.%dar & Wind T(,(4mok~g.v Vol. 2. No. 2, pp. 119 130, 1985. 0741-983X/85 $3.00 + .00 Printed in Great Britain. Pergamon Press Ltd.

MONTHLY AVERAGE DAILY INSOLATION FOR HORIZONTAL AND INCLINED SURFACES

S. M. ABUGHRES Centre for Solar Energy Studies, NASR, Tripoli, P.O. Box 8004, Libya (SPLAJ)

(Received 8 December 1984; accepted 8 March 1985)

AI1ract--The use of more readily available meteorological data in radiation models has always been a limiting factor in many studies. The current paper presents a model originally based on cloud amount to calculate the monthly average daily insolation on horizontal and inclined surfaces___ The cloud amount is replamd by fractional pemible sunshine in accordance with the findings of other research workers. Existing sunshine and radiation records for 10locations are used to estimate the prevailing atmospheric transmittance associated with this type of model. Effects of muldple rellection between ground-surface and atmosphere are also incorporated.

The available data are used for correlation purposes employing Root Mean Square Error (RMSE} as a measure of the model performance. This yielded very satisfactory results, especially for long averaging period simulation. On the average, discrepancies between computed and observed monthly average daily total radiation of lest than 5e/~ and corresponding RMSEs ranging between 2.1 and 3.6~ based on an a~;eraging period of I year are considered to be within the accuracy associated with instrumental measuztments.

Additional teat of the model validity results in calculated ratios of diffuse to total radiation on a horizontal . surface that are in reasomble agreement with well-established methods. However, in view of the low ranking

accuracy of the measuringequipment and the rather short time records used in the analysis, further verification of the model will still be required as more reliable data become available.

INTRODUCTION

A prior requirement to the design e r a solar conversion system and the assessment of its potential, is a knowledge of the insolation data on both a temporal and a spatial basis~ Very often these data are available for a few locations only and, even in those places, they may be either short time records that makes them statistically unreliable or, missing for appreciable periods of time due to the malfunctioning of instruments. Calculation procedures for these data are therefore required to fill temporal gaps at stations where measurements are made and to fill spatial gaps where measurements are not made.

Several mathematical models developed for calcu- lation p u ~ have been reported in the literature [ 1- 10]. Utilization of these models is normally confined to those which use readily available meteorological data. A model, which has now become widely acx~pted, is ba,.~xi on total cloud amount and atmospheric transmittance in addition to incorporating repeated reflection effects between ground surface and atmos- phere, in the absence of cloud amount records, fractional possible sunshine data can be used instead. This is supported by the findings of Hoyt [1 I] and Hay [Z6] who showed that, using long-term averages,

sunshine data can give an estimate of cloud amount which is more compatible with satellite and aircraft observations than does the value reported by a ground- based observer.

In view of the locally available meteorological data, the total cloud based model--with the justification introduced by Hoyt and Hay-- i s used in the present study. The model is incorporated in a mathematical formulation to calculate the monthly average daily insolation on horizontal and inclined surfaces. Repeated reflections coefficients are estimated in accordance with the method proposed by Davies and McKay [10]. Coefficients of atmospheric transmission are empirically determined through a regression technique, using monthly averages of sunshine and radiation data available for a number of locations in Libya. The resulting coefficients are then used to test the model performance employing the root mean square error as a performance measure. A further validity test is carried out by means of comparing the present method of calculating ratios of diffuse to total horizontal radiation with those proposed by Page i l l and Liu and Jordan [12]. Monthly average daily insolation on inclined surface is determined using Liu and Jordan's approach i ! 3].

