~ Pergamon Renewable Energy, Vol. 11, No. 2, pp. 257 262, 1997

© 1997 Elsevier Science Ltd All rights reserved. Printed in Great Britain

PIh S0960-1481(96)00125-5 0960-1481/97 $1%00+0.00

T E C H N I C A L N O T E

Estimation of monthly average daily and hourly solar radiation impinging on a sloped surface using the isotropic sky model

for Dhahran, Saudi Arabia

F. A. AL-SULAIMAN and B. ISMAIL King Fahd University of Petroleum and Minerals, Dhahran, 31261-Saudi Arabia

(Received 10 June 1996; accepted 10 November 1996)

Abstract--In this paper the isotropic sky method of Liu and Jordan is used to theoretically estimate the monthly average daily and hourly solar radiation impinging on an unshaded tilted surface in Dhahran, Saudi Arabia. The surface receiving solar radiation is assumed to be fixed at a tilt angle, fl, equal to the latitude of Dhahran, ~b and oriented such that it is facing south with zero azimuth angle, ~,. The calculation of total radiation on a sloped surface from measurements on a horizontal surface is discussed. Monthly average daily and hourly solar radiation values are then tabulated. The results obtained can be effectively employed in solar process design calculations. © 1997 Elsevier Science Ltd.

INTRODUCTION

Solar radiation data are available in several forms. Most radiation data are available for horizontal surfaces (including direct and diffuse radiation). Two types of solar radiation data are widely used. The first is the monthly average daily total radiation on a horizontal surface//H. The second is the hourly total radiation on a horizontal surface [H for extended periods (one or more years). The monthly average daily, and hourly radiation incident on tilted surfaces/7 ~ and [T (the subscript "T" refers to tilt) are required for solar process design calculations. The theoretical estimation of solar radiation on a sloped surface was carried out for various places, e.g. Greece [1] and Canada [2]. These values are very useful for the analysis of solar process application, such as solar heating for domestic or institutional hot water supplies, designing water heating systems, heat for comfort in buildings, etc. Solar radiation is mainly received on sloped surfaces, and by averaging the data over a month or a year the design procedure upon which the data are needed will reflect more convenient results. Abdelrahman and Hadidi [3] presented a comparative assessment between various methods to model the total radiation on tilted surfaces in Dhahran and the measured values. The results were discussed in terms of implicit statistical parameters such as the root-mean-square error (RMSE) and the mean-bias error (MBE). However, no explicit data were given. As a matter of fact, no such data are made available for researchers and engineers at the present time. In this work, the /1T and Iv values are theoretically estimated for Dhahran, based on/7H data measured by the Energy Research Laboratory at K F U P M [E1-Hadidi, M. and Shaahid, S., Energy Research Laboratory, King Fahd University of Petroleum and Minerals, Personal Communication] (Fig. 1).

RADIATION ON SLOPED SURFACE~-ISOTROPIC SKY MODEL For engineering applications, diffuse radiation may be considered isotropic. There are many

parameters that influence the amount of direct and diffuse solar radiation such as time, data of

257

258

~" 3 0 -

~Z 2s

.~ 20

¢~ 1(

5

?.o

Technical Note

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

Month

Fig. 1. The monthly average daily total radiation on a horizontal surface for Dhahran, Saudi Arabia.

local latitude, altitude, declination angle, zenith angle, atmospheric transmissivity, water vapour, reflectance of the ground (the albedo), cloud condition, sunshine hours, maximum air temperature, relative humidity, etc. It is very difficult to obtain a general formula to estimate direct and diffuse solar radiation by considering all of these parameters simultaneously. On the other hand, some of the parameters have little effect on solar radiation and can be neglected [4]. The isotropic sky method of Lui and Jordan (1962) as extended by Klein (1927) received wide application [5].

Radiation on tilted surfaces includes three components : beam, isotropic diffuse and solar radiation diffusely reflected from the ground. A surface tilted at an angle/~ from the horizontal has a view factor to the sky equal to (1 +cos ~)/2' The surface has a view factor to the ground of (1 -cos B)/2. The surroundings have a diffuse reflectance of pg for the total solar radiation. The reflected radiation from the surroundings on the surface will be I . pg (1-cos/~)/2. The isotropic diffuse model is a simplified scheme and makes the calculation of radiation on a tilted surface easier to carry out.

