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Diffuse and global solar spectral irradiance under cloudless skies
Sohlr Enerey, Vol. 30. No. S pp. 447453. 1983 Printed in Great Britain
O0394)92X/831050447~)7503.00/O © 1983 Pergamon Press Lid
DIFFUSE AND GLOBAL SOLAR SPECTRAL IRRADIANCE UNDER CLOUDLESS SKIES D. T. BRINEand M. IQBAL Department of Mechanical Engineering,University of British Columbia, Vancouver,Canada (Received 1 January 1982;accepted 1 June 1982)
Abstract--A simple empirical model to calculate solar spectral diffuse and global irradiance under cloudless skies is presented here. This formulation takes into account absorption of radiation by molecules such as O~, H20 and the uniformly-mixedgases. Attenuationby Rayleighscattering and aerosol extinction are included. Aerosolattenuation is calculated through Angstr6m'sturbidity parameters a and/3. The diffuse radiation is assumed to be composedof three parts: (1) Rayleigh-scattered diffuse irradiance; (2) aerosol-scattered diffuse irradiance; and (3) irradiance arising out of multiple reflectionsbetween the atmosphere and the ground.The global irradiance is the sum of these three componentsof diffuse irradiance plus the direct irradiance. The input parameters include an extraterrestrial spectrum, zenith angle, turbidity coefficient/3, wavelength exponent a, ground albedo p,, water vapor content and ozone content. The model is shown to yield very good results up to air mass two when compared to accurate theoretical calculations. No comparisons with measured spectra are presented because of a lack of accurate specifications of input parameters. Results are presented to show the effect of variation of certain of the input parameters. INTRODUCTION An accurate solar spectrum, both extraterrestrial and terrestrial, is necessary for the further development of many solar-related energy projects. Chief among these are the photovoltaics for power on satellites, and the continued development of terrestrially-based photovoltaic power cells. The spectrum is used for testing the degradation of materials and for establishing cut-off points in the transparency of solar materials. The extraterrestrial spectrum is attenuated by the earth's atmosphere in two distinct ways: by scattering, and by absorption. Scattering is a continuum process, whereas gas absorption occurs at discrete wavelengths and can very greatly over small wavelength intervals. Scattering by a clean, dry atmosphere is generally referred to as Rayleigh scattering. Aerosols are suspended matter, whether liquid or solid, and can be dust, volcanic ash, pollen, smoke and urban pollution, or smog. Absorption by aerosols appears to be a continuum process and is usually characterized as such. Theoretical analysis of the radiative transfer equation by the Spherical Harmonics Approximation/l] has produced extensive data sets of direct and diffuse solar irradiance. The numerical solution of the highly complex and realistic atmospheres includes all orders of scattering and absorption by the common absorbing gases and aerosols. The theoretical treatment assumes a planeparallel, homogeneous atmosphere of infinite extent horizontally. All nonhomogeneity due to scattering and/or absorption is confined to the vertical direction and is incorporated into a defined atmosphere by dividing the atmosphere into any number of layers and using appropriate amounts of air molecules, water vapor and aerosols for each layer. The defined atmospheres, called "Model atmospheres", progress in complexity from purely Rayleigh scattering, Model A, to Rayleigh scattering plus gas absorption, Model B. Further complexity 447
is added by including nonabsorbing aerosols, Model C, and absorbing aerosols, Model D. The number of data sets generated by the numerical solution is limited due to the large amount of computer time involved and the complexity of the input data. A simple method to obtain the spectral direct and diffuse irradiance is needed by engineers of all disciplines. The desired formulation should be based on a simple homogeneous atmosphere with no layers but incorporating the best available knowledge of the molecular absorption coefficients. Furthermore, the aerosol attenuation could be characterized through the/~ngstrtm turbidity formula. This simple model, once formulated, should then be compared to more accurate data sets [2] and also any experimental spectral data available from the literature [3--6]. The cloudless sky is modelled as this condition generally produces the maximum energy available. This maximum establishes the criteria with which engineers can design equipment for the utilization of solar energy in solar cells, space heating, process water heating and the thermal loading on buildings, etc. The maximum is also necessary in the design of materials for solar applications to prevent early damage or deterioration of plastics. The direct or beam irradiance provides most of the energy available on the ground. The direct spectral irradiance has been studied by several investigators. Moon/7] was probably the first to combine all the individual scattering and absorption characteristics together in a complete model. Thomas and Thekaekara[8] used a formulation developed by Gates [9] to predict the direct irradiance. Hatfield et al [10] also used Gates's formulation and included other effects such as latitude, altitude, surface albedo, slope and surface orientation. Unfortunately, the water vapor transmittance calculated by this method in the 0.8-1.0 #m band
448
D. T. BRINEand M. IQBAL
has been shown[ll] to be low and results in a sharp decrease in irradiance in this band. Also this method requires an interpolation scheme as the extraterrestrial values generally recorded in the literature and the absorption coefficients used are not for the same wavelengths. Leckner's method[12] seems to be the best presently available in simple form. He developed monochromatic transmittance functions using recent data[13] for calculating the direct spectrum. The results of this method show good agreement when compared to measured spectra and theoretical calculations. With Leckner's model, a seemingly adequate yet simple method for calculating a detailed direct spectrum is achieved. Diffuse irradiance results from the interaction of solar radiation with the scattering particles--air molecules and aerosols--of the atmosphere. The scattering by the air molecules is limited to very short wavelengths of less than 1.0~m The scattering by aerosol particles, generally called Mie scattering, is important when particle size is of the order of wavelength. The first impingement of radiation on a particle is called primary scattering. The primary scattered radiation is further scattered by air molecules or aerosol particles and this process is called multiple scattering. The detailed study of diffuse irradiance including the effects of multiple scattering is complex, and consequently, this effect has not been taken into account in this work. A simple formula to predict the scattered diffuse irradiance on a horizontal surface was proposed by Berlage[14]. This formula O~, = 0.5 cos 0 (IoA-/~)
(1)
states that the diffuse irradiance is the difference between the extraterrestrial radiation and the direct irradiance at the ground. For a Rayleigh atmosphere, this does give credible results[15] when compared to more accurate analysis[16] if a subozone layer radiation intensity is substituted for Io~. Thus, such a formulation does not account for all molecular absorption or any aerosol effects directly. Leckner proposed a simple formula for diffuse irradiance Dx = 0.5 Iox cos 0 "ro:g:wA(1 - zrx'r,a).
(2)
This expresses the diffuse irradiance as the difference between the direct irradiance and a "fictitious beam" subject only to molecular absorption. This formula takes into account molecular absorption but assumes that the aerosol and the molecular scattering functions are identical. Also, eqn (2) does not include the effects of multiple reflections between the ground and the atmosphere. Hatfield et al.[10] also present a diffuse radiation model similar to that of Leckner. In this model D~ = Io, cos 0 ro~(1 - r,:~DFk.
