Statistical analysis of solar radiation on variously oriented sloping surfaces

Sohw & [lind 7eihnob~gyVol.4, No. I, pp. 95 108. 1987 Printed iI~Great Britain.

0741 983X/87 $3.00+.00 PergamonJournalsLtd.

SHORT COMMUNICATION Statistical analysis of solar radiation on variously oriented sloping surfaces H.

P. G A R G * a n d

S. N . G A R G

Centre for Energy Studies, Indian Institute of Technology, New Delhi, 110016, India

(Received 25 February 1986 ; accepted 10 March 1986) Abstract--For four years, daily global radiation on a south facing surface with fl = L, L _+ 15' and on four vertical walls namely south wall, north wall, east wall and west wall, has been computed and statistically analysed for each of the four stations New Delhi, Calcutta, Poona and Madras. Daily direct radiation at normal incidence at New Delhi has also been studied. It has been found that maximum global radiation is 3 0 M J m 2day ~for a south facing tilted surface, 2 1 M J m 2 d a y ~for a south wall, 1 8 M J m Zday t for an east/west wall and 12 MJ m 2 day i for a north wall. Maximum direct radiation at normal incidence at New Delhi is also 30 MJ m 2 day ~. For a south facing tilted surface, nearly 80% of the days have energy between 2l and 27 MJ m 2 day ~. Atmospheric transmittance for direct radiation is seen to vary from 20% in July to 52% in November.

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

has been confined to global radiation on horizontal surfaces only. In the present study, statistical analysis of global radiation on variously oriented sloping surfaces has been carried out. Six types of sloping surfaces have been considered. (i) South facing surface with tilt equal to latitude of the place. This tilt is used to collect maximum radiation round the year. (ii) South facing surface with tilt equal to latitude + 15. This tilt is used to collect maximum radiation during the winter months, November, December and January. (iii) South facing surface with tilt equal to latitude - 1 5 . This tilt is employed to collect maximum radiation during the summer months, April, May and June. (iv) Vertical south wall. (v) Vertical east/west wall. (vi) Vertical north wall. Apart from this, daily direct radiation at normal incidence at New Delhi has also been studied and its temporal variability has been carried out. Daily global and diffuse radiation data for 4 years, 1965 68, for each of the four stations, New Delhi, Calcutta, Poona and Madras, was made available from the Indian Meteorological Department, New Delhi. Hourly global and diffuse radiation data for I year, 1967, for New Delhi was also taken from the same source. The latter data was required to compute daily direct radiation at normal incidence. Direct radiation at normal incidence was not computed just after sunrise or just before sunset. The reason is that at these times, cos 0~/cos 01, is very high and (L;h l,a,), i.e. hourly direct radiation on horizontal, is quite unreliable. So computation of direct radiation at normal incidence was started half an hour after sunrise and terminated half an hour before sunset. This situation is very often faced when direct radiation at normal incidence is calculated from measured hourly global and diffuse radiation. Data obtained was based on IPS 56 and it was enhanced by 2.2% to convert it to World Radiomelric Reference (WRR).

Although solar radiation including global and diffuse components, is measured on horizontal surfaces almost universally, it is most often required for sloping surfaces in solar energy applications. The fiat-plate receiver of a solar collector is always tilted towards the equator at an appropriate angle. To design a solar house, it is necessary to know solar radiation falling upon its roof and various vertical walls. Concentrating type collectors and photovoltaic panels, which are always facing towards the sun, require knowledge of direct radiation at normal incidence. To meet all these requirements, one should know the intensity of radiation falling upon the sloping surfaces, and its daily variation over a period of 1 year. With this knowledge, the performance of the system could be predicted quite reasonably. If the distribution of days in a month or year, on the basis of received energy spectrum, is known, the output of the system could be evaluated, and the number of days for which constant output would be maintained, could also be predicted. Solar devices are generally supplemented by other auxiliary devices. Therefore, if the output pattern of the solar device is known, the load pattern of the auxiliary device can be determined. Several years radiation data study also enables one to know yearly w~riation of statistical distribution including the extreme values of solar radiation on the sloping surfaces. Several workers have studied the statistical distribution of solar radiation. Notable work has been done by J. E. Hay [1]. In lndia, Anna Mani [2] has done pioneering work on statistical analysis of global radiation. She has found frequency distribution and cumulative frequency distribution l\)r global radiation at several stations. But their study *Guest Scientist, International Centre for Theoretical Physics (ICTP), Trieste, Italy. 95

