 09 ~2 . ~5 ~8 z~ ~4 27 30 ~° k (~m)

Fig. I. Spectral variations of the solar spectrum, Wien's dis- tribution, and the Rayleigh and aerosol transmittance.

297

298 Technical Note

(4) to give

l 'D={ f i . :exp [ f (o'+aA)dh]d~}/i.¢

- l l~{i. l . lA(exp[-fo'(a,a+ri)dh]}/ iw (6)

The absorption due to aerosols, aA, is always very small and can be neglected. The values of ~r~ and im ~ are evaluated at the band center for each band. The first term of eqn (6) represents the transmittance due to scattering and is denoted by ¢,¢,,. The second term is the contribution of absorption which is the summation over all the band absorptances multiplied by its spectral scattering transmittance and is denoted by r,b,. The solar spectrum values, i,u. ~, are on a wave number basis to be consis- tent with the total band absorptance correlations.

Scattering transmittance The Rayleigh and aerosol transmittance contributions are

denoted by ~'a and ~'A, respectively. Rayleigh scattering is the scattering due to small air molecules with an inverse fourth- power wavelength dependence[9] of

o'R = 8~'~(n, 2 - I)sN/(3No24'). (7)

The particle number density in the atmosphere. N, is assumed to have a distribution with respect to altitude of N-- N0 exp (-h/H). Then the contribution of Rayleigh scattering is given as

ee = exp [ - 8,9 x 10-3m(p/po)/A 4] (8)

where the values of all quantities in eqn (7) are given in Ref. [3]. Aerosol scattering is the scattering due to aerosol and large

dust particles usually having radii between 0.02 and 10/zm which typically are described by a power law radius dependence[10]. Visibility is used as a measure of the aerosol density in the atmosphere. The transmittance due to aerosol scattering is represented by[l I]

~'A = exp (-/3/4 °) (9)

where a varies from 0.5 to according to the size distribution of the aerosols. It has a value generally close to 1.3 -+ 0.2 but seldom below 0.5 or above 2.5. The turbidity coefficient,//, is a measure of aerosol concentration which includes all scattering consti- tuents other than Rayleigh (including water droplets[12]). Methods for calculating their values have been proposed by Bilton et 02.[13], from data obtained at turbidity measuring stations[14]. This method includes the inhomogeneous distribu- tion of the aerosol particles.

Thus, the final expression for the transmittance due to scatter- ing is

~,=., = { fo= i,.n exp [- m,4 -° - O.OO89(mp/po)4 -'] d4 } / i,~. oo)

For the water molecules, even in extremely humid cases, the increase in the total air molecule density is only 3.4 per cent; thus, its effect on the Rayleigh component is negligible. There- fore, the water content in the atmosphere does not contribute significantly to the scattering expression.

Gaseous absorption The important species with absorption bands in the IR region

are (in order of importance) HsO, CO2. 03, N:O. CO, O2, CH4, and N214] where the first three are the most important in at- mospheric absorption calculations. The inhomogeaeous vertical distributions of these species and their pressure broadening in the atmosphere make it necessary to obtain a scaled pressure and scaled water vapor concentration so that the atmosphere can be treated as an equivalent homogeneous medium. The scaled water

vapor concentration, known as precipitable water, has been given by Smith[15] who assumed a water vapor profile in the vertical atmosphere of w = wo(PIPo)" and obtained

U = (PoWo)l[g(4, + I)]. {I 1)

The values of A~ in various locations are given by Smith[15]. The relationship between U and the dew point temperature, Ta. can be expressed as[15]

U = [3.8767/(A, + I)] x l0 c'7"ST't-23a't)lIT4+)95.l). (12)

The scaled pressure is defined as tl6]

, = ( f p d ~ ) / ( f d U ) 03)

and with dU = (w dp)/g, the scaled pressure is

~6 = po(4) + 1)1(41 + 2). (14)

This formula was suggested by Curtis[17] who estimated that the errors introduced by this expression were small for the type of vertical distributions of H20 in the atmosphere.