(3)
They have omitted the absorption due to water vapor and the mixed gases altogether. They include a factor K
for the circumsolar component and define their forward scattering function for both Rayleigh scattering and aerosol scattering as F = 0.5 cos'/3(8)
(4)
which results in F having values 0.5 or less. The preceding studies assume the forward scattering ratio to be 0.5 or less. This is approximately true for a Rayleigh-scattering atmosphere. However, the aerosol scattering is very strongly biased in the forward direction and because of this asymmetry, the effective scattering ratio (toward the ground) is a function of zenith angle. A modification of Leckner's formula seems necessary to account for the dissimilar scattering properties of air molecules and aerosols. One approach, and the one followed here, is to consider a Rayleigh-scattering atmosphere and an aerosolscattering atmosphere as quasi-separate, and calculate the diffuse irradiance as a sum of these two components. Such an approach has been used in broad-band diffuse irradiance calculations [17]. A third component of diffuse irradiance can easily be added to the previously calculated components to account for the multiple reflection effects. The ground albedo must be specified and also a monochromatic atmosphere albedo must be formulated. MATHEMATICALFORMULATION
Direct model The method for calculating the spectral direct irradiance has been adapted from Leckner[12]. This direct model uses explicit transmittances for each of Rayleigh scattering, ozone absorption, water vapor absorption, mixed-gas (CO2, 02) absorption, and aerosol extinction, The transmittance due to Rayleigh scattering is given by zrx = exp ( - 0.0088 mA-4).
(5)
The transmittance due to ozone absorption is Zoa = exp ( - kolm)
(6)
where the monochromatic absorption coefficients, ko, obtained by Vigroux[18] and presented in [19] have been employed. The transmittance due to water vapor absorption is rw = exp ( - 0.2385 kwwm/(1 + 20.07 kwwm)°45)
(7)
The transmittance due to mixed-gas absorption is ~'~A= exp ( - 1.41 k~m[(1 + 118.93 k~m)°45)
(8)
where absorption coefficients, kw and k~, are taken from [12]. The transmittance due to aerosol extinction is given by z,~ = exp ( - ~ m)t -~)
(9)
which is the simple power-law first proposed by ,~ngstr6m [20, 21].
Diffuse and global solar spectral irradiance The air mass, m, is calculated from Kasten's formula [22] m = (cos 0 + 0.15(93.885 - 0)-1253) -1.
(10)
This is the relative air mass, since all calculations in this work are assumed to be at sea level. The direct spectral irradiance on a horizontal surface at sea level at the average earth-sun distance is given by IX = Io~ cos Oz,aroxrg;,r,,az,~
449
upwelling radiation is partially reflected by the atmosphere downward again, and this process continues. The resulting diffuse irradiance after these multiple reflections is given by Din,, = (IX + D,, + D,,,.)pg~,p,,,,/(l - p,,,,p,,~,).
(14)
The ground albedo, pgA, may be taken independent of wavelength. The monochromatic atmospheric albedo, p,A, can be shown t6 be
(11) Paa = r~xr "ar'ga((1 - r'ra)r'xF, + 0.22 (1 - r'A)OJor'rA).
(15) The diffuse irradiance is considered to be composed of three distinct components: those that are (1) Rayleigh The primes on all [he transmittances indicate that they scattered; (2) aerosol scattered; and (3) multiply are all evaluated at m = 1.9. The air mass m = 1.9 is used reflected. The individual transmittances of the direct .as it gives good agreement with theoretical results[2]. model are used to generate equations for each of the two The first term of eqn (15) represents the fraction of the scattering components. The equation for the multiply- upwelling radiation reflected back to the earth by Rayreflected component was derived from basic principles. leigh scattering. The second term represents the fraction The initial idea for the formulation of the atmospheric of the upweiling radiation reflected back to the earth by albedo came from broad-band calculations[17]. The aerosols. The factor 0.22 is the effective back-scatter Rayleigh-scattered diffuse irradiance arriving on a ratio of aerosols evaluated at m = 1.9. In other words, (1 - Fa) = (1-0.78) = 0.22. horizontal surface is The total diffuse irradiance from all sources is given by Dr~ = Io~ cos 0 rO~',~'wX~'ax(l-- rra)F~ (12) D,,, = Dr,, + D,,~, + D,.,,. (16) This is similar to eqn (2) except the factor (1 - r,A~'a~,) has been changed to ( 1 - r,a) and ~',a is taken outside the Similarly, the global irradiance is the sum of the direct bracketed quantity. Analogous to Leckner's approach, irradiance and the total diffuse irradiance. This is given the two beams of radiation are the direct beam and a as beam which includes molecular absorptance and aerosol G; = Ix + D,~,. (17) attenuation. The difference between these is the Rayleigh-scattered diffuse irradiance. Fr is the Rayleigh These monochromatic values of direct irradiance, total forward-scattering ratio, generally taken as 0.5. Similar to eqn (12), the aerosol-scattered diffuse diffuse irradiance and global irradiance can all be summed separately to give the broad-band values of these irradiance on a horizontal surface is quantities. This is accomplished by taking each monochD,~ = Io~ cos O~',~'o~rgxZw~ (1 - r,a)oJoF,. (13) romatic irradiance value in Wm -z (#m)-: and multiplying by the appropriate wavelength interval in #m. These Here, the difference between the direct beam and a beam values are then summed to give a broad-band value for which includes Rayleigh scattering and molecular ab- any wavelength interval. sorbers is the aerosol attenuated radiation. The single scattering albedo of aerosols, OJo, partitions the energy attenuated into scattering and absorption. A value of METHODOLOGYAND VALIDATION To validate this formulation of diffuse spectral irradioJo---1.0 indicates a completely scattering aerosol. The ance, the results obtained must be compared to more single scattering albedo is assumed independent of accurate theoretical calculations and to measured specwavelength. F,, the effective scattering ratio (toward the earth, not necessarily in the forward direction) is a tra. The comparison with theoretical solutions is easy, as extensive numerical results are available[2, 24] and the function of zenith angle, 0. Experimental values of Fo were reported [23] and have been used by other atmospheric input parameters are explicitly known. A similar comparison with measured spectra is difficult. workers [17]. It is to be emphasized that these values of The measured data are either based on long term F,, give only a rough estimate of the effective forward averages[3-5] or are given in graphical form[6] with scattering function. The values of Fo used in this paper are: F, = 0.923 for 0 = 0°, F,, = 0.78 for O= 60°. Like the insufficient information on the atmospheric parameters. Consequently, no comparison with measured spectra is single scattering albedo, the wavelength dependence of F,, is neglected. illustrated in this paper. The direct radiation reaching the ground, and the RayAll theoretical comparisons between Model atmosleigh-scattered and aerosol-scattered diffuse radiation pheres of Dave[2, 24] and the present work use the same extraterrestrial irradiance values[19]; however, the reaching the ground after the first pass through the number of wavelength intervals is different between the atmosphere are partially reflected back by the earth. This Diffuse model
