Improved station-independent correlations between global radiation and sunshine duration

Energy Convers. Mgmt Vol. 30, No. 2, pp. 163-172, 1990 Printed in Great Britain. All rights reserved

0196-8904/90 $3.00+ 0.00 Copyright © 1990 Pergamon Press plc

IMPROVED STATION-INDEPENDENT CORRELATIONS BETWEEN GLOBAL RADIATION AND SUNSHINE DURATION M. HUSSAIN Renewable Energy Research Centre and Department of Physics, University of Dhaka, Dhaka-2, Bangladesh

(Received 17 December 1987; receivedfor publication 28 November 1989) Abstract--Independent regression fits to Angstrom's relation between sunshine duration and global irradiation have been made for monsoon, pre-monsoon and post-monsoon periods. Long-term data of eight stations situated around 20°N lat in India were used to obtain collective fits for each period separately for stations lying in (1) the zone comprising arid and semi-arid regions and (2) the wet-and-dry zone. The root mean square (r.m.s.) errors are < 3% in each case for the estimation of global irradiance of a month and the maximum error found is 6.3% which is significantly smaller than for previous estimates. The present station independent correlations should give precise estimates of G for locations away from pyranometers in a large part of the Indian subcontinent (Fig. 1). The technique of making collective fits depending on seasons and climatic zones may be applied with advantage to other parts of the world. Global irradiation Sunshine duration Turbidity Monsoon Pre-monsoon

Station independent correlations Post-monsoon

Albedo

NOMENCLATURE Go = G= s= S= S' = a, b, c, d = r = ~b = D=

Extra-terrestrial irradiation on horizontal surface (kWh/m: day) Global irradiation on horizontal surface (kWh/m: day) Bright sunshine duration (h) Day length (h) Maximum period over which Campbell-Stokes recorders remain sensitive (h) Regression coefficients Correlation coefficient Geographical latitude (deg) Diffuse irradiation on horizontal surface (kWh/m: day)

INTRODUCTION

The monthly average value of the daily global irradiation, G, is often estimated from the bright sunshine duration, s using the Angstrom relation [1] or one of its variants [2]. As modified by later workers, Angstrom's statistical relation may be written as G Go

--=a+b-

s S

(1)

where Go is the daily extraterrestrial irradiation and S is the day length. Go and S may be computed using standard formulae [3]. The regression constants a and b are determined empirically from data on G and s for each pyranometer station, and both a and b are known to vary from place to place. The uncertainty in the values of a and b makes estimation of global irradiation for locations away from pyranometers rather difficult. To obviate the problem, Ma and Iqbal [2] studied different statistical fits in order to select the most suitable one for estimation of G for any location. They came to the conclusion, using data of stations in Canada and Europe, that Reitveld's relation [2, 4] G s G---~= 0.18 + 0.62 ~

(2)

is generally most accurate. The values of (G/Go) vs (s/S) for four north Indian stations, Ahmadabad, Bhavnagar, Jodhpur and New Delhi (Fig. 1) have been plotted in Fig. 2 along with the linear fit of Reitveld. It immediately appears that the relation, equation (2), gives, in our case, 163

164

HUSSAIN: GLOBALRADIATIONAND SUNSHINEDURATION 72 ° i.

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Fig. 1. Map showingclimaticzonesin Bangladeshand a part of India togetherwith locationsof some pyranometerstations(AW-tropicalwet-and-dryregion;BSh--tropicalsemi-aridregion;BWh--tropical arid region; Ar--tropical wet region). rather unsatisfactory predictions of global irradiation from sunshine duration. It may be mentioned that the data on G and s used are averages over many years (Table 1). For another group of four stations in India north of 17°N, a plot of data (Fig. 3) again leads to the same conclusion. Glover and McCulloch [5] included the latitude effect in their correlation formula G Go

0.29 cos ~b + 0.52 ; ,

~b<60 °

(3)

where ~b is the geographical latitude. Plots using this relation have been shown for New Delhi and Bhavnagar in Fig. 2. Fits for the other stations, Ahmadabad and Jodhpur lie between the two straight lines drawn. It appears that, on the average, the relation gives fair estimates of G. For the period March-May (months 3-5), one obtains underestimates, while for months 10-12, one finds overestimates. The relation has been plotted in Fig. 3 for Calcutta and Vishakhapatnam, while for Nagpur and Bombay, the fits lie in-between. These show that Glover and McCulloch's relation fails to give satisfactory results. Dogniaux and Lemoine [6] arrived at another latitude dependent correlation

°(

Go= 0.00506

;

-0.00313

/

q~ +0.32029

;

+0.37022.