96

S h o r t (_'OlnlntlniczltiOll

2. ANALYSIS

where ./. is the mean solur azmmth angle. It is ~ c n

Khicher's method 13] is employed It) compute global radiation on sloping surfaces when hourly data of global and diffuse radiation on horizontal surface is available. But this method is not applicable when daily global and diffuse radiation on horizontal surface are known. There are several methods to compute global radiation on a tilted surface t'rom measured daily global and diffuse radiation. Liu and Jordan [4], Heywood [5], Norris [6], Klein [7]. and Revfiem [8] have found empirical relationships for this purpose. All of these relationships have been tested against measured values and it has been found that Hay's [I] method gives the results most consistent with the experimental values. A brief description of Hay's method follows. Solar zenith angle, Oh, at any time of the day is given as cos 0~, = cos L cos 6 cos ¢o + sin L sin 6.

( 1)

cos ~ = sill L cos 0,. sin 6 cos L sin (~,, In calculating diffuse radiation on the sloping surlhce, Hay has assumed a portion of the diffuse radiation from the sky as circumsolar and a portion as isotropic. The complele equation for diffuse radiation on an inclined surface is :

+/1

t

!, = lhco s Oh"

2..... )

liI,

Summation of eqns (3), (10) and (11) gives the total radiation, G , on the sloping surface

G~ = SL+D>.+R~.

(i2)

Using eqn (12), global radiation on four types of south facing surfaces for each of the four stations has been calculated. Hay's formula is not applicable for the north or east/west walls. Therefore, Klein's formula has been adopted to calculate global radiation on these walls. Direct radiation on the east/west walt is given as : S$ cos 6 ( I - cos o)>)

(3)

where 0 h is the mean zenith angle for the whole day and 0-, is the mean angle of incidence on the sloping surface. Mean zenith angle, tIh, is defined as :

(,0)

As ground reflected radiation is very small, the usual method

COS ~t

sJ,,

(2e°sLc°s(~sin°)'+

l:Oto'sinLsin6)

(13)

A similar expression for direct radiation on the north wall is:

Daily direct radiation on horizontal surface outside the atmosphere Daily direct radiation at normal incidence outside the atmosphere

~\/l+cos/7'X]

• [I -cos/7\

where 0, is the angle of incidence on the sloping surface. Y" I, gives daily direct radiation on the sloping surface. This way, direct radiation on the sloping surface can be calculated very precisely. But eqn (2) cannot be employed when daily direct radiation on the horizontal surface is known. Ha)' has devised a method to overcome this difficulty. The proposed equation, similar to eqn (2), is :

cos 6h =

....

R,=G,i[ (2)

S$, = S,~ cos 0-1,"

// 52 ~

of Liu and Jordan has been used to calculate it

Hourly direct radiation on the sloping surface is given as COS Or

as

S,, (sin 6 sin L i : 0 ( o ,

(4) S~.,=

cos6 sin Lsin o2~)

(cosLcos(3sin~o,+i~o~O, sinLsinD) . (14,

the equation for day length is : ~,J, =

cos i ( _ t a n L t a n 6 ) . 7.5

(5)

The equation for slope day length, the time for which the sloping surface is exposed to solar radiation, is :

¢o, used in cqns (13) and (14). is in degrees. As diffuse radiation on vertical walls is less compared to that on south facing surfaces, it has been calculated by making an isotropic assumption, i.e.

