Diurnal variation of the hourly hemispherical insolation

Solar & Wind Technology Vol. 5, No. 6, pp. 661-665, 1988 Printed in Great Britain.

0741-983X/88 $3.00+.00 Pergamon Press pie

TECHNICAL NOTE Diurnal variation of the hourly hemispherical insolation K . K . GOPINATHAN Department of Physics, National University of Lesotho, Roma, Lesotho, Southern Africa

(Received 20 January 1988 ; accepted 8 April 1988) Abstract--Ratios of hourly to daily hemispherical insolation are estimated for six locations in Lesotho using the model proposed by Collares-Pereira and Rabl, and the results are presented. The applicability of the model to the Southern African region was first tested by computing the hourly global radiation and comparing it with the measured data for two locations in South Africa. The excellent agreement between the measured and estimated data for the two stations demonstrates the accuracy of the estimation procedure and its applicability to locations in this region.

1. INTRODUCTION All renewable energy sources, including bio-mass, are dependent on climatological data like solar radiation, wind, temperature, humidity and rainfall. Meteorological data collection is therefore the first and most important step in resource survey. Solar radiation data is a necessary basis for the design of solar energy conversion devices and for feasibility studies into the possible uses of solar energy. The total solar energy incident at a given location can be considered to be made up of direct beam radiation and diffuse radiation. Since the direct and diffuse radiation have quite distinct features, it is considered necessary for many purposes to measure the two parts separately and this necessitates two more or less separate measuring systems. Furthermore, the solar energy received at a given location will also change with the time on a given day and long-term average values of hourly hemispherical and diffuse insolation are needed for many applications of solar energy. Many solar devices such as fiat plate collectors require information on the hourly solar insolation. For example, in order to evaluate the transmittivity-absorptivity product of such collectors, knowledge of both hourly beam and diffuse components and their total are needed. In many countries radiation data are recorded at regular hourly intervals. However, because of lack of funds and expert care it is not always possible to measure hourly radiation and its variation with time at all locations. This is more so in developing countries, especially in the African continent. At places where no measurements exist, hourly radiation can be estimated from measured or estimated daily radiation. The first attempt to analyse the hourly radiation data was done by Whillier [1], and Hottel and Whillier [2] who used the data of widely separated locations to obtain the curves of hourly to daily radiation ratio (rt) against the sunset hour angle, for each hour from 9 a.m. to 3 p.m. Liu and Jordan [3] extended the day length of these curves. By using the corrected data for five U.S. locations, Collares-Pereira and Rabl [4] developed an analytical expression for the rt ratio in terms of the sunset hour angle. The motivation for this investigation is the total absence of hourly radiation data for any location in Lesotho and the 661

need to develop a solar energy system for rural applications. At present, Lesotho has only one solar radiation measuring station (Roma, Maseru) where measurements of daily values of total and diffuse radiation are carried out. However, hourly radiation data are not recorded anywhere in the country. In the present study, hourly hemispherical insolation values are calculated for six locations in Lesotho. The correlation suggested by Collares-Pereira and Rabl [4] is used for estimating monthly mean hourly global radiation on a horizontal surface. The applicability of the above model to Lesotho and, in general, to the Southern African region was tested by estimating hourly to daily global radiation ratio, during January and July, for two locations in South Africa. The hourly global radiation values are then estimated for the two locations and the estimated values are compared with the measured data reported in the literature [5]. This was necessary as no measured data on hourly radiation are available for Lesotho for comparison. Lesotho is surrounded on all sides by South Africa and the climatological conditions of Lesotho are more or less the same as nearby South African stations. The two South African locations selected to test the applicability of the Collares-Pereira model are Bloemfontein and Pretoria. Their latitude and elevation are 29.12 °, 1422 m and 25.75 °, 1369 m respectively. Most of the locations in Lesotho selected for study have latitude and elevation near to these values. The Bloemfontein station is only about 100 km away from Maseru, the capital of Lesotho. 2. THE COMPUTATIONAL TECHNIQUES The correlation suggested by Collares-Pereira and Rabl [4] in the form cos W - c o s W, ~V f2nW~'~

rt = ~-~(a+bcos W) sin

(I)

,-- t ~ - - - ) cos W,

was used to find the ratio of hourly to daily global radiation, where rt = the ratio of hourly to daily global radiation, W = hour angle in degrees,

662

Technical Note

4.0 E

Measured - o--o- Estimated

3.5 3.0 -- .~

~,,

._~

2.5

,~

~

2.0

x

"&

- ~

1.5 1.0 0.5

0"/

-

o

I 7

"'~

I 8

I 9

I 10

I 1t

I [ 12 13 Solar time

I 14

I 15

I 16

I 17

I 18

Fig. 1. Comparison of experimental and theoretical values of monthly mean hourly global radiation on a horizontal surface for Bloemfontein.

