Mathcad - Tracking Sunlight Analysis

8/13/2019 Mathcad - Tracking Sunlight Analysis

1/17

Tracking Sunlight AnalysisJavier Ignacio Camacho

Apparent Motion of the SunBecause the rotation and the orbital motion of the Earth around the Sun,

the apparent position of the Sun in the sky changes over time.

In order to utilize the solar energy efficiently we have to perform an analysis and understand the

apparent motion of the Sun.

Rotation of Earth : Definition of Latitude and Longitude

This motion is due to the rotation of the Earth on its axis.

In order to simplify the problem the Earth is considered as a perfect sphere rotating with a constant

angular velocity on a fixed axis.

The axis of rotation of the Ear th crosses the surface of Earth at two points:

North Pole and Soth Pole.The great circle perpendicular to the axis is the Equator.A location on Earth can be specified by two coordinates, the Latitude ( )andthe Longitude().The Longitude specifies a Meridian(a half of great circle passing through the two poles and thelocation),requires an origin as zero point,the Prime Meridian.The zero point of longitude,is defined as the meridian passing through the Royal Greenwich

Observatory.

The Latitude is defined by the poles and equator.

The convention of sign is: eastward is positive and westward is negative.


2/17

VBI 9.53 VBI 55.52

world READPRN "world.prn"( )

world0

world1

Global Position of the VBI Building[Northward Latitude ] [Eastward Longitude]

.VBI

55

52

13.36

DMS 0.975 .VBI

9

53

10.32

DMS 0.173 rad

100 0 10090

60

30

0

30

60

90

Global Position of a VBI Building

VBI

VBI


3/17

Celestial SphereFrom the point of view of an observer on Earth, the Sun is located on a sphere with a large radius.

This imaginary sphere is called Celestial Sphere.There are two commonly used coordinate systems to describe the position of the Sun (or any star) on

the celestial sphere, The Horizont Systemand The Equatorial System.The extension of the center of Earth and an observer O into the sky is poiting to the Zenith Z and theplane perpendicular to that line or the corresponding great circle is the Horizon.The Horizon divides the sphere into two hemipheres. The upper hemisphere is visible to the observer

and the lower is hidden below the horizont.

The angular distance of a celestial body above the horizon is its Height (h)( altitude or elevation).The

height of the North Pole P equals the geographical latitude of the observer ( ).

To completely identify the position of a celestial body X,we need another reference point. The greater

circle connecting the zenith with the North Pole is called the Meridian. It intersects the horizon at

point S, the south point of the horizont.

To identify the position of the celestial body with respect to the south point, we draw a g reat circle

through the zenith and the star, which intersects the horizon at point C.

The angle SC is defined as the Azimuth A, or the horizontal direction of the celestial body. The azimuth

of the Sun always increases over time.

The Horizon Coordinate Systemdefines the position of a celestial body as directly perceived bythe observer. However, because Earth is round and rotating on its axis, those coordinates depend on

the location of the observer and vary overtime. In Equatorial Coordinate System, on the otherhand, the position of the Sun is relatively independent of the location of the observer.

The coordinates of the Sun in the horizon system can be obtained using a coordinate transformation

from its coordinates in the horizontal system.

In the equatorial system, the coordinate equivalent to the latitude of the Earth is the declination and

the coordinate equivalent to the longitude of the Earth for a fixed observer is the hour angle .


4/17

Notation in Positional Astronomyatitude ( ): Geographical coordinate

Longitude():Geographical coordinateHeight(h): Altitude or elevation to the horizontAzimuth(A): Horizontal direction or bearingDeclination(): Angular distance to the equatorHour Angle(): In radians, westwardSunset hour Angle(s):In radians, always positiveEast-West hour Angle(ew):In radians, always positiveRight Ascension(): Absolute celestial coordinateMean Ecliptic Longitude(l):On ecliptic planeTrue Ecliptic Longitude():On ecliptic planeEccentricity of Orbit(e):Currently 0.0167 Obliquity of Eliptic():Currently 23.44

Coordinate Transformation: Spherical TrigonometryTo study the location of the Sun with respect to a specific location on Earth, we will correlate the

coordinates of the location on the terrestrial sphere of Earth with the location of the Sun on the

celestial sphere. The mathematical tool of this study is Spherical Trigonometry.Focus on the spherical triangle PZX, with three arcs:

p=ZX

z=XPx=PZ

The relations between the elements of the spherical triangle and the quantities of interest are:

P=Z=180 - Ap=90 - hz=90 -

x=90 -

First, consider the case of given declination and hour anglein the equatorialsystem to find the height h and the azimuth A in the horizontal coordinates system.

he latitude of the observer's location is a necessary parameter.Cosine Formulacos p( ) cos x( ) cos z( ) sin x( ) sin z( ) cos P( )=