The water absorption of importance in the solar spectrum is in the 0.94, 1.1, i.3, 1.87, 2.3, and 2.7~tm band regions. The ab- sorption characteristics given by Howard et al.[18], are expres- sed with empirically determined constants as functions of pres. sure and optical path for both weak and strong line absorption. Liou et aS. [10], presented a continuous expression for the strong and weak line regions as

A = ks + k4 log [ U(760/~) t'jt' + l0 -IJj~'] (15)

where the values of ks through ks are from Howard et aL[18]. For COx absorption, a constant volume mixing ratio of 2.6 x

10 -4 is assumed; thus, the absorbing path length and scaled pressure are 158cm-atm and 0.5 atm, respectively[16]. The CO2 absorption is greatest for the 2.7/zm band where it overlaps the water vapor band. The overlap band region is expressed as[19]

f A,.uso+co, d~=f A,.u~d~+[f A,.co2 dvJe (16)

where e is the weighting factor given as a decreasing function with increasing f A,,x~o d~. It was found that even for a very dry atmosphere (U=0.2cm). the value of e is zero so that the 2.7~m band for CO2 has a negligible contribution. The total absorption of the 1.4, 1.6, and 2.0p, m bands of CO: was also evaluated[20] and found to be about 0.3 per cent of solar constant. Thus, the absorption due to CO: can, indeed, be neglected.

The O~ absorbs all the energy below 0.31zm in the solar spectrum and has a weak band at 0.6~m. The spectral trans- mittance for O3 is given by Moon[l]. Calculations for a very thick as layer (0.45 cm) and long optical path (m - 5) were made. The result shows the maximum energy absorbed even under this extreme case to be only 4 per cent of the solar constant. However, eqn (6) shows that this value has to be multiplied by the Rayleigh and aerosol scattering spectral transmittances so that the error is reduced to approximately 1.7 per cent of the solar constant. Since the purpose of the present study is to develop an approximate yet accurate expression to predict the solar transmittance and the O~ contribution is generally small, some accuracy is sacrificed by neglecting O3 absorption for simplicity.

Approximate expressions Approximate analytical expressions are desired for the varia-

tions of ~'D with the fundamental measurable parameters. In the exact expression for ~',¢.t, the solar spectrum is approximated by Wien's distribution (Fig. I)

i,u. = 10293.54-s exp ( - 2.614 ). (17)

Technical Note 299

The integral over wavelength in eqn (6) is separated into two regions-from 0 to I gm and from I to ®v.m. For the lower wavelength region, the aerosol transmittance and exp(-2.6]A) are relatively constant in wavelength (Fig. !) and are, therefore, removed from the integral: I n the higher wavelength interval, the aerosol and Rayleigh transmittances are small and are expanded into a two-term series. The resulting expression is

fo' ~',~,, # C, exp [- ml31C;' ] A -s exp (- O.O089mplA "~) d~ .2

+ --L ® 10,293.SA-S exp (-2.61A)

x (I - m~A ° - O.O089mplA 4) dA. (18)

The variation of the scattering contribution with measurable parameters is then

'r~t = ( Cdmp )[exp (- C2mp ) - exp (- 625C~mp)]

x exp (- mBIC:')- C4mp + (I - mBICs °)C6. (19)

The constants are determined by a least squares curve fit with the exact expression given in eqn (I0). The turbidity varies from 0.02 for a clear atmosphere to 0.4 for a very hazy atmosphere[13]. The values of the air mass from 1 to 5 cover most of the daylight hours at most locations in the United States[3]. The scattering transmittance was evaluated for the following values of the various parameters: rap = I (I,0) 5, ml8 = 0.02 (0.02) 2.0, and a = 1.0, 1.3, 1.5, 1.8, and 2.0. The resulting values for the constants are Ct = 5.228, C2 = 0.0002254, C3 = 0.6489, C4=-0.00165, Cs=2.875, and C~=0.2022. This form was quite successful since 80 per cent of the values resulted in a relative error of less than 1 per cent; 90 per cent of the values had an error of less than 2 per cent. On very hazy days (~ = 0.4) and with large optical depths, the relative error is no greater than 10 per cent. Yet the value of the transmittance is typically about 0.08 for this case so that a value within 10 per cent is quite acceptable.