450
D. T. BRINEand M. IQBAL
two studies. The wavelength spectrum investigated extended from 0.29 to 4.0 #m. A comparison between Model A (pure Rayleigh atmosphere) and the present work for air mass one, two and five was carried out. For air mass one, the agreement is excellent; some discrepancy (0.37-0.40/zm) is due to the different wavelength intervals in the two studies. At air mass two, the present work somewhat overestimates the diffuse irradiance at short wavelengths 0.3-0.37 tzm; at longer wavelengths the agreement is quite good. At air mass five, overestimation occurs up to 0.45 #m, after which the agreement is again good. The overestimation at short wavelengths of the present model could be attributed to the fact that Model A included all orders of scattering whereas the present work does not. A comparison of Model B (Rayleigh scattering plus absorbing gases) with the present work for air mass one, two and five was carried out. The results are similar to those presented for Model A at air mass one, the agreement is excellent; at air mass two, the agreement is very good; and at air mass five the overestimation of the present work is again noticed. The foregoing comparisons show that generally the present work accurately depicts scattering by a clean atmosphere. This implies that the formulation and the coefficients used are correct and that the next step-include aerosols--starts from a secure procedure. A comparison of the results of Model C atmosphere (Rayleigh scattering, absorbing gases plus a nonabsorbing aerosol) and the present procedure were carried out as follows. The direct irradiance was calculated for specific ozone and water vapor content corresponding to the values used in Model C. The turbidity parameters a and /3 were varied to give the best fit with the direct spectrum of Model C. In this manner, the values of these parameters were derived. An example of such an exercise is demonstrated in Fig. 1, where it is shown that a = 0.6 and/3 = 0.07 correspond to Model C. The fit of the present direct model and Model C is found to be excellent for air mass one and two. 2000
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Having determined values for a and /3 in the above manner, the diffuse irradiance was calculated using the present formulation, eqns (12-16). Comparison of these results with the diffuse radiation calculations of Model C for air mass one and for air mass two were made. In order to eliminate the effects of multiple reflections between ground and atmosphere, ground albedo is assumed to be zero. For air mass one, up to 0.6 #m, the agreement between Model C and the present work is good. The agreement is also quite good at longer wavelengths, with the present work giving generally slightly lower values than those of Model C. Overall the agreement is very good at air mass one. Leckner's formulation agrees up to 0.40/~m and then drops off abruptly to values substantially below those of Model C. For air mass two, the present work is consistently slightly below the values of Model C. Leckner's formulation is in good agreement with Model C up to 0.42 #m, but thereafter drops significantly below Model C and below the present work. The preceding comparisons indicate that the present formulation consistently produces better results than Leckner's formulation up to air mass two. Many other comparisons have been made and they lead to the same conclusions. When a nonzero ground albedo is considered, the multiple reflections between the ground and the atmosphere add to the total diffuse irradiance. Figure 2 shows a comparison of Model C and the present work for air mass one and ground albedo 0.3. The present work underestimates somewhat in the 0.33-0.40/~m band but is otherwise in excellent agreement with Model C. Also illustrated is the simple formulation of Leckner that has no provision for ground albedo. It is much below the values of Model C and the present work. Figure 3 presents the same comparison as Fig. 2 for air mass two. At this air mass the present work is generally lower than the values of ModN C but still overall much better than Leckner's formulation.
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Fig. 4. The effect of air mass on direct, diffuse and global irradiance on a horizontal surface for/7 = 0.07, a = 1.3, Oe= 0.0, ozone content 0.3 cm. Curve 1, global m = 1.0. Curve 2, direct m = 1.0. Curve 3, global m = 2.0. Curve 4, direct m = 2.0. Curve 5, diffuse m = 1.0. Curve 6, diffuse m = 2.0.
Comparisons of the present study with Hattield et al.[lO] have been made. Due to the previously mentioned formulation of the molecular and aerosol scattering, their results are generally below the values of Model C and of this study.
The turbidity coefficient,/3, has a strong influence on both direct and diffuse irradiance. /3 is an indication of the mass loading of the aerosol and also an indication of the atmospheric visibility for a given aerosol particle size distribution. Figure 5 shows the effect of /3 on direct, diffuse and global irradiance for air mass one and cz = 1.3. For /3=0.2 (Vis= 12km) the broad-band direct irradiance is about 19 per cent less than for /3 = 0.07 (Vis = 50 km). The broad-band diffuse irradiance for/3 = 0.07 is about 49 per cent less than for /3 = 0.2. The broad-band global irradiance for/3 = 0.07 is about 3 per cent greater than for /3 = 0.2. Thus, even for a very turbid atmosphere (/3 = 0.2) the global irradiance available is only marginally affected at air mass one. Spectrally,/3 has the greatest effect on the diffuse irradiance and predominantly at short wavelengths < 1.0 #zm). The
RESULTS In the previous section "Methodology and Validation", the present work was compared to an accurate theoretical model and it was found to produce very good results. The versatility of the present method will be illustrated in this section. The six independent input parameters of this model are: air mass, ozone content, water vapor content, ground albedo, the turbidity coefficient /3 and the wavelength exponent a. These parameters are easily varied, so the effects of each parameter can be investigated independently. The extraterrestrial solar spectrum presented by Thekaekara[25] is used to produce all the results in this section. It is understood that present-day values of the solar constant and its spectral distribution have been proposed[26]. However, the present method is independent of any particular extraterrestrial solar spectrum. Air mass has the greatest effect of any of the parameters on the available direct and global irradiance, whereas the available diffuse irradiance is affected to a much lesser extent. This effect is demonstrated in Fig. 4 for air mass one and two. The broad-band (0.29-4.0/zm) direct irradiance decreases 60 per cent from 925 to 370 W/m 2 and the broad-band global irradiance decreases 56 per cent from 1073 to 471W/m 2. The broad-band diffuse irradiance decreases only 32% from 148 to 100 W/m 2 and for wavelengths greater than about 1.0/~m the diffuse is essentially independent of air mass. The spectral character of the curves for air mass one and two are very similar and it is noticed that the percentage decrease in irradiance is almost uniform at all wavelengths; even though the actual amount attenuated at short wavelengths ( < 1.0 #m) is very large.