I//cos 0;, - sin L sin 6~

c°st ~u', =

)

7.5

,

(6)

where 0; is the solar zenith angle when direct radiation starts falling upon the sloping surface and it is given as : cos 02 =

.//

sin/7 sin 6 ., sin/7sin L~"

e°sLtc°s~'+-

(7)

Equation (1 1) is used to compute ground reflected radiation on the walls. By using eqns (13) (15), daily global radiation on the east/west and north walls has been calculated. Computer programmes were drawn at various stage of the work. Computation of direct radiation at normal incidence from measured hourly global and diffuse radiation is comparatively simple. It is given as :

co-~-l--) El,, = E \

The smaller of the two, o2~and ~o{ is used to compute numerator and denominator of eqn (4). The mean angle of incidence, (I,, on the sloping surface is defined as : cos0~ = cosflcos tgh+sin/7 sin (th cos//,

(8)

cos0,, )"

(16)

Summation is carried out over the length of the day. Due care was taken to avoid I, values near the time ofsunrise or sunset because of the reasons explained earlier. It has been lbund that global radiation on different types

Short Communication of sloping surfaces and direct radiation at normal incidence do not exceed 30 MJ m 2 day- ~. The whole range 0-30 MJ m = day ' has been divided into 10 subranges, 0-3, 3 ~ ... 27 30 in steps of 3 MJ m ~ day ~. The number of days in a m o n t h or a year, were found in each of these 10 subranges and corresponding histograms were drawn as discussed in subsequent sections. 3. R E S U L T S A N D D I S C U S S I O N Figures l ~ show the histograms for number of days (%) against energy on the tilted surface, for each m o n t h and for each of the four stations under study. In this case the tilt of

97

the surface is kept equal to the latitude of the place to collect m a x i m u m energy round the year. As the data is for 4 years for a station, each monthly graph shows the results averaged over 120 days approximately. Figure 1 shows the results for New Delhi. One observes that the m a x i m u m number of days lies either in the '21 24' or in the '24-27' subrange and it depends upon the type of month. For clear summer months, the m a x i m u m lies in the '24-27' subrange, while for clear winter months, the m a x i m u m lies in the'21 24'subrange. During rainy months, July, August and September, the days get distributed in all subranges and the height of the m a x i m u m in the '21 24' subrange also decreases.

NEW DELHI, Tilt =Lat.

JAN

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MAR

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-

DEC

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. . . . . .

~

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. . . . .

Energy on Tilted Surface( Mj / m 2 / d a y

~

)

.

~,

Fig. 1. Monthly distribution of days in different energy subranges for New Delhi.

9~

Short ('on31nunicatiol~ Table 1. Number of days (%) when the energy is above a specific threshold value at New Delhi and Calcutta (tilt = 1at.) No. o f d a y s (%) whcn the energy > x M J m : d a y New Delhi Months

