Solar Energy and Solar Radiations

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LECTURE N° 3

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- Solar Energy and Solar Radiation -

Lecture contributions

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Coordinator & contributor of the lecture:


Prof. Marco Perino, DENERG – Politecnico di Torino, C.so Duca degli Abruzzi
24, 10129 Torino, e-mail: [email protected], http://www.polito.it/tebe/

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1

THE SOLAR ENERGY

Solar energy can be used by three technological processes:




heliochemical,
helioelectrical
heliothermal.

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Heliochemical process, through photosynthesis, maintains life on earth by
producing food, biomass and converting CO2 to O2.

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Helioelectrical process, using photovoltaic converters, provides direct
conversion of radiative energy to electric energy. They are used to power
spacecraft and in many terrestrial applications.

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Heliothermal process, can be used to provide the thermal energy required
for domestic hot water (DHW) production and space heating.

THE SUN- 1

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http://www.pveducation.org/



The sun is a sphere of intensely hot
gaseous matter with a diameter of
1.39 x 109 m and, on an average, at
a distance of 1.5 x 1O11 m from the
earth.



With an effective black body
temperature Ts of 5777 K, the sun is
effectively a continuous fusion
reactor.



This energy is produced in the
interior of the solar sphere, at
temperatures of many millions of
degree.



The produced energy must be transferred out to the surface and then be
radiated into the space (Stefan Boltzmann law: σεT4)



It is estimated that 90 % of the sun's energy is generated in the region from 0 to
0.23R (R being the radius of the sun).

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THE SUN- 2
At a distance of about O.7R from the center, the temperature drops to about
1.3 x 105 K and the density to 70 kg/m3. Hence for r > 0.7 R convection begins to be
important and the region 0.7R < r < R is known as the “convective zone”.
The outer layer of this zone is called the “photosphere”. The edge of the
photosphere is sharply defined, even though it is of low density.

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Above the photosphere is a layer of cooler gases several hundred kilometers deep
called the “reversing layer”.

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Outside that, is a layer referred to as the “chromosphere”, with a depth of about
10,000 km. This is a gaseous layer with temperatures somewhat higher than that of
the photosphere and with lower density.

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Still further out is the “corona”, of very low density and of very high temperature
(about 106 K).

THE SUN- 3

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The photosphere is the source of most solar radiation.

http://www.pveducation.org/

“Solar Energy – Fundamentals, Design, Modelling and Applications” – G.N. Tiwari

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THE SUN- 4

The simplified picture of the sun, its physical structure, temperature and density
gradients indicate that the sun, in fact, does not function as a blackbody radiator
at a fixed temperature.
Rather, the emitted solar radiation is the composite result of the several layers
that emit and absorb radiation of various wavelength.

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The photosphere is the source of most solar radiation and is essentially opaque,
as the gases, of which it is composed, are strongly ionized and able to absorb
and emit a continuous spectrum or radiation.

THE SUN: SPECTRAL IRRADIANCE - 1

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The maximum spectral density occurs at about 0.48 µm wavelength (λ) in the
green portion of the visible spectrum. About 6.4% of the total energy is ultraviolet
(λ < 0.38 µm), 48% is in the visible spectrum and the remaining 45.6% is in the
infrared region (λ > 0.78 µm)

Extra-atmosferic

“Solar Energy – Fundamentals, Design, Modelling and Applications” – G.N. Tiwari

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THE SUN: SPECTRAL IRRADIANCE - 2

http://www.pveducation.org/

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Extra-atmosferic

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THE SUN: SPECTRAL IRRADIANCE - 3

“Solar Energy – Fundamentals, Design, Modelling and Applications” – G.N. Tiwari

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THE SUN: EXTRA-ATMOSPH. IRRADIANCE - 1
The eccentricity of the earth's orbit is such that the distance between the sun and
the earth varies by 1.7%.

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The solar constant, Gsc, is the energy from the sun per unit time [W/m2] received
on a unit area of surface perpendicular to the direction of propagation of the
radiation at mean earth-sun distance outside the atmosphere.

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The World Radiation Center (WRC) has adopted a value of Gsc = 1367 W/m2,
with an uncertainty of the order of 1%.

