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   Introduction  

Vectorial relationships for the geometrical positions of the sun shall be used for quantitative analysis of the effects of solar radiation on building surfaces in this section.

1.2.1 Geometrical Position of the Sun

Solar Time

This is the time based on the physical angular motion of the sun.  Solar noon is the time when the altitude angle of the sun reaches its peak.  Solar time can be calculated from 

ts     =   tl - 4(Lgs - Lgl)  + Eqt      (1.2.1) 

where

ts   = solar time,
tl   = local standard time,
Lgs= standard local longitude,
Lgl = actual longitude, and
Eqt = equation of time (min).

Note that the negative sign for the second term is for eastern longitude. The last two terms in (1.2.1) are in minutes.  The equation of time is actually a correction term for the solar time in reference to the distortion of the sun's position due to the earth's rotation on a wobbling axis.  This is computable from 

Eqt   =   9.87 sin 2B - 7.53 cos B - 1.5 sin B,          min,   (1.2.2)
B     =   360o (jd-81) / 364, 1< jd < 365,   

where jd is Julian date, which is the day number of the year. 

The value for Eqt varies within the limits of +/-15 minutes, and changes with respect to the day of the year.  Figure 1.2.1 shows a graph of Eqt. 


Figure 1.2.1  A graph showing variation of Eqt with different days of the year.




Example 1.2.1 

For Bangkok,   local longitude Lgl  is 100.5° E

                        standard longitude Lgs is 105° E

At 11:00, local time, on 23 May, the solar time can be calculated as in the followings. 

The Julian date jd is obtained as 31 + 28 + 31 + 30 + 23 = 143,

then             B     =   61.32° ,

                   Eqt   =   3.382 minutes     =   3 minutes 22.9 seconds, and

solar time     (ts) =  11:00 - 4(105° - 100.5°) minutes + 3 minutes 22.9 seconds

                          =  10:45:22.9. 

Mathematical Relationships of the Position of the Sun 

The solar altitude angle and the solar azimuth angle can be obtained from a

sinαs    = sin Lt . sinδ + cos Lt. cosδ . cosω   (1.2.3)
sinγs     = cosδ . sinω / cosαs (1.2.4)

where   Lt    =   the latitude of the considered location (For Bangkok this is 13.7° N),

            δ     =   declination angle, the value of which is between +/-23.45°,

            ω     =   solar hour angle, the physical angular position of the sun, and

                   =   π(ts - 12)/12, radian.                                                         (1.2.5)

The declination angle is the angle which the ray of the sun to the center of the earth makes with the plane of the equator.  The declination angle for a given date jd can be obtained from           

                   δ     =   23.45 sin[360(284+jd) / 365]                                   (1.2.6)

Example 1.2.2 

The solar altitude and azimuth angles at 11:00 hour on 23 May in Bangkok can be calculated as in the followings.

            The solar time is obtained in Example 1.2.1 as ts = 10:45:22.9 = 10.7563 (decimal).  The solar hour angle and the declination angle are obtained as 

                   ω    =   π(10.7563 -12)/12,

                          =   -0.3256, radian,

                          =   -18.66° and

       δ     =   23.45 sin[360 (284+143) / 365],

                          =   20.54°. 

The solar altitude angle and azimuth angle are then obtained as  

αs
= sin-1[sin(13.7°) . sin(20.54°) + cos(13.7°) . cos(20.54°).cos(-18.66°)], 
  =   70.9° and
γs =   sin-1[cos(20.54°) . sin(-18.66°) / cos(70.9°)]
  =   -66.3° or -113.7°. 

For May 23, the sun is due north for our location in Bangkok.  Therefore, the angle -113.7° is the correct one.

The mathematical relationships we have used in this example have been used in the calculation of the solar altitude and azimuth angles for the four reference days and the results are as shown in Table 1.1.1. 

Sun-Path Diagram on Rectangular Coordinate 

The relationships (1.2.3) and (1.2.4) can be used to calculate the position of the sun projected onto earth.  If we follow this projection through a day, the path forms a line.

