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## Solar Noon – an overview

4.5 Daily and Seasonal Cycles of Solar Radiation on a Surface

Sunlight falls on outdoor monuments and penetrates through windows: it is composed of direct radiation, which forms the solar beams, and diffuse radiation, which is scattered by the sky in all directions. The diffuse component ranges from 20% to 30% of the total income in clear days to 100% in overcast ones. Scattering is a function of the wavelength (this is the reason why the sky appears blue), and the very energetic photons in the UV band are the most scattered ones, so that they may hit objects not exposed to direct light.

For any latitude ϕ and day of the year, defined by the solar declination δ⊙, direct light can be computed by means of the coordinates of the sun: the altitude H⊙ over the horizon (i.e. solar height) and the azimuth A⊙, i.e. the angular distance between the two vertical circles both containing the zenith and one containing the sun and the other the south point. The astronomical formulae to compute these coordinates are:

(4.14)sinH⊙=sinδ⊙sinϕ+cosδ⊙cosϕcosτ

(4.15)sinA⊙=cosδ⊙sinτ1−(sinδ⊙sinϕ+cosδ⊙cosϕcosτ)2

where the hour angle τ = 180° t/12 is computed from the time t, in hours and tenths of hour, from or to the culmination of the sun, i.e. from or to the true midday. This means that t is negative in the morning, vanishes at noon and is positive in the afternoon. The solar declination δ⊙(j) for the jth day is found in astronomical ephemeris tables or, for our purposes, it can be computed with the simple approximation

(4.16)δ⊙(j)=δ⊙(0)cos2πj365

where the jth day is computed since the winter solstice (i.e. 21 December) and δ⊙(0)=−23° 27′ is the declination at winter solstice. In this formula, the Earth’s orbit around the sun is supposed to be circular. The mean of the Earth orbit eccentricity e⊕ is e⊕ = 0.0167; the eccentricity is found by dividing the distance between the foci by the length of the major axis, and a circle is the limiting case as e⊕ approaches zero. The error in this approximation is small (some primes) and negligible except for precise astronomical calculations.

From the above formulae it follows that:

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at solar noon, when the hour angle τ is zero, cos τ = 1 at any latitude, and the zenith angle Z⊙(defined as Z⊙=90°−H⊙) is Z⊙=ϕ−δ⊙;

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at sunrise and sunset (i.e. H⊙=0), at any latitude (except the poles), cosZ⊙=0, and 2τ is the daytime length expressed in hour angle (h);

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the daylength (the time duration from the moment the upper limb of the sun’s disk appears above the horizon at sunrise to the moment when the upper limb disappears below the horizon at sunset) DL =2τ is computed under the condition cosτ=−tanϕtanδ⊙, i.e. DL = (24/π) arccos(tanϕtanδ⊙) (h).

Solar beams are represented by straight lines originating from the sun and passing through windows; their intercepts on the opposite wall represent the light spot. The envelope of these spots describes the areas affected by the direct solar income during the course of the day as discussed below.

The flux density per unit time or intensity Ip of the solar radiation (also called irradiation) falling on horizontal, vertical or arbitrarily inclined planes (Robinson, 1966; Kondratyev, 1969; Bernardi and Vincenzi, 1994) facing the direction Ap (computed from the moment of the true noon, i.e. the south) and inclined by the angle β with the plane of the horizon is

(4.17)Ip=Io{cosβ(sinδ⊙sinϕ+cosδ⊙cosϕcosτ)+sinβ{cosAp[tanϕ(sinδ⊙sinϕ+cosδ⊙cosϕcosτ)−sinδ⊙secϕ]+sinApcosδ⊙sinτ}}

where Io is the intensity of the solar beam (near the surface) and the irradiation on a horizontal or vertical plane is obtained by setting β = 0° or β = 90°, respectively.

An example of the hourly variation of the flux density of the solar radiation on the horizontal and vertical surfaces with various orientations is here discussed for the latitude ϕ=45°. In order to show the influence of the absorption due to the optical mass, first let us suppose that the atmosphere is perfectly transparent. At the winter solstice, the sun rises at A⊙=124°, i.e. close southeast (135°) and sets at A⊙=236°, i.e. close southwest (225°) and the vertical surfaces facing these two directions receive the maximum flux of energy; at noon, the solar height is low, i.e. H⊙=90°−Z⊙=90°−(ϕ−δ⊙)=22° and the energy income on the vertical surface facing south is greater than that on the horizontal plane. At the equinoxes, sunrise and sunset occur just at east and west, and with a perfectly transparent atmosphere, the vertical surfaces facing these two directions receive the maximum flux of radiation. At this particular latitude, at noon the solar height is H⊙=45°, so that the solar energy that falls on the vertical surface facing south equals that on the horizontal plane. At the summer solstice, sunrise and sunset occur, respectively, at A⊙=57°, i.e. midway between northeast and east–northeast and A⊙=303°, i.e. midway between northwest and west–northwest. At noon the sun is high, i.e. H⊙=68°, so that the solar energy flux that falls on the vertical surface facing south is less than that on the horizontal plane, and at this particular latitude, the flux on the horizontal equals the flux at the winter solstice on the vertical surface facing south, and vice versa.