119

120

MATI IEMATICAL FORMULATION

Terrestrial radiation model The monthly average daily terrestrial insolation on a

horizontal surface is calculated using a total cloud based model. According to Monteith [2], the model may be constructed as follows ;

The direct beam radiation component under normal sky conditions (/'), is expressed by ;

t = t o o - C a ) . (I)

The diffuse radiation component (/)) is given as the sum of diffuse radiation from blue sky, radiation transmitted through cloud and radiation after reflection between ground surface and cloud base

O = ~ , ( I - -CA)+[ io .~ .CA+Ft .pg .p , .CA . (2)

Upon adding the direct and diffuse components given by equations (1) and (2) above;

I--i = Fi , ( ( I - -CA)+T'CA)/ ( I -pg.p , .CA) . (3)

The ground surface and cloud base retlectances appear in the equation as a product which could be expressed b y a single combined reflectance (p) to account for the repeated reflection effects. Also, according to Hoyt [11] and Hay [6], the total cloud amount (CA) could be substituted for by the fractional possible sunshine (n/N) through the following relationship;

CA = 1--(n/N). (4)

Thus using the combined reflectance and fractional possible sunshine as given by equation (4), equation (3) becomes;

( , , , /7 =/70 r+ (1 - - z l - - p ) + p - n

Equation (5) is normally written in the following general form;

/7 = / t o - ~b -f (6)

where (~) and (f) are the transmittance and albedo functions respectively.

The ratio of diffuse to total radiation on a horizontal surface (/~/f/) is given from equations (1) and (4) as;

= t~ u (7)

on introducing the cioudness index (/( = H/H~) as defined by Liu and Jordan [12];

0 To 1 n - - = I (8) H, H K N

where/t~ and [o are, respectively, the monthly average

S. M. AItUGIIRI~:'I

daily values of extraterrestrial and terrestrial clear sky direct beam radiations on a horizontal surface.

Results obtained from this expression are compared with those calculated from the relationships proposed by Page [ I ] and Liu and Jordan [12].

These relationships are :

= fl '00-1-13K" (Page) (9)

/f [1 .390- 4-027/( + 5-53 l K'z - 3-108/~ 3 (Liu and Jordan) (10)

Extraterrestrial and clear sky terrestrial radiations The daily total of extraterrestrial radiation on a

horizontal surface on day (d) of the year, starting January the first, is given by :

(H,),=2~4nS[l+O'33cos(~65d)]

x (cosL.cos f i . s inh , + ( 3 - ~ h , ) s i n L - s i n 6 ) (l l)

where (S) is the solar constant taken as 1353 W m- z, (6) is the solar declination angle which can be approximately expressed as :

/i = 23-45 sin (360(284 +d)/365) (12)

and (h,) is the sunrise hour angle, given as ;

h, = areos ( - tan L- tan 6). (13)

On substituting (6) and (h,) from expressions (12) and (13) in equation (11) and knowing the local latitude angle (L), the daily total extraterrestrial radiation on a horizontal surface can be calculated. The monthly average daily value (//,) is given by the expression;

l d~d2

/f . = ( d 2 - d l ) , x (He) , (14)

where (d t) and (d2) are, respectively, the days of the year at the start and end of the month.

The monthly average daily terrestrial direct beam and total radiations on a horizontal surface under clear sky conditions are calculated, in accordance with reference (14), using the following relationships;

The clear sky instantaneous direct beam and total radiations on a horizontal surface are, respectively;

lo = A" sin g/exp(B/sin ct) (15)

and Ho = A( C + sin ot)/exp( B /sin ct) (16)

where the solar altitude angle (~) is expressed as

sinct = sin L-sin 6+cos L- cos ,6 - cos h (17)

Monthly average daily insolation for horizonlal and inclined surfaces

and the values of the coefficients (A), (B), and (C) are given in a tabular form in reference (14).

The monthly average daily values of clear sky direct beam and total radiations on a horizontal surface are respectively, obtained by numerical integration of equations (15) and (16) in small steps of time* from sunrise to sunset on the recommended day of each month for which the above coefficients are given.