Calcutta

x-18

21

24

27

x-18

21

24

27

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

88 87 91 93 92 76 44 63 74 97 82 80

64 71 85 80 83 52 37 44 70 90 76 60

10 41 70 60 57 15 2 7 24 45 33 6

0 17 11 12 5 0 0 0 0 2 0 0

79 80 88 97 91 65 31 31 41 78 92 70

48 60 70 85 75 24 16 15 20 50 45 16

8 27 33 42 33 6 0 3 6 8 8 0

I 6 6 4 7 1 0 1 I 0 0 0

Annual

82

68

31

4

70

45

t5

2

Table 1 shows the number of days (%) when the energy is above a specific threshold value. Except for rainy months, over 80% of the days have energy more than 18 MJ m 2 day ' in each month. It implies that during clear months, only 10-20% of the days have energy between (~18 MJ m -2 d a y - ~. During rainy m o n t h s like July, only 44% of the days have energy greater than 18 MJ m 2 day t. For Calcutta also (Fig. 2 and Table 1) over 80% of the days have energy greater than 18 MJ m 2 day ~ during clear months. During rainy m o n t h s June, July, August and September, only 30-60% of the days have energy greater than 18 MJ m - 2 d a y - ~. Most of the days have energy between 18 27 MJ m 2 day ~. Days with energy greater than 27 MJ m 2 day Eare comparatively few. For Poona (Fig. 3 and Table 2) during clear months January, February, March, April and May, as m a n y as 100%o of the days have energy greater than 18 MJ m -2 day ~. Therefore, these months are very potent from an energy point of view and a good proportion of energy collection could be carried out during this time. One also notices that the number of days with energy more than 27 MJ m 2 day ~ have increased comparatively. During rainy months, the number of days drops to 23 %0 in July when the energy is greater than 18 MJ m 2 day t. This is because Poona has exceptionally thick clouds during these months. A similar trend is observed for Madras also (Fig. 4 and Table 2). Annually, nearly 72% of the days have energy greater than 18 MJ m 2 day ~. Days with energy greater than 27 MJ m -2 day 1 are almost non-existent. Figure 5 shows a comparison of four cities on an annual basis for the same tilt. New Delhi, Calcutta and Madras have maxima in the '21 24' subrange while the m a x i m u m for Poona is in the '24-27' subrange. A year represents all types of weather including rainy months, therefore, in the annual diagrams, the number of days also appears in lower subranges. The pattern for most of the days appearing in the last four or five subranges is similar for all four stations. Calcutta shows a slightly different pattern for lower subranges. More days lie in lower subranges for Calcutta than those for other stations.

Figure 6 shows the days distribution for the winter months, November, December and January when the south facing surface is tilted at lat. + 15°. Results, shown in Fig. 6, have been averaged over the aforesaid 3 months. During these months, New Delhi, Calcutta and Poona have clear days while Madras has rains during December. That is why the first three stations have very few days in the lower subranges while Madras has comparatively more days in the lower subranges. New Delhi and Poona have the m a x i m u m number of days in the subrange '24-27', 44 and 5 3 % , respectively. Calcutta and Madras have the m a x i m u m number of days in the subrange '21-24', 43 and 26%, respectively. The reason is that Calcutta has dusty weather all year round due to heavy industrialization and Madras has rains during the winter months. Except for Madras, during the winter season, most of the days have energy greater than 18 MJ m 2 day ~. Such days for New Delhi, Calcutta and Poona, are 92, 85 and 94%, respectively. For Madras, such days are only 70%. Figure 7 shows the distribution of days for summer m o n t h s April, M a y and June, when the tilt is kept equal to lat. - 15~. As in Fig. 6, results in this figure have also been averaged over 3 months. It is evident from the figure that all four stations have quite clear days during this period, except for the last week of June during which New Delhi has rains. The m a x i m u m number of days for all four stations lies in the "2427' subrange, such days being 38, 34, 36 and 48%0 for New Delhi, Calcutta, Poona and Madras, respectively. Also most of the days have energy greater than 18 MJ m 2 day t, exact values being 92% for New Delhi, 82% for Calcutta, 90% for Poona and 85% for Madras. Except for Madras, all the three stations have quite a significant number of days with energy greater than 27 MJ m - 2 d a y - ~. Figure 8 shows the days distribution on an annual basis for a vertical south wall. It is evident from this figure that no such 'distinct m a x i m a ' occur for any of the four stations as in the preceding cases. For each station there are several subranges which have a nearly equal number of days. A comparatively mild m a x i m u m is shown in the subrange '6 9' for Calcutta. Poona and Madras, and in the subrange '9 ! 2' for New Delhi. The n u m b e r of days for maxima are 25,

Short Communication

99

CALCUTTA Tilt = Lot.

'iI n

60

JAN

FEB

~

,

,

n

_

.

'°

.

n

~

o

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MAR

°I

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n

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n

APR

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T /.0

0

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z

MAY

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. . . . . .

60

JUN

fl

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20 o

0

[] ~

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= ~ I ], / ]~ . .

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nnn

DEC

t.o ~

[3

n

n

,_

o

,

n,

,

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n

Energy on Tili~l Surface (M j / m 2 / d o y ) Fig. 2. Monthly distribution of days in different energy subranges for Calcutta.

loll

$holl

('omnlunication

POONA

Till = Lat.

60

60

JAN

40

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MAR 20

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2

o ....

an~

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NOV

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Energy Fig.