The variable of solar hour angle ω takes on negative value for the morning period, so the value of sinγs in equation (1.2.4) will always be negative for the morning period.  However, the angle γs as found from inverting sinγs can take a negative value larger than -90° or its complement of -180° as illustrated in Example 1.2.2.

The position of the projection of the sun on a rectangular coordinate can be obtained by considering a unit vector in the direction of the sun.  The distance along the x-axis (corresponding to the east-west direction) and along the y-axis (corresponding to the north-south direction) can be obtained from 

x
=   cosαs . sinγs               
(1.2.7)
  =   70.9° and
and y =   cosαs . cosγs.  (1.2.8)

These relationships can be used to draw the path the projection of the sun makes as the sun travels in the sky from sunrise to sunset.  There are 3 distinct patterns of the resulting sun-path diagram.  The first of these corresponds to the situation when the declination angle is larger than the latitude angle of the location (and daylength is longer than 12 hours).  The second case corresponds to the situation when the declination angle is smaller than the latitude angle (and daylength is less than 12 hours).  The third case corresponds to the situation when the declination angle is smaller than the latitude angle but the daylength is longer than 12 hours.

Daylength corresponds to the total time between sunrise and sunset times.  The sunrise and sunset times can be calculated from Equation (1.2.3) by setting as = 0.  In the event, the value of the solar hour angle is obtained as 

ωs
=   cos-1(-tan Lt . tan δ).           
(1.2.9)

The negative value of ωs corresponds to sunrise time and its positive value corresponds to sunset time.  The time between sunrise and sunset is the daylength. 

Sun-path Diagram for Case 1. Taking Bangkok (with latitude angle 13.7° N) as the reference location, the sun-path diagrams for 27 April and 23 May are shown in Figure 1.2.2 a) and b) respectively.  The day 27 April is the day when solar altitude angle reaches 90° for Bangkok.



Figure 1.2.2    Sun-path diagrams for 27 April and 23 May, when the solar declination angle is larger than the latitude angle of the location.



a) 27 April


b) 23 May



In these cases, the solar azimuth angles are smaller than -90° (these values are always between -90° and -270°). The sun-path is always in the direction north of the location. The period corresponding to this case is 27 April to 15 August. 

Sun-path Diagram for Case 2. Using Bangkok as reference, the sun-path diagrams for 1 January and for 22 March are as shown in Figure 1.2.3 a) and b) respectively.  The date 22 March corresponds to when the sun crosses the equator and daylength just equal 12 hours.

In these cases, the azimuth angles take on values between -90° and +90°.  The sun-path is always south of the location.  The period corresponding to this case is 21 September through December and January to 22 March.

Figure 1.2.3   Sun-path diagram for 1 January and 22 March, when the solar declination angles are smaller than the latitude angle



a) 1 January


b) 22 March


Sun-path Diagrams for Case 3. For Bangkok, the sun-path diagrams for 20 April and 5 September are shown in Figure 1.2.4 a) and b) respectively.  The period between 22 March to 27 April and that between 15 August and 21 September fall into this category.

Figure 1.2.4    Sun-path diagram for 20 April and 5 September.



a) 20 April


b) 5 September



During these two periods the daylength is longer than 12 hours.  The sun rises in the northeastern direction and sets in the northwestern direction.  The sun-path crosses the x-axis, during the day.  The azimuth angle of the sun is less than -90° at sunrise.  As the day progresses, the sun-path traverses the x-axis when the solar hour angle ω reaches the value

ω
= ±cos-1 (tanδ / tanLt)            
(1.2.10)

Beyond this time, the solar azimuth angle will be between 0° and -90° or between 0° and 90°. 

1.2.2 Solar Radiation on Planes 

Calculation of Shading by Shading Devices 

The mathematical relationships of the geometrical position of the sun presented above have been used to calculate the shading effects of shading devices.  Consider the situation in   Figure 1.2.5.

Figure 1.2.5    A window and a long overhang.