THE SUN: EXTRA-ATMOSPH. IRRADIANCE - 2
Two sources of variation in extraterrestrial radiation must be considered:

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a) the variation in the radiation emitted by the sun.
For engineering purposes, in view of the uncertainties and variability of
atmospheric transmission, the energy emitted by the sun can be considered to
be fixed.
a) the variation of the earth-sun distance, that leads to variation of extraterrestrial
radiation flux in the range of ± 3.3%.
A simple equation with accuracy adequate for most engineering calculations is:

(1)

where Gon is the extraterrestrial radiation incident on the plane normal to the
radiation on the nth day of the year

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DEFINITIONS - 1
Beam Radiation – b (e.g. Gb): is the solar radiation received from the sun without
having been scattered by the atmosphere (beam radiation is often referred to as
direct solar radiation; to avoid confusion between subscripts for direct and diffuse,
we use the term beam radiation, subscript “b”).

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Diffuse Radiation – d (e.g. Gd): is the solar radiation received from the sun after its
direction has been changed by scattering by the atmosphere (diffuse radiation is
referred to, in some meteorological literature, as sky radiation or solar sky radiation;
the definition used here will distinguish the diffuse solar radiation from infrared
radiation emitted by the atmosphere. Diffuse radiation, subscript “d”).

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Total Solar Radiation (e.g. G): is the sum of the beam and the diffuse solar
radiation on a surface (the most common measurements of solar radiation are total
radiation on a horizontal surface, often referred to as global radiation on the surface.
When we will make reference to total solar radiation we will not use any subscript).

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G = Gb + Gd

DEFINITIONS - 3

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Irradiance, G [W/m2]: is the rate at which radiant energy (energy flux) is incident on
a surface per unit area of surface. The symbol G is used for solar irradiance, with
appropriate subscripts for beam (b), diffuse (d), or spectral radiation (λ).
Irradiation or Radiant Exposure, I or H [J/m2]: is the incident energy per unit area
on a surface, found by integration of irradiance over a specified period (usually an
hour or a day). Insolation is a term applying specifically to solar energy irradiation.
The symbol H is used for insolation for a day. The symbol I is used for insolation for
an hour (or other period if specified).
The symbols H and I can represent beam, or total and can be on surfaces of any
orientation (with their corresponding subscripts).
Subscripts on G, H, and I are as follows:
 “o” refers to radiation above the earth's atmosphere, referred to as
extraterrestrial radiation,
 “b” and “d” refer, respectively, to beam and diffuse radiation;
 “T” and “n” refer to radiation on a tilted plane and on a plane normal to the
direction of propagation. If neither T nor n appears, the radiation is on a
horizontal plane.

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DEFINITIONS - 4
Radiosity or Radiant Exitance, [W/m2]: is the rate at which radiant energy leaves
a surface per unit area by combined emission, reflection, and transmission.
Emissive Power or Radiant Self-Exitance, [W/m2]: The rate at which radiant
energy leaves a surface per unit area by emission only.

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Any of the above defined radiation fluxes, except insolation, can apply to any
specified wavelength range (such as the solar energy spectrum) or to
monochromatic radiation.
Insolation refers only to irradiation in the solar energy spectrum.

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Solar Time: is the time based on the apparent angular motion of the sun across
the sky, with solar noon defined as the time the sun crosses the meridian of the
observer.

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Standard time: is the time given by local clock.

SOLAR TIME – STANDARD TIME - 1

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Solar time is the time used in all of the sun-angle relationships; it does not coincide
with local clock time (standard time).
It is necessary to convert standard time to solar time by applying two corrections:
 First, there is a constant correction for the difference in longitude between the
observer's meridian (local longitude, Lloc) and the meridian on which the local
standard time is based (longitude of the standard meridian for the local time
zone, Lst).
(to find the local standard meridian, in degree, multiply the time difference in
hour between local standard clock time and Greenwhich Mean Time – GMT,
by 15. In fact, the sun takes 4 min to transverse 1°of longitude).
 The second correction is from the equation of time, which takes into account
the perturbations in the earth's rate of rotation which affect the time the sun
crosses the observer's meridian.
The difference in minutes between solar time and standard time is:

(all equations must be used with degrees and NOT radians)