60~ t

R

oL . . . . ,

,

on Tilfed S u r f a c e

,

n rlH

, "7 ";" T " 7

(MJ/m2/ddy)

13, '7 '~

~

:-

3. M o n t h l y d i s t r i b u t i o n of days in different energy s u b r a n g e s for Poona.

•sl~JpgIA[ Joj sa~'up, J q n s 3 ~ J a u a luoJa~.tp u.i s k i p j o u o ! m q ! a l s ! p 3[tqlIlOIAI "17 "~1~

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102

Short Communication Table 2. Number of days (%) when the energy is above a specific threshold value at Poona and Madras (tilt = lat.) No. o f d a y s ( % ) w h e n t h e e n e r g y > x M J M Poona Months Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec Annual

2day '

Madras

x=18

21

24

27

x=18

2t

24

27

98 94 100 95 99 60 23 33 74 86 87 80

84 93 95 90 96 33 5 7 51 78 74 61

27 73 79 67 76 12 2 1 23 45 27 11

0 7 16 10 4 0 0 0 I 2 0 0

80 97 98 86 78 68 56 69 73 61 58 44

60 90 92 80 62 48 28 46 57 43 32 15

23 66 74 46 17 8 7 13 21 17 3 1

0 3 8 0 0 0 0 0 0 0 0 0

80

66

38

3

72

54

24

1

ANNUAL.

Tilt = Lat.

-L 6O

NEW

CALCUTTA

DELHI

t,O

iP

,

i

n

n

n

,

H

ri

0 I

k nnnl In o

n

G 80--

60

60

r POONA

MADRAS t+o

I

20

2o }-

I

ol E n e r g y on T i l t e d

Surface(MJlm2/day)

- ~

Fig. 5. Annual distribution of days in different energy subranges for four stations.

n

WINTER SEASON

ao;

103

T i l t = Lat..15"

B0

60

60

CALCUTTA

NEW DELHI z.0--

/.o-

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r] I

n

n

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m n

n

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t 1

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so

g "~

6e

MADRAS

POONA

E

L~

i

E n e r g y on Tilted S u r f a c ¢ ( M . ~ m 2 / d a y )

Fig. 6. Distribution of days in different energy subranges during winter months. SUMMER SEASON

Tilt = L a t - 1 5 "

6O NEW DELHI

CALCUTTA 60

20

lo ,.

on~

H

h

=o

m

o , ~ n n f l

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E n e r g y on Tilted S u r f a c e ( MJ / r n 2 / d a y )

Fig. 7. Distribution of days in different energy subranges during s u m m e r months.

r]

N

,,

,,

1 n

-I

SOUTH WALL( a n n u a l )

104

go

80--

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NEW DELHI

CALCUTTA

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Energy ona South Wall (MJ/m 2/day) Fig. 8. Annual distribution of days in different energy subranges for a south wall. EAST/WEST (annual)

I

~o~

6o~-

NEW DELHI

I

CALCUTTA

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I fin Energy arian east/w~t wall (M j/m2/ day) Fig. 9. Annual distribution of days in different energy subranges for an east/west wall.

Short Communication NORTH 80

-

105

WALL(annual) 80 --

-

60 --

60--

NEW

CALCUTTA

DELHI /,oi--

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Energy on a north wall (Mj/m2/day) Fig. 10. Annual distribution of days in different energy subranges for a north wall.

28, 27 and 34% for New Delhi, Calcutta, Poona and Madras, respectively. Because a south wall receives less energy as compared to other south facing tilted surfaces, it shows quite a significant number of days in lower subranges. Figure 9 shows the distribution of days for the east/west wall. On an annual basis, the east wall and west wall receive almost equal energy and that is why the same results are applicable to both the walls. Calcutta has its m a x i m u m in the '%12" subrange, while New Delhi, Poona and Madras have maxima in the '12-15' subrange, the corresponding number of days in the maxima subrange being 38, 36, 45 and 43%, respectively. This figure also shows that m a x i m u m energy falling on an east/west wall is 18 MJ m 2 day 1 approximately. The days distribution for a north wall is shown in Fig. 10. At New Delhi, the sun never goes into the zenith and therefore, at this station a north wall does not receive direct radiation at noon time. But in the s u m m e r months, in early morning and late afternoon when the solar azimuth angle is greater than 90 °, direct radiation falls on the north wall. For the other three stations, the sun goes into the zenith in summer and therefore direct radiation falls on the north wall at noon time also. Apart from direct radiation, the north wall receives diffuse radiation all the time. This Fig. I0 shows that New Delhi, Calcutta and Poona have maxima in the ' 3 6' subrange, while Madras has its m a x i m u m in the "6-9' subrange, the corresponding number of days being 46, 44,