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452
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sharp peak of the direct curve for /3 = 0.07 has disappeared and has become a wide plateau for /3 = 0.2, indicating the pronounced effect of scattering for an increased mass loading of aerosols at short (0.4-0.7/~m) wavelengths Although the global irradiance is decreased only 3 per cent, almost all of this decrease occurs in the visible (0.4-0.75 ~m) band. For air mass two,/3 has a somewhat greater effect on broad-band global irradiance, about 9 per cent decrease, partly due to the decreased value of the effective scatter ratio of the aerosol. The broad-band direct irradiance decreased 31 per cent between/3 = 0.07 and/3 = 0.2 and the broad-band diffuse irradiance decreased 43 per cent between/3 = 0.2 and/3 = 0.07. Spectrally, the peak of the direct irradiance has moved from a plateau between 0.45--0.7/zm for/3 = 0.07 to a peak at 0.7 ~m for/3 = 0.2. The diffuse irradiance has correspondingly shown a large increase at short (0.3-0.6 tim) wavelengths and actually exceeds the direct irradiance for /3 = 0.2 up to 0.6 ~m, illustrating very well that what is scattered from the direct beam is accounted for in the diffuse irradiance. Another influencing factor of aerosols is the wavelength exponent a, which is related to the size distribution of the aerosol particles. Large values of a indicate a relatively higher ratio of small particles to large particles. For most natural atmospheres a = 1,3---0.2. An atmosphere containing a large amount of small particles (large a) scatters more than an atmosphere containing an equivalent mass of large particles (small a). In Fig. 6, it is evident that a = 2.0 scatters more energy from the direct beam, about 8 per cent, than does a = 0.5 The plots of the diffuse irradiance show that the energy scattered from the direct beam reappears as the diffuse irradiance, with the difference between the two broad-band diffuse values being 32 per cent. The global irradiance plots show very little difference (about 2 per cent) between a = 2 . 0 and a =0.5, and the influence of a is seen only at wavelengths less than 0.7 ram. 2
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Fig. 7. The effect of ground albedo on diffuse and global irradiance for air mass one, //= 0.07, a = 1.13, water vapor content 2.0 cm and ozone content 0.3 cm. Curve l, global pg = 0.2. Curve 2, global p= = 0.0. Curve 3, direct. Curve 4, diffuse pe = 0.2. Curve 5, diffuse pz = 0 0. The ground albedo and multiple reflections are considered next. Figure 7 shows the effect of the variation of ground albedo on the diffuse and global irradiance. Two values of ground albedo are illustrated, p= = 0.0 and 0.2. p= =0.2 is representative of most ground covers. Due to multiple reflections between the earth and its atmosphere, a larger ground reflectance produces more diffuse irradiance (about 10 per cent) and corresponding increases in global irradiance (about 2 per cent). The effects of variation of water vapor content and ozone content are limited to specific wavelength bands and primarily to the direct beam. There is very little effect on the diffuse irradiance due to these parameters. The global irradiance indicates that any difference is due mainly to the variation in the direct irradiance. CONCLUSIONS The present method of calculating the diffuse spectral irradiance has been compared to accurate theoretical analysis and found to give good agreement for the model atmospheres investigated here. The major advantage of this formulation is its simplicity. The computer time is minimal and the input parameters are easily varied. The results presented are illustrative of the flexibility and the range of effects that can be studied with this formulation. Comparisons with measured spectra should be undertaken when sufficient input parameters are specified, to complete the validation of this formulation.
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Acknowledgement--Financial support of the NSERC of Canada is gratefully acknowledged. The data tape provided by Dr. J. V. Dave of IBM Scientific Center, Palo Alto is highly appreciated.
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Fig. 6. The effect of a on direct, diffuse and global irradiance for air mass one, ground albedo 0.0, water vapor content 2.0 cm, ozone content 0.3 cm and //= 0.07. Curve l, global a = 0.5. Curve 2, global a = 2.0. Curve 3, direct a = 0.5. Curve 4, direct a = 2.0. Curve 5, diffuse a = 2.0. Curve 6, diffuse a = 0.5.
NOTATION D~ monochromatic diffuse irradiance D~ monochromatic aerosol-scattered diffuse irradiance D,.~ monochromatic multiply-reflecteddiffuse irradiance Dr~, monochromatic Rayleigh-scattered diffuse irradiance Da monochromatic total diffuse irradiance F~ aerosol-effective scattering ratio
Diffuse and global solar spectral irradiance Fr Rayleigh-forward to total-scattering ratio Ga monochromatic terrestrial global irradiance on a horizontal surface Io~ monochromatic extraterrestrial normal irradiance I~ monochromatic terrestrial direct irradiance on a horizontal surface k~ monochromatic mixed gases absorption coefficient ko monochromatic ozone absorption coefficient k,~ monochromatic water vapor absorption coefficient / ozone depth, cm m air mass w water vapor depth, cm
Greek symbols aerosol wavelength exponent 13 turbidity coefficient 0 zenith angle p,,a monochromatic atmospheric albedo p~ ground albedo r~ monochromatic aerosol transmittance %~ monochromatic mixed-gas transmittance ro~ monochromatic ozone transmittance rrA monochromatic Rayleigh transmittance ~-,,~ monochromatic water vapor transmittance ~ single scattering albedo ;t wavelength um
REFERENCES
I. J. V. Dave, Extensive datasets of the diffuse radiation in realistic atmospheric models with aerosols and common absorbing gases. Solar Energy 21,361 (1978). 2. J. V. Dave, A magnetic tape of the direct and diffuse fluxes at the ground-level of the Dave-Braslau models obtained from Dr. Dave, 1530 Page Mill Road, Palo Alto, California. 3. C. J. Kok, Spectral irradiance of daylight at Pretoria. J. Phys. D: Appl. Phys. 5, 1513 (1972). 4. C.J. Kok, Spectral irradiance of daylight for air mass 2. J. Phys. D: Appl. Phy. 5, L85 (1977). 5. C. J. Kok and N. N. Chalmers, Spectral irradiance of daylight at Durban. National Physical Research Laboratory SCIR Research Report 339, (1978). 6. R. E. Bird and R. L. Hulstrom, Solar spectral measurements and modelling. SERI/TR-642-1013, (1981) 7. P. Moon, Proposed standard solar radiation curves for engineering use. J. Franklin Institute 230, 583 (1940). 8. A. P. Thomas and M. P. Thekaekara, Experimental and theoretical studies on solar energy for energy conversion. Sharing the Sun--A Joint Conference of the Am. Section of ISES and the Solar Energy Society of Canada, Inc.. 1. 338 (1976).
