Ener~l) Com. & M~tmt Vol. 20, pp. 213 to 218 0196 8904 80 1101 021350200 0 © Pergamon Press Lid 1980. Printed in Great Britain

M O N T H L Y AVERAGE DAILY INSOLATION ON TILTED COLLECTORS IN LIBYA

MOUSTAFA MORAD HAWAS and TARIQ MUNEER Mechanical Engineering Department, Faculty of Engineering,

Garyounis University, P.O. Box 9476, Benghazi, Libya

(Received 20 February 1980)

Abstract--This work presents monthly average total insolation values calculated for various locations in Libya. The insolation values are for actual sky conditions based on sunshine records. Latitudes ranging from 24 to 32°N and tilt angles from 0 to 90 ° are considered. The presented insolation values show that the optimum value of tilt angle for space heating systems is about latitude plus 15 ~ while that for cooling systems is 0 ° A combination of horizontal roof and vertical south wall gives more or less uniform insolation throughout the whole year. The maximum annual insolation occurs for tilt angle nearly equal to the latitude.

Insolation Solar collectors Solar energy Monthly average Insolation Tilted collectors

N O M E N C L A T U R E

A = constant (eq. 8) al = constant (eq. 9) B = constant (eq. 8) b = tilt angle (deg.)

bl = constant (eq. 9) C = constant (eq. 8) E = annual solar radiation (MJ/m2/year)

E , = annual solar radiation on a horizontal surface (J/mS/year)

H = instantaneous total [direct + diffuse) solar radi- ation on a horizontal surface (W/m 2)

= monthly average daily solar radiation on a hori- zontal surface (J/m2/day)

h = hour angle measured from solar noon (15 ° x No. of h from solar noon) (deg.)

hm = hour angle between sunrise and noon (deg.) I = instantaneous total (direct + diffuse) solar radi-

ation (W/m 2) 7 = monthly average daily solar radiation (J/m2/day)

ID~ =,i-nstantaneous normal direct solar radiation {di- rect solar radiation on a surface normal to the sun's rays). (W/m s)

~c = solar constant (W/m 2) Kd = diffuse ratio KT = cloudness index

L = local latitude (deg.) n = day of the year R = sun-earth distance (m)

R0 = sun-earth mean distance (m) R, = monthly average tilt factor

S = ratio of bright sunshine hours to the length of the day

~t = solar altitude (deg.) 6 = solar declination (deg.)

pg = ground reflectance Subscripts

CS = clear sky D = direct d = diffuse

d B = ground reflected diffuse d~ = sky diffuse O = extraterrestrial

t = daily total

I N T R O D U C T I O N

Knowledge of the a m o u n t of solar rad ia t ion incident on ti l ted surfaces is requi red in solar energy appli-

• I cationS. Wi th the increasing interest in solar energy uti l izat ion in Libya es t imates of insola t ion values are highly desired.

The amoun t of solar r ad ia t ion incident upon a sur- face is dependen t upon the t ime of day, the day of the year, the locat ion of the place [latitude) and the orien- ta t ion and tilt of the surface•

In the design of solar energy systems, knowledge of month ly average d a i l y insola t ion values is of pr ime

impor tance . This work is conce rned with the esti- ma t ion of the mon t h l y average daily insola t ion on surfaces facing south but t i l ted at var ious angles. Six cities are cons idered cover ing a wide range of lat i tude in Libya (Fig. 1) (Lat i tudes of the cities are shown in Table 8). The es t imates are for actual sky condi t ions based on sunshine records dur ing the per iod 1960-1969.

A N A L Y S I S

Monthly average daily extraterrestrial radiation on a horizontal surface

The ins tan taneous extraterrestr ia l solar rad ia t ion incident on a hor izonta l surface (Ha) is

Ha = (R/Ro) 2" Lc ' sin~t (1)

where I~c is the solar cons tan t at the mean sun -ea r th d is tance taken as 4871 KJ /h m 2.

The solar al t i tude (~t) is given as

sin ~t = sin L sin 6 + cos L cos 6 cos h (2)

In tegra t ing equa t ion (1) with respect to t ime f rom sunrise to sunset yields the daily total extraterrestr ia l rad ia t ion (Hal) on a hor izonta l surface.