(4)

HUSSAIN:

GLOBAL RADIATION AND SUNSHINE DURATION

165

B / D

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./

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$ Fig. 2. A plot of (G/Go) vs (s/S) for Ahmadabad, Bhavnagar, New Delhi and Jodhpur along with empirical fits of Reitveld, Glover and McCulloch and Dogniaux and Lemoine, Fits for New Delhi and Bhavnagar are marked D and B, respectively.

Fits using this relation have also been shown in Figs 1 and 2 and the situation is again not a good one. Many other attempts [7, 8] have been made to modify the Angstrom equation. Variables like humidity, atmospheric water vapour content, temperature and number of rainy days in a month have been introduced in order to get fits with station independent parameters. Hay [9] used the surface albedo of the earth as an extra variable and obtained station-independent regression constants for a part of Canada. The fits gave a r.m.s, error of around 5% for estimation of G. Mani and Rangarajan [10] used a constant value for the albedo and found station-dependent fits for Indian stations with a similar accuracy. For north Indian stations, the use of atmospheric water vapour content gave a correlation [8] which yielded estimates within 5% of measured values for Table I. Particulars of stations with period of data on G and D

Station Ahmadabad Bhavnagar Jodhpur New Delhi Bombay Calcutta Nagpur Vishakhapatnam ECM 30/2--G

Climatic zone

Latitute

Annual rainfall (ram)

Zone A (Tropical arid and semi-arid)

23°4 ' 21045 ' 26 ° 18' 28°35 '

823 600 380 714

202 141 132 255

Zone B (Tropical wet-and-dry)

19°7 ' 22°39 ' 21°09 , 17°43 '

2099 1582 1127 974

100 248 218 210

Data period (months)

166

HUSSAIN:

GLOBAL RADIATION AND SUNSHINE DURATION

0.7r-

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niaux 0.6

ol

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Go

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$ Fig. 3. A plot of (G/Go) vs (s/S) for Bombay, Calcutta, Nagpur and Vishakhapatnam along with empirical fits of Reitveld, Glover and McCulloch and Dogniaux and Lemoine. Fits for Calcutta and Vishakhapatnam have been marked C and V, respectively.

two-thirds of the cases with a maximum error of 12%. Garg and Garg [7] found that several station-independent correlations fail miserably for India. Instead of using additional variables, we use in the present work Angstrom's relation to obtain collective fits for a group of stations situated in a similar climatic zone and lying within a small range of latitudes for periods of the year for which the overall effect of the physical variables on the absorption and scattering of solar radiation may be expected to be similar. The stations are a few hundred to over a thousand kilometers apart. The technique proposed earlier [1 l] is developed here using additional data and detailed information on the time of onset and withdrawal of the monsoon which has large effects on the variables. The idea of seasonal fits for individual stations, unlike the present fits for groups of stations, is not new, and recently, Benson et al. [12] showed that, for Atlanta, good fits could be obtained by separating data of the summer and winter months. C L I M A T O L O G I C A L VARIABLES AND THEIR EFFECTS ON SOLAR RADIATION The map in Fig. l shows a part of India and Bangladesh with the climatological classifications of Trewartha [13]. We find that there are arid, semi-arid, wet-and-dry and wet regions with differing features of the soil and its vegetation. The Indian subcontinent, comprised of India, Bangladesh and Pakistan, is characterized by its monsoon rains. At the onset of monsoon, around June-July, the wind direction is almost suddenly

HUSSAIN: GLOBALRADIATION AND SUNSHINE DURATION

167

reversed from the northeasterly to southwesterly direction, and the water vapour carried away from the Indian Ocean, the Bay of Bengal and the Arabian Sea produces rain bearing clouds and heavy rainfall. Around September-October, the wind direction again reverses, the sky clears up, very little precipitation occurs, and a post-monsoon period sets in with fairly stable weather and low atmospheric turbidity. By March, the relatively cool period is over, the pre-monsoon hot weather starts, and the atmospheric turbidity begins to rise. Frequent thunder squalls occur with dust storms or "andhis" in the very west of the northern part of the subcontinent and "norwesters" or storms with heavy showers in the east.