49 and 41%, respectively. The m a x i m u m energy falling on a north wall is 12 MJ m 2 d a y - ~ approximately. Figure I1 shows the direct radiation at normal incidence received at New Delhi during different months. Direct radiation transmission is also shown. One notices that except for the three m o n t h s June, July and August, transmission is quite high, variation being 37% for September to 52% for November. During July and August, including the last week of June, rains set in at New Delhi and there are also some rains in September. Because of this, transmission drops to as low as 20% in July. For the 3 months, June to August, transmission varies from 20 to 26%. During these months, direct radiation energy received varies from 10.6 to 13.6 MJ m 2 day-1. During the other months, its highest value is 24.3 MJ m z day ~ in November (monthly mean value). Figure 12 shows the days' distribution in different energy subranges for direct radiation at normal incidence. In sharp contrast to the global radiation cases, quite a good number of days are distributed in all the subranges. It is not the case that lower subranges have a few days only, the reason being that low direct radiation combined with high diffuse radiation can give high global radiation. This Fig. 12 also shows that m a x i m u m direct radiation at normal incidence does not go beyond 30 MJ m z day ~. One observes that nearly 60% of the days have direct radiation more than 18 MJ m z day '. The m a x i m u m number of days, 20%, lies in the subrange '24 27".

106

Short Communication

1

m

32

28

---¢ :,- Terrestrial Direct Radiation - o - - - - - o - Dinzct Radiation TransrnittQnce

--

70

;2 2.4

-6 E 8 z

5O

20

2

1E

I

\ 12

\\

"6 r~ c3

].~

//

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7o0

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8

1D

0 J

I F

I M

I A

I M

I J Months

I J

....

5--___ A

I S

I O

I N

Fig. l 1. Monthly variation o f direct radiation at normal incidence for New Delhi.

80-

6O

Direct Radiation at Normal

Incider~:e

~6

E 20-z

0 i c)

i t,~

Different Energy Subranges (M J / m 2 / d a y )

Fig. 12. Distribution of days (annual basis) in different energy subranges.

0

107

Short Communication Table 3. Monthly days distributed in different energy subranges. (Direct radiation at normal incidence for New Delhi ; units MJ m 2 day Different energy ranges Months

IL3

3 6

6-9

3 2

-

1 1 2 1 2 5 1 3 -

Jan Feb Mar Apt May Jun Jul Aug Sept Oct Nov Dec

1 2 7 11

1 4

1 4

3

Annual

30

26

19

1 2 3 6 4 5

9 12 12 15 15 18 18-21 21 24 24~27 27 30 Total 2 l 3 2 5 3 1 3 1

21

2 3 4 3

4 3 6

4 6 6

4 3 2 1 3 6 I 2

2 3 2

1 1 7

35

31

The monthly distribution of days in different subranges is shown in Table 3. Values shown are for the number of days out of 30 or 31 days in a m o n t h in a specific subrange. Except for rainy months, the m a x i m u m n u m b e r of days lies in the two subranges '21 24' and '24 27'.