213

214 HAWAS ^ND MUNEER: INSOLATION ON TILTED COLLECTORS

/ f

o J ( \

N

oJ •

S

o

\ e ~ t . . . . . .

i I I I I I I I I I

" s x

%

I I I I

Fig. 1. Location of the cities considered in the Libian Jamahirya.

24 R 2 / Hot = --~ ( / R o ) Isc ~eos L cos 6 sin hm

2nhm \ + ~ sin L sin 6 ) . (3)

The ratio of the sun--earth distance (R) to its mean value (Ro) can be expressed as a function of the day of the year (n) as

(R/Ro) ~ = 1 + 0.033cos(360n/365) (4)

Also the declination (fi) can be approximately expressed as

6 = 23.45 ° sin(360(284 + n)/365) (5)

Table 1. Recommended days for calculating the monthly average daily radiation

Month Day of the year (n) Date

Jan. 17 17 Jan. Feb. 47 16 Feb. Mar. 75 16 Mar. Apr. 105 15 Apr. May 135 15 May Jun. 162 11 Jun. Jul. 198 17 Jul. Aug. 228 16 Aug. Sep. 258 15 Sep. Oct. 288 15 Oct. Nov. 318 14 Nov. Dec. 344 10 Dec.

! The sunrise hour angle (h~) is

hm = arcos ( - tan L tan 6) (6)

Using expressions (4), (5) and (6) total daily extrater- restrial radiation can be conveniently estimated from equation (3)for any day. The 15th or the 16th of each month is mostly used for the estimation of the monthly average values [1, 2]. However, rec- ommended days, for which the extraterrestrial radi- ation is nearly the same as the monthly average, are shown in Table 1 [3]. These recommended days are used in the calculation of the monthly average extra- terrestrial radiation (Ho) in this work.

Monthly averaoe daily terrestrial radiation on a hori- zontal surface

Standard models are available for the estimation of the clear sky instantaneous radiation. According to the model used in this work [4], the clear sky instan- taneous radiation on a horizontal surface is

H = IoN(C + sin ct) (7)

where

ton = A/exp(B/sina) (8)

The values of the constants A, B and C depend on the month. The total daily radiation has been obtained by numerical integration in steps of 15 min from sunrise to sunset. The recommended days have been checked for clear sky terrestrial radiation and were found to give conveniently accurate results. For the estimation

HAWAS AND MUNEER: INSOLATION ON TILTED COLLECTORS 215

Table 2. Monthly average daily insolation (105 J/m 2 day)

Tripoli b = 10 b = 20 b = 30 b = 4 5 b = 90

Month H0 1 194 2 242 3 302 4 359 5 396 6 410 7 402 8 373 9 322

10 259 11 205 12 180

~ 7 Rb 7 Rb 7 Rb 7 gb 7 101 1.29 120 1.54 136 1.75 149 1.95 162 1.76 144 135 1.20 154 1.36 168 1.49 179 1.59 186 1.24 149 164 1.11 176 1.19 185 1.24 188 1.23 186 0.74 125 195 1.04 200 1.05 200 1.03 197 0.94 183 0.35 97 231 0.99 229 0.95 223 0.89 212 0.75 188 0.15 82 237 0.97 231 0.91 222 0.84 208 0.68 181 0.08 74 255 0.98 251 0.93 242 0.86 228 0.71 199 0.11 80 234 1.02 237 1.01 234 0.97 227 0.85 207 0.25 98 186 1.08 196 1.13 202 1.15 203 1.11 196 0.56 120 146 1.17 163 1.30 176 1.40 185 1.46 189 1.07 144 110 1.26 130 1.49 146 1.67 158 1.85 170 1.61 147 93 1.32 112 1.60 129 1.83 142 2.07 156 1.93 142

of the monthly average insolation on a horizontal sur- face under normal sky condition the following well known formula has been used

H = Hcs(al + btS) (9)

where Hcs is the insolation under clear sky conditions and S is the ratio of bright sunshine hours to the

length of the day (sunrise to sunset). The values of at and bl are taken as 0.35 and 0.61 respectively [5].

The values of S were calculated from sunshine records during the period 1960-1969 reported by the meteorological Department in Tripoli [5].