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9. D. M. Gates, Spectral distribution of solar radiation at the earths' surfaces. Science 51,523 (1966). 10. J. L. Hatfield, R. B. Giorgis and R. G. Flocchini, A simple solar radiation model for computing direct and diffuse spectral fluxes. Solar Energy 2% 323 (1981). 11. R. L. Hulstrom, Insolation models, data and algorithms, SERI/TR-36-110 (1978)~ 12. B. Leckner, The spectral distribution of solar radiation at the earth's surface---elements of a model, Solar Energy 20, 143 (1978). 13. R. A. McClatchey, R. W. Fenn, J. E. Selby, F. E. Volz and J. S. Garing, Optical Properties of the atmosphere, 3rd Edn. Air Force Cambridge Research Laboratories, AFCRL-72-0497 (1972). 14. H. P. Berlage, Zur theorie der Beleuchtung einer horizontalen, Flache durch Tageslicht. Met. Zs. 45(5), 174 (1928). 15. S. I. Sivkov, Computations of solar radiation characteristics. Israel Program for Scientific Translations, Jerusalem, 63 (1971). 16. D. Deirmendjian and Z Sekera, Global radiation resulting from multiple scattering in a Rayleigh atmosphere. Tellus VI, 382 (1954). 17. J. A. Davies and J. E. Hay, Calculation of the solar radiation incident on a horizontal surface. Proc. the First Canadian Solar Radiation Data Workshop (Edited by T. Won and J. E. Hay) (1980). 18. E. Vigroux, Contribution ~ 1'rtude experimentale de I'absorption de l'ozone. Annals de Phys. 8, 709 (1953). 19. J. N. Howard, J. I, F. King and P. R. Gast, Thermal radiation. Chap. 16 of Handbook of Geophysics and Space Environment, Rev. Edn. MacMillan, New York (1965). 20. A. Angstrrm, On the transmission of sun radiation and on dust on the atmosphere, Geogr. Ann. 2, 156 (1929). 21. A. ,~ngstrrm, On the atmospheric transmission of sun radiation II, Geogr. Ann. 2-3, 130 (1930). 22. F. Kasten, A new table and approximation formula for the relative optical air mass. Arch. Met. Geophys. Bioklim B14, 206 (1966). 23. G. D. Robinson, Absorption of solar radiation by atmospheric aerosol, as revealed by measurements from the ground. Arch. Met. Geophys. Bioklim. B12, 19 (1962). 24. J. V. Dave, P. Halpern and N. Braslau, Spectral distribution of the dkect and diffuse solar energy received at sea level of a model atmosphere. I.B.M. Palo Alto Scientific Center Tech. Rep. No. G320-3332 (1975). 25. M. P. Thekaekara, Solar Energy outside the earth's atmosphere. Solar Energy 14, 109 (1973). 26. H. Neckel and D. Labs, Improved data of solar spectral irradiance from 0.33 to 1.25 ~. Solar Physics, 74, 231 (1981).
O0394)92X/831050447~)7503.00/O © 1983 Pergamon Press Lid
DIFFUSE AND GLOBAL SOLAR SPECTRAL IRRADIANCE UNDER CLOUDLESS SKIES D. T. BRINEand M. IQBAL Department of Mechanical Engineering,University of British Columbia, Vancouver,Canada (Received 1 January 1982;accepted 1 June 1982)
Abstract--A simple empirical model to calculate solar spectral diffuse and global irradiance under cloudless skies is presented here. This formulation takes into account absorption of radiation by molecules such as O~, H20 and the uniformly-mixedgases. Attenuationby Rayleighscattering and aerosol extinction are included. Aerosolattenuation is calculated through Angstr6m'sturbidity parameters a and/3. The diffuse radiation is assumed to be composedof three parts: (1) Rayleigh-scattered diffuse irradiance; (2) aerosol-scattered diffuse irradiance; and (3) irradiance arising out of multiple reflectionsbetween the atmosphere and the ground.The global irradiance is the sum of these three componentsof diffuse irradiance plus the direct irradiance. The input parameters include an extraterrestrial spectrum, zenith angle, turbidity coefficient/3, wavelength exponent a, ground albedo p,, water vapor content and ozone content. The model is shown to yield very good results up to air mass two when compared to accurate theoretical calculations. No comparisons with measured spectra are presented because of a lack of accurate specifications of input parameters. Results are presented to show the effect of variation of certain of the input parameters. INTRODUCTION An accurate solar spectrum, both extraterrestrial and terrestrial, is necessary for the further development of many solar-related energy projects. Chief among these are the photovoltaics for power on satellites, and the continued development of terrestrially-based photovoltaic power cells. The spectrum is used for testing the degradation of materials and for establishing cut-off points in the transparency of solar materials. The extraterrestrial spectrum is attenuated by the earth's atmosphere in two distinct ways: by scattering, and by absorption. Scattering is a continuum process, whereas gas absorption occurs at discrete wavelengths and can very greatly over small wavelength intervals. Scattering by a clean, dry atmosphere is generally referred to as Rayleigh scattering. Aerosols are suspended matter, whether liquid or solid, and can be dust, volcanic ash, pollen, smoke and urban pollution, or smog. Absorption by aerosols appears to be a continuum process and is usually characterized as such. Theoretical analysis of the radiative transfer equation by the Spherical Harmonics Approximation/l] has produced extensive data sets of direct and diffuse solar irradiance. The numerical solution of the highly complex and realistic atmospheres includes all orders of scattering and absorption by the common absorbing gases and aerosols. The theoretical treatment assumes a planeparallel, homogeneous atmosphere of infinite extent horizontally. All nonhomogeneity due to scattering and/or absorption is confined to the vertical direction and is incorporated into a defined atmosphere by dividing the atmosphere into any number of layers and using appropriate amounts of air molecules, water vapor and aerosols for each layer. The defined atmospheres, called "Model atmospheres", progress in complexity from purely Rayleigh scattering, Model A, to Rayleigh scattering plus gas absorption, Model B. Further complexity 447
is added by including nonabsorbing aerosols, Model C, and absorbing aerosols, Model D. The number of data sets generated by the numerical solution is limited due to the large amount of computer time involved and the complexity of the input data. A simple method to obtain the spectral direct and diffuse irradiance is needed by engineers of all disciplines. The desired formulation should be based on a simple homogeneous atmosphere with no layers but incorporating the best available knowledge of the molecular absorption coefficients. Furthermore, the aerosol attenuation could be characterized through the/~ngstrtm turbidity formula. This simple model, once formulated, should then be compared to more accurate data sets [2] and also any experimental spectral data available from the literature [3--6]. The cloudless sky is modelled as this condition generally produces the maximum energy available. This maximum establishes the criteria with which engineers can design equipment for the utilization of solar energy in solar cells, space heating, process water heating and the thermal loading on buildings, etc. The maximum is also necessary in the design of materials for solar applications to prevent early damage or deterioration of plastics. The direct or beam irradiance provides most of the energy available on the ground. The direct spectral irradiance has been studied by several investigators. Moon/7] was probably the first to combine all the individual scattering and absorption characteristics together in a complete model. Thomas and Thekaekara[8] used a formulation developed by Gates [9] to predict the direct irradiance. Hatfield et al [10] also used Gates's formulation and included other effects such as latitude, altitude, surface albedo, slope and surface orientation. Unfortunately, the water vapor transmittance calculated by this method in the 0.8-1.0 #m band
448
D. T. BRINEand M. IQBAL
has been shown[ll] to be low and results in a sharp decrease in irradiance in this band. Also this method requires an interpolation scheme as the extraterrestrial values generally recorded in the literature and the absorption coefficients used are not for the same wavelengths. Leckner's method[12] seems to be the best presently available in simple form. He developed monochromatic transmittance functions using recent data[13] for calculating the direct spectrum. The results of this method show good agreement when compared to measured spectra and theoretical calculations. With Leckner's model, a seemingly adequate yet simple method for calculating a detailed direct spectrum is achieved. Diffuse irradiance results from the interaction of solar radiation with the scattering particles--air molecules and aerosols--of the atmosphere. The scattering by the air molecules is limited to very short wavelengths of less than 1.0~m The scattering by aerosol particles, generally called Mie scattering, is important when particle size is of the order of wavelength. The first impingement of radiation on a particle is called primary scattering. The primary scattered radiation is further scattered by air molecules or aerosol particles and this process is called multiple scattering. The detailed study of diffuse irradiance including the effects of multiple scattering is complex, and consequently, this effect has not been taken into account in this work. A simple formula to predict the scattered diffuse irradiance on a horizontal surface was proposed by Berlage[14]. This formula O~, = 0.5 cos 0 (IoA-/~)
(1)
states that the diffuse irradiance is the difference between the extraterrestrial radiation and the direct irradiance at the ground. For a Rayleigh atmosphere, this does give credible results[15] when compared to more accurate analysis[16] if a subozone layer radiation intensity is substituted for Io~. Thus, such a formulation does not account for all molecular absorption or any aerosol effects directly. Leckner proposed a simple formula for diffuse irradiance Dx = 0.5 Iox cos 0 "ro:g:wA(1 - zrx'r,a).