5, C O N C L U S I O N S (l) A south facing surface, with tilt equal to latitude of the place, has nearly 80% of the days with energy greater than 18 MJ m 2 day t. For all four stations, the m a x i m u m number of days in a month, lies in the '21-21" or '24~27' subranges depending upon the month. For tilt = lat_+ 15°, 70~92% of the days have energy greater than 18 MJ m 2 day ~. For a south wall, the m a x i m u m number of days lie in the subrange "(~9' MJ m : day n and the m a x i m u m energy received on any day is 21 M J m "-day ~. (2) For an east/west wall, the m a x i m u m number of days lie in the '9 12' or '12 15" subranges. Also the m a x i m u m energy falling on an east/west wall is 18 MJ m -2 day -~. (3) In the case of a north wall, the m a x i m u m number of days lies in '3 6' subrange and the m a x i m u m energy falling on this wall is 12 MJ m 2 day ~. (4) For the direct radiation at normal incidence at New Delhi, transmission varies from 52% in November to 20% in July. Also the m a x i m u m energy received is 24.3 MJ m 2 day t in November and the minimum, 10.6 MJ m -2 day -~ in July. Annually, over 60% of the days have energy greater than 18 MJ m 2 day L.

NOMENCLATURE L fl 6 m p co ~,J~ (~);

latitude of the place (deg.) tilt of the sloping surface (deg.) solar declination (deg.) air mass (dimensionless) reflectance of ground, p - 0.2 (dimensionless) hour angle (radians) hour angle at the time of sunrise (radians). hour angle at the time sunrays start falling upon the sloping surface (radiansl

3 8 3

7 7 4 8 6 l l 4 5 7 7 3

7 3 3 6 I0 1 t I 12 12 8 5

34

60

69

2 4 1

2 5 7 8 6 2 1

31 28 31 30 31 30 31 3l 30 31 30 31

2 5 1 39

365

S,L daily direct radiation on the horizontal surface (MJ m 2day ~) I daily direct radiation at normal incidence (MJ m " day i) Sl, daily direct radiation on the sloping surface (MJ m 2 day l) G daily global radiation on horizontal surface (MJ m ~"day i) D daily diffuse radiation on the horizontal surface ( M J m 2day i) G~ daily global radiation on the sloping surface ( M J m : d a y i) D~ daily diffuse radiation on the sloping surface ( M J M ' d a y i) R, daily reflected radiation on the sloping surPace ( M J m -'day 1) 1,, hourly direct radiation at normal incidence ( M J m : h 1) L, hourly direct radiation on horizontal surface ( M J m 2h 1) It hourly direct radiation on sloping surface ( M J m Zh 1) lr;h hourly global radiation on horizontal surface (MJ m 2 h i) Iji, hourly diffuse radiation on horizontal surface (MJ m -2 h - i ) Oh angle of incidence on horizontal surface or zenith angle (deg.) 0-h mean zenith angle (deg.) 0'/, zenith angle when direct radiation starts falling upon the sloping surface (deg.) 0, angle of incidence on the sloping surface (deg.) mean angle of incidence on sloping surface (deg.) A solar azimuth angle (deg.) mean solar azimuth angle (deg.) REFERENCES 1. J. E. Hay, Study of shortwave radiation on non-horizontal surfaces. Canadian Climate Centre, Report No. 79-12, Ontario (1979). 2. A. Mani and S. Rangarajan, Solar Radiation over India. Allied Publishers, New Delhi (1982).

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Short Communication

3. T. M. Klucher, Evaluation of models to predict insolation on tilted surfaces. Solar Energy, 23, 111 114 (1979). 4. B. Y. H. Liu and R. C. Jordan, Daily insolation on surfaces tilted towards the equator, Trans. Am. Soc. Heat. ReJhig. Air-condit. Engrs. 526-541 (1962). 5. H. Heywood, The computation of solar energy intensities, Part 2. Solar radiation on inclined surfaces. Solar Energy Conf., Phoenix, Arizona, U.S.A. (1965).

6. D. J. Norris, Solar radiation on inclined surfaces. Solar Energy 10, 72-77 (1981). 7. S. A. Klein, Calculation of monthly average insolation on tilled surfaces. Solar Enerqy 19, 325 329 (1977). 8. K. J. A. Revtiem, A simple procedure for estimating global daily radiation on any surface. J. Appl. Met. 17, 112ff 1131 (1978).