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ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 1
ECE 333 – GREEN ELECTRIC ENERGY 12. The Solar Energy Resource
George Gross
Department of Electrical and Computer Engineering
University of Illinois at Urbana–Champaign
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 2
SOLAR ENERGY
q Solar energy is the most abundant renewable
energy source and is considered to be very clean
q Solar energy is harnessed for many applications,
including electricity generation, lighting and
steam and hot water production
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 3
SOLAR RESOURCE LECTURE
q The solar energy source
q Extraterrestrial solar irradiation
q Analysis of solar position in the sky and its
application to the determination of
m optimal tilt angle design for a solar panel
m sun path diagram for shading analysis
m solar time and civil time relationship
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 4
UNDERLYING BASIS: THE SUN IS A LIMITLESS ENERGY SOURCE
Sour
ce :
http
://jo
hnos
ulliv
an.fi
les.w
ordp
ress
.com
/201
2/11
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 5
SOLAR ENERGY q The thermonuclear reactions, as the hydrogen atoms
fuse together to form helium in the sun, are the
source of solar energy
q In every second, roughly 4 billion kg of mass are
converted into energy, as described by Einstein’s
famous mass–energy equation
q This immense energy generated is huge so as to
keep the sun at very high temperatures at all times
E =mc2
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 6
SOLAR ENERGY
q The plentiful solar energy during the past 4 or 5
billion years is expected to continue in the future
q Every object emits radiant energy in an amount
that is a function of its temperature; the sun emits
solar energy into space via radiation
q Insolation or solar irradiation stated in units of
measures the power density of the solar energy
2
Wm
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 7
PLANCK’S LAW
q Physicists use the theoretical concept of a
blackbody – defined to be a perfect emitter, as well
as a perfect absorber – to discuss radiation
q The emissive power intensity of a blackbody is a
function of its wavelength λ and temperature τ as
expressed by Planck’s law
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 8
PLANCK’S LAW
emissive power intensity
K
W / m2 -µm
ρ λ (τ ) = 3.74 × 10 8
λ 5 exp14,400λ τ
⎛⎝⎜
⎞⎠⎟− 1
⎡
⎣⎢
⎤
⎦⎥
µm
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 9
WIEN’S DISPLACEMENT RULE
An important feature of blackbody radiation is
given by Wien’s displacement rule, which
determines the wavelength at which the
emissive power intensity reaches its peak value
λ max =
2, 898τ
µm
λ max
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 10
0
0.5
1
1.5
2
2.5
0 1 2 3
EXTRATERRESTRIAL SOLAR SPECTRUM ir
radi
atio
n in
tens
ity
(kW
/m 2
- µm
)
the total area under the curve is the
extraterrestrial solar irradiation
Sour
ce :
http
://w
ww
.iocc
g.or
g/gr
oups
/mue
ller.h
tml
ultraviolet visible infrared
wavelength (µm)
λ max sun
=2, 8985, 800
= 0.5 µm
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 11
THE SOLAR IRRADIATION
q The sun’s surface temperature is estimated to be
5,800 K and its power density value is assumed as
1.37 kW/m 2 – the value of insolation or solar
irradiation just outside the earth’s atmosphere
q The sun emits maximum energy at the wavelength
λ max sun
=2, 8985, 800
µm = 0.5 µm
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 12
EXTRATERRESTRIAL SOLAR IRRADIATION
Extraterrestrial solar irradiation is defined as the
solar irradiation that strikes an imaginary surface
at the top of the earth’s atmosphere, which lies
perpendicular to the line from the earth’s center
to the sun’s center
extraterrestrial solar irradiation
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 13
STEFAN–BOLTZMANN LAW OF RADIATION
q The total area under the power intensity curve is