Absorption of solar radiation In post-monsoon months, the moisture content of air at the stations is low, and it increases by about a factor of three in the monsoon period. In the wet-and-dry zone, the precipitable water in the atmosphere w changes from ~ 2 to 6 g/cm2 [10]. In the other zone, w increases, generally, from --~1.5 to 5 g/cm2.

Scattering of solar radiation Aerosols, particles suspended in the air, scatter solar radiation. In general, the amount of aerosols increases in the hot pre-monsoon months. The monsoon rains wash down the dust and decreases the Angstrom turbidity. In the pre-monsoon months, Calcutta, in the wet-and-dry zone, has an average turbidity of 0.12, while for New Delhi, in the semi-arid zone, it is 0.11. In the post-monsoon period, the turbidity falls down to 0.08 at Calcutta and 0.06 at New Delhi [14]. Most other locations have somewhat lower turbidities but show a similar variation.

Multiple reflections of solar radiation The ground reflectivity, or the surface albedo of the earth, depends on the colour, structure and humidity of the soil and the nature of the vegetation. These variables differ for different climatic regions. Due to rains, the albedo of moist soil decreases after the monsoon sets in, and in the pre-monsoon period, the earth dries up, and the albedo is generally higher. Measurements show that the average albedo around noon decreases from 0.20 to 0.16 after the monsoon at New Delhi and from 0.18 to 0.12 at Poona near Bombay [15]. The albedo of clouds depends on their types and thicknesses and has seasonal changes. The range of variation is known to be wide, 28-98% [16]. Thus, the amount of multiple reflections varies with the period of the year and may also depend significantly on the location. THE PRESENT TECHNIQUE For the monsoon climate of the Indian subcontinent, the absorption, scattering and multiple reflections of solar radiation have temporal and spatial variations. If the physical variables producing the effects could be introduced along with sunshine duration in equation (1), good correlations with regression constants independent of location may be obtained. Data on variables like the surface albedo of the earth at the station or of the clouds will be hard to find and the correlation formula would lose its simplicity. The amount of precipitable water, the turbidity of the atmosphere and the albedo of the earth's surface and the sky which affect the incoming solar radiation have similar values during the monsoon months, and these have somewhat different values for pre- and post-monsoon periods for each station. Again, there is a regional difference for the variables. We have, therefore, made an attempt to find Angstrom type regression fits for two different zones (Table 1) for each of the three periods, using data of several stations collectively, as shown in Table 2. For our studies, we have used published long-term averaged data on G and s for eight stations in India situated near sea level and lying within a small range of latitudes as shown in Table 1. The data on G are monthly averages of daily values over several years (Table 1) up to 1978 and were obtained by the Indian Meteorological Department using carefully calibrated thermopile pyranometers [15]. Sunshine duration data collected by the same department are averages for 1954-1975 using Campbell-Stokes recorders [10]. According to the classification of Trewartha [13], Ahmadabad and Bhavanagar are near the border between arid and semi-arid regions, New Delhi

HUSSAIN: GLOBAL RADIATION AND SUNSHINE DURATION

168

Table 2. Dates of onset and withdrawal of monsoon and seasonal classification Station