Ws = sunset hour angle, a = 0.409+0.5016 sin (W~ - 6 0 ) , b = 0.6609-0.4747 sin (Ws - 6 0 ) ,

From the rt ratio and the monthly mean daily global radiation, the hourly global radiation can be estimated. The six locations in Lesotho for which rt and hourly global radiation data are estimated are Leribe, Letseng, Maputsoe, Maseru, Qacha's Nek, and Quthing. Monthly mean daily global radiation needed for estimating hourly radiation was obtained from the results published by Gopinathan [6] for these locations.

The sunset hour angle Ws is calculated from the equation W~ = cos J ( - t a n 4 ~ t a n 6 ) where ~b is the latitude of the location and 6 the declination 6 = 23.45 sin (360 284+ n'~ 365 J '

3. RESULTS AND DISCUSSION

where n is the day of the year. The day length So is given by

Figures 1 and 2 give a comparative study of the measured and estimated hourly hemispherical insolation for Bloemfontein and Pretoria during January and July. The solid lines in the figures show the measured data and the dotted

So = ? . cos ~( - tan q5tan 3).

13

4.0 f 3.5

Measured - 0-.-0-- Estimated

3.0 2.5

!

2.0 - .~

~,"

--

January

(.9 1.5 !

1.0 0.5 ! 0

t 7

I 8

I 9

I 10

L [ I 11 12 13 Solartime

I 14

t5

1 16

I 17

I 18

Fig. 2. Comparison of experimental and theoretical values of monthly mean hourly global radiation on a horizontal surface for Pretoria.

Technical Note

663

0.20

0.2(~ --

0.18

0.18 --

0.16 0.14

0.1~

_~'(~""~"~,,,,,¢.~1/2 Hoursfrom solar noon ~

1/2

HOUrs from solar noon

0.14

0.12

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_

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0.08' --

31/2

21/2 0.10

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31/2

--

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~

0.04

0.04

J

0.02

10

~C,

u 0"0-,

~

0.02

11

12

13

14

10

J

11

Hours from s u n r i s e t o sunset

I

12

13

14

Hours from sunrise to sunset

Fig. 3. Ratio of hourly to daily global radiation for Leribe.

Fig.

lines give the theoretical values. The upper pair of curves in both the figures is for January and the lower curves for July. January and July are the typical summer and winter months for the region. The time indicated in the figures is solar time. The local time is converted into solar time using the procedure given by Duttie and Beckman [7]. The hourly

global radiation values presented in the figures are in MJ m- 2. The hourly radiation is plotted at the mid-point of the hour. The remarkable agreement between the measured and estimated values shows the accuracy of the estimated data. The experimental and theoretical curves always move together

5. R a t i o

0.20

0.2(; -

0.18

0.18

0.16

hourly to daily global radiation for Maputose.

0,16 '

,

~

~

Hours from solar noon 0.14 ~

0,14 0.12

~

1/2

Hours f r o m solar noon

0.12 _

21/z

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0.10 0.08

of

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10

31/2

= ~...~_.0.,.~(>--~

11

12

---'°''~'~'°°-

13

14

Hours from sunrise to sunset

Fig. 4. Ratio of hourly to daily global radiation for Letseng.

10

11

12

13

14

Hours from sunrise to sunset

Fig. 6. Ratio of hourly to daily global radiation for Maseru.

664

Technical Note

0.2C --

0.20

0.18 --

0.18 - -

0.18 ~ Lx'o.~ .
0.16 ~ , ~

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tl

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2

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10

14

/

11

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1

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13

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Hours from sunrise to sunset

Hours from sunrise to su nset

Fig. 7. Ratio of hourly to daily global radiation for Qacha's Nek.