the blackbody radiant power density emitted over
all the wavelengths
q The Stefan–Boltzmann law of radiation states that
the total radiant power in W
the surface area of
the blackbody in
Stefan-Boltzmann constant: p
blackbody= σ A τ
4
5.67×10 − 8W / m2 - K
2m
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 14
THE EARTH’S RADIATION
q We consider the earth to be a blackbody with
average surface temperature and area equal
to
q The Stefan–Boltzmann law of radiation states that the
earth radiates
15 oC
5.1×1014 m2
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 15
THE EARTH’S RADIATION
q The wavelength at which the maximum power is
emitted by earth is given by Wien’s displacement rule
p earth = σ A τ 4
= ( 5.67 × 10 − 8 ) (5.1 × 10 14 ) 4(15 +273)
= 2 × 1017 W
λ max earth
=2,898288
µm = 10.1 µm
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 16
05
101520253035
1 49 97 145 193 241 289 337
THE SPECTRAL EMISSIVE POWER INTENSITY OF A 288 – K BLACKBODY
pow
er in
tens
ity (k
W/m
2 - µ
m )
wavelength ( µm )
Sour
ce :
http
://w
ww
.iocc
g.or
g/gr
oups
/mue
ller.h
tml
0 10 20 30 40 50 60 70
λ max earth
= 10.1µm
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 17
EARTH’S ORBIT OVER ITS YEARLY REVOLUTION AROUND THE SUN
winter solstice
vernal equinox
summer solstice
autumnal equinox
Sour
ce :
http
://sc
ijink
s.nas
a.go
v
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 18
EXTRATERRESTRIAL SOLAR IRRADIATION
extraterrestrial solar irradiation
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 19
q In the analysis of all solar issues, we use solar time
based on the sun’s position with respect to the
earth, instead of clock or civil time
q Extraterrestrial solar irradiation depends on the
distance between the earth and the sun and
therefore is a function of the day of the year
EXTRATERRESTRIAL SOLAR IRRADIATION OVER A YEAR
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 20
1.3
1.34
1.38
1.42
1
THE ANNUAL EXTRATERRESTRIAL SOLAR IRRADIATION
extr
ater
rest
rial
sola
r irr
adia
tion
(kW
/m 2 )
1 365
Sour
ce: h
ttp:
//sol
arda
t.uor
egon
.edu
/Sol
arR
adia
tionB
asic
s.htm
l/
day
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 21
EXTRATERRESTRIAL SOLAR IRRADIATION OVER A YEAR
q The extraterrestrial solar irradiation variation over
a day is negligibly small and so we assume that
its value is constant as the earth rotates each day
q We use the approximation for given by:
i0 d= 1,367 1 + 0.034 cos 2π d
365"
#$
%
&'
(
)*
+
,-
d = 1,2, . . .. , 365/366
i0 d
2/W m
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 22
EXTRATERRESTRIAL SOLAR IRRADIATION
q We consider the approximation of extraterrestrial
solar irradiation on January 1: d = 1
q Now, for August 1, d = 213 and the extraterrestrial
solar irradiation is approximately
i
0 1= 1,367 1 + 0.034 cos 2π 1
365"
#$
%
&'
(
)*
+
,- = 1,413 W
m2
i0 213
= 1,367 1 + 0.034 cos 2π 213365
"
#$
%
&'
(
)*
+
,- = 1,326 W
m2
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 23
EXTRATERRESTRIAL SOLAR IRRADIATION
q We observe that in the Northern hemisphere, the
extraterrestrial solar irradiation is higher on a cold
winter day than on a hot summer day
q This phenomenon results from the fact that the
sunlight enters into the atmosphere with different
incident angles; these angles impact greatly the
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 24
EXTRATERRESTRIAL SOLAR IRRADIATION
fraction of extraterrestrial solar irradiation received
on the earth’s surface at different times of the year
q As such, at a specified geographic location, we
need to determine the solar position in the sky to
evaluate the effective amount of solar irradiation at
that location
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 25
THE SOLAR POSITION IN THE SKY
The solar position in the sky varies as a function of:
m the specific geographic location of interest;
m the time of day due to the earth's rotation
