Solar Geometry Solar thermal process mech

Solar thermal process

1. Solar Geometry
Dr. T. Srinivas
Department of Mechanical Engineering
NIT Jalandhar
The best way to predict the future is to create it

2. Outlines

Sun earth relations
S.P. Sukhatme (1996), Solar Energy -Principles of Thermal Collection and Storage, 2nd
Edition, Tata McGraw Hill.
D. Yogi Goswami, Frank Krieth and Jan F. Kreier (2000), Principles of Solar Engineering, 2
nd
Edition, Taylor and Francis, USA.
John A. Duffie and William A. Beckman (2006), Solar Engineering of Thermal Process, 3rd
Edition, John Wiley & Sons.

3. 3
Need
Simulation of solar systems and plants.
Selection of the tracking options.
Design of green buildings.
Evaluate the nature and potential of source.
Shadow analysis.
Frame the national and international time
Discover the location.

4. 4

5. 5
India’s changing energy mix

6. Renewable power installed capacity

7. Solar Energy Technologies
Solar Energy Technologies
Direct methods Indirect methods
Photovoltaic
Water energy
(hydro electric)
Wind
Ocean energy
Biomass
Wave
energy
Thermal
Ocean Thermal
Energy Conversion
Marine
currents

8. Solar facts

The Sun is 3,33,400 times more massive than the earth and contains
99.86% of the mass of the entire solar system.

It consist of 78% Hydrogen, 20% Helium and 2% of other elements.

It is estimated that 90% of the energy is generated in the region of 0 to
0.23 R (where R is the radius of the sun), which contains 40% of the
mass of the sun and density is about 10
5
kg/m
3
.

The sun diameter is 1.39 ×10
6
km and earth diameter is 1.27 ×10
4
km. The mean distance is 1.496 ×10
8
km. (subtended angle: 32 min).

The solar constant 1.367 kW/m
2
.

The power from sun to earth is 1.8 ×10
11
MW

9. 9
Sun Earth Relationship (contd.,)

To understand the sun-earth relationship the following solar geometry is
required.
Latitude, φ
Longitude, L
declination angle, δ
incidence angle, θ
hour angle, ω
day length, S
azimuth angle (solar γ
s
surface γ),
zenith angle, θ
z
altitude angle, α and
tilt angle, β
A site on earth is identified using its Latitude & Longitude

10. Sun Earth Relationship

Sun-earth relationship is used to estimate the solar radiation.

Incidence angle (θ) is the angle between the sun’s direct rays and
the normal to any surface.
Cosθ = Sinδ Sinφ Cosβ
-Sinδ Cosφ Sinβ Cosγ
+ Cosδ Cosφ Cosβ Cosω
+ Cosδ Sinφ Sinβ Cosγ Cosω
+ Cosδ Sinβ Sinγ Sinω
θ = f(δ, ω, φ, β, γ)

11. Sun Earth Relationship(contd.,)
Zenith
Normal to
surface
β
θ
z
θ
γ
γ
s
S
α
https://www.youtube.com/watch?v=QBEEnRNzHvI
https://www.youtube.com/watch?v=F1jbnGl_FV0

12. 12

Declination (δ)is the angular displacement of the sun at solar noon w.r.t. the plane
of equator

(or) is the angular distance north (or south) of the equator of the point, when the
sun is at its zenith

(or) declination is the angle formed by the line extending from the center of the sun
to the center of the earth and the projection of this line on the earth’s equatorial
plane.

Also, when the sun is directly overhead at any location during solar noon, the
latitude of that location gives the declination.
http://esminfo.prenhall.com/science/geoanimations/animations/01_EarthSun_E2.html
Sun Earth Relationship (contd.,)

13. 13
Sun Earth Relationship (contd.,)
Change in declination angle with Julian day during a year





 

365
)284(360
sin45.23
n


14. 14
Sun Earth Relationship (contd.,)
152 Million km 147 Million km
December 21
st
Winter
solstice
δ = -23.5 °
Sun
Earth
March 21
st
Spring
equinox
δ = 0 °
June 21
st
Summer solstice
δ = 23.5 °
September 21
st
Autumn
equinox δ = 0 °
Fig. Seasonal variation in declination angle

15. 15

Hour Angle (ω)
is the angular displacement of the sun,
east or west of the local meridian due
to the rotation of earth on its axis (15
degrees per hour or 4 min per deg).