Monthly averaffe daily insolation on tilted surfaces

The solar radiation incident upon a tilted surface is composed of the direct beam radiation, the sky diffuse radiation and that part of radiation reflected from the ground"

Month Ho 1 198 2 246 3 305 4 361 5 396 6 409 7 402 8 373 9 324

10 263 11 209 12 184

Table 3. Monthly average daily insolation (105 J/m 2 day)

Benghazi b = l0 b = 2 0 b = 30 b = 4 5 b = 9 0

Rb 7 gb 7 gb 7 Rb 7 Rb 7 110 1.28 131 1.53 149 1.72 163 1.92 176 1.71 156 131 1.20 148 1.35 161 1.47 171 1.56 177 1.21 140 167 1.11 179 1.19 187 1.23 191 1.22 188 0.72 125 200 1.04 205 1.04 205 1.02 201 0.93 187 0.33 98 240 0.99 238 0.95 231 0.89 219 0.75 194 0+14 82 241 0.97 236 0.91 226 0.83 211 0.67 183 0.08 73 262 0.98 257 0.93 247 0.85 233 0.70 202 0.10 79 242 1.02 244 1.00 242 0.96 234 0.84 212 0.24 98 194 1.08 205 1.13 211 1.14 212 1.09 204 0.55 123 149 1.17 165 1.30 178 1.39 187 1.44 191 1.04 144 121 1.26 143 1.48 161 1.65 175 1.82 188 1.57 162 92 1.31 110 1.58 125 1.81 137 2.04 149 1.88 135

Month Ho 1 210 2 256 3 312 4 364 5 396 6 407 7 401 8 375 9 330

10 272 11 220 12 196

Table 4. Monthly average daily insolation (105 J/m 2 day)

GHADAMES b = 10 b = 2 0 b = 30 b = 4 5 b = 9 0

Rb 7 Rb 7 Rb 7 Rb 7 ~ 7 128 1.26 152 1.49 173 1.66 189 1.84 203 1.60 178 160 1.18 180 1.33 197 1.43 209 1.51 216 1.13 169 180 1.10 192 1.17 200 1.20 203 1.18 199 0.67 129 208 1.03 212 1.03 212 1.00 207 0.90 191 0.30 96 238 0.98 235 0.94 227 0.87 215 0.72 189 0.11 77 244 0.96 237 0.90 226 0.82 211 0.65 181 0.05 69 258 0.97 252 0.92 241 0.84 226 0.68 195 0.08 74 241 1.01 242 0.99 239 0.94 230 0.82 207 0.21 92 197 1.07 207 1.11 212 1.12 212 1.06 203 0.50 119 162 1.15 179 1.27 192 1.35 201 1.39 204 0.97 151 132 1.24 155 1.44 173 1.60 187 1 .74 200 1.47 170 117 1.29 140 1.54 160 1.74 177 1.94 192 1.75 172

216 HAWAS AND MUNEER: INSOLATION ON TILTED COLLECTORS

Table 5. Monthly average daily insolation (10 s J/m 2 day)

JALO b = 10 b = 2 0 b = 30 b = 4 5 b = 9 0

Month Ho H Rb 7 Rb 7 Rb 7 Rb 7 Rb 7 1 215 122 1.25 143 1.47 161 1.64 175 1.80 187 1.55 161 2 261 155 1.18 174 1.31 189 1.41 199 1.48 206 1.10 158 3 316 186 1.10 198 1.16 206 1.19 209 1.16 204 0.65 131 4 366 201 1.03 205 1.03 204 0.99 198 0.89 182 0.28 91 5 396 234 0.98 230 0.93 222 0.86 210 0.71 184 0.10 75 6 406 248 0.96 241 0.90 229 0.81 213 0.64 181 0.04 68 7 400 261 0.97 255 0.91 244 0.83 228 0.67 195 0.07 72 8 376 247 1.01 248 0.98 244 0.93 234 0.81 210 0.19 91 9 333 225 1.07 236 1.10 242 1.11 243 1.05 232 0.48 132

10 277 165 1.15 182 1.26 195 1.34 203 1.37 206 0.94 150 11 226 137 1.23 159 1.43 177 1.58 192 1.71 2 0 4 ! 1.42 171 12 202 115 1.28 136 1.52 155 1.71 170 1.90 183 1.69 162

