A universal relationship for estimating clear sky insolation

Energy Conversion and Management 45 (2004) 27–52www.elsevier.com/locate/enconman

A universal relationship for estimating clear sky insolation

Chakradhar Lingamgunta, T. Nejat Veziroglu *

Department of Mechanical Engineering, Clean Energy Research Institute, College of Engineering,

University of Miami, P.O. Box 248294, Coral Gables, FL 33124-0622, USA

Received 16 December 2002; accepted 12 May 2003

Abstract

An investigation of estimating clear sky insolation 1 has been conducted. Some discrepancies have been

found between the conventional method of estimation and the data. It has been noticed that the con-

ventional method underestimates the insolation. A simpler, more elegant and more accurate ‘‘universal’’

relationship has been derived for estimating the annual mean daily clear sky insolation as a function of the

latitude and altitude of a given location. As part of this derivation, two simple and accurate correlations

have been obtained, i.e. for extraterrestrial insolation and for atmospheric attenuation.� 2003 Elsevier Ltd. All rights reserved.

Keywords: Clear sky insolation; Extraterrestrial insolation; Annual mean clear sky insolation; Atmospheric

attenuation; Mean atmospheric attenuation

* Corresponding author. Tel.: +1-305-284-4666; fax: +1-305-284-4792.

E-mail address: [email protected] (T.N. Veziroglu).1 For ‘‘Solar Radiation’’, the term ‘‘Insolation’’ will be used.

0196-8904/$ - see front matter � 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0196-8904(03)00111-0

mail to: [email protected]

Nomenclature

A altitude of locationA dimensionless altitude, altitude of location divided by height of tallest building in

world, Petronas Towers (452 m) in Kuala Lumpur, Malaysia (dimensionless)a0 constant (dimensionless)a1 constant (dimensionless)Daa dimensionless annual mean daily insolation loss due to atmosphere; difference be-

tween dimensionless calculated annual mean daily extraterrestrial insolation onhorizontal surface, H ao, and dimensionless measured annual mean daily clear skyinsolation on horizontal surface, H acm (dimensionless)

Daae estimated dimensionless annual mean daily insolation loss due to atmosphere (di-mensionless)

Daam measured dimensionless annual mean daily insolation loss due to atmosphere (di-mensionless)

Eaa percentage error for atmospheric attenuation (dimensionless)Eao percentage error for extraterrestrial insolation (dimensionless)Ep percentage error (dimensionless)Elv percentage error for clear sky insolation (dimensionless)Gc clear sky solar radiation on horizontal surface (Wm�2)Gcb clear sky beam solar radiation on horizontal surface (Wm�2)Gcd clear sky diffuse solar radiation on horizontal surface (Wm�2)Gcnb clear sky beam solar radiation normal to surface (Wm�2)Gcnd clear sky diffuse solar radiation normal to surface (Wm�2)Gon solar radiation outside atmosphere on surface normal to itself (Wm�2)Gsc solar constant; energy from sun, per unit time, received on unit area of surface

perpendicular to direction of propagation of radiation, at mean earth–sun distanceoutside of the atmosphere. (Value ¼ 1367 Wm�2)

Hac annual mean daily clear sky solar radiation on horizontal surface (Jm�2)Hacm measured annual mean daily clear sky solar radiation on horizontal surface (Jm�2)H ac annual mean daily clear sky solar radiation on horizontal surface non-dimensiona-

lised with respect to solar constant times 24 h, H ac ¼ Hac24�3600�Gsc

(dimensionless)H ace estimated annual mean daily clear sky solar radiation on horizontal surface non-

dimensionalised with respect to solar constant times 24 h (dimensionless)H aclv Lingamgunta–Veziroglu correlation for annual mean daily clear sky solar radiation

on horizontal surface non-dimensionalised with respect to solar constant times 24 h,H aclv ¼ Hac

24�3600�Gsc(dimensionless)

H acm measured annual mean daily clear sky solar radiation on horizontal surface non-dimensionalised with respect to solar constant Gsc (dimensionless)

Hao annual mean daily extraterrestrial radiation on horizontal surface (Jm�2)H ao annual mean daily solar radiation outside the atmosphere on horizontal surface non-

dimensionalisedwithrespect tosolarconstant times24h,H ao¼ Hao24�3600�Gsc

(dimensionless)

28 C. Lingamgunta, T.N. Veziroglu / Energy Conversion and Management 45 (2004) 27–52

H aoe estimated annual mean daily solar radiation outside atmosphere on horizontal sur-face (dimensionless)

H aom measured annual mean daily solar radiation outside atmosphere on horizontal sur-face (dimensionless)

Hcb daily clear sky beam solar radiation on horizontal surface (Jm�2)Hcd daily clear sky diffuse solar radiation on horizontal surface (Jm�2)Hmc monthly mean daily clear sky insolation (Jm�2)Hmo dimensionless monthly mean daily extraterrestrial radiation on horizontal surface