Monthly average daily radiation on inclined surfaces The monthly average daily total radiation on a tilted

surface((3} can be estimated by individually considering the direct beam, diffuse, and reflected components of the radiation incident on the tilted surface. Assuming diffuse and reflected radiation to be isotropic, Liu and Jordan [13"i have proposed that ¢~ can be expressed as;

d = ( R - ~ ) ~ + 6 ( I +cos ~)/2

+R.p,(l-cos ~/2 (18) where(~)is the tilt factor defined as the ratio of monthly average daily direct beam radiation on the tilted surface to that on a horizontal surface, ~ ) is the angle of inclination of the surface from the horizontal, and (Ps) is the ground reflectance ranging from 0-2 to 0-7 depending, upon the extent of snow cover. For the ground surface in Libya, a reflectance of 0-2 is a most likely value. Liu and Jordan [13] have also suggested

121

that/~ can be estimate, d to be the ratio ofext ratcrrestrial radiation on the tilted surface to that on a horizontal surface for the month. For surfaces facing directly towards the equator;

Khj /~___ cos ( L - ' ) cos J" sin h" + G-8"0)sin ( L - fl' sin 6

{nh, \ cos L'cos~'sinh, +~]-~)sin L.sin~

(19)

where (h,) is the sunset hour angle for the tilted surface which is given by;

h, = min {h,, arcos [ - tan ( L - ll) tan J]}. (20)

MODEL ANALYSIS AND RESULTS

Model analysis The analysis of the model was carried out using

radiation and sunshine records available for 10 widely separated locations in the country as shown in Fig. 1. The climatic conditions prevailing in the country vary considerably from one place to another. In the northern region, there are three types of climates; mediterranean climate as in Shahhat, semi-mediterranean climate as in Tripoli, and semi-desert climate as in Sirt, Ajdabiyah and Tubruq. The rest of the country has a

Med. Sea

Ghadames Jaghbub

• Sabha

Ghat L I B Y A . Kufrah o

(S.P.L~A.J.)

-W'-.-_

Fig. I. Distribulion of the locations selected for Ihe present study.

* A lime step of 4 rain was u.s~ in the pre~nt Study. "

122 S.M.

predominantly desert climate except a small region at the mid-southern frontier which has an unclassified mountain climate. I n the presentanalysis, the country is broadly divided into two main climatic regions; a coastal region overlooking the mediterranean and an inland region constituting the rest of the country. This broad classification came as a result of the following analysis.

The analysis of the terrestrial radiation model requires a knowledge of the combined ground surface- atmosphere reflectance and cloud (atmospheric) transmittance. The former was estimated according to a relationship proposed by Davies and McKay [10]. Neglecting aerosol effects, the relationship gives the atmospheric reflectance as the sum of components due to molecular scattering, assumed to apply only to the cloudless portion of the sky, and the product of average cloud albedo and total cloud amount. The molecular scattering effects have been accounted for by Lacis and Hanses [15] as equivalent to a clear sky reflectance of 0~685. The average cloud reflectance has been taken by earlier investigators [6, 10] as equal to about 0-6. Hence, assuming a ground surface reflectance of 0-2 as a most likely value for the Libyan soil, the combined reflectance can be expressed as;

I I p = pg(0-0685(1 - - CA) + 0-6CA) = 0-12 - - 0.1063 ~ .

(21) Following Hay [6],it was decided in the beginning to

estimate the cloud transmittance by means of correlation and regression statistics for relationships between fractional possible sunshine (n/N) and atmospheric transmissivity, Y, defined as; (H/He) ((1 - p) +p(n/N)). Preliminary attempts to find a single value

A i~t J(;H RI-";

for the transmittance failed since it appeared that there is more than one value for the ten investigated locations as it could be noticed from the graphical presentation in Fig. 2. A careful examination of the graph reveals that the cloud transmittance in the ingland region is higher than that in the coastal region. In addition, the transmittance appears to decrease with decreasing fractional possible sunshine which suggests a seasonal change in its value.

