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Estimate the solar radiation on a tilted surface for a collector field located in Quito (Ecuador) Jessica Mariño Salguero, Henry Delgado Betancourt EUETIB, Barcelona Industrial Engineering College POLYTECHNIC UNIVERSITY OF CATALONIA Barcelona, Spain [email protected], [email protected]

Abstract—this report estimates the solar radiation on a tilted surface for a collector field located in Quito (Ecuador) for September 15 and December 15. Applying different theoretical equations the instantaneous radiation from the available data is calculated. The calculations were performed to different slope values: 3º, 15º, 30º and 45º. The result of our analysis showed that the optimum slope is 3º for the not tracking surface along of the year. The values of mean daily radiation on a horizontal surface for the two days of study were contrasted with the sum of instantaneous radiation, obtaining a relative error of 1.2% for both cases. The quantity of instantaneous incident radiation on a tilted surface with optimum slope (β=3º) is 95.67% in September and 94.09% in December, of the total available radiation.

H0 Kt Go I Ib Id It It,b It,d It,ρ

extraterrestrial radiation [MJ/m2day] clear index [] solar constant 1367 [W/m2] total instantaneous radiation [MJ/m2h] beam instantaneous radiation [MJ/m2h] diffuse instantaneous radiation [MJ/m2h] total tilted surface instantaneous radiation [MJ/m2h] tilted beam instantaneous radiation [MJ/m2h] tilted diffuse instantaneous radiation [MJ/m2h] ground reflected instantaneous radiation [MJ/m2h] II.

Keywords—thermal solar collector; tilted surface; solar radiation.

I. θz ϒs Φ δ n ω ts LCT ωs ϒ N θ β βopt Lloc Lst D EOT H Hd Hb

NOMENCLATURE

zenith angle [º] solar azimuth angle [º] latitude angle [º] declination angle [º] number day of the year hour angle [º] solar time [h] local time [h] sunset hour angle [º] azimuth angle [º] day length [h] incidence angle of radiation [º] slope angle [º] optimum slope [º] local longitude [º] longitude of standard time zone meridian [º] daylight savings time equation of the time (min) mean daily radiation [MJ/m2day] mean daily diffuse radiation [MJ/m2day] mean daily bean radiation [MJ/m2day]

INTRODUCTION

The quantity of solar radiation incident on surface depends on its orientation and slope. The data available of radiation in the location is often the mean daily global radiation and its components, beam and diffuse radiation received on horizontal surface, but the data on tilted surfaces are not available and they are also estimated with different models from those measured on horizontal surfaces [1]. The solar thermal collectors are normally installed with constant slope. For this reason is necessary to know the quantity of total radiation incident on the tilted surface to design the solar systems and evaluate its performance.

III.

PROBLEM DEFINITION

The objectives of this study are: to estimate the solar radiation on a tilted surface for a collector field located in Quito (0.17ºW-78.48ºS), Ecuador, and to calculate the hourly distribution of both the total radiation on the horizontal and on the tilted solar collector field for September 15 and for December 15. The following aspects were considered:  The data available (Fig.1) of mean daily radiation corresponds to the radiation for the central day of the month H(15)

𝑡𝑎𝑛⁡(𝛾𝑠 ) =

18 17

𝑠𝑖𝑛 𝜔

𝛿 = 23.45𝑠𝑖𝑛 (360

284+𝑛 365

MJ/m2/day

16 15

(3) (4)

24

14

𝐿𝐶𝑇 = 𝑡𝑠 −

)

360º

𝜔1 = (𝑡𝑠 − 12)

13

(2)

𝑠𝑖𝑛 𝛿 𝑐𝑜𝑠 𝜔−𝑐𝑜𝑠 𝜙 𝑡𝑎𝑛 𝛿

𝐸𝑂𝑇 60

+

1 15

(𝐿𝑠𝑡 − 𝐿𝑙𝑜𝑐 ) + 𝐷

(5)

12

𝐸𝑂𝑇 = 0.258 𝑐𝑜𝑠 𝐵 – 7.416 𝑠𝑖𝑛 𝐵 −3.648 𝑐𝑜𝑠 2𝐵 ⁡– 9.228⁡𝑠𝑖𝑛⁡2𝐵

11 10

(6)