around its tilted axis; and,
m the day of the year that the earth is on its
orbital revolution around the sun
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 26
LATITUDE AND LONGITUDE
q A geographic location on earth is specified fully
by the local latitude and longitude
q The latitude and longitude pair of geographic coordi–
nates specify the North–South and the East–West
positions of a location on the earth's surface; the
coordinates are expressed in degrees or radians
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 27
LATITUDE AND LONGITUDE
parallels
equator
Tropic of Cancer at 23.45 °
each parallel is an imaginary east-west circle that passes through all the locations that are at the corresponding latitude
o90 60o
o30
o0
− 30o
− 60o
− 90o
Tropic of Capricorn at − 23.45 o
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 28
LATITUDE AND LONGITUDE
prime meridian meridians
o180 o150o120
o90
o60
o30o0 − 30o
− 60o
− 90o
− 120o − 150o
a meridian is an imaginary arc on the earth's surface that connects the North and South poles
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 29
THE SOLAR IRRADIATION VARIES BY THE GEOGRAPHIC LOCATION
sunlight
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 30
EARTH’S ROTATION
http
://up
load
.wik
imed
ia.o
rg/w
ikip
edia
/com
mon
s/d/
d7/E
arth
%27
s_A
xis.g
if
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 31
EARTH’S ROTATION
q Although the sun’s position is fixed in space, the
earth’s rotation around its tilted axis results in the
“movement” of sun from east to west during each
day’s sunrise–to–sunset period
q The “movement” of the sun’s position in the sky
causes variations in the solar irradiation received
at a specified location on the earth’s surface
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 32
THE SOLAR IRRADIATION VARIES AS A FUNCTION OF THE TIME OF A DAY
east
azimuth angle
North South
altitude angle
East
West
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 33
THE SOLAR POSITION IN THE SKY AT ANY TIME OF THE DAY
q The solar position in the sky at any time of the
day – sunrise–to–sunset period – is expressed in
terms of the altitude angle and the solar azimuth angle
q The altitude angle is defined as the angle between
the sun and the local horizon, which depends on
the location’s latitude, solar declination angle and
solar hour angle
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 34
q The solar declination angle refers to the angle
between the plane of the equator and an
imaginary line from the center of the sun to the
center of the earth
q The solar declination angle variation during a day is
sufficiently small and so we assume it to remain
constant and represent it as a function of d by
SOLAR DECLINATION ANGLE
δ d
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 35
SOLAR DECLINATION ANGLE
Sour
ce: h
ttps
://en
cryp
ted-
tbn3
.gst
atic
.com
/imag
es?q
=tbn
:AN
d9/
solar rays
equator
solar declination angle
solar declination angle
δ
d= 0.41 sin 2π
365d − 81( )
$
%&'
() radians
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 36
SOLAR HOUR ANGLE
q Solar noon is the time at which the solar position in the sky is vertically over the local meridian, i.e., the line of longitude; in other words, the sun is due South (North) of the location in the Northern (Southern) Hemisphere
q Solar hour angle refers to the angular rotation in radians the earth must go through to reach the solar noon; h is positive before the solar noon – ante meridiem – and negative after solar noon – post meridiem
( )hθ
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 37
SOLAR HOUR ANGLE
q We consider the earth to rotate each hour; so
q At 11 a.m. in solar time
and at 2 p.m. in solar time
2π24
θ ( h) =
π12
h radians
θ (1) =
π12
θ ( − 2) = −
π6
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 38
ALTITUDE ANGLE
Then, the relation of altitude angle and the
location’s latitude, solar declination angle and solar
