LECTURE N° 3

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- Solar Energy and Solar Radiation -

Lecture contributions

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Coordinator & contributor of the lecture:

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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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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°).

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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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- 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/

2

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

ID

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).

2

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.

U

Above the photosphere is a layer of cooler gases several hundred kilometers deep

called the “reversing layer”.

-E

D

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.

ES

Still further out is the “corona”, of very low density and of very high temperature

(about 106 K).

THE SUN- 3

ID

The photosphere is the source of most solar radiation.

http://www.pveducation.org/

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

3

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.

ES

-E

D

U

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

ID

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

4

-E

D

U

THE SUN: SPECTRAL IRRADIANCE - 2

http://www.pveducation.org/

ES

Extra-atmosferic

ID

THE SUN: SPECTRAL IRRADIANCE - 3

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

5

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%.

-E

D

U

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.

ES

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:

ID

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

6

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”).

U

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”).

-E

D

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).

ES

G = Gb + Gd

DEFINITIONS - 3

ID

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.

7

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.

U

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.

-E

D

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.

ES

Standard time: is the time given by local clock.

SOLAR TIME – STANDARD TIME - 1

ID

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)

8

SOLAR TIME & EQUATION OF TIME - 1

Sun

12 PM

day “i+1”

U

12 PM

day “i”

Sun path

swept in 24 h

12 PM

day “i+1”

12 PM

day “i”

ES

-E

D

Sun path

swept in 24 h

ID

SOLAR TIME & EQUATION OF TIME - 2

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

9

LATITUDE (φ ) & LONGITUDE (L)

http://academic.brooklyn.cuny.edu/

ES

-E

D

U

http://www.kowoma.de/

SOLAR TIME & EQUATION OF TIME - 3

(3)

ID

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)

10

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:

U

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.

-E

D

Time is usually specified in hours and minutes,

ES

Care must be exercised in applying the corrections between standard time and

solar time, which can total more than 60 min.

ANGLE DEFINITIONS – 1

ID

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:

11

ES

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

-E

D

U

SUN – EARTH POSITION - 1

ID

SUN – EARTH POSITION - 2

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

12

-E

D

U

SUN – EARTH POSITION - 3

ES

http://www.kowoma.de/

ANGLE DEFINITIONS – 2

φ, Latitude: is the angular location north or south of the equator, north positive:

-90° ≤ φ ≤ +90°

ID

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.

13

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:

U

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)

-E

D

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 )

ES

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°)

ID

ANGLES SCHEME – 1

Zenith

Sun

N

θz

αs

Earth

α s = 90° − θ z

S

14

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)

U

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.

ES

-E

D

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) .

ID

ANGLE DEFINITIONS – DECLINATION, δ - 5

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

15

ANGLE DEFINITIONS – Monthly Mean Days & n

ES

-E

D

U

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

ID

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).

16

ANGLE RELATIONS – Solar Zenith angle, θz

The assessment of the Zenith angle of the sun, θz , can be done using the

following equation:

(10)

U

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

ES

-E

D

reduces to:

ANGLE RELATIONS – Solar Azimuth angle, γs

ID

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).

17

ANGLE RELATIONS – Special cases (γ = 0)

Tilted surfaces sloped due south (typical for north hemisphere)

ES

-E

D

U

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

ID

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)

18

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:

ES

-E

D

U

(13)

SHADING - 1

Three types of shading problems typically occur:

ID

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.

19

SHADING - Solar plots

ES

-E

D

U

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.

ID

(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.

20

SHADING - Type a) – Use of Solar plots

(rectangular coordinate plot)

-E

D

γs

U

s

α

(that is day and month)

ES

“Solar Energy Pocket Refernce” –

C.L. Martin, D.Y. Goswami – ISES

ID

(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

21

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

U

50.0

40.0

30.0

20.0

-E

D

10.0

0.0

-180.0

-135.0

-90.0

-45.0

0.0

45.0

90.0

135.0

180.0

ES

azimuth

SHADING - Type a) – Use of Solar plots

ID

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.

22

SHADING - Solar plots in polar coordinate

-E

D

U

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°)

ES

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

ID

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

23

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.

-E

D

U

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

ES

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

ID

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.

24

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

U

Atmosphere

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

-E

D

And substituting cosθz (that is eq. 10):

(15)

ES

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

BEAM RADIATION ON HOR. AND TILTED

SURFACE – Angle scheme

ID

Sun

Gb,n

r

n

θz Gb

r

n

θ

Gb,n

β

25

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.

U

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

-E

D

(16)

ES

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

MONTHLY MEAN DAILY EXTRATERRESTRIAL

RADIATION ON A HORIZONTAL SURFACE, Ho

ID

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

26

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):

-E

D

U

(17)

ES

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

ID

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)

27

BEAM RADIATION TRANSMISSION - scheme

Sun

Atmosphere

GO,n

r

n'

θ

Gb,n

-E

D

Gb

U

r

n

ES

β

RATIO OF BEAM RADIATION ON TILTED SURFACE

TO THAT ON HORIZONTAL SURFACE - 2

ID

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.

ID

ES

-E

D

U

Solar Energy Fundamentals: Fundamentals, Design, Modeling and

Applications, Tiwari GN, CRC Press Inc, 2002, ISBN 0849324092.

29