Extraterrestrial radiation received on an inclined plane
The previous section deals with the extraterrestrial radiation received on a horizontal plane, total or spectral. This section deals with an inclined plane whose inclination is /? and azimuth is a {cf. Figure 2.4, Chapter 2).
The extraterrestrial irradiance Eoi_{nc}0> 0 received on this inclined plane at time
t is given by:
where в is the angle of incidence of the sun rays, formed by the normal to the plane and the direction of the sun, given by equation (2.12). The extraterrestrial irradiation a) received during a period of time [//, U]> expressed in TST, is obtained by integrating equation (3.23), after a change of variable from t to the hour angle со, on the corresponding interval [coj, CO2]:
Using equation (2.14), it comes:
where the parameters A, В, and C are given by equation (2.15).
Given an opaque plane, the daily extraterrestrial irradiation H_{oda} (Д «) is given by:
where the integral limits со/ and CO2 are solutions to cos(#)=0, equation that is resolved by means of equations (2.14) and (2.15).
Let take a few special cases. In the case of a vertical plane such as windows or walls of building oriented to the east, i.e., /?=л/2 and o. = nl2, the limits co_{t} and a>2 are respectively соsunrise ^{an}d 0. The hour angle at sunrise co_{sunrise} (= -co_{sunsel}) is given by equation (2.19), and it comes:
The integral limits for a vertical plane oriented to the west (а = Зд/2) are respectively 0 and co_{sunset}. It follows that the plane oriented to the west receives the same daily irradiation than the plane oriented to the east:
In the case of a vertical plane oriented to the north or the south, the first step is the calculation of the hour angle co^-w corresponding to the time when the sun passes in the east-west plane containing the vertical plane. It happens for 1$=п12, and the solution is obtained by equation (2.11) for tan(|/s):
if co_{swtset} > Jt/2, i.e., S and Ф are of opposite signs, co_{E}~_{0}=
For a vertical plane oriented to the south (й=л) and located in the northern hemisphere, the limits are -(o_{E}-w‘^{An}& a>_{E}-w■ It comes:
For a plane located in the northern hemisphere oriented southward (й=л) and whose inclination/!is equal to the latitude Ф (/?=Ф), the limits are -coe-wand coe-iv■ It comes:
Plants such as sunflowers and other radiation collectors can change orientation (tilt and azimuth) during the day. For example, concentration systems can rotate about an east-west, or north-south axis, with continuously adjusted positions in order to minimize the angle of incidence. There are also tracking systems fitted with two axes so that the plane is always normal to the direction of the sun. In these cases, the angle of incidence в is given by:
Since time in equation (2.2) is expressed in h, H_{0da}J/3, a) is expressed in Wh m^{-2} in the previous equations. The conversion to J m^{-2} is done by multiplying the result of the previous equations by 3600 s. The daily average of extraterrestrial irradiance Eodayifi> ^{a}) ^{m} W m^{-2} is obtained by dividing H_{0day}{fi, a) resulting from one of the previous equations by 24 h.
Figure 3.7 exhibits the annual profile of the daily average of extraterrestrial irradiance E_{0day}{Д a) at latitude +45° as a function of the number of the day in the year received by a plane (i) horizontal, (ii) inclined at 45° facing south, and (iii) vertical facing south. The daily average irradiance received on a vertical plane E_{0da} (J3=n/2, a = n) is the same than that received on a horizontal plane E_{0da}Jfi=0, a) during about half of the year including the boreal winter and is much less than during the other half of the year including the boreal summer. With an inclination of 45° (/?=л/4), the plane facing south receives more radiation in annual average than the horizontal plane: 398 versus 308 W m^{-2}. The variation throughout the year is much smaller for the inclined plane than for the horizontal plane. The inclined plane exhibits a minimum of 348 W m^{-2 }and two maxima of 439 W m^{-2}, while the horizontal plane exhibits a minimum of 121 W m^{-2} and a maximum of 485 W m^{-2}.
Figure 3.7 Annual profile of the daily mean of the extraterrestrial total irradiance at latitude 45° received by a plane (i) horizontal, (ii) inclinedat45° facing south, and (iii) vertical facing south.
Spectral distribution of the extraterrestrial radiation
Any object whose surface temperature is greater than О К emits electromagnetic radiation. Radiated energy is in the form of waves of different wavelengths. The distribution of the irradiance with the wavelength is called the spectral distribution of the irradiance. Its integral is total irradiance. Note that the usual notation for the wavelength is A which should not be confused with the longitude.
Let Eo,v(A) be the extraterrestrial spectral irradiance received on a plane at normal incidence at a given day at wavelength A and Ersi(A) be its annual average. It comes:
There is no dependence on the solar angles and the hour angle on the wavelength. If E()(X) and #_{0}(A) denote respectively the spectral extraterrestrial irradiance and the spectral extraterrestrial irradiation received on a horizontal plane at wavelength A, then:
The previous equations are valid both for total irradiance E_{0} or total irradiation H_{0 }and for their spectral counterpart £o(A) or #_{0}(A).
The spectral irradiance is entirely determined by the emission properties and the temperature of the surface of the object, according to the laws of Kirchhoff and Planck. The radiation emitted by the sun is approximately that of a black body, that is to say, of a perfect emissive body, whose surface temperature is extremely high around
5780 К (about 5500 °C). Figure 3.8 exhibits a typical distribution of the extraterrestrial spectral irradiance E_{0},v(ty. The spectral distribution indicates the amount of power at each wavelength. The area under the curve between two wavelengths 2/ and 2^ is the irradiance integrated over this interval.