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SOLAR TIME & EQUATION OF TIME - 1

Sun

12 PM
day “i+1”

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12 PM
day “i”

Sun path
swept in 24 h

12 PM
day “i+1”

12 PM
day “i”

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Sun path
swept in 24 h

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SOLAR TIME & EQUATION OF TIME - 2

http://www.artesolare.it/tempo_solare_medio.htm

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LATITUDE (φ ) & LONGITUDE (L)
http://academic.brooklyn.cuny.edu/

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http://www.kowoma.de/

SOLAR TIME & EQUATION OF TIME - 3
(3)

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 Lst is the standard meridian for the local time zone (in °),
 Lloc is the longitude of the location in question (in °) ,
 longitudes are in degrees west, that is 0°< L < 360 °.
The parameter E is the equation of time (in minutes)
(4)

(all equations must be used with degrees and NOT radians)

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SOLAR TIME & EQUATION OF TIME - 4
Where B is given by (and “n” is the day of the year, 1 ≤ n ≤ 365)

(5)

In using eq. 3 it has to be remembered that:

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 the equation of time, E, and displacement from the standard meridian (i.e. first
term of right hand side of equation 3) are both in minutes,
 there is a 60-min difference between daylight saving time and standard time.

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 Time is usually specified in hours and minutes,

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 Care must be exercised in applying the corrections between standard time and
solar time, which can total more than 60 min.

ANGLE DEFINITIONS – 1

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The geometric relationships between a plane of any particular orientation relative
to the earth at any time and the incoming beam solar radiation, that is, the position
of the sun relative to that plane, can be described in terms of several angles:

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http://www.esrl.noaa.gov/

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SUN – EARTH POSITION - 1

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SUN – EARTH POSITION - 2

http://www.esrl.noaa.gov/

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SUN – EARTH POSITION - 3

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http://www.kowoma.de/

ANGLE DEFINITIONS – 2

φ, Latitude: is the angular location north or south of the equator, north positive:
-90° ≤ φ ≤ +90°

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δ, Declination: is the angular position of the sun at solar noon (i.e., when the sun is
on the local meridian) with respect to the plane of the equator (e.g. is the angle
between the equator plane and a line drawn from the centre of the earth to the centre
of the sun), north positive:
-23.45° ≤ δ ≤ + 23.45°
β, Slope: is the angle between the plane of the surface in question and the
horizontal (β > 90°means that the surface has a downward-facing component ):
0° ≤ β ≤ 180°

γ, Surface azimuth angle: is the deviation of the projection on a horizontal plane of
the normal to the surface from the local meridian, with zero due south, east negative,
and west positive:
-180° ≤ γ ≤ +180°
θ, Angle of incidence: is the angle between the beam radiation on a surface and
the normal to that surface.

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ANGLE DEFINITIONS – 3
ω, Hour angle: is the angular displacement of the sun east or west of the local
meridian due to rotation of the earth on its axis at 15°per hour; morning negative
afternoon positive (solar time must be in hours, hour angle in degrees):

ω = (Solar Time - 12) ⋅15°

(6)

Additional angles are defined that describe the position of the sun in the sky:

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θz, Zenith angle: is the angle between the vertical and the line to the sun, that is,
the angle of incidence of beam radiation on a horizontal surface (0° ≤ θz ≤ 90°
when the sun is above the horizon)

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αs, Solar altitude angle: is the angle between the horizontal and the line to the
sun, that is the complement of the zenith angle (αs = 90°- θz )

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γs Solar azimuth angle: the angular displacement from south of the projection of
beam radiation on the horizontal plane. Displacements east of south are negative
and west of south are positive (-180° ≤ γs ≤ +180°)

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ANGLES SCHEME – 1

Zenith

Sun

N

θz
αs

Earth

α s = 90° − θ z
S

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ANGLE DEFINITIONS – DECLINATION, δ - 4
The declination δ can be found from the approximate correlation of Cooper (for
more precise relations see Duffie & Beckman) :
(7)

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Variation in sun-earth distance, the equation of time, E, and declination, δ, are all
continuously varying functions of time of year.

It is customary to express the time of year in terms of “n”, the day of the year, and
thus as an integer between 1 and 365.