Fig. 8. Ratio of hourly to daily global radiation for Quthing.

and overlap for most hours of the day. This is true for both the locations during January and July, demonstrating the applicability of the procedure to locations in Southern Africa. Equation (1) is then used to estimate the ratio of hourly to daily hemispherical radiation for six locations in Lesotho. The estimated r t ratios for Leribe, Letseng, Maputsoe, Maseru, Qacha's Nek and Quthing are shown in Figs 3-8 respectively. The r t ratios are plotted as a function of day length (or sunset hour angle), for each month, for each o f t h e hours ½, 1~, 1 2~, t 3~, 4~ and 51 measured from solar noon. A smooth curve can be observed for each hour and

thus a set of curves for each station. Using these ratios and the reported values o f m o n t h l y m e a n daily global radiation [6] for each of these locations, hourly global radiations are calculated. The estimated hourly global radiation for the six locations are presented in Tables 1-3. Table 1 gives the hourly global radiation values for Leribe and Letseng. Table 2 presents hourly radiation for Maputsoe and Maseru and Table 3 for Qacha's Nek and Quthing. The hourly radiation values presented in the tables are in MJ m - 2 on a horizontal surface. The latitude (q~) and the elevation (h) of the locations are included in the tables. The highest r t ratio is for Quthing

Table 1. Estimated monthly mean hourly global radiation on a horizontal surface for Leribe and Letseng Global radiation in MJ m - 2 Station

Leribe = 28.53os

h = 1670 m

Letseng ~b = 29.0°S h = 3085 m

Hours

J

F

M

A

M

J

J

A

S

O

N

D

11-12, 10--11, 9-10, 8-9, 7-8, 6-7,

12 1 1-2 2-3 3-4 4-5 5--6

3.24 2.99 2.53 1.94 1.29 0.66

3.13 2.88 2.40 1.79 1.13 0.50

2.87 2.6l 2.13 1.52 0.88 0.29

2.81 2.52 2.00 1.35 0.67 0.08

2.26 2.00 1.55 0.97 0.40 . .

2.17 1.91 1.45 0.87 0.30 .

2.31 2.04 1.56 0.96 0.36 .

2.41 2.15 1.69 1.11 0.51

2.93 2.66 2.15 1.50 0.83 0.22

3.10 2.84 2.36 1.73 1.06 0.44

3.22 2.97 2.51 1.90 1.25 0.62

3.28 3.03 2.58 1.99 1.34 0.71

11-12, 10--11, 9-10, 8-9, 7-8, 6-7,

12-I 1-2 2-3 3--4 4-5 5~6

2.65 2.45 2.07 1.59 1.06 0.55

2.54 2.33 1.95 1.45 0.92 0.41

2.48 2.25 1.84 1.32 0.76 0.25

1.87 1.68 1.33 0.90 0.45 0.05

1.36 1.21 0.93 0.58 0.23 . .

1.46 1.28 0.97 0.58 0.20 .

2.33 2.06 1.57 0.96 0.36 .

1.81 1.62 1.27 0.83 0.38

2.46 2.23 1.80 1.26 0.69 0.18

2.93 2.68 2.23 1.64 1.01 0.42

2.67 2.46 2.08 1.58 1.04 0.52

2.43 2.25 1.92 1.48 1.00 0.53

Technical Note

665

Table 2. Estimated monthly mean hourly global radiation on a horizontal surface for Maputsoe and Maseru Global radiation in MJ m - 2 Station Maputsoe ~b = 28.89°S h = 1670 m

Maseru t~ = 29.32°S h = 1571 m

Hours

J

F

M

A

M

J

J

A

S

O

N

D

11-12, 10-11, 9-10, 8-9, 7-8, 6-7,

12-1 1-2 2-3 3-4 4-5 5q5

3.23 2.99 2.35 1.93 1.29 0.67

3.03 2.78 2.32 1.73 1.09 0.49

2.96 2.69 2.20 1.58 0.91 0.30

2.62 2.36 1.87 1.26 0.63 0.07

2.21 1.96 1.51 0.95 0.38 . .

2.08 1.83 1.38 0.83 0.28 .

1.97 1.74 1.33 0.82 0.30 .

2.47 2.21 1.73 1.13 0.52

2.91 2.64 2.13 1.49 0.82 0.21

3.03 2.77 2.30 1.69 1.04 0.43

3.21 2.96 2.50 1.90 1.25 0.63

3.22 2.98 2.54 1.96 1.32 0.71

11-12, 10-11, 9-10, 8-9, 7-8, 6-7,

12-1 1-2 2-3 3-4 4-5 5-6

3.28 3.03 2.57 1.97 1.31 0.68

3.24 2.98 2.49 1.86 1.17 0.53

2.90 2.63 2.15 1.54 0.89 0.29

2.50 2.24 1.78 1.20 0.59 0.06

2.20 1.95 1.50 0.94 0.38 . .