(2)
This expresses the diffuse irradiance as the difference between the direct irradiance and a "fictitious beam" subject only to molecular absorption. This formula takes into account molecular absorption but assumes that the aerosol and the molecular scattering functions are identical. Also, eqn (2) does not include the effects of multiple reflections between the ground and the atmosphere. Hatfield et al.[10] also present a diffuse radiation model similar to that of Leckner. In this model D~ = Io, cos 0 ro~(1 - r,:~DFk.
(3)
They have omitted the absorption due to water vapor and the mixed gases altogether. They include a factor K
for the circumsolar component and define their forward scattering function for both Rayleigh scattering and aerosol scattering as F = 0.5 cos'/3(8)
(4)
which results in F having values 0.5 or less. The preceding studies assume the forward scattering ratio to be 0.5 or less. This is approximately true for a Rayleigh-scattering atmosphere. However, the aerosol scattering is very strongly biased in the forward direction and because of this asymmetry, the effective scattering ratio (toward the ground) is a function of zenith angle. A modification of Leckner's formula seems necessary to account for the dissimilar scattering properties of air molecules and aerosols. One approach, and the one followed here, is to consider a Rayleigh-scattering atmosphere and an aerosolscattering atmosphere as quasi-separate, and calculate the diffuse irradiance as a sum of these two components. Such an approach has been used in broad-band diffuse irradiance calculations [17]. A third component of diffuse irradiance can easily be added to the previously calculated components to account for the multiple reflection effects. The ground albedo must be specified and also a monochromatic atmosphere albedo must be formulated. MATHEMATICALFORMULATION
Direct model The method for calculating the spectral direct irradiance has been adapted from Leckner[12]. This direct model uses explicit transmittances for each of Rayleigh scattering, ozone absorption, water vapor absorption, mixed-gas (CO2, 02) absorption, and aerosol extinction, The transmittance due to Rayleigh scattering is given by zrx = exp ( - 0.0088 mA-4).
(5)
The transmittance due to ozone absorption is Zoa = exp ( - kolm)
(6)
where the monochromatic absorption coefficients, ko, obtained by Vigroux[18] and presented in [19] have been employed. The transmittance due to water vapor absorption is rw = exp ( - 0.2385 kwwm/(1 + 20.07 kwwm)°45)
(7)
The transmittance due to mixed-gas absorption is ~'~A= exp ( - 1.41 k~m[(1 + 118.93 k~m)°45)
(8)
where absorption coefficients, kw and k~, are taken from [12]. The transmittance due to aerosol extinction is given by z,~ = exp ( - ~ m)t -~)
(9)
which is the simple power-law first proposed by ,~ngstr6m [20, 21].
Diffuse and global solar spectral irradiance The air mass, m, is calculated from Kasten's formula [22] m = (cos 0 + 0.15(93.885 - 0)-1253) -1.
(10)
This is the relative air mass, since all calculations in this work are assumed to be at sea level. The direct spectral irradiance on a horizontal surface at sea level at the average earth-sun distance is given by IX = Io~ cos Oz,aroxrg;,r,,az,~
449
upwelling radiation is partially reflected by the atmosphere downward again, and this process continues. The resulting diffuse irradiance after these multiple reflections is given by Din,, = (IX + D,, + D,,,.)pg~,p,,,,/(l - p,,,,p,,~,).
(14)
The ground albedo, pgA, may be taken independent of wavelength. The monochromatic atmospheric albedo, p,A, can be shown t6 be
(11) Paa = r~xr "ar'ga((1 - r'ra)r'xF, + 0.22 (1 - r'A)OJor'rA).