hour angle is given by
where ℓ is the local latitude
β(h)
d
sin β(h)d
"#$
%&'
= cos(ℓ ) cos δd
"
#$
%
&' cos θ (h)( ) + sin(ℓ ) sin δ
d
"
#$
%
&'
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 39
EXAMPLE: ALTITUDE ANGLE AT CHAMPAIGN
q Champaign’s latitude is 0.7 radians
q October 22 corresponds to d = 295; the solar
declination angle is computed to be
q At 1 p.m. solar time, the hour angle is
δ
295= 0.41 sin 2π
365295 − 81( )$
%&'
()= − 0.21 radians
θ ( − 1) =
π12
⋅ ( − 1) = −π12
radians
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 40
EXAMPLE: ALTITUDE ANGLE AT CHAMPAIGN
q We compute the altitude angle at Champaign from
and so
sin β( − 1)295( )
= cos(0.7) cos(− 0.21 )cos −π12
$
%&
'
() + sin(0.7) sin(− 0.21 )
= 0.59
β( − 1) 295 = sin − 1 (0.59) = 0.623 radians
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 41
SPECIAL CASE: THE ALTITUDE ANGLE AT SOLAR NOON
equator
local horizon
δ d
β (0) dℓ"
"
ℓ"N
local horizon β (0) d
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 42
SPECIAL CASE: ALTITUDE ANGLE AT SOLAR NOON
q The altitude angle at solar noon of day d satisfies
q However, a more direct expression for is
obtained from the geometric relationship
sin β (0) d( )
= cos(ℓ) cos δ d( )cos θ (0)( ) + sin(ℓ) sin δ d( )
β (0) d
β (0) d =π2 − ℓ + δ d radians
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 43
EXAMPLE: ALTITUDE ANGLE AT SOLAR NOON
q We determine the altitude angle for Champaign at
ℓ = 0.7 radians, at solar noon on March 1 (d = 60)
q The solar declination angle is
q The altitude angle at solar noon is
δ
60= 0.41 sin
2π365
60 − 81( )$
%&'
()= − 0.15 radians
β (0) 60 =π2 − ℓ + δ 60 = 0.72 radians
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 44
THE SOLAR AZIMUTH ANGLE
q The solar azimuth angle is defined as the angle
between a due South line in the Northern Hemi-
sphere and the projection of the line of sight to
the sun on the earth surface
q We use the convention that is positive when
the sun is in the East – before solar noon – and
negative when the sun is in the West – after noon
φ
φ
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 45
THE SOLAR AZIMUTH ANGLE
φ β (h)
d
φ > 0before solar noon
E
φ < 0after solar noon
S
W
sunrise
sunset
solar noon
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 46
THE SOLAR AZIMUTH ANGLE
q The equation for the solar azimuth angle is
determined from the relationship
q Since the sinusoidal function is given to
ambiguity because , we need to
φ (h)
d
sin φ (h)d
"
#$
%
&' =
cos δd( ) sin θ (h)( )
cos β(h)d( )
sin x = sin (π − x)
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 47
THE SOLAR AZIMUTH ANGLE
determine whether the azimuth angle is greater or
less than :
π2
if cos θ (h)( ) ≥tan δ d( )
tan (ℓ) then φ (h)
d≤π
2
else φ (h)d>π
2
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 48
EXAMPLE: WHERE IS THE SUN IN THE SKY
q Determine the altitude and the solar azimuth angles
at 3 p.m. in Champaign with latitude ℓ = 0.7 radians
at the summer solstice – d = 172
q The solar declination is
q The hour angle at 3 p.m. is
δ 172= 0.41 radians
θ ( − 3) = −
π
4
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 49
EXAMPLE: WHERE IS THE SUN IN THE SKY
q Then we compute the altitude angle:
q Then
β ( − 3)172
= 0.85 radians
sin β ( − 3)172( )
= cos(0.7) cos 0.41( )cos −π
4
$
%&
'
() + sin(0.7) sin 0.41( )
= 0.75
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 50
EXAMPLE: WHERE IS THE SUN IN THE SKY
q The sine of the azimuth angle is obtained from
q Two possible values for the azimuth angle are
or
φ ( − 3)
172= sin − 1 (− 0.9848) = − 1.4 radians
sin φ ( − 3)172
#$%
&'( =
cos 0.41( ) sin −π
4
#
$%
&
'(
cos 0.85( )= − 0.9848
φ ( − 3)
172= π − sin − 1 (− 0.9848) = 4.54 radians
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 51
EXAMPLE: WHERE IS THE SUN IN THE SKY
q Since
q Then we can determine
q Thus
cos θ (− 3)( ) = 0.707
tan δ172
$
%&
'
()
tan(ℓ)= 0.515and