It is considered as –ve in the morning
and +ve in the afternoon.

At solar noon, ω = 0.

ω = 15 (solar time -12), deg.
Sun Earth Relationship (contd.,)
Sun rays
ω
Equatorial
plane
N
S
Longitude on noon
Local longitude on
forenoon
ω
Locatio
n
Hour angle measured as an angle between local
longitude and longitude ahead of sun

16. 16
Sun Earth Relationship (contd.,)
EastWest
Place
Sunset, ω = -90
°
Solar noon, ω = 0
FN, ω + ve
AN, ω -ve
Sunrise ω = 90
°
Changes in hour angle from sunrise to sunset

17. 17

Latitude (φ)defined as the
angular distance of a point
(on the surface of the earth)
from the equator.

Latitude zero is the equator

Tropic of Cancer: Latitude 23.5°N

Tropic of Capricorn: Latitude 23.5°S
Sun Earth Relationship (contd.,)
Equator, 0 °
Polar axis 66.5 °to vertical
Arctic circle, 66.5 °N
Antarctic circle, 66.5 °S
Latitude, Ø
Tropic of Cancer, 23.5 °N
Tropic of Capricorn, 23.5 °S
North pole
South pole
Ø
Location
Local longitude
Latitude on earth with principle meridians

18. 18
Slope or Tilt angle (β)
is the angle the surface makes with the horizontal plane.
A south-facing collector tipped up to an angle equal to its latitude is perpendicular to
the sun’s rays at solar noon during the equinoxes
Sun Earth Relationship (contd.,)
Tilt angle of a solar collector
β = Ø –δ
Average
β = Ø ±13°

19. 19

Surface Azimuth Angle (γ)
is the angle measured on the horizontal plane from due south to
the horizontal projection of the normal to the surface.
It is also given as the angle between the local meridian and the
horizontal projection of the normal to the surface

Surface azimuth angle is zero due south, east -ve and west +ve.
(It is measured from south in clock direction).
Sun Earth Relationship (contd.,)

20. 20

Solar Azimuth angle (γ
s
)
is the angle on the horizontal plane measured from south to the
horizontal projection of the sun’s rays

Solar azimuth angle is zero due south, east -ve and west +ve.
(Measured from north in clock direction).

It gives the direction of the shadow cast in the horizontal plane by a
vertical rod.
cos γ
s
= (cosθ
z
sinφ –Sinδ) / (sinθ
z
cosφ)







 





cossin
sinsincos
cos
1
Z
Z
s
Sun Earth Relationship (contd.,)

21. 21
Sun Earth Relationship (contd.,)
Solar azimuth angle
(a) Solar azimuth angle on (a) three dimensional view and (b) plan







 





cossin
sinsincos
cos
1
Z
Z
s

22. 22

Zenith angle (θ
z
)
is the angle subtended by a vertical line to the
zenith (i.e. the point directly overhead) and
the line of sight to the sun.
C
osθ
z
= Sinδ Sinφ + Cosδ Cosφ Cosω
θ
z
= f(δ, ω, φ)
Sun Earth Relationship (contd.,)

23. 23

Altitude angle (α)
is the angle on the vertical plane between the sun’s
rays and the projection of the sun’s rays on the
horizontal plane.
α = 90 -(θ
z
)
Sun Earth Relationship (contd.,)
Zenith
θ
z
γ
s
S
α

24. 24

Air Mass (m)
is the ratio of the optical thickness of the atmosphere through
which beam radiation passes to the optical thickness if the sun
were at the zenith.
At sea level, when the sun is at the zenith, m = 1 and when the
zenith angle is 60º, m = 2.
Sun Earth Relationship (contd.,)
Zenith
θ
z
O
Sun ray through atmospheric
layer
ZS
Air mass is the ratio of OZ and OS
z
OZ
OS
mmassAir sec, 

25. 
Longitude/ Meridian: The longitude
angle can be defined as the angle between
the prime meridian and the meridian
passing through the location.