The sum of the total band absorptances weighted with the fractional solar energy at each band is calculated from eqns (6) and (15) (Fig. 2). The values of i,~,.~i** and *',~,t.~ are taken at band center. The value of At in the scaled pressure is chosen to be 3 as an average value for the United States and P0 is assumed to be latin. The Rayleigh scattering contributions for all the bands were found to be negligible (Fig. I). Figure 2 shows that the I.I and 1.38 v.m bands are the dominant absorption contribu- tors with similar H20 concentration profiles. Therefore, considering the superposition of similar expressions with the small contributions of Rayleigh scattering and known pressure value, the approximate analytical expression for ~',b* is

~,b, -" [C~ + C~ log (m U + C~)] exp ( - m#lC~o). (20)

The exact values of v,b, are calculated from eqns (6) and (15) by varying mU from 0.2 to 50cm where 50cm represents a humid day with a long optical path. Constants C~ through Cto are

028

0 2 4 63 #m Oond -

020

r E 016

'~ 0 '2 DOriC

ooe ~ ' :" T bon(~ '

004 / " !38H.m bon(~

i mU (cm)

Fig. 2. Band absorptances of water vapor.

determined by the least squares curve fit as C7 = 0.1055, Ca= 0.0?053, C~=0.07854, and Cto = 1.519. The relative errors are generally less than I per cent and go up to 5 per cent in the extremely long optical depth case. it is also found that a reason- able deviation of P0 (up to +-20 per cent) and A, values have negligible effects on ~'~b,.

DISCUSSION McClatchey et aL[4], assembled data on the scattering and

absorption coefficients as a function of altitide for the 1962 standard atmosphere and five other climate models. Two types of haze models are employed--the 23 km visibility representing a clean atmosphere and the 5 km visibility as a hazy atmosphere. Hottel[5] presented a model of total direct transmittance and it is considered to be exact since the error is less than 0.3 per cent. In order to make comparisons with Hottel's model, it is necessary to obtain the precipitable water to express the visibility in terms of the turbidity parameters. The total extinction coefficient at 0.55 ~m is inversely proportional to the visual range (visibility) and can be written as[21]

o'A + ~ = 3.19/vis (21)

with ~ = 0.1162 km "t [22]. From the definition of turbidity, the equivalent length for aerosol can be defined as

HA = (/31,~ °)I[o-A(0)]. (22)

The values of /~s are evaluated from the data of McClatchey et ui. as/']'A = 1.577 Iran for vis = 23 km and/~A = 1.132 km for vis = 5 kin. The data on ~rA also show a wavelength dependence of a = I which results in the following: /3 = 0.1373 for vis = 23 km and /3 = 0.4797 for vis = 5 kin. The precipitable H20 was also evaluated for all climates.

Comparisons were made with the exact spectral integrations as well as the present model for all climate models[23]. The specific results for the 1962 standard atmosphere are shown in Fig. 3. All variations with air mass, visibility, and H20 content are ac- curately predicted. The largest differences occur at large optical thicknesses where the value of the direct transmittance is small (,-~ < 0.05). This result also shows that although values for m,8 > 2 have not been included in the constant evaluation, this expres- sion quite accurately predicts the variations.

IO L [ I I I 1962 STANDARD ATMOSPHERE

09 Nottel [5]

. . . . . Exoct Colculohon

08 ------ R-esent Model

06 ~ V l S : 2 3 k m h:2kmkm

04 k k

0 2

OI

VIS : S k i n . h = 0 ~,~"="~"~ O 0 I I I t t I *

I0 15 20 25 30 35 40 45 50 rrl

Fig. 3. Effect of air mass and altitude on the direct transmittance for the 1962 standard atmosphere.

300 Technical Note

The comparisons with Hottel's work for various climate models at an altitude of 2 km were also made. The value of turbidity at an elevation of 2 km is calculated from the data of Ref. [4] with

= 0.55 f2 ® o'A(h) dh.

Comparisons were made for all climate models[23] with typical results shown in Fig. 3. The present model and exact calculation are only 3-4 per cent higher than Hatters values. This result is not necessarily in error but is due to the uncertainty and assumptions involved in interpreting the parameters of McClat- chey et at.