Table 6. Monthly average daily insolation (10 5 J/m 2 day)

SEBHA b = 10 b = 2 0 b = 3 0 b = 4 5 b = 9 0

Month Ho 1 228 2 272 3 323 4 369 5 395 6 403 7 398 8 378 9 339

10 286 11 238 12 215

141 1.23 165 1.43 185 1.59 200 1.72 213 1.44 180 172 1.16 192 1.29 208 1.38 218 1.43 224 1.02 169 199 1.09 212 1.14 219 1.16 221 1.13 215 0.60 133 207 1.02 210 1.01 208 0.97 202 0.86 184 0.24 88 236 0.97 232 0.92 223 0.85 209 0.69 182 0.07 71 253 0.95 244 0.88 231 0.79 213 0.62 180 0.02 64 264 0.96 256 0.90 244 0.82 226 0.65 193 0.04 67 248 1.00 248 0.97 242 0.92 232 0.78 206 0.16 86 204 1.06 213 1.09 216 1.08 215 1.02 204 0.44 113 171 1.14 188 1.24 199 1.30 207 1.33 208 0.87 148 148 1.21 171 1.39 189 1.53 203 1.64 215 1.32 177 127 1.26 150 1.48 169 1.65 184 1.82 198 1.57 172

7 = ID + Ls + la , (10)

The direct component (in) is related to the direct component on a horizontal surface (HD) by the tilt factor (Rb)

-Rb = ID/HD (11)

Actually (Rb) depends on the atmospheric cloudiness, water vapor and particulate concentration. However, it was suggested ['6] that (Rb) can be estimated to be

the ratio of extraterrestrial radiation on a tilted sur- face to that on a horizontal surface for the month. For surfaces facing south

cos (L - b) cos 6 sin h~ + (n/180)h" sin (L - b) sin 6

and

cos L cos 6 sin hm + (n/180)hm sin L sin 3 (12)

h,, = min(hm and arcos ( - tan(L - b) tan 6) (13)

Table 7. Monthly average daily insolation (10 s J/m 2 day)

KUFRA b = 10 b = 2 0 b = 2 5 b = 4 0 b = 9 0

Month Ho H Rb ? Rb 7 Rb 7 Rb 7 Ru 7 1 243 154 1.21 177 1.39 197 1.46 205 1.61 221 1.31 183 2 285 181 1.15 201 1.26 215 1.30 221 1.37 229 0.93 166 3 332 206 1.08 217 1.12 223 1.13 224 1.11 220 0.53 128 4 372 219 1.01 221 1.00 217 0.98 214 0.87 197 0.20 85 5 393 231 0 . 9 7 226 0.91 216 0.87 209 0.72 183 0.04 65 6 399 251 0.95 242 0.87 227 0.82 218 0.65 186 0.00 60 7 395 261 0.96 252 0.89 238 0.84 229 0.68 197 0.02 62 8 379 253 0.99 252 0.96 245 0.93 239 0.81 215 0.12 79 9 345 219 1.05 227 1.07 229 1.07 229 1.01 219 0.38 112

10 298 188 1.12 204 1.21 216 1.24 220 1.28 224 0.79 153 11 253 166 1.19 190 1.35 210 1.42 218 1.54 233 1.21 188 12 231 142 1.23 166 1.43 186 1.51 194 1.69 211 1.43 181

HAWAS AND MUNEER: INSOLATION ON TILTED COLLECTORS 217

en

.=,

e~

"6 [- 06

d ~

[-

II

II

I

t~

, .d

. ~ ~

. ~ ~

o a o o o o ¢.r~ ¢ q t 'q ¢'q

~ t ~ Q 'x:~ . .~ ¢=

The values of (Rb) have been calculated at the rec- ommended day in each month for various tilt angles (b)

The direct component (Io) is then

7 O = (H - Ha)R b

= H(I - Ka)Rb (14)

where Kd is the diffuse ratio defined as

Kd = Hd/H (15)

The diffuse ratio was found by many investiga- tors [7, 8, 9] to depend on the cloudness index (KT) defined as

K,T = H/Ho (16)

and many relations are proposed. The relation exten- sively used is that suggested by Liu and Jordan [7]

Kd = 1.39 -- 4.027 KT + 5.531 12 _ 3.108 13 (17)

The diffuse component on a tilted surface can be calculated, assuming isotropic diffuse radiation, by

7d, = Hd(l + cosb)/2

= KdH(1 + cosb)/2 (18)

and

I d g ~ " p g n ( g - - c o s b)/2 (19)

RESULTS AND DISCUSSION

The main object of this work is to calculate and present the values of insolation incident upon solar collectors situated in different cities in Libya. The col- lectors face south and are tilted at various angles to the horizontal.