Onset

Withdrawal

Pre-monsoon

Monsoon

Post-monsoon

period

period

period

June-September June-September July-August July-September

October-February October-February September-February October-February

June--September June-September June-September June-September

October-February October-February October-February October-February

Zone A

Ahmadabad Bhavnagar Jodhpur New Delhi

13 June 13 June 1 July 26 June

20 September 22 September 7 September 18 September

March-May March-May March-June March-June

Bombay Calcutta Nagpur Vishakhapatnam

10 June 8 June 11 June 5 June

25 September 10 October 7 October 15 October

March-May March-May March-May March-May

Zone B

borders semi-arid and wet-and-dry regions, and Jodhpur is in an arid area while Bombay, Calcutta, Nagpur and Vishakhapatnam belong, clearly, to the wet-and-dry region which we call zone B (Fig. 1). We group together stations in the arid and semi-arid regions in north India and classify them to belong to zone A. In an earlier attempt[l l] at obtaining independent correlations for monsoon, pre- and post-monsoon periods, we considered, for the sake of simplicity, that, for any zone, each of the three periods consists of the same months. The r.m.s, error was found to be 3.6% with a maximum error of 12% when the data of all the six stations considered were used collectively. When the data of the two zones were used separately, the r.m.s, error became 3.2% with little change in maximum error. In this work, we accept the fact, within a zone, there may be differences between the lengths of the monsoon period. Meteorologists have fixed the normal dates of the onset and the withdrawal of the monsoon with reference to the rather sharp increase and decrease, respectively, seen in 5 day means of rainfall and changes in circulation. According to the isochrones given by Rao [17] for the onset and withdrawal of the southwest monsoon, the dates, as read from the curves, are shown in Table 2. We find that, for all the four stations in zone B, the period between onset and withdrawal of the monsoon covers most of June and the whole month of September. We classify June-September as monsoon months, considering that a month belongs to the monsoon period if the monsoon climate persists for more than half of the month. For Ahmadabad and Bhavnagar, we observe that again June-September is to be regarded as the monsoon period. For Jodhpur and New Delhi, the monsoon appears later, and we classify July-September as the monsoon months for New Delhi, and July-August for Jodhpur. Table 2 shows the resulting classification considering that the pre-monsoon period starts around March. DATA AND RESULTS

The data on G (and D) for the eight stations (Table 1) used in our studies have fairly small errors, as these are averages over a few hundred measured daily values. The standard deviation for daily data G is 0.5-1.5 kWh/m 2 [15]. The standard deviation of the mean data is very much smaller. For a few of the mean data, the standard deviations in G/Goare shown in Figs 4 and 5. Information on the r.m.s, error of sunshine duration has not been available but a long term mean of this, too, should have a fairly small statistical error. For each of the three periods, for each zone, we have obtained the values of the regression constants a and b in equation (1) using least-square techniques. Table 3 gives the values along with the correlation coefficient r and the r.m.s, errors of the fits (RMSE). We used data of all the stations (Table 1) in each zone to obtain the four-station fits. We have also obtained three-station fits by not taking into account the data of Jodhpur and Vishakhapatnam. This is because Jodhpur has a somewhat different climate than the other three (Fig. 1), while for Vishakhapatnam, the latitude is rather different from the rest. It may be seen that, in each case, the correlation coefficient is high, and the standard error is small. Table 4 shows that, for zone A, three-station fits (G3) give satisfactory predictions for Jodhpur as well (maximum error 6.1%).

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Fig. 5. A plot o f (G/G0) vs (s/S) for Bombay, Calcutta, NaBpur and Vishakhapatnam. Present fits using data o f the four stations in zone B and the c o l l ~ t i v e fit o f

S

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Fig. 4. A plot of (G/Go) vs (s/S) for Ahmadabad, Bhavnagar, New Delhi and Jodhpur. Present fits using data of the four stations in zone A and the collective fit of Garg and Garg for 1i stations in India are shown.

Go

6

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170

HUSSAIN:

GLOBAL RADIATION AND SUNSHINE DURATION

Table 3. Linear regression fits between (G/Go) and (s/S) Zone

Period

a

b

r

RMSE (%)

Four-station fits f

Monsoon Pre-monsoon Post-monsoon

0.276 0.464 0.423

0.505 0.298 0.316

0.97 0.87 0.78

2.9 2.7 2.3

f

Monsoon Pre-Monsoon Post-monsoon

0.270 0.238 0.232

0.510 0.548 0.516

0.97 0.89 0.95

2.6 1.8 2.0

f A

Monsoon Pre-monsoon Post-monsoon

0.275 0.478 0.443

0.502 0.278 0.285

0.97 0.90 0.84

2.2 2.8 2.4

B

Monsoon Pre-monsoon Post-monsoon

0.263 0.253 0.235

0.530 0.529 0.507

0.98 0.90 0.96

2.5 1.8 1.8

A

B

Three-station fits

Table 4. Global radiation data and their estimated values for zone A. G4 and G~ represent estimates using four- and three-station fits, respectively Stations

Ahmadabad Gexo G4 G3

I

2

3

4

5

Months 6 7

8

9

10

I1

12

4.90 4.90 4.85

5 . 7 9 6 . 7 3 7 . 3 3 7 . 6 1 6.39 4.85 4.52 5 . 7 6 6.54 7 . 2 9 7.74 6 . 5 7 4 . 9 1 4.56 5 . 7 0 6 . 5 3 7 . 2 8 7 . 7 2 6.54 4.89 4.54