It is a known fact that the transmission of sunlight is affected by the length of the sun path or air-mass and obstacles in the way. An investigation by Haurwitz [16], showed that the transmittance could be exponentially expres~d in terms of the air-mass with constant parameters depending on the type of obstruction. His tabulated findings provide simple means ofcalculating the transmittance once the types of clouds and other obstructions are known. In its general form, the expression he arrived at may be written as follows ;

v = a" e - b,, (22)

where a and b are constant parameters which depend on the type of obstruction and m is the air-mass.

However, as the relevant data were not available, the transmittance parameters were estimated using the correlation between air-mass and the variable In [Y -(n/N)]/[1--(n/N)] derived from equations (5) and (22). Results of the correlation are presented graphically in Fig. 3 giving parameters' values which depict transmittance changes both regional-wise and seasonal-wise. It may be noted that, although, the seasonal changes amount to only about 10~, the increase in the transmittance from the coast to the

1.0

O.8

Y

0 . 6

O.h

o

.h

! i

.

× o ~ , ~ Ca) Coast

o • ; T r i p o l i

o ; S h a h h a t

x ; S i F t

; T u b r u q

• ; A j d a b i y a h

l i

0 . 6 0 .8

.0

.8

D.6

D. 14

.0 .h

( b ) I n l a n d

• ; K u f r a h

o ; S a b h a

x ; ( ] ha t

~ ; G h a d a m e s

• ; J a g h b u b

i i

0.6 0.8 .0

I I / N IIIN

trig. 2. Relalionship between fractional possible sunshine(n/N) and atmospheric transmissivily Y delincd as : (It / l l . )(( 1 - p ) + p ( . / N ) ) .

n ( Y - '~ )

l1 ( i - ~)

Monthly average daily insolation for horizontal and inclined surfaces

M o n t h l y ave, r a g e ' a i r m a s s , m

-! .0

-2.0

2.0 , i i ,

,~,,. ,~ o ,o " , ,o %,

eX • X •

3.0 t I

( a ) C o a s t

a = 0 . 3 3 3

b = 0 . 1

123

| n

n (Y- ~)

n ( 1 - -~)

- I .0

-2 .0

M o n t h l y a v e r a g e a i r m a s s ; ni

2.0 i . l i t

o o

. 3r 0

(b) lellaeld

a = 0.606

b = 0.1

Fig. 3. Graphical determination of the constants (a) and (b) in t = a- e --~m'.

inland region is almost 100%. This, perhaps, would be expected due to the marked diference in the climate and in the cloud types and other obstructions.

P erfocmance and results The data used to verify the model were collected

during 1981 at the locations selected for the present

study. Exceptions are those of Tripoli and Kufrah where averaged records of 8 and 5 years ~ v e l y were available. The sunshine and radiation records were both obtained using CampbelI-Stockes Recorders in conjunction with 3087/TI Casella Bimetalic Actinographs. The relatively large mass of the bimetalic strips in the actinograph results in long

- 8

o

"o 6 o ~ ° T r i p o l i 3.5~5 ~ si f t 3.6~

~- ~ A jdabiyah 2.3~ ~ S h a h h a t 3~%

m 4 ~,,~e Tubruq 3.0~; >,,- ,c~ E -~ ~ ~ Ghadames 3- I

~ ~ Sabha 3-4% E

2 ~ " G h a t 2 . 4 ~ ~ ~3 / Kufrah 2.6~

/ 2 -- "o

t. 0 ' * i . t t

2 4 6 8 )0

Measured f~nthly average daily total radiation KW.hr/m 2 day

Fig. 4. Comparison of calculated aod measured monthly average daily total radiations on a horizontal surface.