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All equations could also be used with non integer values of “n”, but the use of
integer values is adequate for most engineering calculations (the maximum
rate of change of declination is about 0.4°per day) .

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ANGLE DEFINITIONS – DECLINATION, δ - 5

http://www.esrl.noaa.gov/

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ANGLE DEFINITIONS – Monthly Mean Days & n

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Tabular data (see table below) help the assessment of “n” and supply information
about which day of the month must be used as the representative “average day of
the month” (to be used in some formulas - to be seen later)

ANGLE RELATIONS – Angle of incidence, θ

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To assess θ, angle of incidence of beam solar radiation on a surface whatever
oriented and tilted at a certain time during the year, the following equations can be
used:

(8)

and

(9)

The angle θ may exceed 90°, which means that the sun is behind the surface.
When using equation (8), it is necessary to ensure that the earth is not “blocking”
the sun (i.e., that the hour angle, ω, is between sunrise and sunset).

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ANGLE RELATIONS – Solar Zenith angle, θz
The assessment of the Zenith angle of the sun, θz , can be done using the
following equation:
(10)

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The value of θz must be between 0°and 90°.
It has to be remembered that for horizontal surfaces (β = 0) the angle of incidence of
the beam radiation is equal to the solar zenith angle, that is θz = θ and eq. 8

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reduces to:

ANGLE RELATIONS – Solar Azimuth angle, γs

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The solar azimuth angle γs , can be assessed by means of:

(10a)

γs can have values in the range of 180°to - 180°.
γs is negative when the hour angle, ω, is negative and positive when the hour angle
is positive. The sign function in the above equations is therefore equal to +1 if ω is
positive and is equal to - 1 if ω is negative.
For north or south latitudes between 23.45°and 66.4 5 °, γs will be between 90°
and -90°for days less than 12 h long; for days with more than 12 h between
sunrise and sunset, γs , will be greater than 90°or less than -90°early an d late in
the day when the sun is north of the east-west line in the northern hemisphere (or
south of the east-west line in the southern hemisphere).

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ANGLE RELATIONS – Special cases (γ = 0)
Tilted surfaces sloped due south (typical for north hemisphere)

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For this case γ = 0 and the angle of incidence of surfaces sloped due south (or due
North) can be derived from the fact that surfaces with slope β to the north or south
have the same angular relationship to beam radiation as a horizontal surface at an
artificial latitude of (φ - β) :

ANGLE RELATIONS – Sunset hour angle, ωs

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Equation 10 can be solved for the sunset hour angle ωs , that is when θz = 90°:

(11)

the sunrise hour angle is the negative of the sunset hour angle.
From this it follows the number of daylight hours, N :

(12)

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ANGLE RELATIONS – Profile angle, αp
An additional angle of interest is the profile angle, αp , of beam radiation on a
receiver plane R that has a surface azimuth angle of γ.
It is the angle through which a plane that is initially horizontal must be rotated
about an axis in the plane of the surface in question in order to include the sun.
The profile angle is useful in calculating shading by overhangs and can be
determined from:

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(13)

SHADING - 1

Three types of shading problems typically occur:

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a) shading of a collector, window, or other receiver by nearby trees, buildings or
other obstructions. The geometries may be irregular and systematic
calculations of shading of the receiver in question may be difficult.
Recourse is made to diagrams of the position of the sun in the sky, for example
plots of solar altitude, αs, versus solar azimuth γs, on which shapes of
obstructions (shading profiles) can be superimposed to determine when the
path from the sun to the point in question is blocked.

b) The second type includes shading of collectors in other than the first row of
multi-row arrays by the collectors on the adjoining row.
c) shading of windows by overhangs and wingwalls.
When the geometries are regular, shading problems can be assessed through
analytical calculation, and the results can be presented in general form.

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SHADING - Solar plots

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Solar plots are a 2D representation of the sun paths over the sky dome.
These paths are plotted for different periods of the year and are the projection of
the sun orbits over the horizontal plane.
Each solar plot is draw for a specific location (that is, for a certain latitude, φ) and
allows to assess the sun position for every hour of the day and for every day of
the year, by means of the solar altitude angle, αs, and of the solar azimuth, γs.
Solar plots may be plotted in either polar or rectangular coordinate charts.