~HMAlt¥ The direct solar transmittance for a clear sky that accurately

predicts the variation with fundamental quantities is presented. The expression is

ro= {5.228[exp (- 0.0002254mp)- exp (- 0.1409mp)]

× exp [- m/3/(O.6489)']/mp - 0.2022m0/2.875 •

+ 0.00165rap + 0.2022- [0.1055 + 0.07053

x log (mU + 0.07854)] exp ( - m~1.519") (23)

where U can be determined from eqns (11) or (12). If no turbidity data exists near the locations of interest, the following expression can be used[21,22]

= ( 0 . 5 5 ) ° ( 3 . 9 1 / v i s - 0.01162)

x [(I .577 - I. 132Xvis - 5)/(23 - 5) + I. 132]. (24)

The values of visibility and dew point temperature or absolute humidity are typically available from local weather statiens. Equation (24) assumes that the equivalent length for aerosol varies linearly with visibility. The value of a can be taken as 1.3 since this value has been most frequently observed[! i].

Acknowledgement--This work was supported in part by the University of Illinois Research Board.

NO~CLATURE A total band absorptance, cm -~

A, fractional absorption within a band at wave number a absorption coefficient, km -s

Cs-Cm constants determined by the present model g gravitational constant, m s -2

H atmosphere scaled height, Pal(pug), km /-1~ aerosol scaled height, km

h altitude, km i,,,~ normal spectral intensity of the sun, W m -2 p.m -~ i,~ solar constants, Wm -2

k3-ks constants in eqn (15) m air mass N particle number density, cm -3 n, index of refraction of air p pressure, arm 0 scaled pressure with respect to H20, atm

T., dew point temperature, °F U precipitable water, cm

vis visibility, km w absolute humidity a wavelength dependence for aerosol scattering

turbidity coefficient • weighting factor for 2.7 ttm CO2 and H20 overlapped

band, defined in eqn (16) e zenith angle

KL optical thickness A wavelength,/~m

A~ power of water content profile v wave number, cm-' p density, kg m -3

cr scattering coefficient, kin- transmittance

Subscripts A aerosol

abs absorption CO2 carbon dioxide

D direct g gas

1"120 water vapor R Rayleigh

scat scattered 0 at earth's surface A wavelength dependent

wave number

RKFlgRgNCF~ i. P. Moon, Proposed standard radiation curves for engineering

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and J. S. Gating, Optical Properties of the Atmosphere. Air Force Cambridge Research Laboratories, AFCRL-72-0497, Envir. Res. Paper No. 411 (1972).

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6. ASHRAE Technical Committee, ASHRAE Handbook of Fundamentals, American Society of Heating, Refrigerating, and Air Conditioning Engineers, New YoA (1972).

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13. T. Bilton, E. C. Flowers, R. A. McCormick and K. R. Kirfis, Atmospheric turbidity with the dual wavelength sun- photometer. Solar Energy Data Workshop, Silver Spring, Maryland, 61-67 (1973).

14. E. C. Flowers, R. A. McCormick and K. R. Kirfis, At- mospheric turbidity over the United States, 1961-1966. J. Appl. Meteor. 8, 955-962 (1969).

15. W. Smith, Note on the relationship between total precipitable water and surface dew point. J. Appl. Meteor. S, 726-727 (1966).

16. W. Roach, The absorption of solar radiation by water vapor and carbon dioxide in a cloudless atmosphere. Quart. J. R. Met. Sac. r/, 364-373 (1961).

17. A. Curtis, Di~ussion of a statistical model for water vapour absorption by R. M. Goody. Quart. J. R. Met. Sac. 78, 638..640 (1952).

18. J. N. Howard, D. E. Burch and D. Williams, Infrared trans- mission of synthetic atmosphere. IlL Absorption by water vapor. J. Opt. Sac. Am. 416(4), 242-245 (1956).

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21. R. A. McClatchey and J. E. Selby, Atmospheric trans-

Technical Note 301

mittance from 0.25 to 28.5 ~m: computer code LOWTRAN- 2. Air Force Cambridge Research Laboratories, AFCRL-72- 0745, Environ. Res. Paper No. 427 (1972).

22. L. Elterman, Vertical Attenuation Model with Eight Surface Meteorological Ranges 2-13kin. Air Foce Cambridge

Research Laboratories, AFCRL-70-0200, Environ. Res. Paper No. 318 0970).

23. R. O. Buckius and R. King, Direct and Diffuse Solar Trans- mission for a Clear Sky, University of Illinois UILU ENG 78-4004 (1978).