Numerical on Calculation of Incident Radiation

1. Calculations of Incident Radiation on
different walls and roof of a Building
Working Examples

2. Example -1: Calculate the total solar radiation incident on a south facing vertical
wall (i.e. β = 900) at solar noon (i.e. ω = 00)on June 21st and Dec. 21st for ɸ = 230 and
the reflectivity of the ground is 0.60.
Given:
Latitude angle (ɸ) = 230
The hour angle (ω) = 00 (at solar noon).
The declination (δ) = +23.50 for June 21st .
= - 23.50 for Dec 21st .
Surface azimuth angle (γ) is = 00 (south facing)
Tilt angle (β) is = 900 (vertical)
Reflectivity (R) is = 0.60
Angle of incidence (θi) is given by:
NOTE: At horizontal surface θi = θz .
It means at horizontal surface, the
angle of incidence the zenith angle
are equal.

3. Example -2: Calculate the total solar radiation incident on a south facing vertical wall at 6
am, 7 am, 8 am, 9 am, 10 am, 11 am, 12:00 noon, 1 pm, 2 pm, 3 pm, 4 pm, 5 pm, 6 pm on
June 21st and Dec. 21st for ɸ = 28.70410 and the reflectivity of the ground is 0.60.
Angle of incidence (θi) is given by:
S. No. Time of
the day
Angle of
incidence (θi)
1. 6 am 110.47
2. 7 am 103.63
3. 8 am 97.44
4. 9 am 92.19
5. 10 am 88.18
6. 11 am 85.66
7. 12 Noon 84.796
S. No. Time of
the day
Angle of incidence
(θi)
1. 1 pm 85.66
2. 2 pm 88.18
3. 3 pm 92.19
4. 4 pm 97.44
5. 5 pm 103.64
6. 6 pm 110.47
June 21st

4. S. No. Time of
the day
Angle of
incidence (θi)
1. 6 am 69.525
2. 7 am 62.368
3. 8 am 55.256
4. 9 am 48.606
5. 10 am 43.012
6. 11 am 39.176
7. 12 Noon 37.794
S. No. Time of
the day
Angle of incidence
(θi)
1. 1 pm 39.176
2. 2 pm 43.012
3. 3 pm 48.606
4. 4 pm 55.249
5. 5 pm 62.368
6. 6 pm 69.53
Dec. 21st
I = IDIN Cos θ + IDθ+ IRθ

5. S.no Time ω Sin αs αs
1 6 am -90 0.1915 11.04
2 7 am -75 0.4000 23.58
3 8 am -60 0.5937 36.42
4 9 am -45 0.7603 49.49
5 10 am -30 0.8881 62.64
6 11 am -15 0.9684 75.56
7 12 Noon 0 0.9959 84.81
8 1 pm 15 0.9685 75.56
9 2 pm 30 0.8881 62.64
10 3 pm 45 0.7603 49.49
11 4 pm 60 0.5937 36.42
12 5 pm 75 0.4000 23.58
13 6 pm 90 0.1915 11.04
S.no Time ω Sin αs αs
1 6 am -90 -0.1915 -11.04
2 7 am -75 0.0167 0.9552
3 8 am -60 0.2107 12.16
4 9 am -45 0.3773 22.17
5 10 am -30 0.5050 30.33
6 11 am -15 0.5854 35.83
7 12 Noon 0 0.6129 37.80
8 1 pm 15 0.5854 35.83
9 2 pm 30 0.5050 30.33
10 3 pm 45 0.3773 22.17
11 4 pm 60 0.2107 12.16
12 5 pm 75 0.0167 0.9552
13 6 pm 90 -0.1915 -11.04
June 21st (+23.50 ) Dec. 21st (-23.50 )

6.  Total solar Irradiation
• IDIN = A exp (−
𝐵
Sin αs
)
{ A=1230 for Dec & Jan, and 1080 for mid sem whereas
B= 0.14 and 0.21 for winter and Summer respectively}
• IDθ = C*IDIN * FWS (C = 0.135 & 0.058 for Summer & Winter respectively
• IRθ = (IDIN + IRθ ) ρgFWG
• FWS =
1+cos β
2
• FWG =
1−cos β
2
IHθ = IDIN Cosθ + IDθ+ IRθ

