Estimate the solar radiation on a tilted surface for a

collector field located in Quito (Ecuador)

Jessica Mario Salguero, Henry Delgado Betancourt

EUETIB, Barcelona Industrial Engineering College

POLYTECHNIC UNIVERSITY OF CATALONIA

Barcelona, Spain

jezbel1989_18@hotmail.com, henry.dbetancourt@gmail.com

Abstractthis 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: 3, 15, 30 and 45. The result of our analysis showed that

the optimum slope is 3 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 (=3) is 95.67% in September and 94.09% in December, of the total available radiation.

Keywordsthermal solar collector; tilted surface; solar radiation.

I. NOMENCLATURE

z zenith angle [] s solar azimuth angle [] latitude angle [] declination angle [] n number day of the year

hour angle [] ts solar time [h]

LCT local time [h]

s sunset hour angle [] azimuth angle [] N day length [h]

incidence angle of radiation [] slope angle [] opt optimum slope [] Lloc local longitude []

Lst longitude of standard time zone meridian []

D daylight savings time

EOT equation of the time (min)

H mean daily radiation [MJ/m2day]

Hd mean daily diffuse radiation [MJ/m2day]

Hb mean daily bean radiation [MJ/m2day]

H0 extraterrestrial radiation [MJ/m2day]

Kt clear index []

Go solar constant 1367 [W/m2]

I total instantaneous radiation [MJ/m2h]

Ib beam instantaneous radiation [MJ/m2h]

Id diffuse instantaneous radiation [MJ/m2h]

It total tilted surface instantaneous radiation [MJ/m2h]

It,b tilted beam instantaneous radiation [MJ/m2h]

It,d tilted diffuse instantaneous radiation [MJ/m2h]

It, ground reflected instantaneous radiation [MJ/m2h]

II. 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 (0.17W-78.48S), 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)

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

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 =180) because the city is in the south hemisphere.

Four slopes, =3, =15, =30 and =45 were analyzed for the determination of the optimum slope for a non-

tracking 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.

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

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

IV. METHODOLOGY

A. Location of the Sun

Location of the Sun is given at any time by (z, s), which were evaluated with the following equations:

cos z = cos cos cos + sin sin (1)

() =

(2)

= 23.45 (360284+

365) (3)

1 = ( 12)360

24 (4)

=

60+

1

15( ) + (5)

= 0.258 7.416

3.648 2 9.2282 (6)

= ( 1)360

365 (7)

= (8)

Thus, the day length is:

=

24 (9)

B. On an arbitrary surface, incidence angle of radiation:

= (, , , , ) cos() = sin() sin() cos() sin() cos() sin() cos() +

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

cos()sin()sin()cos()cos() (10)

The radiation received on a tilted surface depends on this solar

incident angle.

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

= (11)

C. Radiation on a horizontal surface and its components:

= + (12)

= () (13)

=

0 (14)

0 =86400

(1 + 0,033 (

360

365)) (cos cos sin +

180 sin ) (15)

1 Hour angle () 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].

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

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

10

11

12

13

14

15

16

17

18

MJ/

m2

/day

0

1

2

3

4

5

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

kWh

/m2 /

day

Month

Tilt 0 Tilt 15 Tilt 90

for0.17 < Kt 0.75:

= 1.188 2.272 + 9.473

2 21.8653 +

14.6484 (16)

D. Instantaneous radiation

The mean daily radiation is correlated with the

instantaneous radiation of the following statistical relations:

= + (17)

=

(18)

=

(19)

Where:

=

24( + cos)

coscos

sin180

cos (20)

=

24

coscos

sin180

cos (21)

= 0,409 + 0,5016sin( 60) (22)

= 0,6609 0,4767sin( 60) (23)

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:

= , + , + , (24)

, = (25)

, = 1

2 (26)

, = (1+

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 is solar incidence angle on a tilted plane and z is solar zenith angle.