C A L C U L A T I O N P R O C E D U R E

Solar radiation impinging on a surface can be maximized by tilting the surface with an angle fl, equal to the latitude q~ of the surface location and facing south (for northern hemisphere) at which the azimuth angle 7 is set to zero. In this work, the isotropic model of Liu and Jordan [5] is implemented to estimate the monthly average daily radiation H~, for Dhahran (~b = 26.13 ° N). The monthly average daily solar radiation on an unshaded tilted surface is given by Ref. [5] as :

/7~ = H . ( I - / - 7 ~ \ _ _ ~ - H ) Rb + Hdn / I - - COS/3\ _ / I - - COS fl\ (1)

The diffuse reflectance to the ground pg for Dhahran is estimated as 0.35 [E1-Hadidi, M. and Shaahid, S., Energy Research Laboratory, King Fahd University of Petroleum and Minerals, Personal Communication]. Now, in order to calculate HT for every month in the year, several terms in eq. (1) need to be calculated a priori. The fraction

H. J

is correlated with the average clearness index, /~ (defined as the ratio of the/~H to the average daily extraterrestrial solar radia t ion, / to . ). The governing equations pertaining to this correlation are as follows [5] : for co s ~< 81.4 and 0.3 ~</£T ~< 0.3

H . ] = 1.391 - 3.560/(T +4.189/£2--2.137/~ 3, (2a)

and for ~Os > 81.4 ° and 0.3 ~</£T ~< 0.8

Technical Note 259

= 1.311-3.022&+3.427Z+l.82lK+ (2b)

where, the sunset hour angle ws, is given as

cos ws = -tan 4 tan 6,

4 and 6 are the latitude and declination angles, respectively, and 6 is given by eq. (4)

6 = 23.45sin (360=),

where n is the day of the year based on A the average day of the month. For Dhahran S, n and A are given in Table 1.

Extraterrestrial radiation for the average day of the month on a horizontal surface, t-ioH, can be calculated for each month using eq. (5) [5] :

l?oH = 1.169* 10’

n l+O.O33cos~

>( cos~cos~sinms+~sin~sin6

> (5)

I?, can be calculated using eqs (6) and (7) :

I& = cos 6 sin w&

cos 4 cos 6 sin ws + (n/l 80)~s sin 4 sin 6 ’ (6)

where w& is the sunset hours angle for the tilted surface for the average day of the month, which is generally given by :

w$ = min L toss ’ (-tan 4 tan 6)

90” (7)

where means smaller

59+i 90$-i

12O+i 15l+i 18l+i 212+i 243+i 273+i 304+i 334+i

17 17 - 20.9 15 46 -13.3 16 75 -2.4 15 105 9.4 15 135 18.8 9 160 22.9

18 199 21.0 17 229 13.1 16 259 1.8 16 289 - 10.0 15 319 -19.1 11 345 -23.1

* [El-Hadidi and Shaahid, Energy Research Laboratory, King Fahd Uni- versity of Petroleum and Minerals, Dhahran, Personal Communication].

260 Technical Note

Table 2. Results of calculated values of HT*

/-Ion /_71 /~dH /~T Month (M J/m2) (M J/m2) /£T /-TdH//-Tn /~b (MJ/m 2) (MJ/m 2) /q = /-TT/Hn

January 23.28 13.4 0.58 0.32 1.50 4.29 17.98 1.34 February 27.44 14.4 0.52 0.41 1.30 5.86 16.78 1.17 March 32.59 18.79 0.58 0.36 1.15 6.76 20.58 1.09 April 36.99 20.72 0.56 0.37 1.00 7.67 20.70 1.00 May 39.45 26.72 0.68 0.27 0.88 7 .21 24.48 0.92 June 40.19 26.83 0.67 0.28 0.84 7.51 23.83 0.89 July 39.67 25.42 0.64 0.30 0.84 7.63 22.64 0.89 August 37.70 24.72 0.66 0.29 0.94 7.17 23.74 0.96 September 33.93 22.19 0.65 0.29 1.08 6.44 23.52 1.06 October 28.81 18.49 0.64 0.30 1.28 5.55 22.16 1.20 November 24.12 15.04 0.62 0.28 1.46 4 .21 20.07 1.33 December 21.98 12.08 0.55 0.34 1.56 4.11 16.55 1.37

* [E1-Hadidi and Shaahid, Energy Research Laboratory, King Fahd University of Petroleum and Minerals, Dhahran, Personal Communication].

of HT are calculated, and the results are tabulated in Table 2. The difference between /~T and /?H values are graphically presented in Fig. 2. The next objective is to calculate the monthly averge hourly solar radiation, IT, on a sloped surface. This is done using the isotropic model given by eq. (8) [5] as follows :

/T = KTHonI(rt_lqos\ /~'dH /'1 + c o s f l \ ~ f H ) R b + ~ r a ~ ) + p g r t ( ~ ) ] '

where the terms rd and rt are given by eqs (9a) and (9b), respectively :

(8)

r~ cos co-cos cos (9a) rd = 24 rCCOs

sin COs - 180 cos COs

7T COS CO -- COS COs rt = ~ ( a + b c o s c o )

TCCO S sin cos - ~ cos COs

(9b)

5O

40

"~ 3o ..q

20. ....~7 _OH ~ +

+ H H F-, ~0~ I 0 ! X H'r

f I I r I I I I 0 1 1 4 5 6 7 8 9 1 0 1 1 Month

Fig. 2. Monthly average daily total radiation.

I 12

Technical Note

The coefficients a and b are given by

a = 0.409 + 0.5016 sin (~Os - 60)

and

b = 0.6609 - 0.4767 sin (co s - 60).