cos θ (− 3)( ) >
tan δ172
$
%&
'
()
tan(ℓ)
φ ( − 3)
172= − 1.4 radians
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 52
IMPORTANCE OF THE ANALYSIS ON SUN’S POSITION IN THE SKY
q We are now equipped to determine the sun’s
position in the sky at any time and at any location
q To effectively design and analyze solar plants, the
sun’s position in the sky analysis has some highly
significant applications, including to
m build sun path diagram and do shading analysis
m determine sunrise and sunset times
m evaluate a solar panel’s optimal position
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 53
SUN PATH
sun’s path at the summer solstice
sun’s path at an equinox
sun’s path on winter solstice
zenith
N S
W
E
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 54
SUN PATH DIAGRAM
q The sun path diagram is a chart used to illustrate the
continuous changes of sun’s location in the sky at
a specified location over a day’s hours
q The sun’s position in the sky is found for any hour
of the specified day d of the year by reading the
azimuth and altitude angles in the sun path diagram
corresponding to that hour
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 55
SUN PATH DIAGRAM FOR 40 °N
6 a.m.
7 a.m.
8 a.m.
9 a.m.
10 a.m.
11 a.m. solar noon
1 p.m.
2 p.m.
3 p.m.
4 p.m.
5 p.m.
6 p.m.
sola
r alti
tude
ang
le
solar azimuth angle
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 56
SUN PATH DIAGRAM FOR SHADING ANALYSIS
q In addition to the usefulness of sun path diagrams to
help us find the sun’s position in the sky, they also
have strong practical application in shading
analysis at a site – an important issue in PV design
due to the strong shadow sensitivity of PV output
q Modification of the sun path diagram for shading
analysis requires a determination of the azimuth
and altitude angles of the obstructions
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 57
EXAMPLE: SUN PATH DIAGRAM FOR SHADING ANALYSIS
sola
r alti
tude
ang
le
solar azimuth angle
6 a.m.
7 a.m.
8 a.m.
9 a.m.
10 a.m.
11 a.m. solar noon
1 p.m.
3 p.m.
4 p.m.
5 p.m.
2 p.m.
6 p.m.
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 58
IMPORTANCE OF SHADING ANALYSIS
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 59
SHADING ANALYSIS USING SHADOW DIAGRAM
q In the set-up of a solar field, it is important to
design the arrays so that the solar panels do not
shade each other
q In addition to the application of sun path diagrams,
there are other graphical and analytic approaches
for shading analysis; such topics are outside the
scope of the course
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 60
SUNRISE AND SUNSET
q An important issue is the determination of the
sunrise/sunset times since solar energy is only
collected during the sunrise to sunset hours
q We estimate the sunrise/sunset time from the
equation used to compute the solar altitude angle,
which is zero at sunrise and sunset
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 61
SUNRISE AND SUNSET
q The relationship for the solar angle results in:
q Now we can determine the sunrise solar hour
angle and the sunset hour angle to be:
cos θ (h)( ) = −sin(ℓ ) sin δ
d
$
%&
'
()
cos(ℓ ) cos δd
$
%&
'
()= − tan(ℓ ) tan δ
d
$
%&
'
()
sin β (h)
d
"#$
%&' = 0
κ + d
κ − d
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 62
SUNRISE AND SUNSET
q The corresponding sunrise and sunset angles are
so that the solar times for sunrise/sunset are at
κ + d= cos − 1 − tan(ℓ ) tan δ
d( )( )
κ − d= − cos − 1 − tan(ℓ ) tan δ
d( )( )
12 : 00 −
κ + d
π / 12 and 12 : 00 −
κ − d
π / 12
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 63
SUNRISE TIME IN CHAMPAIGN
q Champaign is located at ℓ = 0.7 radians
q On October 22, the solar declination angle is – 0.21
radians and the sunrise solar hour angle is :
q The sunrise expressed in solar time is at
κ + 295
= cos − 1 − tan(0.7) tan − 0.21( )( ) = 1.39 radians
12 : 00 − 1.39
π / 12 = 6 : 27 a.m.