The radiated power lies within a fairly limited range between 200 and 4000 nm, from X-rays to far infrared. This interval contains 99 % of the total solar irradiance TSI Exsh ^{as} it ^{w}iH be seen below. The spectral distribution is not uniform over this range and exhibits a marked peak around 500 nm. Half of the received power lies in the visible range, the rest being in the ultraviolet and in the near and middle infrared.
The distribution is not as smooth as that predicted for a perfect black body by Planck’s law. There are several small troughs especially at wavelengths less than 800 nm. These troughs are called lines and are due to absorption and emission processes taking place in the sun and dependent on the wavelength. The sun does not emit as much energy in these absorption lines as in the neighboring wavelengths.
Figure 3.9 exhibits an excerpt of the spectral distribution, between 200 and 1500 nm for better readability of the lines; note their large number and their large amplitude at wavelengths between 200 and 400 nm, that is to say, in the ultraviolet (UV) range.
The spectral distribution Eon(E) is not constant over time. 1 wrote previously that the inter-daily variations of Eon can reach 5 W m^{-2}. These variations do not affect EonQl) equally. The visible and infrared domains are the least affected, while the ultraviolet is the most affected. The amplitude of the variations is 1-3 orders of magnitude greater in the UV range than in the visible or the infrared range. Knowledge and measurements of these variations in UV are still incomplete, although it is known that UV radiation is an important factor in the physical and chemical processes in the upper atmosphere.
Table 3.3 gives some integrated irradiances EjsiQ-h h)over certain spectral intervals [2/, 2^]. It reads that around 99 % of the total solar irradiance TSI Ejsi lies between 200 and 4000 nm. The UV-B irradiance in the interval [280, 315] nm is very small and is about 1 % of E-psi- The irradiance in the UV-A interval [315, 400] nm is much greater and is about 6 % of E-fsi- About half of E^si lies in the visible part of the spectrum,
Figure 3.8 Typical distribution of the extraterrestrial spectral irradiance Eon(2) received on a plane at normal incidence between 0 and 4000 nm.
Figure 3.9 Typical distribution of the extraterrestrial spectral irradiance Eg^(A) received on a plane at normal incidence between 200 and 1500 nm.
i.e., in the range [380, 780] nm. The range of radiation for photosynthesis by green foliage of plants is [400, 700] nm; this radiation is called photosynthetically active radiation, abbreviated as PAR, and is on average equal to 39 % of E_{TSI}. If pyranometers were installed at the top of the atmosphere, they would measure approximately 93 % of E_{TSI}.
Table 3.3 Typical values of extraterrestrial irradiance received at normal incidence in various spectral intervals and their fraction of the total solar irradiance TSI
Spectral range (nm) |
Irradiance in the interval (W m ^{2}) |
Fraction of Ejsi (%) |
250-20,000 |
1361 |
100 |
250-4000 |
1350 |
99 |
380-2100 |
1202 |
88 |
400-1100 |
907 |
67 |
1100-4000 |
335 |
25 |
1100-20,000 |
346 |
25 |
4000-20,000 |
II |
1 |
250-400 |
109 |
8 |
400-800 |
661 |
49 |
280-315 (UV-B) |
18 |
1 |
315-400 (UV-A) |
87 |
6 |
480-485 (blue) |
II |
1 |
510-540 (green) |
57 |
4 |
620-700 (red) |
123 |
9 |
380-780 (visible CIE) |
660 |
48 |
400-700 (PAR) |
534 |
39 |
330-2200 (typical spectral range of pyranometers) |
1263 |
93 |
PAR stands for photosynthetically active radiation. CIE stands for International Commission on Illumination. Fractions are rounded up to the nearest integer.
Figure 3.10 Typical distribution of the extraterrestrial spectral irradiance Eo(X) received on a horizontal plane between 200 and 1500 nm for three solar zenithal angles: 0°, 30°, and 60°.
The definition, or designation, of the spectral intervals can vary according to the field of application. For example, the visible range can denote the interval [380,780], or [400, 700], or [400, 780], or [400, 800] nm in the scientific literature. The interval [380, 780] nm indicated here is that retained by the International Commission on Illumination - usually abbreviated as CIE for its French name, Commission Internationale de l'Eclairage. Another example is the use of the term broadband, which is not precisely defined and is often used to denote a wide spectral range. It is often used in the measurement of radiation by pyranometers that generally operate in the interval [300,2500] nm approximately and whose outputs are called broadband measurements. This term can be used a little differently to denote a wide interval relative to a domain. For example, it can be read in scientific articles and other documents broadband UV that opposes spectral UV, or UV-B or UV-A.
As for E() and H_{0} during a day, and H_{0}(X) reach their maximum when the solar zenithal angle is at its minimum, i.e., at 12:00 TST. Eg(X) and H_{0}(X) vary with 6s and offer the same bell profile than Eg and H_{0} during the day. Figure 3.10 shows a typical spectral distribution of Eg(X) for three solar zenithal angles в$ 0°, 30°, and 60°, for wavelengths from 200 to 1500 nm. The influence of is clearly seen. The greater 0$, the lower the spectral irradiance. The general shape is naturally the same for all angles since the distributions differ only by a factor cos(#s) (equation 3.35) that does not depend on X.