(dimensionless)Ho daily extraterrestrial insolation on horizontal surface (Jm�2)Ho dimensionless daily extraterrestrial radiation on horizontal surface; extraterrestrial

insolation on horizontal surface divided by 24 h times solar constant, Ho ¼ Ho24�3600�Gsc

(dimensionless)i hemisphere power: i ¼ 1 for northern hemisphere and i ¼ 2 for southern hemisphereIcb hourly clear sky beam solar radiation on horizontal surface (Jm�2)Icd hourly clear sky diffuse solar radiation on horizontal surface (Jm�2)j month of year (dimensionless)k constant (dimensionless)l hour of day (dimensionless)n day of year assuming first day of year to be one (dimensionless)N number of locations (dimensionless)r0 correction factor (km�2)r1 correction factor (km�2)rk correction factor (km�2)

Greek symbolsd declination, angular position of sun at solar noon (i.e. when sun is on the local

meridian) with respect to plane of equator, north positive (degrees)/ latitude, angular location north or south of the equator (degrees)hz zenith angle, angle between vertical and line to sun (degrees)sb atmospheric transmittance for beam solar radiation; ratio of clear sky beam solar

radiation normal to surface to extraterrestrial radiation normal to surface, i.e.sb ¼ Gcnb

Gon(dimensionless)

sd atmospheric transmittance for diffuse solar radiation; ratio of clear sky diffuse solarradiation normal to surface to extraterrestrial radiation normal to surface, i.e.sd ¼ Gcnd

Gon(dimensionless)

x hour angle, angular displacement of sun east or west of local meridian due to ro-tation of Earth on its axis at 15� per hour, morning negative, afternoon positive(degrees)

xs sunset hour angle (degrees)

C. Lingamgunta, T.N. Veziroglu / Energy Conversion and Management 45 (2004) 27–52 29


1. Introduction

In many applications of solar energy, the solar radiation incident on the surface of the earth atthe location of interest is an important input parameter. Knowledge of the solar radiation enablesus to derive information about the performance of solar energy systems. The purpose of thepresent study is to derive a universal relationship that is simpler than these set of conventionalequations in existence in order to estimate the annual mean daily clear sky insolation at a givenlocation on the earth�s surface with better accuracy.Fig. 1 schematically shows a certain location P as well as the position of the sun. Fig. 1(a) indicates

the solar radiation received at location P with latitude / on the surface of the earth. It should benoted that the solar radiation is subject to attenuation in the atmosphere, which depends on thecomposition of the atmosphere and the distance traveled through it. The amount of solar radiationreceived by a location varies with the position of the sun. The position of the sun with respect to theearth can be determined through the declination angle d and the hour angle x as shown in Fig. 1(b).Also, the zenith angle hz made by the sun with a horizontal surface at location P is shown in Fig. 1(c).The solar radiation received at the surface of the earth is subject to variations due to changes in

the solar radiation outside the atmosphere and to two additional and more significant pheno-

S

N

EW

SUN

O

PMeridianLocation

Equator

Equatorialplane

Latitude

Atmosphere

Earth

W

S

N

O

V

U

Projection of Sun's ray in equatorial plane

Sun's ray(a)

(b)

N

(c)

W

Q

SHorizontal plane at P (tangential surface at p to the earth surface)

P

Perpendicular to the horizontal plane P (coming out of the

plane of the paper)

Projection of sun's ray in a horizontal plane

U

V

E

E

Projection of NS meridian in equatorial plane

U

θ

φ δ

ω

z

Fig. 1. Schematic diagram of insolation for a given location, showing the pertinent geometry and parameters

(U indicates sun�s direction).


mena, viz. (a) atmospheric scattering by air molecules, dust etc. and (b) atmospheric absorptionby O3, H2O, CO2 and pollution. The effects of the atmosphere in scattering and absorbing solarradiation are variable with space and time as atmospheric conditions and air mass change.Therefore, the attenuation of the solar radiation by the atmosphere is a function of the latitude,altitude and pollution, as well as time.In order to predict the insolation for a given location, it is useful to define a standard ‘‘clear

sky’’ and calculate the radiation that would be received on a horizontal surface under no-cloudconditions. Good prediction of the clear sky solar radiation for a given location requires, inprincipal, long term average meteorological data, which are still scarce for many locations aroundthe world. It is, therefore, not always possible to predict the value of clear sky solar radiation witha reasonable accuracy for many locations.For a given location on the surface of the earth, Hottel [1] has presented a method for esti-

mating the clear sky beam solar radiation Gcb which takes into account zenith angle, hz and al-titude A for a standard atmosphere and for four different climate types using a set of six equations.Liu and Jordan [2] developed an empirical relationship which includes the transmission coeffi-cients for beam solar radiation sb and diffuse solar radiation sd for clear days. The clear sky diffusesolar radiation Gcd, estimated by Liu and Jordan, can be added to the clear sky beam solarradiation Gcb, as predicted by Hottel, to obtain the total clear sky solar radiation, Gc.There are several correlations and/or models for estimating clear sky and cloudy sky insolations.