Figure 1.2.5    A window and a long overhang.(continued)





On a window of height hw, there is a sloped overhang (here assumed to be much longer than the width of the window) of width W over the window.  The slope angle of the overhang is β1.  The projection of the edge of the overhang is at a distance h from the upper edge of the window.  The shade of the edge of overhang is at a distance hs from the projection of the edge of the window on the plane of the wall.  The angle β2 is the angle of projection of the ray of the sun on a plane perpendicular to the plane of the window.

The shaded portion of the window is equal to hs-h.  This shaded portion can be calculated from the given geometry as follows.  From the figure we obtain        

hs
=   W cos β1 . tan β2,         
(1.2.11)

then the proportion R of the shaded area to the total area of the window is given as 

R
=   (hs - h) / hw       
  =   (W cosβ1 . tanβ2 - h) / hw. (1.2.12)

The angle b2 is related to the position of the sun and the direction the window faces.  If these are given, then the ratio R can be calculated. 

Example 1.2.3 

Given the configuration of the window and its overhang in Figure 4.2.5.  Suppose the following values are given: 

                   W    =   0.5 m,

                   h     =   0.25 m,

                   β1   =   15°, and

                   hw   =   1 m. 

If the window faces East with an azimuth angle γp = -90o, the extent of the shade at 10:00 on  22 May can be calculated as follows. 

The solar time, and the angular position of the sun can be calculated and given as: 

                   ts     =   9.758 hr,

                   αs   =   57.22°,     

                   γs    =   -106.40° 

The azimuth angle of the sun differs from that of the window by 

                   (-90o) - (-106.40°)    =    16.40°, 

The relationship between these angles is illustrated in Figure 4.2.6, in which a coordinate is also erected.

Figure 1.2.6    Details of the shading configuration of Example 1.2.3.




The angle β2 is related to the other quantities as follow 

                   tanβ2    =   hs / |bc|

                               =   |cd| / (W cosβ1) 

But              |cd|       =   |ab|

                               =   |oa| tanαs, and

                  |oa| cos(γs - γp) =   |ad|

                                           =   |bc|. 

Therefore    |cd|       =   |bc| tanαs / cos(γs - γp),

                   tanβ2    =   tanas / cos(γs - γp), and

                   hs         =   W cosβ1 . tanαs / cos(γs - γp). 

In the above |bc| is used to mean the length or distance between points b and c in Figure 1.2.6, etc. 

For this example, hs is evaluated as: 

                   hs    =   0.5 (cos 15°) (tan 57.22°) / (cos 16.4°)

                          =   0.782 m. 

And the proportion of shaded area R is: 

                   R     =   (0.782-0.25)/1  =   0.532. 

For a vertical shading device, similar use of the relationships (1.2.11) and (1.2.12) can be made. Such applications have been made in calculation of the shading coefficients of shading devices as part of the evaluation of the thermal performance of a building envelope. 

In the technical part of the Ministerial Regulation for Energy Conservation in Buildings (which is a by-law of the Energy Conservation Promotion Act) in Thailand, this method is used to calculate the shading effects of standard shading devices.  Some building energy simulation programs also utilize these relationships. 

Reference Solar Radiation 

Chapter 3 introduces the beam and the diffuse components of solar radiation reaching earth. In this and the other chapters, there will be occasions to consider practical aspects of solar radiation.  It is appropriate to introduce a set of solar radiation data, prepared from a record of solar radiation measurements taken originally from a station in the King Mongkuts' Institute of Technology Thonburi, Bangkok as in table 1.2.1.

Table 1.2.1      Reference solar radiation data for 4 reference days.

