Young have developed a good, practical formula which is well suited for use with small solar elevation angles [8]
. _ ms ga а gtnv-i (2)
_ з tк* V 4 sin^V-t (S. ffiLGS 9&a ri tT V 4 GJGGCaca?
This equation approaches the simple expression 1/sin V asymptotically for angles greater than 250. (The reader may wish to try using the angle V = 900 in the Kasten-Young equation. Note that L = 1 with the sun directly overhead.
2.4 Enhanced global irradiance model
One would expect the diffuse contribution IF to the global irradiance to increase with increasing atmospheric turbidity (increasing values of a) and that these two quantities are related to one another and to the solar elevation angle. This connection has been examined in earlier work, and the graphical result is shown in Figure 3. The Linke turbidity factor TL enters the analysis in the term for the direct irradiance and is equal to unity for a pure Rayleigh atmosphere (no aerosols - only molecular scattering). The term aL is in this formulation replaced by:
a1 = «ф [-0.B662 ■ Тя ■ L ■ (3)
where Dr(L) is the Rayleigh optical depth as a function of the air mass L. A very useful empirical equation for 1/Dr has been developed by Louche, Peri and Iqbal and modified by Fritz Kasten [8]:
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-7T = 6J&296+ 1.7 515 ■ L - 0.12Є2 ■ - (ШВ65 ■ £a - 0JD0Q13 ■ L4 <4)
The data of Figure 3 has been used to find an expression for the diffuse irradiance on a horizontal surface IF as a function of the elevation angle V and the turbidity factor TL:
£r = (4ВД4 ■ Г, - Ш2)- (1-ехр[аі - Г, - 0.P905- Y]') (5)
2.5 Equation for finding TL from observations
In view of the foregoing remarks it is now possible to write an algorithm for the determination of the Linke turbidity factor from observations of the global solar irradiance on a horizontal surface and with knowledge of the solar elevation angle. The elevation angle is readily computed when the latitude, longitude and time of day are known.
tc = 2» ■ Щ - sinF ■ дС-азьи-*!. - l-вд + (4^4,^ _ 4^33) ■ (1 - (6)
IG is the global irradiance measured, and V can be found from the time and position data. Knowledge of V yields the air mass L which in turn permits the optical depth DR to be determined. The only unknown parameter in the equation is the Linke turbidity factor TL which can then be calculated. This program has been carried out for clear days at a wide range of locations during the voyage from the Arctic to the Antarctic.
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For all of the good, clear days during the eight month expedition the Linke turbidity factor was computed using the algorithm described above. For these same days the mid day temperature and relative humidity were obtained by examination of the Galathea III database. From the temperature and humidity data it is straightforward to compute the amount of water present in a cubic meter of surface air. These calculations were performed with the data shown in Figure 4 as the result. The regression shows that a moisture content close to zero should yield a Linke turbidity factor near unity as expected. The value 1,19 may reflect the fact that some aerosols will typically be present in the maritime environments from which data is available in addition to water vapor.
This analysis permits the estimation of the Linke turbidity factor in maritime environments based upon knowledge of the temperature and relative humidity. Compute the water content of a cubic meter of surface air, and apply the equation shown in Figure 4 to find TL. With TL in hand a good prediction of the global irradiance on the horizontal on a clear day, including the distribution of direct and diffuse irradiance, can be made using Equation 6. As local temperature and relative humidity are standard meteorological parameters, no special equipment or data is needed to do the calculations. Visibility conditions are also derivable from knowledge of the turbidity factor as discussed elsewhere [7]. This method does not take account of other aerosol which may be present, but it can be applied in typical maritime conditions without a modest amount of dry atmospheric aerosol particles.
[1] Galathea 3 Expedition portal in English: http://galathea3.emu. dk/eng/index. html
[2] Frank Bason; Solar Irradiance Measurements from the Danish Galathea 3 Expedition, ISES Solar World Conference Beijing 2007, Proceedings.
[3] Pierre Ineichen, Richard Perez; A new airmass independent formulation for the Linke turbidity coef
ficient, Solar Energy 73, no. 3, pp 151-157, 2002.
[4] Linke, F.; Transmissions-Koeffizient und Trubungsfaktor, Beitr. Phys. fr. Atmos. 10, 91-103 (1922).
[5] Frank Bason; Solar Radiation Measurements from Greenland to Antarctica - Optics Table Data from the Danish Galathea III Expedition (2006-2007) , North Sun Conference 2007, Riga,
Latvia, May 2007
[6] John Duffie, William Beckman, Solar Engineering of Thermal Processes, Wiley & Sons, New York, 1991.
[7] Frank Bason; Diffuse Solar Irradiance and Atmospheric Turbidity, EuroSun 2004 Conference
Proceedings, Freiburg, Germany, June 2004.
[8] F. Kasten, A. T. Young; Revised optical air mass tables and approximation formula, Applied Optics,
28, no. 22, pp 4735-4738, 15 Nov 1989.