These can be found in reports of Task IX of the International Energy Agency (IEA) [3], Bener [4],Davies et al. [5], Festa and Ratto [6] and Viorel Badescu [7]. Out of these review papers, pertainingspecifically to clear sky insolation estimates, five correlations were noted to be of better accuracy, viz.the Hourwitz correlation [8,9], the Daneshyar–Paltridge–Proctor correlation [10,11], the Berger–Duffie correlation [12], the Adnot–Bourges–Campana–Gicquel correlation [13] and the Kasten–Czeplak correlation [14]. These correlations are presented in the Appendix A. As can be seen fromthe equations given in the Appendix A, these correlations give the clear sky insolation estimates as afunction of the zenith angle hz alone. Although hz is important in estimating the clear sky insolation,there is one shortcoming of these correlations. They neglect the altitude effect, although it is wellknown that the higher is the altitude of a given location, the higher is the clear sky insolation.If we are interested in the annual mean daily clear sky insolation, then we must firstly calculate

the hourly insolations, secondly sum them for a given day and finally sum them for 365 days.These all result in tedious and long calculations.

2. Experimental data

The experimental data for monthly mean daily clear sky solar radiation on a horizontal surfaceHmc have been obtained and/or calculated for 120 locations around the world [15].

2 These

2 Anon. Site URL: http://eosweb.larc.nasa.gov/sse [15] indicates that their 1· 1 degree cell averaged clear sky datamay have up to 10% cloudiness. Relative to ground site data, these satellite based data have biases between )2% to

+3.3% and RMS uncertainties in the range of 14% as noted in both the Methodology and Accuracy sections of the

above website.

http://eosweb.larc.nasa.gov/sse


locations have been selected to cover the location variables (viz. latitudes, longitudes, altitudes)more or less randomly. For all the locations considered, the annual mean daily clear sky solarradiation on a horizontal surface Hac has been calculated from the monthly mean daily clear skysolar radiation Hmc by taking their arithmetic mean, i.e.,

Hac ¼1

12

X12j¼1

Hmc ð1Þ

For the 120 locations, Table 1 presents their names, countries, latitudes, longitudes, altitudes andthe annual mean daily clear sky solar radiation on a horizontal surface Hac, calculated using Eq.(1). As can be seen from Table 1, the locations selected represent 48 countries in the five conti-nents. The latitude of the selected locations in the northern hemisphere ranges from 0.05� near theequator for Entebbe, Uganda, to 64.82� north of the equator for Fairbanks, USA. In the southernhemisphere it ranges from )1.3� near the equator for Nairobi, Kenya, to )41.28� south of theequator for Wellington, New Zealand. The altitudes vary from 1 m above sea level for Venice,Italy, to 3085 m above sea level for Eldoret, Kenya. The annual mean daily clear sky insolationranges from a minimum of 13.28 · 106 Jm�2 for Archangel, Russia, to a maximum of 26.10 · 106Jm�2 for Addis Ababa, Ethiopia. Table 1 also presents cloud cover data for the 120 locations forusefulness.

3. Conventional way of estimating clear sky insolation

A non-dimensional annual mean daily clear sky solar radiation on a horizontal surface H ac canbe defined as:

H ac ¼Hac

24� 3600� Gscð2Þ

where Hac is the annual mean daily clear sky solar radiation and Gsc is the solar constant. If, for agiven day, Hcb is the daily clear sky beam solar radiation on a horizontal surface and Hcd is thedaily clear sky diffuse solar radiation on a horizontal surface, then Hac can be calculated from:

Hac ¼1

365

X365n¼1

ðHcb þ HcdÞn ð3Þ

In Eq. (3), Hcb and Hcd can be calculated from the following relationships:

Hcb ¼X24l¼1

Icb ð4Þ

and

Hcd ¼X24l¼1

Icd ð5Þ

Table 1

Geographical position of the 120 locations considered, the measured annual mean daily clear sky insolation, Hac, andcloud cover

No. Location Country Latitude

(degrees) /Longitude

(degrees)

Altitude

(m) AAnnual mean

daily clear sky

insolation

(measured,

106 Jm�2) Hac

Cloud

cover (%)