7. S. No. Time Sin αs A B IDIN IDθ IDIN + IDθ IRθ
1 6 am 0.1915 1080 0.21 360.73 24.35 385.08 115.53
2 7 am 0.4000 1080 0.21 638.88 43.13 682.01 204.60
3 8 am 0.5937 1080 0.21 758.24 51.18 809.42 242.83
4 9 am 0.7603 1080 0.21 819.35 55.31 874.66 262.40
5 10 am 0.8881 1080 0.21 852.57 57.55 910.12 273.04
6 11 am 0.9684 1080 0.21 869.45 58.69 928.14 278.44
7 12 Noon 0.9959 1080 0.21 874.68 59.04 933.72 280.12
8 1 pm 0.9685 1080 0.21 869.45 58.69 928.14 278.45
9 2 pm 0.8881 1080 0.21 852.57 57.55 910.12 273.04
10 3 pm 0.7603 1080 0.21 819.35 55.31 874.66 262.40
11 4 pm 0.5937 1080 0.21 758.24 51.18 809.42 242.83
12 5 pm 0.4000 1080 0.21 638.88 43.13 682.01 204.60
13 6 pm 0.1915 1080 0.21 360.73 24.35 385.08 115.53
June 21st (+23.50 )

8. S. No. Time Sin αs A B IDIN IDθ IDIN + IDθ IRθ
1 6 am -0.1915 1230 0.14 2555.08 766.524 3321.604 996.48
2 7 am 0.0167 1230 0.14 0.2812 0.08434 0.36554 0.1097
3 8 am 0.2107 1230 0.14 632.90 189.87 822.77 246.83
4 9 am 0.3773 1230 0.14 848.71 254.62 1103.33 331
5 10 am 0.5050 1230 0.14 932.19 279.657 1211.847 363.55
6 11 am 0.5854 1230 0.14 968.372 290.512 1258.884 377.67
7 12 Noon 0.6129 1230 0.14 978.819 293.65 1272.469 381.74
8 1 pm 0.5854 1230 0.14 968.372 290.51 1258.882 377.67
9 2 pm 0.5050 1230 0.14 932.19 279.657 1211.847 365.55
10 3 pm 0.3773 1230 0.14 848.71 254.62 1103.33 331
11 4 pm 0.2107 1230 0.14 632.90 189.87 822.77 246.83
12 5 pm 0.0167 1230 0.14 0.2812 0.08434 0.36554 0.1097
13 6 pm -0.1915 1230 0.14 2555.08 766.524 3321.604 996.48
Dec 21st (- 23.50 )

9. For June 21st IHθ = IDIN Cos θi + IDθ+ IRθ
S. no Time of
the day
Angle of
incidence
(Cos θi)
IDIN IDIN Cos θi IDθ IRθ IHθ
1 6 am -0.3497 360.73 -126.15 24.35 115.53 13.73
2 7 am -0.2357 638.88 -150.58 43.13 204.60 97.15
3 8 am -0.1295 758.24 -98.192 51.18 242.83 195.818
4 9 am -0.03821 819.35 -31.307 55.31 262.40 286.403
5 10 am 0.03175 852.57 27.069 57.55 273.04 357.659
6 11 am 0.0757 869.45 65.8174 58.69 278.44 402.9474
7 12 Noon 0.0907 874.68 79.334 59.04 280.12 418.494
8 1 pm 0.0757 869.45 65.8174 58.69 278.45 402.9574
9 2 pm 0.0318 852.57 27.069 57.55 273.04 357.659
10 3 pm -0.0382 819.35 -31.307 55.31 262.40 286.403
11 4 pm -0.1295 758.24 -98.192 51.18 242.83 195.818
12 5 pm -0.2357 638.88 -150.58 43.13 204.60 97.15
13 6 pm -0.3497 360.73 -126.15 24.35 115.53 13.73