There is a standard longitude for a
country/region with which standard/
watch times are defines

For India, the standard meridian = 82.5°E
(Allahabad)

Hence, time difference between
Greenwich and India = 82.5/15 = 5.5
hours ahead of GMT
Earth speed = 15º per hr = 1º per 4 min
Sun Earth Relationship (contd.,)
Polar axis 66.5 °to
vertical
Longitudes
North pole
South pole
Representation of latitude and
longitude at a location on earth

26. 
Solar time
is based on the apparent angular motion of the sun across the sky.
Solar noon is the time the sun crosses the local meridian of the
observer. At solar noon, the sun will be at zenith of the observer.
This is used in all solar energy calculations.

Watch time
is based on the longitude and depends on standard meridian of that
country.
The rotational speed is therefore, 4 minutes per meridian
(longitude) or 15°per hour.
So, the time difference between the standard time (based on
standard meridian) and local time (based on the local meridian) will
be 4 (L
st
–L
lo
), in minutes,
Sun Earth Relationship (contd.,)

27. 
Solar time can be calculated
from
the watch
time, for locations east of Greenwich (say,
Asian, African and Australian locations), as
Solar time = Watch Time -4 (L
st
–L
lo
) + EoT
(in min)

While for locations west of Greenwich (say, the
Americas) it is
Solar time = Watch Time + 4 (L
st
–L
lo
) + EoT
(in min)
EoT = 9.87 sin 2B -7.53 cos B -1.5 sin B
where, B = 360 (n-81)/364 (in degrees)
Sun Earth Relationship (contd.,)

28. 
Sunrise (and sunset) hour angle (ω
sr
and ω
ss
)
is the hour angle at the time of sunrise (or
sunset) (or) it is the hour angle when the
altitude angle is zero.
Sun Earth Relationship (contd.,)
ωsr = ωss = Cos-1 (-tanδ tanØ)
15
)tantan(cos
1





sssrw

29. Sun Earth Relationship (contd.,)
Zenith
θ
z
S
N
Summer
Winter

30. 30

Day length
is the time between sunrise and sunset.

The relationship for the sunrise and sunset hour angles and
day length could be obtained from the relation of zenith
angle (θ
z
), which gives the angle between the sun's rays
and the normal to the horizontal surface.
Sun Earth Relationship (contd.,)

31. 31
Sun Earth Relationship (contd.,)
Region LatitudeShortest dayLongest day
Tropics
00 12.07 12.07
10 11.32 12.42
20 10.56 13.20
Temperate
30 10.14 14.04
40 09.20 15.00
50 08.05 16.21
60 05.54 18.49
Polar
70 00.00 24.00
80 00.00 24.00
90 00.00 24.00
Day length with longitude

32. 
Incidence angle (θ)
is the angle between the sun’s direct rays and the normal to any
surface.
Cosθ = Sinδ Sinφ Cosβ
-Sinδ Cosφ Sinβ Cosγ
+ Cosδ Cosφ Cosβ Cosω
+ Cosδ Sinφ Sinβ Cosγ Cosω
+ Cosδ Sinβ Sinγ Sinω
For β = 0,
Cosθ
z
= Sinδ Sinφ + Cosδ Cosφ Cosω
Sun Earth Relationship (contd.,)

33. Sun Earth Relationship (contd.,)
Cosθ= SinδSinφCosβ
-SinδCosφSinβ Cosγ
+ CosδCosφCosβ Cosω
+ CosδSinφSinβ CosγCosω
+ CosδSinβ SinγSinω
For γ=0,
= sinδ (sinφ cosβ -cosφ sinβ) + cosδ (cosφ cosβ + sinφ sinβ) cosω
After the simplification
Cosθ = Sinδ Sin(φ-β) + Cosδ Cos (φ-β) Cosω
33
 cossinsincoscoscoscoscossincossincossinsincos 

34. 34
Formulation for Hour angle, ω
s
at sunrise or sun set time
Cosθ = Sinδ Sinφ Cosβ
-Sinδ Cosφ Sinβ Cosγ
+ Cosδ Cosφ Cosβ Cosω
+ Cosδ Sinφ Sinβ Cosγ Cosω
+ Cosδ Sinβ Sinγ Sinω
For horizontal surface, β = 0 and θ = θ
z
Cosθ
z
= Sinδ Sinφ + Cosδ Cosφ Cosω
During sunrise or sunset time, θ
z
=90 °
ω
s
= cos
-1
(-tanδ tanφ), deg
Maximum sunshine hours or day length (in hours)
hrS ),tantan(cos
15
2
1
max