=,

=

(28)

V. 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.

TABLE I. POSITION OF SUN AT SOLAR NOON IN QUITO

Date n Local

time

()

()

z ()

s ()

s () N

(hours)

Sep, 15 258 12:08:56 2,22 0 2,38 0,00 89,99 11,78

Dec, 15 349 12:09:23 -23,34 0 23,17 0,00 90,07 11,79

TABLE II. MEAN DAYLY RADIATION AND COMPONENTS

Date H Ho Kt Hd/H Hd Hb

MJ/m2/day

% MJ/m2/day MJ/m2/day

Sept, 15 15,34 37,12 0,413 0,75 11,519 3,817

Dec, 15 14,33 35,58 0,403 0,77 10,984 3,344

TABLE III. INSTANTANEOUS RADIATION ON HORIZONTAL SURFACE

Date

a b rt rd I Id Ib

- - - - MJ/m2h MJ/m2h 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

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 2 and 23 along the year in order to

achieve the best performance of the installation.

TABLE IV. OPTIMUM MONTLY SLOPE FOR A TILTED SURFACE

Month Jan Feb Mar Apr May Jun

opt () 21,10 13,12 2,65 9,58 18,96 23,48

Month Jul Ago Sep Oct Nov Dec

opt () 21,68 13,95 2,38 9,43 18,98 23,17

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 =0 and =3, but we chose the slope of 3 because with the

400

450

500

550

600

650

Jan

Feb

Mar

Ap

ri

May Jun

Jul

Ago

st

Sep

t

Oct

No

v

DecInst

anta

neo

us

rad

iaci

on

(W

/m2

)

Months

= 0 =3 =15 =30 =45

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

the panel.

TABLE V. INSTANTANEOUS RADIATION ON A TILTED SURFACE WITH =3

Date Rb Ib,t I,t Id,t IT

- 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

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.

Fig. 4. Hourly radiation distribution in September 15.

Fig. 5. Hourly distribution in December 15.

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

Fig. 7. Instantaneous radiation and its components on horizontal surface in December 15

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.

0,00

0,50

1,00

1,50

2,00

2,50

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Rad

iati

on

MJ/

h.m

2

Hours

=0 =3 =45

0,00

0,50

1,00

1,50

2,00

2,50

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Rad

iati

on

MJ/

h.m

2

Hours

=0 =3 =45

0

100

200

300

400

500

600

700

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Rad

iati

on

W/m

2

Hours

I Id Ib

0,00

100,00

200,00

300,00

400,00

500,00

600,00

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Rad

iati

on

W/m

2

Hours

I Id Ib

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

Fig. 9. Instantaneous radiation and its components on tilted surface (=3) 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 3. 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.

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

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

As it can be seen in the Figure 10 and 11, the total

instantaneous radiation for a slope of 45 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 3 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.

0

100

200

300

400

500

600

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Rad

iati

on

W/m

2

Hours

It Ib,t I,t Id,t

0

100

200

300

400

500

600

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Rad

iati

on

W/m

2

Hours

It Ib,t I,t Id,t

0

100

200

300

400

500

600

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Rad

iati

on

W/m

2

Hours

It Ib,t I,t Id,t

0

100

200

300

400

500

600

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Rad

iati

on

W/m

2

Hours

It Ib,t I,t Id,t

TABLE VI. INSTANTANEOUS RADIATION ON HORIZONTAL AND TILTED SURFACE ON SEPTEMBER 15 (n=258)

Slope angles =0 =3 =45

LCT ts z I It It

(hour) (hour) MJ/m2h MJ/m2h 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 0,00 0,00 0,00

Total 15,15 14,67 12,60

TABLE VII. INSTANTANEOUS RADIATION ON HORIZONTAL AND TILTED SURFACE ON DECEMBER 15 (n=349)

Slope angles =0 =3 =45

LCT ts z I It It

(hour) (hour) MJ/m2h MJ/m2h 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 14,16 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 =0, in September was 15,15MJ/m

2h and in December was 14.16

MJ/m2h, which have a relative error of 1.2% compared with

raw data used. When =3 the mean daily radiations were 14.67 MJ/m

2h for September and 14.43 MJ/m

2h 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 =45, 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] Burlon S, Bivona S, Leone C. Instantaneous hourly and daily

[2] 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/3-SEMESTRE/2-SOLAR%20THERMAL/Ecuador-climatolog%EDa/NASA%20Surface%20meteorology%20and%20Solar%20Energy%20-%20QUITO.htm

[3] Rodriguez, I. (2014). Topic 1. Introduction. Solar Energy from Sun., (pg. 43). Barcelona.

[4] William B. Stine, M. G. (2001). Power From The Sun. Obtenido de 3. The Suns Position: http://www.powerfromthesun.net/Book/chapter03/ chapter03.html.