10. For Dec 21st IHθ = IDIN Cos θi + IDθ+ IRθ
S. no Time of
the day
Angle of
incidence
(Cos θi)
IDIN IDIN Cos θi IDθ IRθ IHθ
1 6 am 0.3498 2555.08 893.767 766.524 996.48 2656.771
2 7 am 0.4637 0.2812 0.1303 0.08434 0.1097 0.32434
3 8 am 0.5700 632.90 360.753 189.87 246.83 797.453
4 9 am 0.6612 848.71 561.167 254.62 331 1146.787
5 10 am 0.7312 932.19 681.617 279.657 363.55 1324.824
6 11 am 0.7752 968.372 750.682 290.512 377.67 1418.864
7 12 Noon 0.7902 978.819 773.463 293.65 381.74 1448.853
8 1 pm 0.7752 968.372 750.682 290.51 377.67 1418.862
9 2 pm 0.7312 932.19 681.617 279.657 365.55 1326.824
10 3 pm 0.6612 848.71 561.167 254.62 331 1146.787
11 4 pm 0.5700 632.90 360.753 189.87 246.83 797.453
12 5 pm 0.4638 0.2812 0.1303 0.08434 0.1097 0.32434
13 6 pm 0.3498 2555.08 893.767 766.524 996.48 2656.771

11. Solution:
Altitude angle at solar noon (αs = αmax) = sin-1{cos ɸ cos ω cos δ + sin ɸ sin δ}
Also at solar noon give αs = αmax = pi/2-(ɸ- δ)
At solar noon, Solar azimuth angle (γs) = cos -1[
{sin ɸ cos ω cos δ − sin ɸ sin δ}
cos αs
]
Similarly at sun-set and sun-rise, the Altitude angle is minimum αs = αmin = 0, then:
0= sin-1{cos ɸ cos ω cos δ + sin ɸ sin δ} which gives the value of ω
0 = cos ɸ cos ω cos δ + sin ɸ sin δ i.e. cos ω = - (tan ɸ tan δ) and hence,
ω = cos-1 {- tan ɸ tan δ}
NOTE: In some test books the hour angle (ω) at which αs = 0, is denoted by ωo, which
ultimately gives the total sun-shine hours in a day at a year any location and is very
useful to estimate the heating/cooling requirement of the buildings.

12. Exmple -1: Calculate the total solar radiation incident on a south facing wall
(vertical surface) at solar noon on June 21st and Dec. 21st for ɸ = 230 and the
reflectivity of the ground is 0.60.
Given:
Latitude angle (ɸ) = 230
The hour angle (ω) = 00 (at solar noon).
The declination (δ) = +23.50 for June 21st .
= - 23.50 for Dec 21st .
Surface azimuth angle (γ) is = 00 (south facing)
Tilt angle (β) is = 900 (vertical)
Reflectivity (R) is = 0.60
Angle of incidence (θi) is given by:
NOTE: At horizontal surface θi = θz .
It means at horizontal surface the
angle of incidence is equal to the
zenith angle

13. Solar Azimuth Angle
Solar azimuth angle (gs ) is given by:
gs = cos -1 {(cos ɸ sin δ - sin ɸ cos δ cos ω)/(cos αs )}
= sin -1 {cos δ cos ω)/(cos αs )}
At Solar Noon (ω = 0), therefore,
gs = 1800 for ɸ > δ and gs = 00 for ɸ < δ
gs = is not defined from the above equation for solar noon.
The angle on a horizontal
plane between the line due
to south and the projection
of the sun’s rays on the
horizontal place

14. Angle of incidence
Angle of incidence (θi) is given by:
At horizontal surface θi = θz
It means at horizontal surface the
angle of incidence is equal to zenith
angle
αs
θi = θmax = θz = π/2 - αs

15. Surface Azimuth Angle (γ): It is angle on a horizontal plane between the line
due south and projection of the normal to the surface on the horizontal plane.
γ
Surface Azimuth Angle

17. Solution:
Altitude angle at solar noon (αs = αmax) is = sin-1{cos ɸ cos ω cos δ + sin ɸ sin δ}
{
𝝅
𝟐
− mod(ɸ-δ)} = 89.5
At solar noon, Solar azimuth angle (γs) is = 0 as ɸ < δ

19. Notations
Latitude angle (ɸ) = 230
The hour angle (ω) = 00 (at solar noon).
The declination (δ) = +23.50 for June 21st .
= - 23.50 for Dec 21st .
Wall azimuth angle (γ) is = 00 (south facing)
Tilt angle (β) is = 900 (vertical)
Reflectivity (R) is = 0.60