Incident solar radiation
2.3 Diurnal variation of solar irradiance
For a given set of atmospheric conditions, the irradiance at any point on the Earth’s surface is determined by the solar elevation, b. This rises during the day from zero (or its minimum value in the Arctic or Antarctic during the summer) at dawn to its maximum value at noon, and then diminishes in a precisely symmetrical manner to zero (or the minimum value) at dusk. The exact manner of the variation ofbwith time of day depends on the latitude, and on the solar declination,d, at the time. The declination is the angle through which a given hemisphere (north or south) is tilted towards the Sun (Fig. 2.5). In summer it has a positive value, in winter a negative one; at the spring and autumn equinoxes its value is zero. Its maximum value, positive or negative, is 23270. Pub-lished tables exist in which the solar declination for the northern
hemisphere may be found for any day of the year. Alternatively, a relation derived by Spencer (1971) may be used
d¼0:3963722:9133 coscþ4:02543 sinc0:3872 cos 2c
þ0:052 sin 2c ð2:6Þ
where c is the date expressed as an angle (c¼360d/365; d¼day number, ranging from 0 on 1 January to 364 on 31 December): bothd andcare in degrees. The declination for the southern hemisphere on a given date has the same numerical value as that for the northern hemi-sphere, but the opposite sign.
The solar elevation,b, at any given latitude,g, varies with the time of day,t, (expressed as an angle) in accordance with the relation
sinb¼singsindcosgcosdcost ð2:7Þ wheretis 360t /24 (tbeing the hours elapsed since 00.00 h). If we write this in the simpler form
sinb¼c1c2cost ð2:8Þ wherec1andc2are constants for a particular latitude and date (c1¼sing sind,c2¼cosgcosd), then it becomes clear that the variation of sinb with the time of day is sinusoidal.Figure 2.6shows the variations in both b and sin b throughout a 24-hour period corresponding to the longest summer day (21 December) and shortest winter day (21 June) at the latitude of Canberra, Australia (35S). For completeness, the values of band sin b during the hours of darkness are shown: these are negative and correspond to the angle of the Sun below the horizontal plane.
21 March d= 0°
22 December d= –23° 27´
d= 0°
23 September 22 June
d= 23° 27´
Northern summer
d
Northern winter
Fig. 2.5 Variation of the solar declination throughout the year.
2.3 Diurnal variation of solar irradiance 39
The variation of sinbis sinusoidal with respect to time measured within a 24-hour cycle, but not with respect to hours of daylight.
The sine of the solar elevationbis equal to the cosine of the solar zenith angley. Irradiance due to the direct solar beam on a horizontal surface is, in accordance with the Cosine Law, proportional to cos y. We might therefore expect that in the absence of cloud, solar irradiance at the Earth’s surface will vary over a 24-hour period in much the same way as sinb(Fig.
2.6), except that its value will be zero during the hours of darkness. Exact conformity between the behaviour of irradiance and sin b is not to be expected since, for example, the effect of the atmosphere varies with solar
Fig. 2.6 Variation in solar elevation (b) and sinbduring a 24-h period on the longest summer day and the shortest winter day at 35latitude.
elevation. The greater attenuation of the direct beam at low solar elevation due to increased atmospheric pathlength, although in part balanced by a (proportionately) greater contribution from skylight, does reduce irradi-ance early in the morning and towards sunset below the values that might be anticipated on the basis ofeqn 2.8. This has the useful effect of making the curve of daily variation of irradiance approximately sinusoidal with respect to daylight hours, even though, as we have seen, the variation of sin b is strictly speaking only sinusoidal with respect to hours since 00:00 h.
Figure 2.7gives some examples of the diurnal variation of total irradiance at different times of year and under different atmospheric conditions.
A smooth curve is of course only obtained when cloud cover is absent, or is constant throughout the day. Broken cloud imposes short-term irregularities on the underlying sinusoidal variation in irradiance (Fig. 2.7). If E(t) is the total irradiance at timeth after sunrise, then by integrating E(t) with respect to time we obtain the total solar radiant energy received per unit horizontal area during the day. This is referred to as thedaily insolationand we shall give it the symbol,Qs
Qs¼ no, or constant, cloud cover the diurnal variation in E(t) is expressed approximately by
EðtÞ ¼Emsinðpt=NÞ ð2:10Þ
whereEmis the irradiance at solar noon.928By integrating this expression in accordance witheqn 2.9, it can be shown that the daily insolation on such a day is related to the maximum (noon) irradiance by
Qs¼2NEm=p ð2:11Þ Thus if, as for example inFig. 2.7c(16 March), the maximum irradiance is 940 W m2in a l2-hour day, then the total energy received is about 26 MJ.
On days with broken cloud, the degree of cloud cover is varying continuously throughout any given day. Nevertheless, in many parts of the world, the cloud cover averaged over a period as long as a month is approximately constant throughout the day.1399 Therefore, the diurnal variation in total irradiance averaged over a month will be approximately sinusoidal and will conform witheqn 2.10. From the average maximum irradiance, the average daily insolation can be calculated usingeqn 2.11.
2.3 Diurnal variation of solar irradiance 41