Normally measurements are made for the insola- tion on a horizontal plane; the insolation on a tilted surface is then calculated using the measured horizon- tal insolation.

In the absence of measured data, the insolation on a horizontal surface can be conveniently estimated for actual sky conditions subject to the availability of cloud cover data [2]. Different standard models are available in the literature.

The values of (H) in the present work have been calculated through the numerical integration of the instantaneous insolation for a clear sky condition (Equation 7) and then corrected for actual sky con- ditions using the measured mean-monthly average bright sunshine hours.

The values of (H) for the various cities are shown in Tables 2-7.

Insolation available on tilted surfaces of various angles of tilt (b) is shown in Tables 2-7. The calcula- tions are made for surfaces due south according to the accepted practise of locating fiat plate solar collectors in the northern hemisphere, due south.

218 HAWAS AND MUNEER: INSOLATION ON TILTED COLLECTORS

300

280

260

240

zZo

i 180 160 140 12(3

80

4O 20 ~ =i ~ =

/ S

Month

Fig. 2. Seasonal variation of insolation for different tilt angles (Benghazi).

'30C

b 28£

x,,, ', x - - - x ' - '~-~ \ 4 ' , / , / / / '"

-8~ ,oo

__~ 60

heating season coding season ~;

I 0 N D d F M A M J J A S S

( E~e~o)/2 Mean value= 5 6 5

ONDJFMAMJJAS Month

Fig. 3. Combined insolation on horizontal and south-fac- ing vertical surfaces.

The reason for choosing the considered values of the tilt angles (b) is as follows. Tilt angles of 10 and 20 ° are chosen since these are conservative roof slopes, 30 ° since this is the average latitude of most of the cities considered, 45 ° due to the reason that lati- tude plus 15 ° is recommended for heating systems and 90 ° since the south wall can also be used as an ad- ditional solar collector•

Table 8 shows the total annual insolation. It can be seen that tilt angle of 30 ° (latitude) gives the maxi- mum total annual insolation, between 20 and 30 ° is very small (1%).

The seasonal change of monthly-average insolation is shown in Fig. 2. Three values of tilt angle, namely 0, 20 and 45 ° are considered. It can be seen that for winter heating systems a tilt angle of 45 ° (Lati- tude + 15 °) is the optimum in the sense that it col- lects the maximum amount of insolation, while a horizontal collector is the optimum for summer cool- ing systems. Fig. 3 shows the variation of insolation for a combination of horizontal and south facing ver- tical surfaces. This combination gives a more or less uniform insolation tbrouglaout the whole year. l'lats

suggests that horizontal roof-south wall combination gives nearly uniform insolation throughout the year.

It should be noted that the ground reflectance was taken as 0.2 which is the most suitable value for Libyan soil.

REFERENCES

[1] B. Y. H. Liu and R. C. Jordan, availability of solar energy for flat plate solar heat collectors, ASHRAE, Low Temperature Engineering Application of Solar Energy (1967)•

[2] G. W. Paltridge and D. Proctor, Solar Energy 18,(3) 235-243 (1976).

[3] S. A. Klein, Solar Energy 19,(4) 325-329 (1977). [4] ASHRAE Handbook of Fundamentals (1976). [-5] S.E. Hassan, Solar radiation. Meteorological Depart-

ment. Tripoli Libya, Technical Note No 4, (1972). [-6] B. Y. H. Liu and R. C. Jordan, Solar Energy 4,(3)

(1960). [-7] B. Y. H. Liu and R. C. Jordan, Trans ASHRAE

526-541 (1962). [-8] J. K. Page, Proc. UN Conf. on New Sources of

Energy, Paper No. 35/5/98 (1961). [-9] N. K. O. Choudhury, Solar Energy 7,(2) 44-52 (1963).