If the same calculations are made taking into account the attenuation of the solar radiation through the optical air mass, which increases with decreasing solar height, the results are quite different, as shown in Fig. 4.2. At the winter solstice (Fig. 4.2(a)), the declination δ⊙(0)=−23° 27′ is at its minimum and the optical thickness of the atmosphere is maximum. When the sun is very low on the horizon and the light beams are perpendicular to vertical surfaces facing the sun, the fraction of energy reaching the surfaces is small. The incoming intensity becomes relevant only when the sun is relatively high. At noon, the vertical surface facing south receives the maximum flux density and the horizontal plane receives much less.

FIGURE 4.2. Hourly variation of the flux density of the solar irradiation Ip on the horizontal H (dotted line) and vertical surfaces (full lines) with various orientations (principal compass directions, thick colour lines; secondary, thin black lines) for latitude ϕ = 45°. (a) Winter solstice, (b) equinoxes and (c) summer solstice.

At equinoxes (Fig. 4.2(b)), δ⊙=0 and the atmospheric absorption is smaller. Near sunrise and sunset, the atmosphere reduces the solar energy (but less than in winter) and then the maximum flux density on the vertical surfaces facing the sun remains more or less the same, reaching the minimum at noon where the geometric effect of the slant beam dominates over the smaller atmospheric attenuation. On vertical surfaces not facing the solar motion (northern sector from west to east), the flux density is much smaller. The flux density on the vertical surface facing south equals the flux density on the horizontal plane. This is obvious at noon, when the solar altitude is H⊙=45°, forming the same angle with the horizontal plane and the vertical surface. On the other hours, for the vertical surface, the advantage of receiving the beam with an angle approaching the normal is substantially compensated for by atmospheric absorption.

At the summer solstice (Fig. 4.2(c)), the declination δ⊙ is at its maximum and the optical thickness of the atmosphere is further reduced. However, the height of the sun over the horizon is the dominant factor, except near sunrise and sunset. In fact, the flux density increases from sunrise to east, then decreases and reaches the minimum at south and continues symmetrically till sunset. The elevated solar altitude makes the solar income maximum on the horizontal plane. The flux on the south surface is small and at noon it is close to the value reached at 6.30 a.m. on the north–northeast or at 17.30 p.m. on the north–northwest.

A comparison between the plots at the two solstices shows that at noon the summer flux on the vertical south surface is greater than the winter flux on the horizontal plane at the same hour and the summer flux on the horizontal plane at noon is even greater than the winter flux on the vertical surface facing south.

This example shows how the solar income is affected by the atmospheric optical length, although in clear sky. If the atmospheric attenuation is affected by the local weather (e.g. haze persistency, cloud cover) or pollution, the departures are even larger.

It is possible to integrate over the whole daytime the flux density of solar radiation for an arbitrarily oriented or slant surface (Fig. 4.3). At the winter solstice (Fig. 4.3(a)), the dominance of the southern sector is evident. At the equinoxes (Fig. 4.3(b)), the maximum instantaneous values of irradiation are similar for the whole southern sector from east–southeast to west–southwest, with a slight minimum in south, and the surfaces facing east and west having slightly less irradiation. At the summer solstice (Fig. 4.3(c)), all surfaces receive a similar flux, except for the northern sector. However, the daily total solar income on these vertical surfaces (i.e. β = 90°) shows a different situation: the south surface receives the maximum total and the east and west ones receive half of it. On the other hand, when β tends to zero, all the slant surfaces tend to the horizontal plane and the differences tend to vanish. For slant surfaces facing south and southeast or southwest, the total income reaches the maximum, respectively, at the slopes β = 45° and β = 40°. The whole southern sector has convex plots with the maximum at a sloping angle that tends to vanish at the east and west orientations. The east and west plots are still convex but close to a straight line. The northern sector has concave plots, with the total income decreasing when the slope increases. The surface exposed to north is no more reached by direct solar radiation when β ≥ 45°. At the summer solstice, all the integral values, except for those of the northern sector, are quite similar between them.

FIGURE 4.3. Daily total income It of the solar radiation on a slant surface oriented in the eight cardinal directions is represented against the sloping angle for the latitude ϕ = 45°. (a) Winter solstice, (b) equinoxes and (c) summer solstice.

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