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 64
SOLAR TIME AND CIVIL TIME
q So far, we used exclusively solar time measured
with reference to solar noon in all our analysis of
insolation and its impacts
q However, in our daily life we typically use civil or
clock time, which measures the time to align with
the earth’s daily rotation over exactly 24 hours
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 65
SOLAR TIME AND CIVIL TIME
q The difference at a specified location on the earth
surface between the solar time and the civil time
arises from the earth’s uneven movement along
its orbit of the annual revolution around the sun
and the deviation of the local time meridian from
the location longitude
q As such, two distinct adjustments must be made
in order to convert between solar time and civil time
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 66
SOLAR DAY AND 24–HOUR DAY
q We examine the difference between a solar day and
the corresponding civil 24–hour day
q A solar day is defined as the time elapsed between
two successive solar noons
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 67
HOW LONG IS A SOLAR DAY ht
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earth’s orbit around the sun
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ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 68
SOLAR DAY q The earth’s elliptical orbit in its revolution around the
sun results in a different duration of each solar day
q The difference between a solar day and a 24-h day
is given by the deviation in minutes
e | d = 9.87sin 2(b | d )( ) −7.53cos(b | d )−1.5sin (b | d ),
where,
b | d = 2π364
d − 81( ) radians
e | d
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 69
DIFFERENCE BETWEEN A SOLAR AND A 24–HOUR DAY OVER A YEAR
-20 -15 -10
-5 0 5
10 15 20
1
min
utes
1
365
day
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 70
LOCAL TIME MERIDIAN AND LOCAL LONGITUDE
q There are 24 time zones to cover the earth, each
with its own time meridian with 15 ° longitude gap
between the time meridians of two adjacent zones
q The second adjustment deals with the longitude
correction for the fact that the clock time at any
location within each time zone is defined by its
local time meridian which differs from the time
zone meridian
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 71
LOCAL TIME MERIDIAN AND LOCAL LONGITUDE
q For every degree of longitude difference, the solar
time difference corresponds to
q The time adjustment due to the degree longitude
difference between the specified location and the
local time meridian is the product of 4 times the
longitude difference expressed in minutes
24 hour ⋅ 60 m / hour360o = 4
mdegree longitude
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 72
LOCAL TIME MERIDIAN AND LOCAL LONGITUDE
q The sum of the adjustment and the longitude
correction results in:
q This relationship allows the conversion between
solar time and civil time at any location on earth
| de
solar time = clock time + e | d + 4 ×1803.14
×
(local time meridian − local longitude)
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 73
EXAMPLE: SOLAR TIME AND CLOCK TIME
q Find the clock time of solar noon in Springfield on
July 1, the 182nd day of the year
q For d = 182, we have
b
182 = 2π
364182 − 81( ) = 1.72 radians
e 182 = 9.87sin (2 x1.72)−7.53cos (1.72)−1.5sin (1.72)
= − 3.51 mins
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 74
EXAMPLE: SOLAR TIME TO CLOCK TIME
q For Springfield, IL, with longitude 1.55 radians, the
clock time in the central time zone is:
solar time − e | d − 4 ×1803.14
×
(local time meridian − local longitude)
= solar noon−(−3.51)− 57 ( − 1.44 ) − (− 1.55 )( )
= 11: 38 a.m.
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 75
WORLD TIME ZONE MAP
Source: http://www.physicalgeography.net/fundamentals/images/world_time2.gif
five time zones span across China’s territory, but by
government decree the entire country uses the time zone
at the location of the capital as the single standard time
ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 76
CONCLUSION
q With the conversion scheme between the solar
and clock times, the analysis of solar issues, e.g.,
the expression of sunrise/sunset on civil time
basis, makes the results far more meaningful for
use in daily life
q Such a translation renders the analysis results to
be much more concrete for all applications