Jan Feb Mar Apri May Jun Jul Ago Sept Oct Nov Dec

𝐵 = (𝑛 − 1)

H 14,8 15,6 16,3 15,5 14,8 14,4 15,3 16,0 15,3 15,2 15,4 14,3

360

(7)

365

Fig. 1. Mean daily values in a month [MJ/m2day], for Quito (NASA, 2014)





𝑐𝑜𝑠𝜔𝑠 = − 𝑡𝑎𝑛𝜙 𝑡𝑎𝑛𝛿 The collector field is oriented with the optimum azimuth according to our location. Quito is in the equatorial line, the sun path is from east to west drawing a straight line in the middle of north and south hemisphere [3], we have chosen a north orientation (azimuth ϒ=180º) because the city is in the south hemisphere. Four slopes, β=3º, β=15º, β=30º and β=45º were analyzed for the determination of the optimum slope for a nontracking surface working the whole year. The fig. 2 shows approximately the incident radiation on a tilted surface in Quito according to the atmospheric science data center of NASA. Tilt 0

Tilt 15

kWh/m2/day

Thus, the day length is: 𝑁=

𝜋 24

𝜔𝑠

(9)

B. On an arbitrary surface, incidence angle of radiation: 𝜃 = 𝑓(𝜙, 𝛿, 𝜔, 𝛽, 𝛾) cos(𝜃) = sin(𝛿) sin(𝜙) cos(𝛽) − sin(𝛿) cos(𝜙) sin(𝛽) cos(𝛾) + ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡cos(𝛿) cos(𝜙) cos(𝛽) cos(𝜔) + cos(𝛿) sin(𝛽) sin(𝛾) + ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡cos⁡(𝛿)sin⁡(𝜙)sin⁡(𝛽)cos⁡(𝛾)cos⁡(𝜔)⁡⁡

(10)

The radiation received on a tilted surface depends on this solar incident angle.

Tilt 90

5

For solar noon ω=0 (ts=12) and a solar incident angle equal to zero, the optimum slope is obtained from the equation 10:

4 3 2

𝛽 = ⁡𝜙 − 𝛿

(11)

1

C. Radiation on a horizontal surface and its components:

0 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

𝐻 = 𝐻𝑏 + 𝐻𝑑

Month

𝐻

Fig. 2. Monthly averaged radiation incident on an equator-pointed tilted surface. Graph made with data from the NASA (NASA, 2014)



(8)

𝐻𝑑

𝐾𝑡 =

For the estimation of the solar radiation it is used the isotropic model. 𝐻0 = IV.

𝜋𝜔𝑠

METHODOLOGY

180

A. Location of the Sun Location of the Sun is given at any time by (θz, ϒs), which were evaluated with the following equations: cos θz = cos Φcos δ cos ω+ sin Φ sin δ

(1)

= 𝑓(𝐾𝑡 )

1

86400𝐺𝑆𝐶 𝜋

𝐻

𝑠𝑖𝑛𝜙 sin 𝛿)

(13) (14)

𝐻0

(1 + 0,033𝑐𝑜𝑠 (

(12)

360𝑛 365

)) (cos 𝜙 cos 𝛿 sin 𝜔𝑠 + (15)

Hour angle (ω) describes the earth rotation around its polar axis and is the angular distance between the meridian of the observer and the meridian whose plane contains the sun.[4].

TABLE I.

for⁡0.17⁡ < ⁡Kt ≤ 0.75: 𝐻𝑑

= 1.188 − 2.272𝐾𝑡 + 9.473𝐾𝑡2 − 21.865𝐾𝑡3 + 𝐻 ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡14.648𝐾𝑡4 ⁡⁡⁡⁡⁡

(16)

n

Sep, 15

258

Local time 12:08:56

Dec, 15

349

12:09:23 -23,34

TABLE II.