124 S . M . AI~UGItRI-LS

>. ¢D T9

~o J= 8 >

>.. .-_ g

0J -c~

~ 4

R.M.S.E. /

• - 5-5;' / /

x - 3 . 2 : ' o ~ o o o o

° ~ ) J~e

/ , ~ o o o-Tripoli

~¢" x- Ku f r a h C" ° , - ~ ,

6 8 Io

Measured daily average total

radiation KW.hr/m 2 day

r0 o

5 > e0

~ 4

- u , , ° ,

R.M.S.f. o - 8 . 5 : ' - - / ~ x

o o o

o - ] ~ - i H o l i

x-Kul r~ i I i

l 2 3 4 5

Measured d a i l y average total radiation KW.hr/m 2 day

Fig. 5. C o m p a r i s o n o f c a l c u l a t e d a n d m e a s u r e d d a i l y a v e r a g e to ta l r a d i a t i o n s o n a h o r i z o n t a l surface .

time response and thus low sensitivity. Also, due to aging and lack of periodic reealibration, the accuracy of the instruments is not highly ranked. This together with the human error associated with reading offthe charts make the data rather unreliable for analysis based on short averaging periods.

The performance of the model in simulating downwards solar radiation at the earth's surface for

periods of a day or longer was measured using the Root Mean Square Error computed by;

R M S E = y" ( P - - M ) 2

where P and M are the calculated and measured radiations, and x is the number of comparisons. The parameters P and M are both expressed as a percentage

g

L ID

--- O

>. >.

g t _

g c c O O :E

t l

0.8

r .6

0.4

13.2

4

~ Liu and Jordan

~ Page

~ ~ i n t model

I

0.2 :a !~ 9.6 0 o

;kmlhl : a v e r a q c dai Iv trJza] rad ia l i tm

E x l r a l , . , r e : ~ : ~ ; . I d n i l y i n s ~ J l a t i o n

l I 0

I:ig. 6. Comparison of mclhods to ¢ompulc she rali~ of monlhly average daily values of diffuse to tolal f ~ l d i ; l l i O I I S t ) l | ; I [ lo t ' iZOl | l . : . l l surface.

Monlhiy average d~il}" ir~so|~ition/'or horizo.~al and incli~',:.d surfaces 125

o

0 Z

"0 0

I

o

"0

L

I _

b~

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. i o

I-/'% Ir'PI N ~ .,.~" ~ ..,,1" ~ I'.~.

Z

O

II

...I

0

L.

i - -

" - - 0

. J

o ~ n ~ ~ . . ~ - r ~ , . ~ , ~ - ~ - .

• o . . . . . . ,

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• . , • . . . . . .

41" h i P . N ~ I~J N ~ C~

-.-1" 0 ~ 0 0 O~ I..~ ;"'.- ( ~ -~r ,,,,o

~ n

f ' r " u E ~

I I ~ " I I I I I

I ' T ' l ' l r " ~ . . I ~ -..I ( ~

O0 f "~ i ' l l r".,,, t,,.r~ ,p"~ (~J ~ M ~

I . . . . . . • • .

-=r o - t i,.ir~ oO ,,,~D O ~ N o ~ • o o o . o o • ,

¶

C~ l i t U't 0 - - 3 " ~ t ~ ~ t ~

. . . . . , . o .

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I I "1 - I I I I I

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i

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t r ~O ' t --1" I ~ N

r,"'l ~,.,f~ ,,,£)

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U ' ~ N N

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I : :E U ::IE

;'~'~O° 0 ~ ~'~ O O ' ~

g! " t " I I l ! !

C~l -.-~ (~E:Z --.1 (]E:L 'L-J

126 S . M . ABU,~;nKI~S

i

0

~J

0

O .

Cn

Z ~ Z ~ Z

0 0 , 0

• " l i I I I U

oi

to

JO u s ' ' ' ~

k .

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0

C

" 3

• . ! . , ! o ,

~ o , . o ~ o o r - . , . o ~ ~ ~ r - , , . o , . .r, ,- , ' , , N c o , . o o o , . c , , . r , ~ ~ . . ~ o - i • - ! • •

U "1" I1 II I II 4" U U I I I "1" I I I I I

(~ l ,~ '~ ~ ~ . - - / I - I r ' l ' l " C~ I . , . J (~L.--I ( ~ L j i ~ l " i - : C~._J I ~ _ _ (~-__J

Monthly average daily insolation for horizontal and inclined surfaces 127

1

U 0

> 0 Z

o

0

Z 0

-:3 - - ) • ; ( ~

" I I r " . , J

E

I -

r,~

E

E 0

o - l i ~ l ( . ~ l i ~ l ( . ~ l ~ l ( = ~ tJ ,.n

| " l r I "T" ~ * J ~ . ~ l

Z 0

~ 0

II . J

O"

I .