The hour angle, oJ, is given by eq. (10) as :

~o = 15(N-- 12)

where N, the number of daylight hours is given by :

2 - - I N = ~ c o s ( - tan q5 tan 6).

Rb, which appears in eq. (8) is calculated using eq. (12)

cos 3 cos (9 R b =

cos ~b cos 6 cos co + sin ~b sin 6 '

The results of IT are presented in Table 3.

261

(9c)

(9d)

(10)

(11)

(12)

RESULTS AND D I S C U S S I O N

The monthly average daily, and hourly solar r ad i a t i on / /v and IT are estimated for Ohahran (using the isotropic sky model). The results obtained (uisng existing data on a horizontal surface) [Tables 2 and 3] are useful for solar engineering design calculations, such as solar collectors sloped at a fixed angle. The results give an average trend for the solar radiation received. Figure 2 compares //H and //T values in the months of the year. Mos t ly / /T exceeds /qH, as expected. It should be noted that the estimated values are not verified with any measurements due to the inaccessibility of available data. Table 3 shows that the maximum radiation value of 3.349 M J / m 2 occurs in the hour pair 11 12 for the month of May. The calculated [v values in Table 3 represent average values over the year. These values can be used effectively wherever needed.

Table 3. Computer results pertaining Iv for Dhahran in M J / m 2

Pair hour of the average day in the month (solar time)

6-7 am 7-8 am 8-9 am 9-20 am 10-11 am t 1-12 am Month (5-6 pm)* (4-5 pm) (3~4 pm) (2-3 pm) (1-2 pro) (12-1 pro)

January - - 0.599 1.266 1.914 2.385 2.620 February - - 0.600 1.194 1.790 2.237 2.519 March 0.805 1.505 2.186 2.687 2.960 April 0.312 0.886 1.531 2.124 2.602 2.832 May 0.413 1.119 1.855 2.537 3.065 3.349 June 0.439 1.094 1.829 2.445 2.956 3.233 July 0.415 1.628 1.734 2.367 2.835 3.121 August 0.375 1.084 1.793 2.492 2.937 3.299 September 0.284 0.968 1.736 2.473 2.994 3.301 October - - 0.819 1.572 2.298 2.838 3.151 November 0.678 1.391 2.107 2.616 2.901 December 0.634 1.174 1.741 2.179 2.464

* Solar time is taken to be symmetrical around noon time.

262 Technical Note

CONCLUSION

The monthly average daily and hourly solar radiation impinging on an unshaded tilted surface are estimated for Dhahran, Saudi Arabia, using the isotropic sky method of Liu and Jordan. The surface receiving solar radiation is assumed to be fixed at a tilt angle,/~, equal to the latitude of Dhahran, ~b and oriented such that it is facing south with zero azimuth angle, 7. Monthly average daily and hourly solar radiation values are then tabulated. The results obtained can be effectively employed in solar process design calculations. The results obtained are reasonable and follow the expected intuitive trend that / lv is normally higher than/tH. It should be noted that the estimated values are not verified with any measurement due to the lack of accessible data.

Acknowledgement--The authors wish to acknowledge the Energy Research Laboratory (ERL) at King Fahd University of Petroleum and Minerals (KFUPM) for providing the solar radiation data that made this research possible.

NOMENCLATURE

ldH JqH

IT

N

S

monthly average daily diffuse radiation on a horizontal surface, MJ/m 2 monthly average daily total radiation on a horizontal surface, MJ/m 2 extraterrestrial radiation for the average day of the month on a horizontal surface, MJ/m 2 monthly average daily total radiation on a tilted surface, MJ/m 2 monthly average hourly radiation on a tilted surface, MJ/m 2 average clearness index number of daylight hours average day of the month average beam geometric factor declination angle of the surface in degrees

Greek symbols'

7 Pg O9

(D s

tilt angle of the surface in degrees latitude with respect to the location of the surface in degrees azimuth angle diffuse reflectance to the ground hour angle in degrees sunset hour angle in degrees sunset hour angle for the tilted surface in degrees.

REFERENCES

1. Pisimanis, D., and Notaridou, V., Estimating direct, diffuse and global solar radiation on an arbitrarily inclined plane in Greece. Solar Energy 1987, 39, 159-172.

2. Hay, J. E., and Davies, J. A., Calculation of the solar radiation incident on an inclined surface. Proc. First Canadian Solar Radiation Data Workshop, Ministry of Supply and Services Canada, 1980, Vol. 59.

3. Abdelrahman, M. A., and Hadidi, M., Comparison of calculated and measured values of total radiation on tilted surfaces in Dhahran, Saudi Arabia. Solar Energy 1986, 37(3), 239-243.

4. Ashjaee, M., Roomina, M. R., and Ghafouri-Azar, R., Estimating direct, diffuse, and global solar radiation for various cities in Iran by two methods and their comparison with the measured data. Solar Energy 1993, 50, 441445.

5. Duffle, J., and Beckman, W., Solar Engineerin9 and Thermal Processes, 2rid edn. John Wiley, New York, 1991.