(15) The diffuse irradiance is considered to be composed of three distinct components: those that are (1) Rayleigh The primes on all [he transmittances indicate that they scattered; (2) aerosol scattered; and (3) multiply are all evaluated at m = 1.9. The air mass m = 1.9 is used reflected. The individual transmittances of the direct .as it gives good agreement with theoretical results[2]. model are used to generate equations for each of the two The first term of eqn (15) represents the fraction of the scattering components. The equation for the multiply- upwelling radiation reflected back to the earth by Rayreflected component was derived from basic principles. leigh scattering. The second term represents the fraction The initial idea for the formulation of the atmospheric of the upweiling radiation reflected back to the earth by albedo came from broad-band calculations[17]. The aerosols. The factor 0.22 is the effective back-scatter Rayleigh-scattered diffuse irradiance arriving on a ratio of aerosols evaluated at m = 1.9. In other words, (1 - Fa) = (1-0.78) = 0.22. horizontal surface is The total diffuse irradiance from all sources is given by Dr~ = Io~ cos 0 rO~',~'wX~'ax(l-- rra)F~ (12) D,,, = Dr,, + D,,~, + D,.,,. (16) This is similar to eqn (2) except the factor (1 - r,A~'a~,) has been changed to ( 1 - r,a) and ~',a is taken outside the Similarly, the global irradiance is the sum of the direct bracketed quantity. Analogous to Leckner's approach, irradiance and the total diffuse irradiance. This is given the two beams of radiation are the direct beam and a as beam which includes molecular absorptance and aerosol G; = Ix + D,~,. (17) attenuation. The difference between these is the Rayleigh-scattered diffuse irradiance. Fr is the Rayleigh These monochromatic values of direct irradiance, total forward-scattering ratio, generally taken as 0.5. Similar to eqn (12), the aerosol-scattered diffuse diffuse irradiance and global irradiance can all be summed separately to give the broad-band values of these irradiance on a horizontal surface is quantities. This is accomplished by taking each monochD,~ = Io~ cos O~',~'o~rgxZw~ (1 - r,a)oJoF,. (13) romatic irradiance value in Wm -z (#m)-: and multiplying by the appropriate wavelength interval in #m. These Here, the difference between the direct beam and a beam values are then summed to give a broad-band value for which includes Rayleigh scattering and molecular ab- any wavelength interval. sorbers is the aerosol attenuated radiation. The single scattering albedo of aerosols, OJo, partitions the energy attenuated into scattering and absorption. A value of METHODOLOGYAND VALIDATION To validate this formulation of diffuse spectral irradioJo---1.0 indicates a completely scattering aerosol. The ance, the results obtained must be compared to more single scattering albedo is assumed independent of accurate theoretical calculations and to measured specwavelength. F,, the effective scattering ratio (toward the earth, not necessarily in the forward direction) is a tra. The comparison with theoretical solutions is easy, as extensive numerical results are available[2, 24] and the function of zenith angle, 0. Experimental values of Fo were reported [23] and have been used by other atmospheric input parameters are explicitly known. A similar comparison with measured spectra is difficult. workers [17]. It is to be emphasized that these values of The measured data are either based on long term F,, give only a rough estimate of the effective forward averages[3-5] or are given in graphical form[6] with scattering function. The values of Fo used in this paper are: F, = 0.923 for 0 = 0°, F,, = 0.78 for O= 60°. Like the insufficient information on the atmospheric parameters. Consequently, no comparison with measured spectra is single scattering albedo, the wavelength dependence of F,, is neglected. illustrated in this paper. The direct radiation reaching the ground, and the RayAll theoretical comparisons between Model atmosleigh-scattered and aerosol-scattered diffuse radiation pheres of Dave[2, 24] and the present work use the same extraterrestrial irradiance values[19]; however, the reaching the ground after the first pass through the number of wavelength intervals is different between the atmosphere are partially reflected back by the earth. This Diffuse model
450
D. T. BRINEand M. IQBAL
two studies. The wavelength spectrum investigated extended from 0.29 to 4.0 #m. A comparison between Model A (pure Rayleigh atmosphere) and the present work for air mass one, two and five was carried out. For air mass one, the agreement is excellent; some discrepancy (0.37-0.40/zm) is due to the different wavelength intervals in the two studies. At air mass two, the present work somewhat overestimates the diffuse irradiance at short wavelengths 0.3-0.37 tzm; at longer wavelengths the agreement is quite good. At air mass five, overestimation occurs up to 0.45 #m, after which the agreement is again good. The overestimation at short wavelengths of the present model could be attributed to the fact that Model A included all orders of scattering whereas the present work does not. A comparison of Model B (Rayleigh scattering plus absorbing gases) with the present work for air mass one, two and five was carried out. The results are similar to those presented for Model A at air mass one, the agreement is excellent; at air mass two, the agreement is very good; and at air mass five the overestimation of the present work is again noticed. The foregoing comparisons show that generally the present work accurately depicts scattering by a clean atmosphere. This implies that the formulation and the coefficients used are correct and that the next step-include aerosols--starts from a secure procedure. A comparison of the results of Model C atmosphere (Rayleigh scattering, absorbing gases plus a nonabsorbing aerosol) and the present procedure were carried out as follows. The direct irradiance was calculated for specific ozone and water vapor content corresponding to the values used in Model C. The turbidity parameters a and /3 were varied to give the best fit with the direct spectrum of Model C. In this manner, the values of these parameters were derived. An example of such an exercise is demonstrated in Fig. 1, where it is shown that a = 0.6 and/3 = 0.07 correspond to Model C. The fit of the present direct model and Model C is found to be excellent for air mass one and two. 2000
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Having determined values for a and /3 in the above manner, the diffuse irradiance was calculated using the present formulation, eqns (12-16). Comparison of these results with the diffuse radiation calculations of Model C for air mass one and for air mass two were made. In order to eliminate the effects of multiple reflections between ground and atmosphere, ground albedo is assumed to be zero. For air mass one, up to 0.6 #m, the agreement between Model C and the present work is good. The agreement is also quite good at longer wavelengths, with the present work giving generally slightly lower values than those of Model C. Overall the agreement is very good at air mass one. Leckner's formulation agrees up to 0.40/~m and then drops off abruptly to values substantially below those of Model C. For air mass two, the present work is consistently slightly below the values of Model C. Leckner's formulation is in good agreement with Model C up to 0.42 #m, but thereafter drops significantly below Model C and below the present work. The preceding comparisons indicate that the present formulation consistently produces better results than Leckner's formulation up to air mass two. Many other comparisons have been made and they lead to the same conclusions. When a nonzero ground albedo is considered, the multiple reflections between the ground and the atmosphere add to the total diffuse irradiance. Figure 2 shows a comparison of Model C and the present work for air mass one and ground albedo 0.3. The present work underestimates somewhat in the 0.33-0.40/~m band but is otherwise in excellent agreement with Model C. Also illustrated is the simple formulation of Leckner that has no provision for ground albedo. It is much below the values of Model C and the present work. Figure 3 presents the same comparison as Fig. 2 for air mass two. At this air mass the present work is generally lower than the values of ModN C but still overall much better than Leckner's formulation.
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Fig. 4. The effect of air mass on direct, diffuse and global irradiance on a horizontal surface for/7 = 0.07, a = 1.3, Oe= 0.0, ozone content 0.3 cm. Curve 1, global m = 1.0. Curve 2, direct m = 1.0. Curve 3, global m = 2.0. Curve 4, direct m = 2.0. Curve 5, diffuse m = 1.0. Curve 6, diffuse m = 2.0.
Comparisons of the present study with Hattield et al.[lO] have been made. Due to the previously mentioned formulation of the molecular and aerosol scattering, their results are generally below the values of Model C and of this study.