1 Abu Dhabi UAE 24.43 54.65 13.0 23.67 43.48

2 Accra Ghana 5.60 )0.17 54.0 24.16 64.58

3 Addis Ababa Ethiopia 8.98 38.80 2450.0 26.10 58.40

4 Adelaide Australia )34.93 138.52 43.0 21.71 63.55

5 Albuquerque USA 35.05 )106.62 1620.0 22.46 55.52

6 Ankara Turkey 39.95 32.28 861.0 20.77 62.21

7 Antalya Turkey 36.87 30.73 40.0 21.46 48.24

8 Archangel Russia 64.58 40.50 4.0 13.28 69.53

9 Arusha Tanzania )3.33 36.62 154.0 25.96 68.63

10 Atlanta USA 33.65 )84.43 315.0 21.56 62.17

11 Auckland New Zealand )37.02 174.80 49.0 21.17 68.34

12 Bangkok Thailand 13.73 100.57 2.0 20.48 65.97

13 Bechar Algeria 31.62 )2.23 26.0 22.27 42.01

14 Belfast Britain 54.65 )6.22 67.0 15.84 80.64

15 Bogota Colombia 4.70 )74.13 2556.0 25.60 79.63

16 Bombay India 19.12 72.85 6.0 24.16 47.18

17 Bordeaux France 44.83 )0.70 47.0 18.76 67.68

18 Boston USA 42.37 )71.03 192.0 19.66 71.70

19 Brisbane Australia )27.43 153.08 41.0 23.36 54.97

20 Buenos Aires Argentina )34.58 )58.48 25.0 21.31 59.10

21 Bulawayo Zimbabwe )20.15 28.62 1356.0 24.73 48.90

22 Buraimi Oman 24.23 55.78 284.0 23.62 45.40

23 Cairo Egypt 30.08 31.28 95.0 22.82 38.18

24 Calcutta India 22.65 88.45 6.0 23.26 56.03

25 Cambridge Britain 52.22 0.10 12.0 16.16 77.25

26 Canberra Australia )35.30 149.18 629.0 21.38 63.03

27 Caracas Venezuela 10.50 )66.88 1035.0 24.37 55.99

28 Casablanca Morocco 33.57 )7.67 49.0 22.23 48.04

29 Charleston USA 32.90 )80.03 12.0 22.14 62.73

30 Chittagong Bangladesh 22.35 91.82 27.0 23.22 59.10

31 Cleveland USA 41.40 )81.85 266.7 19.80 74.31

32 Colombo Sri Lanka 6.90 79.87 7.0 25.67 75.13

33 Copenhagen Denmark 55.67 12.30 9.0 15.19 69.77

34 Dakar Senegal 14.73 )17.50 23.0 23.72 54.69

35 Dallas USA 32.90 )97.03 146.0 22.36 58.51

36 Dar Es Salaam Tanzania )6.87 39.20 14.0 25.99 58.61

37 Dhaka Bangladesh 23.97 90.38 8.0 22.61 56.53

38 Djakarta Indonesia )6.18 106.83 8.0 25.13 71.92

39 Edmonton USA 53.35 )114.10 206.0 17.50 73.80

40 Eldoret Kenya 0.53 35.28 3085.0 25.67 64.42

41 Entebbe Uganda 0.05 32.45 1146.0 25.06 68.61

42 Fairbanks USA 64.82 )147.87 133.0 13.79 68.29

(continued on next page)


Table 1 (continued)



(degrees)

Altitude

(m) AAnnual mean

daily clear sky

insolation

(measured,

106 Jm�2) Hac

Cloud

cover (%)