SHADING - Type a) – Use of Solar plots
(rectangular coordinate plot)

Solar position plot of θz and αs, versus γs , for latitudes of ± 45°is shown in Figure.
Lines of constant declination, δ, are labeled by dates of mean days of the months
(see Table 1.6.1). Lines of constant hour angles, ω, are labeled by hours.

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(plots for latitudes from 0 to ± 70°are included in App endix H of Duffie & Beckman).

The angular position of buildings, wingwalls, overhangs, or other obstructions can
be entered on the same plot (the angular coordinates corresponding to altitude and
azimuth angles of points on the obstruction - the object azimuth angle, γo, and
object altitude angle, αo) can be calculated from trigonometric considerations and
drawn on the plot). For obstructions such as buildings, the points selected must
include corners or limits that define the extent of obstruction. It may or may not be
necessary to select intermediate points to fully define shading.

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SHADING - Type a) – Use of Solar plots
(rectangular coordinate plot)

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γs

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s

α

(that is day and month)

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“Solar Energy Pocket Refernce” –
C.L. Martin, D.Y. Goswami – ISES

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(rectangular coordinate plot)

C

A
B

The shaded area represents the existing building
as seen from the proposed collector site.
The dates and times when the collector would be
shaded from direct sun by the building are evident

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USE OF SOLAR PLOTS - Example
(rectangular coordinate plot)

percorso solare: latitudine 37 °N
giugno

maggio - luglio

aprile - agosto

marzo - settembre

febbraio - ottobre

gennaio - novembre

dicembre

90.0
80.0
70.0

altezza zenitale

60.0

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50.0
40.0
30.0
20.0

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10.0

0.0

-180.0

-135.0

-90.0

-45.0

0.0

45.0

90.0

135.0

180.0

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azimuth

SHADING - Type a) – Use of Solar plots

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Implicit in the preceding discussion is the idea that the solar position at a point in
time can be represented for a point location.
Collectors and receivers, in reality, have finite size, and what one point on a large
receiving surface “sees” may not he the same as what another point sees.
The hypotheses of a “point” collector/receiver can be made if the distance between
collector and obstruction is larger compared to the size of the collector itself
(this also implies that the collector is either completely shaded or completely
lighted).
For partially shaded collectors (that is no “point” hypotheses”), it can be considered
to consist of a number of smaller areas, each of which is shaded or not shaded.
Besides solar plots in Rectangular Cartesian coordinates, quite common are also
solar plots in polar coordinates.

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SHADING - Solar plots in polar coordinate

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The concentric circumferences
represent point at the same solar
eight (same solar eight angle, αs).
For the circumference of max radius
it is αs = 0° (horizon), for the
circumference centre αs = 90° (sun
at zenith).
Each circle is spaced of 10°.
Radiuses represent points having
the same azimuth (again interval
between radius is 10°)

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(Note: In this chart the sign of the azimuth angle is reversed)

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SHADING - Solar plots in polar coordinate – example

Linea dell’ora: 15

Linea del mese:
marzo - settembre

Results:
αs = 30°
γs = 55°

βS = 30°

ΦS = -55° (= 305°)

Assess the sun position at 15.00 o’clock of 23rd September at a latitude of 46°N

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SHADING - Type c) – Solar plots and overhangs
or wingwalls
The solar position charts can be used to determine when points on the receiver are
shaded. The procedure is identical to that of the previous example; the obstruction
in the case of an overhang and the times when the point is shaded from beam
radiation are the times corresponding to areas above the line. This procedure
can be used for overhangs of either finite or infinite length. The same concepts can
be applied to wingwalls.

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Alternatively the concept of shading planes and profile angle, αp, can be used:

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If the profile angle , αp, is less than (90 - ψ) the receiver surface will “see” the sun
and it will not be shaded).

SHADING - Type b) - Collectors in row

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Shading calculations are needed when flat-plate collectors are arranged in rows.
Normally, the first row is unobstructed, but the second row may be partially shaded
by the first, the third by the second, and so on.