(18)

𝐻 𝐼𝑑

Ho

Hb

2

MJ/m /day

MJ/m2/day

37,12

0,413

0,75

11,519

3,817

35,58

0,403

0,77

10,984

3,344

INSTANTANEOUS RADIATION ON HORIZONTAL SURFACE

𝐼𝑏,𝑡 = 𝐼𝑏 𝑅𝑏

(25)

-

-

-

MJ/m h

MJ/m h

MJ/m2h

Sept, 15

0,660

0,423

0,142

0,131

2,17

1,51

0,66

Dec, 15

0,660

0,422

0,142

0,131

2,03

1,44

0,59

TABLE IV.

OPTIMUM MONTLY SLOPE FOR A TILTED SURFACE

Month

Jan

Feb

Mar

Apr

May

Jun

βopt (º)

21,10

13,12

2,65

9,58

18,96

23,48

Month

Jul

Ago

Sep

Oct

Nov

Dec

βopt (º)

21,68

13,95

2,38

9,43

18,98

23,17

2

)

(27)

The amount of direct radiation on a tilted surface, from the horizontal one, can be calculated by multiplying the direct horizontal irradiation by the ratio Rb. where θ is solar incidence angle on a tilted plane and θz is solar zenith angle. =

𝑐𝑜𝑠 𝜃 𝑐𝑜𝑠 𝜃𝑧

(28)

RESULTS AND DISCUTION

The table I, II and III show the different parameter evaluated for September 15 and December 15 at solar noon (ts=12:00) in Quito.

β=3

β=15

β=30

β=45

650 600 550 500 450 400 Jan

1+𝑐𝑜𝑠𝛽

Instantaneous radiacion (W/m2)

𝐼𝑑,𝑡 = 𝐼𝑑 (

2

The table IV shows the optimum monthly slope considering the day 15 of each month as a reference one. The slope angle rank could vary between 2º and 23º along the year in order to achieve the best performance of the installation.

(26)

2

Ib

-

β=0 1−𝑐𝑜𝑠𝛽

Id 2

Dec

(24)

I

Nov

𝐼𝑡 = 𝐼𝑏,𝑡 + 𝐼𝑑,𝑡 + 𝐼𝜌,𝑡

rd

Oct

E. Tilted surface irradiace model The total radiation on the tilted surface is the sum of beam, diffuse and ground reflection components. The calculation is performed using the following equation:

rt

Sept

(21)

b

Date

Agost

𝜋𝜔𝑠 cos 𝜔𝑠 180

(23)

V.

Hd

Jul

cos 𝜔−cos 𝜔𝑠

(20)

𝑏 = 0,6609 − 0,4767sin⁡(𝜔𝑠 − 60)

𝐼𝑏

11,79

%

Jun

𝜋𝜔𝑠 cos 𝜔𝑠 180

(22)

𝐼𝑏,𝑡

Hd/H

May

cos 𝜔−cos 𝜔𝑠 sin 𝜔𝑠 −

𝑎 = 0,409 + 0,5016sin⁡(𝜔𝑠 − 60)

𝑅𝑏 =

90,07

14,33

TABLE III.

24 sin 𝜔𝑠 −

𝐼𝜌,𝑡 = 𝐼𝜌

Kt

Apri

𝜋

89,99

(19)

𝐻𝑑

(𝑎 + 𝑏 cos 𝜔) 𝑟𝑑 =

0,00

15,34

Mar

𝜋

23,17

Dec, 15

a 24

0

N (hours) 11,78

ωs (º)

Sept, 15

Where: 𝑟𝑡 =

ϒs (º) 0,00

MJ/m /day

Feb

𝑟𝑑 =

θz (º) 2,38

2

(17)

𝐼

ω (º) 0

MEAN DAYLY RADIATION AND COMPONENTS

H Date

𝑟𝑡 =

δ (º) 2,22

Date

D. Instantaneous radiation The mean daily radiation is correlated with the instantaneous radiation of the following statistical relations: 𝐼 = 𝐼𝑏 + 𝐼𝑑

POSITION OF SUN AT SOLAR NOON IN QUITO

Months Fig. 3. Incident instantaneous radiation on different tilted surfaces along the year.