43 Freiburg Germany 48.00 7.85 278.0 17.21 70.72

44 Genova Italy 44.82 8.85 54.0 18.97 54.89

45 Goa India 15.48 73.82 1022.0 24.16 53.38

46 Goteborg Sweden 57.70 12.00 31.0 14.54 67.57

47 Hamburg Germany 53.63 10.00 14.0 15.26 75.06

48 Hanoi Vietnam 21.03 105.85 16.0 22.39 73.38

49 Harare Zimbabwe )17.83 31.02 1500.0 24.91 52.65

50 Havana Cuba 23.17 )82.35 24.0 23.90 53.50

51 Indianapolis USA 39.73 )86.27 241.0 20.59 69.11

52 Irkutsk Russia 52.27 104.35 468.0 17.03 71.13

53 Izmir Turkey 38.43 27.17 28.0 20.20 60.84

54 Jerusalem Israel 31.78 35.22 757.0 22.43 43.60

55 Karachi Pakistan 24.90 67.13 4.0 23.04 43.48

56 Kasama Zambia )10.12 31.13 1380.0 25.60 56.40

57 King�s Park Hong Kong 22.32 114.17 33.0 21.64 70.97

58 Kinshasa Zaire )4.37 15.25 358.0 23.76 69.48

59 Kodaikanal India 10.23 77.47 2150.0 24.23 69.03

60 Kunming China 25.02 102.68 400.0 20.27 63.53

61 Lahore Pakistan 31.55 74.33 214.0 21.67 42.88

62 Lodwar Kenya 3.12 35.62 506.0 25.20 57.94

63 London Britain 51.52 )0.12 5.0 16.42 76.48

64 Los Angeles USA 33.93 )118.40 103.0 22.79 59.51

65 Luanda Angola )8.85 13.23 45.0 24.88 65.82

66 Lusaka City Zambia )15.42 28.32 1272.0 25.16 50.95

67 Macon France 46.30 4.80 47.0 18.04 68.48

68 Madison USA 43.13 )89.13 270.0 19.94 71.70

69 Madras India 13.00 18.18 16.0 24.59 43.79

70 Madrid Spain 40.45 )3.72 667.0 19.66 58.58

71 Mansa Zambia )11.10 28.85 1178.0 25.45 53.68

72 Marseille France 43.45 5.23 3.0 19.33 57.79

73 Melbourne Australia )37.82 144.97 35.0 20.88 66.23

74 Mendoza Argentina )32.83 )68.78 769.0 22.79 60.70

75 Milan Italy 45.43 9.28 147.0 18.68 63.48

76 Mombasa Kenya )4.03 39.62 55.0 25.78 59.94

77 Mongu Zambia )15.25 23.15 1047.0 24.98 48.33

78 Montreal Canada 45.47 )73.75 17.0 19.62 74.48

79 Moscow Russia 55.75 37.57 156.0 15.34 70.03

80 Naha Japan 26.20 127.68 36.0 23.62 74.93

81 Nairobi Kenya )1.30 36.75 1661.0 26.03 68.58

82 Nancy France 48.68 6.22 212.0 17.06 71.03

83 Ndola Zambia )13.00 28.65 1262.0 25.34 52.88

84 New Delhi India 28.58 77.20 239.0 22.43 43.06

85 New York USA 40.78 )73.97 96.0 19.73 72.28

86 Nice France 43.65 7.20 5.0 19.30 56.90


Table 1 (continued)



(degrees)

Altitude

(m) AAnnual mean

daily clear sky

insolation

(measured,

106 Jm�2) Hac

Cloud

cover (%)

87 Omsk Russia 54.93 73.40 105.0 16.49 65.47

88 Ottawa Canada 45.32 )75.67 38.0 19.66 74.53

89 Paris France 48.82 2.33 52.0 16.96 70.75

90 Passau Germany 48.58 13.47 409.0 17.39 70.87

91 Perth Australia )31.93 151.95 64.0 22.03 58.70

92 Phoenix USA 33.93 )112.02 340.0 22.18 53.11

93 Quebec Canada 43.65 )70.32 23.0 19.91 72.59

94 Reno USA 39.50 )119.78 1342.0 21.28 56.91

95 Rome Italy 41.80 12.55 46.0 19.87 54.80

96 Salt Lake City USA 40.78 )111.20 1286.0 21.46 60.42

97 Santiago Cuba 20.05 )75.82 178.0 23.22 47.94

98 Santiago Chile )33.45 )70.70 520.0 22.93 63.78

99 Sapporo Japan 43.05 141.33 17.0 19.80 75.99

100 Seattle USA 47.45 )122.30 4.0 19.04 74.40

101 Seoul South Korea 37.57 126.97 86.0 20.95 64.62

102 Seville Spain 37.42 )5.90 30.0 20.92 50.63

103 Shanghai China 31.17 121.43 5.0 20.70 70.65

104 Strasbourg France 48.55 7.63 149.0 16.74 70.72

105 Tashkent Uzbekistan 41.27 69.27 479.0 20.30 60.83

106 Tucuman Argentina )26.85 )65.10 481.0 24.05 63.08

107 Tunis Tunisia 36.83 10.23 3.0 21.27 53.64

108 Turin Italy 45.50 12.33 238.0 18.43 63.28

109 Ulan Bator Mongolia 47.93 106.98 1325.0 18.79 65.98

110 Valparaiso Chile )33.03 )71.60 41.0 23.08 61.59

111 Vancouver Canada 43.67 )79.38 2.0 20.05 75.00

112 Venice Italy 45.50 12.37 1.0 16.81 63.28

113 Vienna Austria 48.25 16.37 203.0 16.81 71.11

114 Visakha-

patnam

India 17.72 83.33 4.6 24.66 61.38

115 Vladivostok Russia 43.12 131.90 138.0 19.04 67.04

116 Warsaw Poland 52.28 20.97 107.0 15.41 72.68

117 Wellington New Zealand )41.28 174.77 2.0 19.98 67.85

118 Windhoek Namibia )22.57 17.10 1728.0 24.73 36.69

119 Winnipeg Canada 49.90 )97.23 254.0 18.43 72.28

120 Zurich Switzerland 47.48 8.53 569.0 17.86 68.66

Means 22.90 15.92 385.1 21.28 62.28


where Icb is the hourly clear sky beam solar radiation on a horizontal surface and Icd is the hourlyclear sky diffuse solar radiation on a horizontal surface. They can be obtained from the followingrelationships:

Icb ¼ 3600� Gonsb cos hz ð6Þ


and

Icd ¼ 3600� Gonsd cos hz ð7Þ
where Gon is the extraterrestrial solar radiation on a surface normal to itself, sb is the atmospherictransmittance for beam solar radiation, hz is the zenith angle and sd is the atmospheric trans-mittance for diffuse solar radiation. The atmospheric transmittance for beam solar radiation sb,the zenith angle hz and the atmospheric transmittance for diffuse solar radiation sd are evaluatedbased on the midpoint of the hour.From Eqs. (3)–(7), one obtains:
Hac ¼3600

365

X365n¼1

X24l¼1

Gon cos hzðsb þ sdÞ ð8Þ

The solar radiation outside the atmosphere on a surface normal to itself Gon can be calculatedfrom:

Gon ¼ Gsc 1�

þ 0:033 cos 360n365

�ð9Þ

where Gsc is the solar constant and n is the day of the year. Substituting Eq. (9) in Eq. (8) and theresulting equation in Eq. (2), the dimensionless annual mean daily clear sky solar radiationbecomes:

H ac ¼1

24� 365X365n¼1

1

�þ 0:033 cos 360n

365

�X24l¼1

sbð þ sdÞ cos hz ð10Þ

where cos hz can be calculated from:

cos hz ¼ cos/ cos d cosx þ sin/ sin d ð11Þ
where / is the latitude of the location, d is the declination angle and x is the hour angle. In Eq.(11), the declination angle d can be calculated from the equation by Cooper [16]:
d ¼ 23:45 sin 360284þ n365

� �ð12Þ

The atmospheric transmittance for beam solar radiation sb can be calculated from the equationby Hottel [1]:

sb ¼ a0 þ a1 expð�k= cos hzÞ ð13Þ
where a0, a1 and k are constants. These constants for the standard atmosphere with 23 km visi-bility and altitudes of the location less than 2.5 km can be calculated from the following equationsproposed by Hottel [1]:
a0 ¼ f0:4237� 0:00821ð6� AÞ2g r0 ð14Þ

a1 ¼ f0:5055þ 0:00595ð6:5� AÞ2g r1 ð15Þ

k ¼ f0:2711� 0:01858ð2:5� AÞ2g rk ð16Þ


where r0, r1 and rk are correction factors and A is the altitude of the location in km. The correctionfactors for four different climate types, viz. Tropical, Midlatitude Summer, Subarctic Summer andMidlatitude Winter are given in Table 2 [17]. In order to eliminate the difficulty (or the impos-sibility) of determining the seasons for the data in Table 1, we have combined the climate typesMidlatitude Summer and Midlatitude Winter into one, viz. Midlatitude by taking the averages(see Table 3).The atmospheric transmittance for diffuse solar radiation sd can be calculated from the

equation by Liu and Jordan [2]:

Table

Corre

Clim

Tro

Mid

Sub

Mid

Table

Corre

Clim

Tro

Mid

sd ¼ 0:271� 0:294sb ð17Þ

Using Eqs. (10)–(17) and Tables 1 and 3, the non-dimensional annual mean daily clear sky solarradiation H ac has been evaluated for the 120 locations considered using a FORTRAN computerprogram. The results, non-dimensionalised actual data (from Table 1) and the percentage error,together with location information, have been tabulated in Table 4. The percentage error Ep hasbeen defined as follows:

Ep ¼H ace � H acm

H acm

� 100% ð18Þ

where H ace is the dimensionless estimated annual mean daily clear sky insolation on a horizontalsurface and H acm is the dimensionless measured annual mean daily clear sky insolation on ahorizontal surface. It can be observed that the conventional method underestimates the insolationfor 104 locations out of the 120 locations considered and overestimates for 16 locations. The errorbetween the estimated and actual insolation varies from a minimum of )0.13% for King�s Park,Hong Kong, to a maximum of )22.33% for Fairbanks, Alaska, USA, with a mean absolute errorof 8.81%. When we take into account that the actual data may include cloudiness up to 10%(mean cloudiness of 5%), then the mean underestimation becomes about 14%. The conventionalmethod of estimation is not only too time consuming, involving thousands of calculations, butalso has a relatively high error and, in general, underestimates the insolation. Therefore, in the

2

ction factors for different climate types [17]

ate type r0 r1 rk

pical 0.95 0.98 1.02

latitude Summer 0.97 0.99 1.02

arctic Summer 0.99 0.99 1.01

latitude Winter 1.03 1.01 1.00

3

ction factors used for the 120 locations considered

ate type r0 r1 rk

pical 0.95 0.98 1.02

latitude 1.00 1.00 1.01

Table 4

Comparison of clear sky insolation data with conventional estimation

No. Location Annual mean daily

clear sky insolation

(measured, dimen-

sionless) H acm

Annual mean daily


(conventional,

dimensionless) H ace

Percentage errorHace�Hacm

Hacm� 100%

1 Abu Dhabi 0.2004 0.1790 )10.682 Accra 0.2046 0.1999 )2.313 Addis Ababa 0.2208 0.2302 4.27

4 Adelaide 0.1838 0.1632 )11.215 Albuquerque 0.1902 0.1871 )1.606 Ankara 0.1757 0.1656 )5.767 Antalya 0.1816 0.1588 )12.538 Archangel 0.1125 0.0895 )20.489 Arusha 0.2198 0.2029 )7.6810 Atlanta 0.1825 0.1713 )6.1311 Auckland 0.1792 0.1586 )11.5012 Bangkok 0.1735 0.1929 11.18

