Solar Geometry

Presented by Runsheng Tang

1. Solar constant

The sun is a sphere of diameter 1.39 109m and its average distance 1.495 1011m. We regard the sun as a blackbody, the effective blackbody temperature is Ts=5762K. The solar constant I0 is defined as solar radiation on an extraterrestrial surface normal to the solar rays. So we have:

20

24 4 dIDTs

I0=1350 W/m2

Measurements by NASA indicated that the mean I0 is 1373 W/m2

If all radiation from the sun is absorbed, and then is dissipated by the earth to sky dome by thermal radiation, the earth temperature can be calculated as follows:

CKI

T

TrIr

o62794

4

4 0

420

2

In fact. Only 70% of incident radiation is absorbed, other 30% is directly reflected to the sky again by the atmosphere of the earth. So we have:

CI

T o184

7.04 0

I0=1373 is an average value, the actual radiation incident on the earth changes with the day of a year.

2. Sun-earth Geometry

The earth moves about the sun in an elliptical orbit, completing one revolution per year. In addition, the earth rotates once per day about its polar axis. The polar axis is inclined at an angle of 23.45o from the normal to the plane of the earth’s orbit (ecliptic)

First, the change of distance between earth and sun in a year cause the change in the solar radiation incident on the earth surface. This effect can be accounted for by assuming that the effective solar constant varies with time of year according to the expression.

0,0 )]25.365

360cos(033.01[ I

nI eff

Where n is the day of a year counted from first day on January, the peak value of I0,eff occurs 10 days after winter solstice

Second, there is a difference between real time (local time instead of standard time) and solar time.

Real time (local time) is the time based on the local longitude and rotation of the earth around its polar axis. However, as viewing from the earth, we can assume that on any single day the sun completes one circular orbit around the earth. This motion is described in solar time

Solar time = local time + E

Local time = standard time - 4(Lst-Lloc)

Hence, solar time = standard time –4(Lst-Lloc)+E

E=9.87sin2B-7.53cosB-1.5sinB (in minutes)

Where B=360(n-81)/364, Lst is the standard meridian for the local time zone, and Lloc is the longitude of the location in question in degrees east.

Example: Finding at what time (clock time) is the solar noon in Zhanjian, on July 21.

Solar time is the time of day measured from the solar noon, and solar noon is defined as that time of day when the sun appears due south (north) for a given point on the earth’s surface.

Solution:

For Zhanjian, Lloc=110o28”E, Lst=120E

On July 21, n=202, and B=119.67

Hence, we have: E=-6.07 mins

Solar time=standard time-4(120-110.47)-6.07=standard time – 44.19

Setting solar time to be 12:00 (solar noon) obtains:

standard time= 12:44’11”

If a summer time is executed, the clock time for solar noon at Zhanjian on July 21 should be 13:44’11”

One point must be emphasized that the time in the relevant calculations in solar energy must be solar time instead of local standard time.

The solar orbit makes angle with the plane of the earth’s equator, and this is called declination and varies with day of year

To derive declination, let us introduce into Fig.1 (x,y,z) coordinates that are fixed in the sun. The z axis is normal to the plane of the ecliptic and x axis points toward the earth at winter solstice. The unit vector ne from the earth to sun is given by:

)10(25.365

360

)0,sin,(cos

n

with

ne

The unit vector of polar axis is

np= (sin23.45o, 0, cos23.45o)

The declination is defined as the angle of the sun and the equatorial plane. Hence, the angle between ne and polar axis np is 90- , so we have:

)25.365

)10(360cos(45.23sinsin

sin)90cos(

n

nn

o

pe

For spring and autumn equinoxes, = 0o, and for winter solstice,

=-23.45, and for summer solstice =23.45.

3. Incidence angles on fixed surfaces

Fig.2: Coordinate system fixed at the earth surface. x axis points towards solar noon, y axis points east, z axis = axis of rotation (north is +)

In this coordinate system, the unit vector ns from the earth to sun is given by

)sin,sincos,cos(cos sn

Where is the hour angle (afternoon is +), related to solar time t by =15t, and unit normal vector nh of the ground surface (a horizontal surface) is

)sin,0,(cos hn

Hence, the incidence angle of the sun on a horizontal surface is

sinsincoscoscos

cos

hsz nn

This angle is frequently called the zenith angle. Sunset occurs when the zenith angle reaches 90o, and the corresponding sunset hour angle is

tantancos s

And the sunset time ts is For example at Zhanjian ( ) on June 21( ), the corresponding hour angle is

15/s 02.2145.23

hrtso

s 64.6,6.99

Namely, the day length is 13hr and 17mins, the sunrise time is 5:22 in solar time.