For the case where the collectors are long in extent so the end effects are
negligible, the profile angle provides a useful means of determining shading.
As long as the profile angle is greater than the angle CAB, no point on row N will
be shaded by row M (with M = N-1).
If the profile angle at a certain time is CA’B’ and is less than CAB, the portion of
row N below point A’ will be shaded from beam radiation.

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EXTRATERRESTRIAL RADIATION ON A
HORIZONTAL SURFACE, Go
Several types of radiation calculations are done using normalized radiation levels,
that is, the ratio of radiation level to the theoretically possible radiation that would
be available if there were no atmosphere.

G o = G o,n ⋅ cosθ z
Go,n

θz Go

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Atmosphere

(Go,n is assed by means of eq. 1)

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And substituting cosθz (that is eq. 10):

(15)

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where Gsc, is the solar constant and “n” is the day of the year.

BEAM RADIATION ON HOR. AND TILTED
SURFACE – Angle scheme

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Sun

Gb,n

r
n

θz Gb

r
n
θ
Gb,n

β

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DAILY EXTRATERRESTRIAL RADIATION
ON A HORIZONTAL SURFACE, Ho
It is often necessary for calculation of daily solar radiation to have the integrated
daily extraterrestrial radiation on a horizontal surface, Ho.
This is obtained by integrating Go over the period from sunrise to sunset.

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If Go is expressed in W/m2 and Ho, in J/m2 , it is possible to write:

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(16)

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where ωs, is the sunset hour angle (in °)

MONTHLY MEAN DAILY EXTRATERRESTRIAL
RADIATION ON A HORIZONTAL SURFACE, Ho

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The monthly mean daily extraterrestrial radiation:

Ho

for latitudes in the range +60°to -60°can be calcul ated, with good approximation,
with the same equation (16), using “n” and δ corresponding to the “mean day of the
months” from Table 1.6.1

An overbar is typically used to indicate a “monthly average quantity”.
The monthly mean day is a day which has the Ho closest to H o

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HOURLY EXTRATERRESTRIAL RADIATION
ON A HORIZONTAL SURFACE, Io
It is also of interest to calculate the extraterrestrial radiation on a horizontal surface
for an hour period. Integrating equation 15 for a period between hour angles ω1 and
ω2 which define an hour (where ω2 is the larger):

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(17)

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The limits ω1 and ω2 may define a different time other than an hour.

RATIO OF BEAM RADIATION ON TILTED SURFACE
TO THAT ON HORIZONTAL SURFACE - 1

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It is often necessary to calculate the hourly radiation on a tilted surface of a
collector from measurements or estimates of solar radiation on a horizontal
surface.
The most commonly available data are total radiation for hours or days on the
horizontal surface, whereas the need is for beam and diffuse radiation on the plane
of a collector.
The geometric factor Rb, the ratio of beam radiation on the tilted surface to that on
a horizontal surface at any time, can be calculated exactly as:
(14)

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BEAM RADIATION TRANSMISSION - scheme
Sun

Atmosphere
GO,n

r
n'

θ

Gb,n

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Gb

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r
n

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β

RATIO OF BEAM RADIATION ON TILTED SURFACE
TO THAT ON HORIZONTAL SURFACE - 2

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The symbol G is used to denote rates [W/m2], while I [J/m2], is used for energy
quantities integrated over 1 hour (and H over one day).

Rb =

G b,T
Gb

Rb =

I b,T
Ib

The typical development of Rb, was for hourly periods; in such case to assess
angles with the previous equations angles assessed at the midpoint of the hour
must be used (e.g. for the assessment of Rb for the hour comprised between 10
and 11 am the evaluation of the angles must be done at the time 10.30).

The optimum azimuth angle for flat-plate collectors is usually 0°in the northern
hemisphere (or 180°in the southern hemisphere).
Thus it is a common situation that γ = 0°(or 180°).

28

References and relevant bibliography

 2005 ASHRAE Handbook of Fundamentals - ASHRAE, Atlanta, USA,
2005.
 Solar Energy Thermal Processes, John A. Duffie - William A. Beckman,
John Wiley & Sons Inc , New York, US, 2006, ISBN: 0471223719.

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 Solar Energy Fundamentals: Fundamentals, Design, Modeling and
Applications, Tiwari GN, CRC Press Inc, 2002, ISBN 0849324092.

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