13 Bechar 0.1886 0.1696 )10.0714 Belfast 0.1341 0.1148 )14.4515 Bogota 0.2166 0.2332 7.66

16 Bombay 0.2045 0.1868 )8.6517 Bordeaux 0.1587 0.1397 )11.9718 Boston 0.1665 0.1485 )10.7919 Brisbane 0.1978 0.1748 )11.6220 Buenos Aires 0.1804 0.1636 )9.3421 Bulawayo 0.2094 0.2082 )0.5722 Buraimi 0.2001 0.1851 )7.4823 Cairo 0.1933 0.1719 )11.0824 Calcutta 0.1970 0.1819 )7.6625 Cambridge 0.1369 0.1202 )12.2126 Canberra 0.1811 0.1732 )4.3327 Caracas 0.2064 0.2149 4.17

28 Casablanca 0.1882 0.1660 )11.7929 Charleston 0.1873 0.1669 )10.9230 Chittagong 0.1967 0.1828 )7.0831 Cleveland 0.1677 0.1523 )9.1632 Colombo 0.2172 0.1982 )8.7433 Copenhagen 0.1287 0.1112 )13.6134 Dakar 0.2008 0.1924 )4.1935 Dallas 0.1894 0.1696 )10.4236 Dar Es Salaam 0.2201 0.1984 )9.8337 Dhaka 0.1914 0.1799 )5.9938 Djakarta 0.2128 0.1986 )6.6539 Edmonton 0.1481 0.1204 )18.6840 Eldoret 0.2174 0.2359 8.53

41 Entebbe 0.2121 0.2210 4.23

42 Fairbanks 0.1167 0.0906 )22.3343 Freiburg 0.1456 0.1357 )6.8244 Genova 0.1606 0.1399 )12.92


Table 4 (continued)



(measured, dimen-

sionless) H acm

Annual mean daily


(conventional,



Hacm� 100%

45 Goa 0.2045 0.2098 2.58

46 Goteborg 0.1231 0.1063 )13.6947 Hamburg 0.1293 0.1166 )9.8848 Hanoi 0.1895 0.1844 )2.6549 Harare 0.2109 0.2131 1.03

50 Havana 0.2024 0.1815 )10.3451 Indianapolis 0.1743 0.1559 )10.5352 Irkutsk 0.1443 0.1273 )11.7553 Izmir 0.1709 0.1550 )9.3254 Jerusalem 0.1897 0.1831 )3.5055 Karachi 0.1952 0.1784 )8.6256 Kasama 0.2167 0.2199 1.49

57 King�s Park 0.1832 0.1829 )0.1358 Kinshasa 0.2012 0.2067 2.77

59 Kodaikanal 0.2052 0.2274 10.83

60 Kunming 0.1715 0.1860 8.48

61 Lahore 0.1834 0.1738 )5.2762 Lodwar 0.2134 0.2100 )1.5863 London 0.1390 0.1219 )12.3264 Los Angeles 0.1931 0.1666 )13.7265 Luanda 0.2106 0.1979 )6.0266 Lusaka City 0.2131 0.2134 0.14

67 Macon 0.1526 0.1360 )10.8568 Madison 0.1690 0.1480 )12.3869 Madras 0.2083 0.1940 )6.8770 Madrid 0.1663 0.1614 )2.9171 Mansa 0.2155 0.2165 0.47

72 Marseille 0.1637 0.1423 )13.0573 Melbourne 0.1768 0.1565 )11.4774 Mendoza 0.1929 0.1810 )6.1875 Milan 0.1582 0.1400 )11.4776 Mombasa 0.2182 0.2005 )8.1177 Mongu 0.2115 0.2104 )0.5278 Montreal 0.1661 0.1376 )17.2179 Moscow 0.1297 0.1133 )12.6880 Naha 0.1999 0.1769 )11.5381 Nairobi 0.2204 0.2273 3.14

82 Nancy 0.1445 0.1328 )8.0883 Ndola 0.2146 0.2159 0.62

84 New Delhi 0.1900 0.1767 )7.0085 New York 0.1671 0.1507 )9.8386 Nice 0.1634 0.1419 )13.1887 Omsk 0.1395 0.1146 )17.8488 Ottawa 0.1664 0.1383 )16.8789 Paris 0.1436 0.1297 )9.68



Table 4 (continued)