Sun Earth Relationship (contd.,)

35. Convert watch time into solar time
Input: location, surface orientation,
date and time
Find declination at given day number
Find hour angle from solar time
Find sunrise angle or sunset angle from declination and location
Find day length by multiplying 2/15 to sunrise angle
Find sunrise time and sun set time from half of the day length
Find zenith angle from location, declination and hour angle
Find air mass from zenith angle
Start
Find incident angle, θ = f(δ, Ø, β, γ and ω)
Find solar azimuth angle from zenith angle, location and declination
Flow chart showing the
methodology for solar
geometry solutions
Solar time = Watch Time -4 (L
st
–L
lo
) + EoT (in minutes)
ω= 15 (solar time -12) (in degrees)
deg,
365
)284(360
sin45.23 in
n





 




cossin
sinsincos
cos
Z
Z
s


) tan tan(-cos
15
2
1-
S
End
) tan tan(-cos
-1

s
)coscoscossin(sincos
1
 

z
)(cos
1
zm 



36. 36
Sun path diagram
Sun path diagram at 0°latitude
Solar Geometry (contd.,)

37. 37
Minimum incidence angle
(i) Fixed inclined surface
Fortiltedsurface(),
Inthismode,thesolarcollectorplaneistiltedtowards
south(γ=0).
Aftersubstitutingtheconditionofγinincidenceangle,
θ=cos-1(sinδsin(φ-β)+cosδcos(φ-β)cosω)
Sinceφ-β=0,
Incidenceangleisthefunctionofdayandtimeforfixed
tiltedsurface.

)cos(coscos
1




38. 38
Minimum incidence angle (contd.,)
(
ii) E-W horizontal axis rotation
The incidence angle equation can be differentiated with respect to β and equate to
zero to get the condition for the minimum incidence angle at γ = 0.
Forγ=0,theincidenceangle,
θ=cos-1(sinδsin(φ-β)+cosδcos(φ-β)cosω)
After substituting the φ-β, the minimum incidence angle can be resulted as a
function of declination and hour angle.











cos
tan
tan
1











cos
tan
tan
1













































 cos
cos
tan
tancoscos
cos
tan
tansinsincos
111

39. 39
Minimum incidence angle (contd.,)
(iii) North south (N-S) horizontal axis rotation
In this case the surface azimuth angle, γ
s
is -90°before noon and +90°
after noon. Therefore incidence angle can be differentiated with
reference to β with the condition in surface azimuth angle,
In the above equation, the positive sign in forenoon and negative is in
afternoon.
Incidence angle can be determined after substituting this condition in the
incidence angle.















coscoscossinsin
sincos
tan
1

40. 40
Minimum incidence angle (contd.,)
(
iv) North south (N-S) fixed inclined axis with azimuth
angle rotation
It is also called as azimuth axis tracking.
In this case, β = φ and γ = γ
s
where







 





cossin
sinsincos
cos
1
Z
Z
s

41. 41
Minimum incidence angle (contd.,)
(v) Dual axis tracking
In this case,
β = θ
z
and γ = γ
s

42. 42
Summary on solar geometry

For single axis tracking (linear concentrating collector), N-S
horizontal axis rotation is recommended to result lowest incidence.

There is no much benefit with E-W horizontal axis rotation.

The incidence at equinox is lower than summer and winter.

For point focusing collector, the tilt angle should be equal to zenith
and the surface and solar azimuths to be equal.

Based on the results, N-S horizontal axis tracking has been
recommended for the future work of self tracking machine.

43. Work assignment

Draw solar azimuth angle (γ
s
) with shadow effect and plot the
measured and estimated values for a day length.

Find and plot the measured and estimated values of incident angle
by tilting a plain surface with surface azimuth angle (γ) and tilt
angle (β) to obtain zero incident (θ= 0).

Develop a 3D model demonstrating the variable nature of sun
earth relations using any suitable software such as solid works etc.

Influence of building shape and architecture with year round sun
earth relations.
Sun Earth Relationship (contd.,)

44. 44
Sun Earth Relationship (contd.,)

Derive solar azimuth angle.

Derive generalized expression for incidence
angle.

Derive minimum incidence angle for different
cases.