0 h -

l~'tl.~Gr~ oo o ~ o o ~ o

~ ID~ ; I TM ~0-.~- .~'.Q~ . ~ r t ~ Z

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o

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( 3

U

II + II I1 I i ' r - I " r CO..=./ ~ ,=.=/ ~ - J

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~ r ~ l ~ O L ~ I O O l ~ l 0 ~ 0 0 ¸

• o,

I ~ 0 0 0 0 0 O 0 ~ r o j

o .

i ' r " ~ 0.1 U ~

I ~F-I "T- I ~ v m I I + II II I

CO. --J ~00 . - - . I ~ . J

128 S . M . AI~UGHRI~- ~;

° ~

1-

u

o

0 z

u

0

e~

.

t -

z 0

I'M c 4

°

cM

i _

o

C:

--.)

C

0 ~Z

O'~-:t" I ~ C"~-..I" C'v'~ ~ ~

"*.0 -:3- -~, ~" ~ ~.0 ~ ~,D ~L#'~

U ~rl

II I I ! I ' I ' f ZZZ C~.__J II _.J fl~__l

Monthly average daily insolation

of the measured daily radiation averaged over the prescribed period.

Simulation results are shown in Figs. 4 and 5 with the RMSEs calculated using averaging periods of 1 year and ! month respectively. The correlation between calculated and measured monthly average daily total radiations, Fig. 4, appears to be very good. On the average, monthly deviations between calculations and measurements are within 5~o. The RMSEs range from 2" I to 3-6~ suggesting reliable model performance for long period simulation. As the averaging period is

for horizontal and inclined surfaces 129

that a tilt angle equal to local latitude gives maximum annual radiation, whereas a tilt angle equal to local latitude plus 15 ° and minus 15 ° are more suitable for winter and summe.r applications respectively.

CONCLUSIONS AND RECOMMENDATIONS

in general, t he model performed quite well, especially for long period simulation. Deviations between computed and observed monthly average daily total radiations of less than 5~o are within the accuracy

reduced to I month, Fig. 5, the RMSEs increase but at a/~'generally associated with instrumental measurements. somewhat relatively low rate. Detailed studies [4,10-1of More reliable correlation will be possible when long- the relationship between RMSEs and averaging periods, showed that the RMSEs for total radiation decrease quickly as the averaging period increases from 2 to 20 days and then more slowly for longer periods.

Figure 5 also shows the simulation capability of the model during the summer season and during the winter season. During summer (Fig. 5a) the maximum daily deviations between computed and measured daily average total radiations are within 9 and 12~o whereas the RMSEs are 3-2 and 5-5~o for Kufrah and Tripoli respectively. Corresponding values during winter, Fig. 5b, are 13 and 25~o for deviations, and 4-6 and 8-5~ for R MSEs. The rather lower model performance in winter may well be attributed to the greater frequency of overcast conditions and the greater variability of sky conditions during this season specially in the coastal region.

The model was also tested against other methods developed for estimating ratios of diffuse to total radiations on a horizontal surface. Many methods have been reported in the literature, the most commonly used are those proposed by Page [1], and Liu and Jordan [12-1. Using data available in the 10 selected locations, the present model was used to calculate these ratios and the results are compared with the other two methods in Fig. 6. Although it was not possible to cover the whole scale, the limited results appear to be in reasonable agreement with these two methods.

A practical requirement in many solar energy applications is a knowledge of the monthly average daily total radiation on inclined collectors. Table I gives the results of calculations, using the relationships proposed by Liu and Jordan [:131, for the 10 selected locations. The table also gives the measured and calculated monthly average daily total radiations on a horizontal surface obtained from the present model. Three tilt factors and their corresponding radiations are given for each location. These are intended to cover preliminary design requirements for seasonal and annual solar energy applications. Although these factors were not prolmrly optimized, it could be seen

time records of reasonably accurate measurements become available.