1.99 1.75 1.32 0.79 0.26 .

2.14 1.89 1.44 0.88 0.32 .

2.55 2.27 1.78 1.17 0.54

3.00 2.72 2.20 1.54 0.85 0.22

3.05 2.79 2.32 1.71 1.05 0.44

3.24 2.99 2.53 1.92 1.27 0.64

3.41 3.16 2.69 2.07 1.40 0.75

Table 3. Estimated monthly mean hourly global radiation on a horizontal surface for Qacha's Nek and Quthing Global radiation in MJ m-2 Station

Qacba's Nek = 30.07°S h = 1970 m

Quthing = 29.41°S h = 1650 m

Hours

J

F

M

A

12-1 1-2 2-3 3-4 4-5 5-6

2.99 2.76 2.53 1.80 1.21 0.63

2.78 2.55 2.13 1.59 1.01 0.46

2.79 2.53 2.07 1.48 0.86 0.28

2.62 2.35 1.86 1.25 0.62 0.06

11-12, 12-1 10-I 1, 1-2 9-10, 2-3 8-9, 3-4 7-8, 4-5 6-7, 5--6

3.19 2.95 2.51 1.93 1.29 0.68

2.90 2.67 2.23 1.67 1.06 0.48

2.97 2.70 2.21 1.58 0.91 0.30

2.57 2.30 1.83 1.23 0.60 0.06

11-12, 10-11, 9-10, 8-9, 7-8, 6-7,

during June, between 12-13 hours, and the lowest rt ratio for the same time interval for Quthing is during December. The hourly global radiation values for Lesotho are, in general, quite high for all the locations. 4. CONCLUSION The correlation suggested by Collares-Pereira and Rabl, for estimating the ratio of hourly to daffy global radiation, can be applied accurately to locations in the Southern African region. The results of the hourly to daffy global radiation ratio and the hourly global radiation values presented here can serve as a useful reference on hourly radiation for future solar energy applications in Lesotho. REFERENCES 1. A. Whillier, The determination of hourly values of total solar radiation from daily summation. Arch. Meteorol. Geophys. Bioklimatol. Ser. B. 7, 197-204 (1956).

M

J

J

A

S

O

N

D

2.04 1.81 1.39 0.87 0.34 . .

2.04 1.77 1.35 0.80 0.26 .

2.23 1.97 1.49 0.91 0.32 .

2.37 2.12 1.66 1.08 0.49

2.87 2.60 2.10 1.47 0.81 0.21

2.76 2.53 2.10 1.55 0.96 0.40

3.00 2.77 2.34 1.79 1.18 0.60

3.27 3.03 2.58 2.00 1.36 0.74

2.04 1.81 1.39 0.86 0.34 . .

1.97 1.73 1.30 0.77 0.25 .

2.06 1.82 1.38 0.84 0.29 .

2.10 1.87 1.46 0.95 0.43

2.86 2.59 2.10 1.47 0.80 0.21

2.90 2.65 2.21 1.63 1.00 0.42

2.46 2.27 1.92 1.47 0.97 0.50

3.21 2.98 2.54 1.97 1.34 0.73

2. H. C. Hottel and A. Whilh'er, Evaluation of flat plate solar collector performance, Trans. Conf. on the use of Solar Energy: The Scientific Basis Vol. II(1), Section A, pp. 74-104 (1955).

3. B.Y.H. Liu and R. C. Jordan, The interrelationshipand characteristicdistributionof direct,diffuseand totalsolar radiation.Sol. Energy 4, 1-19 (1960). 4. M. Collare~-Pereira and A. Rabl, The average distribution of solar radiation-correlationbetween diffuse and bemispherical and between daily and hourly insolationvalues.Sol. Energy 22, 155-164 (1979). 5. A. J. Drummond and E. Vowinckel, The distributionof solarradiationthroughout Southern Africa. J. Meteorol. 14, 343-353 (1957). 6. K. K. Gopinatban, Distributionof global and sky radiation throughout Lesotho. Solar Wind Technol. 5, 103106 (1988). 7. J. A. DUI~¢ and W. A. Beckman, So/ar Engineering of Thermal Processes, Wiley, N e w York (1980).