Time(hour)  
Solar radiation (W/m2)
  Mar 21
 June 21
Sept 21
Dec 21
Beam Diffuse Beam Diffuse Beam Diffuse Beam Diffuse
  7 202.1   81.1   82.0   87.8 116.2   49.9 146.7   45.6
  8 266.8 156.2 127.3 158.0 184.3 112.0 265.0 101.0
  9 326.6 226.0 197.1 215.0 239.0 163.8 352.0 145.0
10 380.6 270.7 284.5 266.0 278.0 207.0 420.0 173.0
11 414.0 293.5 349.1 305.0 305.0 240.9 472.3 188.0
12 424.9 303.4 367.5 318.0 320.0 258.0 489.7 194.0
13 414.0 293.5 349.1 305.0 305.0 240.9 472.3 188.0
14 380.6 270.7 284.5 266.0 278.0 207.0 420.0 173.0
15 326.6 226.0 197.1 215.0 239.0 163.8 352.0 145.0
16 266.8 156.2 127.3 158.0 184.3 112.0 265.0 101.0
17 202.1   81.1   82.0  87.8 116.2   49.9 146.7  45.6

Figure 1.2.7    Time plots of the solar radiation for the four reference days.

The following figure shows time plots of the radiation for reference days.

Solar Radiation on Inclined Planes 

Consider the situation in Figure 1.2.8, where a plane is inclined with an angle β with respect to the horizontal plane.         

The projection of the normal of the plane makes an azimuth angle γp with respect to the reference southern direction.  For this unshaded plane, the total solar radiation on the plane Eetq  is given as 

(1.2.13)

where  

EeS 

=   the beam normal component of the solar radiation, in the direction of the sun, W/m2
cosθ
 =   cosine of angle between the plane normal and the solar vector,
Eed 
 =   diffuse component of the solar radiation on a horizontal plane, W/m2,
Eeg
 =   total solar radiation on a horizontal plane (global radiation), W/m2, and
pg
 =   reflectivity of the horizontal surface of the ground around the inclined plane.

The angle θ can be obtained from consideration similar to that used in Example 1.2.3.  It is convenient to calculate cosθ  from the following relationship. 

cosθ

=   sinδ . sin Lt . cosβ - sinδ . cos Lt . sinβ . cosγp

+ cosδ . cos Lt . cosβ . cosw + cosδ . sin Lt. sinb . cosγp . cosω
+ cosδ. sinβ . sinγp . sinω                                                           (1.2.14)

Applying (1.2.13) and (1.2.14) using the data in Table 1.2.1 to calculate total radiation for a horizontal plane, and for 4 vertical planes in each cardinal directions, the results are shown in Figure 1.2.9.  For vertical plane β = 90°, cosb  = 0, sinβ  = 1, and (1.2.14) simplifies to

 

           cosθ  =   -sinδ . cos Lt. cosγp + cosδ . sin Lt. cosγp . cosw + cosδ . sinγp . sinω.

The average solar radiation on each surface in Figure 1.2.9 for the four reference days can be taken to approximate the annual average solar radiation of each surface.  The results are shown in Table 1.2.2.


Figure 1.2.8    A plane with inclination angle β and azimuth angle γp



 


Figure 1.2.9    Total radiation on horizontal and vertical planes.


a) Total radiation on horizontal plane


b) Total radiation on vertical plane - North

c) Total radiation on vertical plane - South

d) Total radiation on vertical plane - East

e) Total radiation on vertical plane - West
 


Table 1.2.2 Approximate annual average solar radiation on each surface of Figure 4.2.9.

 

Quantity

 

Surface

North

South

East

West

Horizontal

Annual average radiation (W/m2 )

141.47

204.6

197.98

197.98

383.4



Solar Radiation on Buildings

The size of the total solar radiation on all the surfaces of buildings of different shapes is of interest. Here, buildings of rectangular shape will be considered. Using the approximate annual average solar radiation on different surfaces in Table 1.2.2, Table 1.2.3 exhibits the calculated-annual-average solar radiation on the buildings illustrated. Tall building seems to receive the least solar radiation per unit surface area and per unit volume.


Table 1.2.3 Annual average solar radiation on rectangular building.

 

Building shape

Building type

Base dimension

Varied dimension

Total radiation on all surface

per unit surface area (W/m2)

per unit building volume
(W/m3 )

Cubicle

-

-

225

1125

 

Tall, Square

 

Width and length

 

Height, H 5m
10m

 


195
190

 


819
780

Long, along
N-S

N&S walls

Length, L 5m
10m


250
254


849
814


Long, along
E-W


E&W walls


Length, L
5m
10m



238
240



809
769