Currently, solar radiation measurements are sparse . . . . . inaccurate. At selected locations within the

country, the accuracy should be improved in the future. However, there will still be large areas without any solar radiation measurements where a numerical model based on standard meteorological data, such as the one described above, can be used to fill the spatial gaps and any missing temporal gaps. Furthermore, the model can be used, after further justifications, to compute ratios of diffuse to total radiations for use in tilted surface applications.

Acknowledgements--The author wishes to thank the National Academy for Scientific Research for the computer facilities and the Meteorological Department for the provision of data. Thanks are also due to Dr. It'shad Abroad for helping with the computations.

NOMENCLATURE

a COnStant

,4 apparent solar radiation at air mass 1 b constant B atmospheric extinction coefficient C diffuse radiation factor

CA total cloud amount d day of the year starting 1 January O diffuse radiation on horizontal surface f albedo function for multiple reflections between

ground and atmosphere G total radiation on inclined surface h solar hour angle

H total radiation on horizontal surface 1 direct radiation on horizontal surface

K cloudness index L latitude angle

m air mass n bright sunshine duration

N day length R inclination (tilt) factor S solar constant

~, solar ahitud¢ angle // surface inclination angle 6 solar declination angle

130 ~. M. ABUGIIRIk~

p reflectance z transmittance ~b cloud transmittance function

Subscripts 1 and 2 day of the year at the start and end of the month

respectively c cloud base e extraterrestrial g ground surface o clear sky r sunrise s sunset t _daily total

Superscripts - monthly average daily value

R E F E R E N C E S

1. J.K. Page, Thcestimation ofmonthly mean values ofdaily total short-wave radiation on vcrtical and inclined surfaces from sunshine records for latitudes 40"N--40"S. Proc. UN Co~. on New Sources of Energy, Paper No. 35/5/98 (1961).

2. J. L. Monteith, Attenuation of solar radiation: A climatological study. (2. J. R. met. Soe. 88, 508-2521 (1962).

3. S. E. Hassan, Solar radiation. Technical Note No. 4, Meteorol. Dep~ Tripoli, Libya (1972).

4. M. A_ Atwater and J. T. Ball, A numerical solar radiation model based on standard meteorological observations. Solar Energy 21, 163-170 (1978).

5. H. R. Rietveld, A new method to estimate the regression cx~eflicients in the formula relating radiation to sunshine. Agric. Met. 19, 243-252 (1978).

6. J.E. Hay, Calculatingofmonthly nv,,,an solar radiation fo r horizontal and inclined surfaces. Solar Energy 23, 301-307 (1979).

7. D. Maure and N. Galanis, Solar radiation data for Quebec. Solar Energy 23, 3119-314 (1979).

8. M. M. Hawas and T. Muncer, Monthly average daily insolation on tilted collectors in Libya. Energy Cony. Mgmt. 20, 213-218 (1980).

9. M.A. Muntasser and R. D. Garg, A solar radiation data map for the SPLAJ. Int. Report, Dept. of Mech. Eng., Univeristy of AI-Fateh, Tripoli P.O. Box; 1098, Libya (May, 1981).

10. J.A. Daviesand D.C. McKay, Estimatingsolar irradiance and components. Solar Energy 29, 55-64 (1982).

I i. D. V. Hoyt, Percent of possible sunshine and total cloud cover. Monthly Weather Roy. 105, 648-652 (1978).

12. B. Y. H. Liu and R. C. Jordan, The interrelationship and characteristic distribution of direct, difus¢, and total solar radiation. So/at Energy 4, 1-19 (1960).

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14. ASH RAE, Handbook of Fundamentals, Chapter 2 7 (1981). 15. A.A. Lads and J. E. Hansert, A parameterization for the

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