The fig. 3 shows the incident instantaneous solar radiation for solar noon of day 15 belonging to each month. We observe that the optimum slopes to our selected location, Quito - Ecuador, is β=0 and β=3, but we chose the slope of 3º because with the

β=0 the rainwater can become stagnant on the surface of flat solar thermal collector and damage the physical structure of the panel.

I

Id

Ib

700

INSTANTANEOUS RADIATION ON A TILTED SURFACE WITH β=3º Rb

Ib,t

Iρ,t

Id,t

IT

Date -

W/m2

W/m2

W/m2

W/m2

Sept, 15

1,00

184,87

0,08

418,59

603,54

Dec, 15

0,98

160,47

0,08

398,84

559,39

Radiation W/m2

600 TABLE V.

500 400 300 200 100

In the table V and figures 4 and 5, it is observed that the total instantaneous radiation is similar for the both cases, September and December. Also it is shown the variations using different slope values: the horizontal, optimum and the worst situation. β=0

β=3

0 5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

Hours Fig. 6. Instantaneous radiation and its components on horizontal surface in September 15

β=45 I

Id

Ib

2,50

500,00

Radiation W/m2

Radiation MJ/h.m2

600,00 2,00 1,50

400,00 300,00

1,00

200,00

0,50

100,00

0,00

0,00 5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

Hours

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

Hours

Fig. 4. Hourly radiation distribution in September 15. Fig. 7. Instantaneous radiation and its components on horizontal surface in December 15

β=0

β=3

β=45

Fig. 6 and fig.7 show that the beam radiation was significantly smaller than the diffuse one for the two cases. The peak of the instantaneous radiation was 603.59 W/m2 and 563.93 W/m2 for September and December respectively, at solar noon (ts=12:00h). As it is seen in the graph all the components get their maximum value at the same hour, 12:00, in the case of beam radiation, it gets 184.72 W/m2 in September and 164.50 W/m2 in December.

Radiation MJ/h.m2

2,50 2,00 1,50 1,00 0,50 0,00 5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

Hours Fig. 5. Hourly distribution in December 15.

It

Ib,t

Iρ,t

Id,t

It

600

Ib,t

Iρ,t

Id,t

600

Radiation W/m2

Radiation W/m2

500 500 400 300 200

400 300 200 100

100 0

0 5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

Hours

Hours

Fig. 8. Instantaneous radiation and its components on tilted surface (β=3º) in September 15

Fig. 10. Instantaneous radiation and its components on tilted surface (β=45º) in September 15

It It

Ib,t

Iρ,t

Ib,t

Iρ,t

Id,t

600

Id,t

600

Radiation W/m2

500

Radiation W/m2

500 400 300 200

400 300 200 100

100 0 5

0 5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

Hours Fig. 9. Instantaneous radiation and its components on tilted surface (β=3º) in December 15

The Fig. 8 and Fig.9 are the results of evaluating the instantaneous radiation and its components when the surface is tilted an angle of 3º. The maximum total radiation is 603.54 W/m2 in September 15 and 577.92 W/m2 in December 15. A small variation exists between the tilted and the horizontal surface. Besides, the reflected radiation is almost zero, so you can even neglect it. Also it is seen than the values of the components, if it is compared tilted and horizontal surfaces in the same month are almost the same because of that the total one has not varied.

6

7

8

9 10 11 12 13 14 15 16 17 18 19

Hours Fig. 11. Instantaneous radiation and its components on tilted surface (β=45º) in December 15

As it can be seen in the Figure 10 and 11, the total instantaneous radiation for a slope of 45º is 511.26 W/m2 in September and 511.96 W/m2 in December. Both values are smaller if they are compared with the cases of a tilted surface with a slope of 3º and the horizontal surface case. Furthermore the reflected radiation is a high value comparing with the optimal case, where this reflected radiation is almost zero.

TABLE VI.