The turbidity coefficient,/3, has a strong influence on both direct and diffuse irradiance. /3 is an indication of the mass loading of the aerosol and also an indication of the atmospheric visibility for a given aerosol particle size distribution. Figure 5 shows the effect of /3 on direct, diffuse and global irradiance for air mass one and cz = 1.3. For /3=0.2 (Vis= 12km) the broad-band direct irradiance is about 19 per cent less than for /3 = 0.07 (Vis = 50 km). The broad-band diffuse irradiance for/3 = 0.07 is about 49 per cent less than for /3 = 0.2. The broad-band global irradiance for/3 = 0.07 is about 3 per cent greater than for /3 = 0.2. Thus, even for a very turbid atmosphere (/3 = 0.2) the global irradiance available is only marginally affected at air mass one. Spectrally,/3 has the greatest effect on the diffuse irradiance and predominantly at short wavelengths < 1.0 #zm). The
RESULTS In the previous section "Methodology and Validation", the present work was compared to an accurate theoretical model and it was found to produce very good results. The versatility of the present method will be illustrated in this section. The six independent input parameters of this model are: air mass, ozone content, water vapor content, ground albedo, the turbidity coefficient /3 and the wavelength exponent a. These parameters are easily varied, so the effects of each parameter can be investigated independently. The extraterrestrial solar spectrum presented by Thekaekara[25] is used to produce all the results in this section. It is understood that present-day values of the solar constant and its spectral distribution have been proposed[26]. However, the present method is independent of any particular extraterrestrial solar spectrum. Air mass has the greatest effect of any of the parameters on the available direct and global irradiance, whereas the available diffuse irradiance is affected to a much lesser extent. This effect is demonstrated in Fig. 4 for air mass one and two. The broad-band (0.29-4.0/zm) direct irradiance decreases 60 per cent from 925 to 370 W/m 2 and the broad-band global irradiance decreases 56 per cent from 1073 to 471W/m 2. The broad-band diffuse irradiance decreases only 32% from 148 to 100 W/m 2 and for wavelengths greater than about 1.0/~m the diffuse is essentially independent of air mass. The spectral character of the curves for air mass one and two are very similar and it is noticed that the percentage decrease in irradiance is almost uniform at all wavelengths; even though the actual amount attenuated at short wavelengths ( < 1.0 #m) is very large.
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452
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sharp peak of the direct curve for /3 = 0.07 has disappeared and has become a wide plateau for /3 = 0.2, indicating the pronounced effect of scattering for an increased mass loading of aerosols at short (0.4-0.7/~m) wavelengths Although the global irradiance is decreased only 3 per cent, almost all of this decrease occurs in the visible (0.4-0.75 ~m) band. For air mass two,/3 has a somewhat greater effect on broad-band global irradiance, about 9 per cent decrease, partly due to the decreased value of the effective scatter ratio of the aerosol. The broad-band direct irradiance decreased 31 per cent between/3 = 0.07 and/3 = 0.2 and the broad-band diffuse irradiance decreased 43 per cent between/3 = 0.2 and/3 = 0.07. Spectrally, the peak of the direct irradiance has moved from a plateau between 0.45--0.7/zm for/3 = 0.07 to a peak at 0.7 ~m for/3 = 0.2. The diffuse irradiance has correspondingly shown a large increase at short (0.3-0.6 tim) wavelengths and actually exceeds the direct irradiance for /3 = 0.2 up to 0.6 ~m, illustrating very well that what is scattered from the direct beam is accounted for in the diffuse irradiance. Another influencing factor of aerosols is the wavelength exponent a, which is related to the size distribution of the aerosol particles. Large values of a indicate a relatively higher ratio of small particles to large particles. For most natural atmospheres a = 1,3---0.2. An atmosphere containing a large amount of small particles (large a) scatters more than an atmosphere containing an equivalent mass of large particles (small a). In Fig. 6, it is evident that a = 2.0 scatters more energy from the direct beam, about 8 per cent, than does a = 0.5 The plots of the diffuse irradiance show that the energy scattered from the direct beam reappears as the diffuse irradiance, with the difference between the two broad-band diffuse values being 32 per cent. The global irradiance plots show very little difference (about 2 per cent) between a = 2 . 0 and a =0.5, and the influence of a is seen only at wavelengths less than 0.7 ram. 2
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Fig. 7. The effect of ground albedo on diffuse and global irradiance for air mass one, //= 0.07, a = 1.13, water vapor content 2.0 cm and ozone content 0.3 cm. Curve l, global pg = 0.2. Curve 2, global p= = 0.0. Curve 3, direct. Curve 4, diffuse pe = 0.2. Curve 5, diffuse pz = 0 0. The ground albedo and multiple reflections are considered next. Figure 7 shows the effect of the variation of ground albedo on the diffuse and global irradiance. Two values of ground albedo are illustrated, p= = 0.0 and 0.2. p= =0.2 is representative of most ground covers. Due to multiple reflections between the earth and its atmosphere, a larger ground reflectance produces more diffuse irradiance (about 10 per cent) and corresponding increases in global irradiance (about 2 per cent). The effects of variation of water vapor content and ozone content are limited to specific wavelength bands and primarily to the direct beam. There is very little effect on the diffuse irradiance due to these parameters. The global irradiance indicates that any difference is due mainly to the variation in the direct irradiance. CONCLUSIONS The present method of calculating the diffuse spectral irradiance has been compared to accurate theoretical analysis and found to give good agreement for the model atmospheres investigated here. The major advantage of this formulation is its simplicity. The computer time is minimal and the input parameters are easily varied. The results presented are illustrative of the flexibility and the range of effects that can be studied with this formulation. Comparisons with measured spectra should be undertaken when sufficient input parameters are specified, to complete the validation of this formulation.
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WAVELENGTH , ~m
Fig. 6. The effect of a on direct, diffuse and global irradiance for air mass one, ground albedo 0.0, water vapor content 2.0 cm, ozone content 0.3 cm and //= 0.07. Curve l, global a = 0.5. Curve 2, global a = 2.0. Curve 3, direct a = 0.5. Curve 4, direct a = 2.0. Curve 5, diffuse a = 2.0. Curve 6, diffuse a = 0.5.
NOTATION D~ monochromatic diffuse irradiance D~ monochromatic aerosol-scattered diffuse irradiance D,.~ monochromatic multiply-reflecteddiffuse irradiance Dr~, monochromatic Rayleigh-scattered diffuse irradiance Da monochromatic total diffuse irradiance F~ aerosol-effective scattering ratio
Diffuse and global solar spectral irradiance Fr Rayleigh-forward to total-scattering ratio Ga monochromatic terrestrial global irradiance on a horizontal surface Io~ monochromatic extraterrestrial normal irradiance I~ monochromatic terrestrial direct irradiance on a horizontal surface k~ monochromatic mixed gases absorption coefficient ko monochromatic ozone absorption coefficient k,~ monochromatic water vapor absorption coefficient / ozone depth, cm m air mass w water vapor depth, cm
Greek symbols aerosol wavelength exponent 13 turbidity coefficient 0 zenith angle p,,a monochromatic atmospheric albedo p~ ground albedo r~ monochromatic aerosol transmittance %~ monochromatic mixed-gas transmittance ro~ monochromatic ozone transmittance rrA monochromatic Rayleigh transmittance ~-,,~ monochromatic water vapor transmittance ~ single scattering albedo ;t wavelength um
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