For convenience, usually we adopt such coordinate system that x’ axis is normal to the horizontal surface (points to top of the sky dome), y’ points toward east, and z’ points toward north. The relationship of coordinate systems between (x,y,z) and (x’,y’,z’) is shown in Fig.3

Fig.3: the coordinate system (x’,y’z’) is obtained by rotating coordinate system (x,y,z) through an angle around Y axis

Based on the exchange principle of coordinate systems, we have:

cossin'

'

sincos'

zxz

yy

zxx

Hence, in the coordinate system (x’,y’,z’), the unit vector ns’ of sun can

be expressed as:

])cossinsincoscos[

,sincos],sinsincoscos([cos'

sn

A collector is installed with a tilted angle and orientation angle ( west is +). The unit vector nc

’ of the normal to the collector can be expressed as:

)cossin,sinsin,(cos' cn

cossincossincossinsincoscos

sinsinsincoscossinsin

coscoscoscoscos ''

cs nn

As a special case, when ,we have: 0

coscoscos

:

sin)sin(coscos)cos(cos

If:

Setting , we can obtain the solar time when the sun rays strike on the collector as follows:

90

4. Applications of solar geometry

First, on knowing the incidence angle of solar rays on a horizontal surface, the solar radiation incidence on the extraterrestrial horizontal surface can be calculated by:

)cos(coscoscoscos ,0,0 seffzeff II

Integrating over time of day from sunrise to sunset, one obtains the daily extraterrestrial solar radiation on a horizontal surface

)cos(sincoscos

cos

,0

,00

ssseffday

t

t

zeff

I

dtIHs

s

Where is the length of day. hrsday 24

For a south-north faced building, in the (x’,y’,z’) coordinate system, the unit normal vector of south walls is (0,0,-1), and is (0,1,0) for east walls, hence the incidence angle of solar ray on the south and east walls can be expressed as:

sincoscos

cossinsincoscoscos

e

s

If cos is negative, the radiation incidence on the south wall is zero, at this moment, the solar rays will strike at the north walls.

For a building and solar collector, we usually need to know the shading of on obstacle in order to avoid collectors being shaded. As a simple case, the south facing collectors, the shade of the collector can easily be calculated by finding the profile angle.

s

As shown in Fig.4, the angle formed between the projection of solar rays on the plane X’OZ’ and Z’ axis is the profile angle of solar rays on the east-west faced vertical plane, and expressed as:

p

sinsincoscoscos

cossinsincoscostan

',

',

xs

zsp n

n

Once knowing the the profile angle of solar ray, the length of the shading of an obstacle on the horizontal surface can be found. For example, how much distance must be spaced between south-faced collectors so as to the collectors in the front line will not shade the collectors in the behind line at 10:00 on Dec.21.

For Kunimg , o01.25 o30,45.23

We have: , i.e.: the the distance between collectors must be 1.26 times of the height of collectors

26.1tan p

A window with height of h is installed on the south wall of a building located at . Now we design a overhang that is required to allow the sun rays incidence on the entire window at the solar noon from Oct.21 to Feb.21 and no solar rays incidence on the window at the solar noon from May 21 to July 21.

o31

At solar noon, p

For Oct.21, 085.11w

For May 21, 91.19s

Hence,

09.11

085.42

s

w

and

hy

x

y

x

s

w

tan

tanSo, we have x=0.25h, y=0.277h

If the building is not oriented with south-north facing, we must change the coordinate system (x’,y’,z’) to (x”,y”,z”) that makes y”-axis and z”-axis agree with the orientation of building by rotating (x’,y’,z’) around x’-axis. Then the profile angle can be easily calculated as in coordinate system (x’,y’,z’), i.e.:

'',

'',tan

xs

zsp n

n

The most important in solar geometry is to how to suggest coordinate system of the sun based on the requirement in question.

The most essential coordinate systems are (x,y,z) and (x’,y’,z’), all other coordinate system can be obtained from both system by rotating coordinates

Thanks for your attendance