5 . 5 5 5 . 7 7 5 . 0 0 4.59 5 . 4 7 5 . 9 2 5 . 1 9 4.70 5 . 5 1 5 . 8 7 5 . 1 3 4.65

5.14 5.11 5.05

5 . 9 6 6 . 8 7 7 . 2 7 7 . 6 0 6 . 0 3 4.62 4 . 3 1 5 . 4 6 5 . 9 1 5 . 2 2 4.78 5 . 8 7 6 . 7 7 7.44 7 . 9 2 6.17 4.52 4 . 3 1 5 . 2 3 6 . 0 0 5 . 4 0 4.81 5 . 8 0 6.76 7 . 4 2 7 . 9 0 6 . 1 5 4 . 5 1 4.30 5 . 3 5 5 . 9 5 5 . 3 3 4.76

3.99 4.02 4.00

5 . 0 0 6.14 6.94 7 . 2 9 6.54 4.99 5 . 9 4 6.89 7 . 1 5 6.74 4.96 5 . 9 5 6 . 8 8 7 . 1 8 6.80

4.72 4.48 4,44

5 . 5 7 6 . 5 5 7 . 2 3 7 . 5 5 7 . 0 7 5 . 9 8 5 . 5 4 6.10 5 . 8 3 4.90 4.43 5 . 3 3 6 . 2 8 7 . 1 8 7 . 6 2 7 . 4 2 5 . 7 1 5 . 5 6 6.14 5 . 7 3 4 . 8 7 4.30 5 . 3 3 6 . 2 8 7 . 1 7 7 . 6 2 7 . 4 4 5 . 8 7 5 . 5 4 6 . 1 3 5 . 6 7 4 . 8 1 4.25

Bhavnagar Gexp G4 G3

New Delhi Gexo 6;4 G3

5 . 3 3 5 . 0 5 5 . 6 0 5 . 3 6 4.52 3.84 5 . 4 3 5 . 3 6 5 . 4 6 5 . 3 9 4 . 5 5 3.86 5 . 4 1 5 . 3 4 5 . 5 2 5 . 3 6 4.50 3.84

Jodhpur Ge~o G4 G3

For zone B, we find a similar result for Vishakhapatnam. These may be regarded as tests of our fits. We estimated G for Goa (lat 15°29'), using the four-station fits (G4) , which is just outside zone B but which has a wet-and-dry climate. It was found that the maximum error in the estimate of G for a month is 5.5% only for the station. Monthly G was estimated also for Poona (Fig. 1) which is situated 559 m above sea level and the maximum error was 6.2%. It appears that, for such a station height, our fits give good predictions as the atmospheric transmission of radiation is not very different from sea-level stations. These results confirm that errors in estimation for any location using the fits given in Table 3 should be small for the northern part of the Indian subcontinent, except possibly for hill stations. DISCUSSION

Figures 4 and 5 show fits for the regression relations between (G/Go) and (s/S) for zones A and B, respectively, obtained using data of four stations in either case. These appear to be excellent and the r.m.s, errors are < 3% in each case. They also show the Angstrom fit obtained by Garg and Garg [18] using collectively monthly data over the year for 11 Indian stations. We find that our classification of north and central Indian stations into two zones and three periods appears to be quite satisfactory. The monsoon regression constants are similar for zones A and B, but the pre-monsoon and post-monsoon fits differ for the two zones. It appears that seasonal variations in the transmission and scattering of solar radiation between monsoon and

HUSSAIN:

GLOBAL RADIATION AND SUNSHINE DURATION

171

post-monsoon periods for zone A and between monsoon and pre-monsoon periods for zone B are not very strong as indicated by the corresponding straight lines. Pre-monsoon fits for zone A and post-monsoon fits for zone B are, on the other hand, quite different from those of other periods. It is to be noted that the correlations for each period, in our case, are valid only for the ranges of values of (s/S) used in the regression analysis in accord with the idea presented by Benson [12] earlier. It is 0.2 <~(s/S)<~ 0.6 for monsoon and 0.4 ~<(s/S)<~0.9 for pre-monsoon and postmonsoon periods. One observes that the correlation constants and the r.m.s, errors of four- and three-station fits are nearly the same (Table 3) for each zone and season. For predicting solar radiation in the arid zone in India, or nearby areas of Pakistan, one may expect the best results from the four-station correlations as data of an arid station, Jodhpur, are taken into account. For semi-arid locations, the three-station correlations may be used. The latitude of Vishakhapatnam is somewhat lower than others in zone B, and therefore, for stations at higher latitudes, say for Bangladesh, one may prefer to use the coefficients of three-station fits. Computations show that it does not matter much whether one uses the three- or four-station fit (Table 4). Garg and Garg [18] found that, for their individual station fits, i.e. fits obtained for each station using monthly data over the year, 6% predictions of G lie in the 8-10% error range and 1% of predictions have errors > 10%, while for their collective fit using data over the year of 11 stations together the figures for the two ranges of errors are 8 and 3%, respectively. In the present work, the maximum error found is 6.3% only. Mani's estimates [10], too, show that errors above 10% appear. The occasional large errors in predictions of G from correlations obtained by others disappear in the present case, making the fits very useful. Moreover, the systematic high or low estimated values of (G/Go) for a number of succeeding months in other fits, as Figs 2-5 indicate, are removed by the present approach. In the present method, for a particular season, the same pair of correlation constants (Table 3) give highly satisfactory estimates of global radiation for different locations within a climatic zone, even if these are situated as far as 1000 km away. This is explained by the fact that for cloud free conditions, the transmission and scattering of radiations from the sun depend to a large extent on the climatic characteristics of a place and the season of the year, as the turbidity of the atmosphere, the moisture content of air and the albedo of the earth's surface are similar over a climatic zone for any season. Again, the cloud characteristics may change with the season and the climate. Computed values of G/Go from siS for Ahmadabad and New Delhi have r,m.s, errors of 0.026 and 0.019, respectively, for single station fits over the year, while for our correlations, these are 0.010 and 0.007 for the same set of data. One expects that such should be the case for locations where the turbidity of the atmosphere, the moisture content in the air and the albedo of the sky and the earth vary strongly with the season. An attempt was made to use S', the average daily period over which Campbell-Stokes recorders remain sensitive instead of S, the day length for the correlations. No improvement in r.m.s, errors was found, and one may use either of them. Our technique of making climatological and seasonal partitions of data for regression analysis has proved to be very successful for a large part of India and may also be found suitable for estimating G from sunshine duration in many lands. For countries where there is a shortage of pyranometer stations, it should be particularly useful to employ the method for assessing solar energy availability.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

A. Angstrom, Q. JI R. met. Soc. 50, 121 (1924). C. C. Y. Ma and M. Iqbal, Sol. Energy 33, 143 (1984). J. A. Duffle and W. A. Beckman, Solar Engineering of Thermal Processes. Wiley, New York (1980). M. R. Rietveld, Agric. Met. 19, 243 (1978). J. Glover and J. S. G. McCulloch, Q. Jl R. met. Soc. 84, 172 (1958). R. Dogniaux and M. Lemoine, Proc. Int. Daylighting Conf., Phoenix, Ariz. (1983). H. P. Garg and S. N. Garg, Energy Convers. Mgmt 23, 113 (1983). M. Hussain, Sol. Energy 33, 217 (1984). J. E. Hay, Sol. Energy 23, 301 (1979). A. Mani and S. Rangarajan, Solar Radiation Over India. Allied Publishers, New Delhi (1982).

172 1!. 12. 13. 14. 15. 16. 17.

HUSSAIN:

GLOBAL RADIATION AND SUNSHINE DURATION

M. Hussain, Proc. ENERGEX Conf., Regina, Canada, pp. 393-396 (1984). R. B. Benson, M. V. Paris, J. E. Sherry and C. G. Justus, Sol. Energy 32, 523 (1984). G. T. Trewartha and L. H. Horn, An Introduction to Climate, 5th edn. McGraw-Hill, Tokyo (1980). A. Mani, O. Chacko and S. Hariharan, Tellus XXI, 6 (1969). A. Mani, Handbook of Solar Radiation Data for India. Allied Publishers, New Delhi (1980). WMO Technical Note No. 172, Chap. 2, Secretariat of the World Meteorological Organization (1981). Y. P. Rao, in Worm Survey of Climatology (Edited by K. Takahashi and H. Arakawa), Vol. 9. Elsevier, Amsterdam (1981). 18. H. P. Garg and S. N. Garg, Energy Convers. Mgmt 25, 409 (1985).