Substitution from the relations between the elements yield

sin h( ) sin ( ) sin ( ) cos ( ) cos ( ) cos ( )= EQ 1


5/17

Sine Formulasin Z( )

sin z( )

sin P( )

sin p( )=

Substitution from the relations between the elements yield

cos h( ) sin A( ) sin ( ) cos ( )=

Applying Formula Csin x( ) cos Z( ) cos z( ) sin p( ) sin z( ) cos p( ) cos X( )= EQ 2Substitution from the relations between the elements yield

cos h( ) cos A( ) sin ( ) cos ( ) cos ( ) cos ( ) sin ( )= EQ 3Next, we consider the case of given height h and azimuth A in the horizon coordinatesystem to find declination and hour angle in the equatorial system.Again the

atitude of the obsever's location is a necessary parameter.Cosine Formulacos z( ) cos p( ) cos x( ) sin p( ) sin x( ) cos Z( )=

sin ( ) cos h( ) cos A( ) cos ( ) sin h( ) sin ( )= EQ 4cos ( ) sin ( ) cos h( ) sin A( )=

sin z( ) cos P( ) cos p( ) sin x( ) sin p( ) cos x( ) cos Z( )= EQ 5cos ( ) cos ( ) cos h( ) coa A( ) sin ( ) sin h( ) cos ( )= EQ 6

Treatment in Solar TimeThe Solar Time ( t

), which is based on the hour angle of the Sun, is an intuitive measure of time and

was used for thousands of years in all cultures of the world.

Becasue the apparent motion of the Sun is nonuniform and depends on location, it is not an accurate

measure of the time.

The difference between solar time and standard time as used in everyday life, even with a proper

alignment, can be more than 15 min.

To make an estimation of solar radiation, sometimes high accuracy is not required.

s

t

12

12

=

t

12 12s

=


6/17

Obliquity and Declination of the Sun

The orbital plane of the Earth arround the Sun, the Ecliptic, is at an angle called the Obliquity ()from the Equator.

Currently: 23.44 (This is what causes the seasons)Related to the motions of the Sun over a calendar year, there are four cardinal points.

Vernal Equinox, the trajectory of the Sun intersects the celestial equator, heading north.

Summer Solstice, the trajectory of the Sun reaches its northernmost point, which is about23.44 above the celestial equator.

Autmnal Equinox, the trajectoyr of the Sun intersects the celestial equator, heading south.

Winter Solstice, the trajectory of the Sun reaches its lowest point, which is about 23.44 belowthe celestial equator.

In line with the concept of solar time, the motion of the Sun along its orbital can be described by

the Mean Longitude (l).Vernal Equinox l=0Summer Solstice l=/2=90Autmnal Equinox l==180Winter Solstice l=3/2=270


7/17

Declination Approximation FormulaThe next formula is a simple approximation by assuming that the declination varies sinusoidally

with the mean longitude l and consequently is linear according to the number of the days in a year.

The error could be as large as 1.60, but it is accceptable in this application.

23.44=

sin l( ) sin2 N 80( )[ ]

365.2422

=

N INT275 M

9

K INTM 9

12

D 30=

Where :

M=Number of month

N= Number of day

D= is the day of the month

K=2(common year)/K=1(leap year)INT=means taking the integer part of the number

The number 80 is the of the day of the vernal equinox(March 20 or 21)


8/17

Sunrise and Sunset TImeSunrise (or Sunset)The condition of sunrise is the time when the height of the Sun is zero

sin ( ) sin ( ) cos ( ) cos ( ) cos ( ) 0=

cos s tan ( ) tan ( )( )=

In terms of angle s acos tan ( ) tan ( )( )( )

=

In terms of of the 24-h solar timets 12

12

acos tan ( ) tan ( )( )( )

1212

acos tan ( ) tan ( )( )( )

=


9/17

Direct Solar Radiation on South Face of VBI BuildingGeneral Formualtion

cos()=sin()(sin( )cos()-cos( )sin()cos())+cos()(cos( )cos()cos()

+sin( )sin()cos()cos()+sin()sin()sin()

SimplificationsVBI 0deg Azimuth anglecos ( ) sin .VBI PVC sin ( ) cos .VBI PVC cos ( ) cos ( )=

PVC

2= Polar Angle

cos VBI cos .VBI sin ( ) sin .VBI cos ( ) cos ( )=

Direct Daily Solar Radiation EnergyAn important application in terms of solar time is the computation of the direct solar radiation

energy HD on a surface on a clear day.

The effect of clouds and scattering is not considered in this case..

On a clear day, on a surface perpendicular to the sunlight, the power is 1kW/m2 and the total

radiation energy in an hour is 1 kWh/m2.

When the sunlight is tilted with an angle , the radiation energy is reduced to cosx 1kWh/m2.

Therefore, the daily direct solar radiation energy in units of kWh/m2is the integration of cosover

24h.