INSTANTANEOUS RADIATION ON HORIZONTAL AND TILTED SURFACE ON SEPTEMBER 15 (n=258) Slope angles LCT

ts

ω

θz

β=0

β=3

β=45

I

It

It

2

2

(hour)

(hour)

º

º

MJ/m h

MJ/m h

MJ/m2h

5,68

6

-90

90,01

0,00

0,00

0,00

6,68

7

-75

75,02

0,40

0,39

0,35

7,68

8

-60

60,03

0,87

0,81

0,72

8,68

9

-45

45,05

1,36

1,27

1,11

9,68

10

-30

30,09

1,78

1,72

1,47

10,68

11

-15

15,18

2,07

2,05

1,74

11,68

12

0

2,38

2,17

2,17

1,84

12,68

13

15

15,18

2,07

2,05

1,74

13,68

14

30

30,09

1,78

1,72

1,47

14,68

15

45

45,05

1,36

1,27

1,11

15,68

16

60

60,03

0,87

0,81

0,72

16,68

17

75

75,02

0,40

0,39

0,35

17,68

18

90

90,01

Total

0,00

0,00

0,00

15,15

14,67

12,60

TABLE VII.

INSTANTANEOUS RADIATION ON HORIZONTAL AND TILTED SURFACE ON DECEMBER 15 (n=349) Slope angles LCT

ts

ω

θz

β=0

β=3

β=45

I

It

It

(hour)

(hour)

º

º

MJ/m h

MJ/m h

MJ/m2h

5,69

6

-90

89,93

0,00

0,00

0,00

6,69

7

-75

76,18

0,37

0,37

0,36

7,69

8

-60

62,60

0,82

0,83

0,73

8,69

9

-45

49,43

1,27

1,29

1,11

9,69

10

-30

37,22

1,67

1,70

1,47

10,69

11

-15

27,37

1,93

1,98

1,74

11,69

12

0

23,17

2,03

2,08

1,84

12,69

13

15

27,37

1,93

1,98

1,74

13,69

14

30

37,22

1,67

1,70

1,47

14,69

15

45

49,43

1,27

1,29

1,11

15,69

16

60

62,60

0,82

0,83

0,73

16,69

17

75

76,18

0,37

0,37

0,36

17,70

18

90

89,93

0,00

0,00

0,00

Total

2

14,16

2

14,43

12,67

The two tables VI and VII show the values of total incident radiation on a horizontal and tilted surface by hour. The sum of all the instantaneous radiations in the case of β=0º, in September was 15,15MJ/m2h and in December was 14.16 MJ/m2h, which have a relative error of 1.2% compared with raw data used. When β=3º the mean daily radiations were 14.67 MJ/m2h for September and 14.43 MJ/m2h for December. These values show that the 95.67% and 94.09% of total available radiation fall upon the tilted surface. The worst case was β=45º, because the mean incident daily radiation is 83.2% of the total radiation in the two days. I.

CONCLUSSIONS

The values of mean daily radiation for September 15 and December 15 (Table II) are almost similar due to the special geographical characteristics of the Ecuadorian location where the climatic conditions remain relatively constant along the year without extreme changes. In a solar collector using the horizontal and optimum slope, the variation of the radiation is slightly small, receiving the maximum of the incident energy and at the same time working in the best conditions and efficiency for this particular case. The clear index for this location is about 0.4. For this reason the diffuse radiation is more important than the beam one and a flat plate collector is a very good solution to take advantage of the situation. The mean daily radiation of the raw data used is in the same order of magnitude of the results of the instantaneous radiation integration along the time on a horizontal surface.

REFERENCES [1] [2]

[3] [4]

Burlon S, Bivona S, Leone C. Instantaneous hourly and daily NASA. (1 de 10 de 2014). Atmospheric science data center. Obtenido de NASA Surface meteorology and Solar Energy - Available Tables: file:///D:/1.-MASTER%20EN%20ENERGIA%20TERMICA/3SEMESTRE/2-SOLAR%20THERMAL/Ecuadorclimatolog%EDa/NASA%20Surface%20meteorology%20and%20Solar %20Energy%20-%20QUITO.htm Rodriguez, I. (2014). Topic 1. Introduction. Solar Energy from Sun., (pág. 43). Barcelona. William B. Stine, M. G. (2001). Power From The Sun. Obtenido de 3. The Sun’s Position: http://www.powerfromthesun.net/Book/chapter03/ chapter03.html.