VBI Building Direct Daily Solar Radiation EnergyGlobal Position of the VBI Building[Northward Latitude ] [Eastward Longitude]

.VBI 0.975 .VBI 0.173

PVC

2 Polar Angle

VBI

0 deg Azimuth angle


10/17

Year 2013January

D1 1 1 1

31

M1 1 K 2 23.44 0.409 ObliquitySun DeclinationNumber of Day

N1D1floor

275 M1

9

K floor

M1 9

12

D1 30 1D1

sin

2 N1D180

365.2422

Sunset hour Angle(s)s1D1

acos tan 1D1 tan .VBI

Direct Daily Solar Radiation EnergyHD1D1

24

cos 1D1

sin .VBI sin s1D1 s1D1

sin 1D1 cos .VBI

FebruaryD2 1 1 1 28 M2 2

Number of Day Sun Declination

N2D2floor

275 M2

9

K floor

M2 9

12

D2 30 2D2

sin

2 N2D280

365.2422




24

cos 2D2


sin 2D2 cos .VBI

MarchD3 1 1 1 20 M3 3Number of Day Sun Declination

N3D3floor

275 M3

9

K floor

M3 9

12

D3 30 3D3

sin

2 N3D380

365.2422


11/17




24

cos 3D3

sin .VBI sin s3D3 s3D3 sin 3D3 cos .VBI

Vernal EquinoxDVX 21 21 1 31

Number of Day

NVXDVXfloor

275 M3

9

K floor

M3 9

12

DVX 30

Sun DeclinationVXDVX

sin

2 NVXDVX80

365.2422

Sunset hour Angle(ew)

ewDVXacos

tan VXDVX

tan .VBI

Direct Daily Solar Radiation EnergyHDVXDVX

24

cos VXDVX

sin .VBI sin ewDVX ewDVX

sin VXDVX cos .VBI

AprilD4 1 1 1 30 M4 4

Number of Day

N4D4floor

275 M4

9

K floor

M4 9

12

D4 30

Sun Declination4D4

sin

2 N4D480

365.2422


12/17


ew4D4acos

tan 4D4

tan .VBI


24

cos 4D4

sin .VBI sin ew4D4 ew4D4

sin 4D4 cos .VBI

MayD5 1 1 1 31 M5 5

Number of Day

N5D

5

floor275 M5

9

K floorM5 9

12

D5 30

Sun Declination

5D5sin

2 N5D580

365.2422


ew5D5acos

tan 5D5

tan .VBI


24

cos 5D5


sin 5D5 cos .VBI

JuneD6 1 1 1 30 M6 6

Number of Day

N6D

6

floor275 M6

9

K floorM6 9

12

D6 30

Sun Declination

6D6sin

2 N6D680

365.2422


13/17


ew6D6acos

tan 6D6

tan .VBI


24

cos 6D6


sin 6D6 cos .VBI

JulyD7 1 1 1 30 M7 7

Number of Day

N7D

7

floor275 M7

9

K floorM7 9

12

D7 30

Sun Declination

7D7sin

2 N7D780

365.2422


ew7D7acos

tan 7D7

tan .VBI


24

cos 7D7


sin 7D7 cos .VBI

AugustD8 1 1 1 31 M8 8

Number of Day

N8D

8

floor275 M8

9

K floorM8 9

12

D8 30

Sun Declination

8D8sin

2 N8D880

365.2422


14/17


ew8D8acos

tan 8D8

tan .VBI


24

cos 8D8


sin 8D8 cos .VBI

SeptemberD9 1 1 1 22 M9 9

Number of Day

N

9D9floor

275 M9

9

K floor

M9 9

12

D

9

30

Sun Declination

9D9sin

2 N9D980

365.2422


ew9D9acos

tan 9D9

tan .VBI


24

cos 9D9


sin 9D9 cos .VBI

Autumn EquinoxDAX 23 23 1 30

Number of Day

NAXDAXfloor

275 M9

9

K floor

M9 9

12

DAX 30

Sun Declination

AXDAXsin

2 NAXDAX80

365.2422


15/17

Sunset hour Angle(ew)sDAX

acos tan AXDAX tan .VBI

Direct Daily Solar Radiation EnergyHDAXDAX

24

cos AXDAX

sin .VBI sin sDAX sDAX

sin AXDAX cos .VBI

OctoberD10 1 1 1 31 M10 10

Number of Day

N10D10floor

275 M10

9

K floor

M10 9

12

D10 30

Sun Declination10D10

sin

2 N10D1080

365.2422

Sunset hour Angle(s)

s10D10acos tan 10D10

tan .VBI


24

cos 10D10

sin .VBI sin s10D10 s10D10 sin 10D10 cos .VBI

NovemberD11 1 1 1 30 M11 11

Number of Day

N11D11floor

275 M11

9

K floor

M11 9

12

D11 30

Sun Declination

11D11sin

2 N11D1180

365.2422


16/17




24

cos 11D11


sin 11D11 cos .VBI

DecemberD12 1 1 1 31 M12 12

Number of Day

N12D12floor

275 M12

9

K floor

M12 9

12

D12 30

Sun Declination

12D12sin

2 N12D1280

365.2422




24

cos 12D12


sin 12D12 cos .VBI

Daily Solar Radiation on a Vertical Surface Facing South


17/17

0 100 200 3000

1

2

3

4

5

6

7

8

9

10

VBI Direct Daily Solar Radiation Energy(HD(kWh/m^2))

Months

kWh/m^2

Mathcad - Tracking Sunlight Analysis

Documents