(measured, dimen-

sionless) H acm

Annual mean daily


(conventional,



Hacm� 100%

90 Passau 0.1472 0.1363 )7.4391 Perth 0.1865 0.1700 )8.8992 Phoenix 0.1877 0.1712 )8.7993 Quebec 0.1685 0.1422 )15.6194 Reno 0.1801 0.1731 )3.9195 Rome 0.1682 0.1472 )12.4696 Salt Lake City 0.1816 0.1689 )7.0397 Santiago 0.1967 0.1892 )3.8598 Santiago 0.1942 0.1755 )9.6199 Sapporo 0.1676 0.1436 )14.32100 Seattle 0.1612 0.1323 )17.91101 Seoul 0.1773 0.1581 )10.81102 Seville 0.1771 0.1573 )11.15103 Shanghai 0.1751 0.1702 )2.79104 Strasbourg 0.1418 0.1320 )6.86105 Tashkent 0.1719 0.1563 )9.09106 Tucuman 0.2036 0.1843 )9.48107 Tunis 0.1801 0.1579 )12.29108 Turin 0.1562 0.1415 )9.41109 Ulan Bator 0.1591 0.1502 )5.56110 Valparaiso 0.1954 0.1672 )14.41111 Vancouver 0.1699 0.1418 )16.56112 Venice 0.1423 0.1372 )3.62113 Vienna 0.1423 0.1337 )6.04114 Visakhapatnam 0.2087 0.1885 )9.68115 Vladivostok 0.1611 0.1457 )9.57116 Warsaw 0.1305 0.1216 )6.81117 Wellington 0.1692 0.1476 )12.72118 Windhoek 0.2094 0.2086 )0.36119 Winnipeg 0.1561 0.1303 )16.50120 Zurich 0.1512 0.1417 )6.29

Means 0.1796 0.1669 8.81

Note: After the pertinent computer calculations, the data in columns 3 and 4 have been rounded to four digits after the

period, and to two digits in column 5.


following section, a new Universal Relationship will be developed that will give the annual meandaily clear sky insolation in a straightforward manner and also with better accuracy.

4. Universal relationship for estimating clear sky insolation

It will be convenient to divide this section into three parts, viz. (1) atmospheric attenuation, (2)extraterrestrial insolation and (3) clear sky insolation.


4.1. Atmospheric attenuation

The difference between the calculated annual mean daily extraterrestrial insolation on a hori-zontal surface Hao and the measured annual mean daily clear sky insolation on a horizontalsurface Hacm for a given location on the surface of the earth is the insolation loss due to atmo-spheric attenuation. In dimensionless form, this difference becomes:

Table

Recom

Mo

Jan

Feb

Ma

Apr

Ma

Jun

July

Aug

Sep

Oct

Nov

Dec

Daa ¼ H ao � H acm ð19Þ

where H ao is the dimensionless calculated annual mean daily extraterrestrial insolation and H acm isthe dimensionless measured annual mean daily clear sky insolation. The daily extraterrestrialradiation on a horizontal surface Ho is given by [17]:

Ho ¼24� 3600� Gsc

p1

�þ 0:033 cos 360n

365

�cos/ cos d sinxs

�þ pxs

180sin/ sin d

�ð20Þ

where xs is the sunset hour angle. A non-dimensional daily extraterrestrial insolation on a hori-zontal surface Ho can be defined as:

Ho ¼Ho

24� 3600� Gscð21Þ

Substituting Eq. (20) in Eq. (21), we get

Ho ¼1

p1

�þ 0:033 cos 360n

365

�cos/ cos d sinxs

�þ pxs

180sin/ sin d

�ð22Þ

The dimensionless monthly mean daily solar radiation outside the atmosphere on a horizontalsurface Hmo can be calculated from Eq. (22) using n and d for the mean day of the month fromTable 5 [17]. Then, the dimensionless annual mean daily extraterrestrial insolation on a horizontalsurface H ao can be calculated from the monthly mean daily clear sky solar radiation Hmo by takingtheir arithmetic mean for a year, i.e.

5

mended average days for months and values of n and d by months [17]

nth Average day for the month Day of the year n Declination angle d (degrees)

uary 17 17 )20.9ruary 16 47 )13.0rch 16 75 )2.4il 15 105 9.4

y 15 135 18.8

e 11 162 23.1

17 198 21.2

ust 16 228 13.5

tember 15 258 2.2

ober 15 288 )9.6ember 14 318 )18.9ember 10 344 )23.0

Table 6

Dimensionless atmospheric attenuation

No. Location Annual mean

daily extrater-

restrial insola-

tion (calcu-

lated, dimen-

sionless) H ao

Annual mean

daily clear sky

insolation

(measured,

dimensionless)

H acm

Annual mean

daily insola-

tion loss due

atmosphere

(measured,

dimensionless)

Daam

Annual mean

daily insola-

tion loss due

atmosphere

(estimated,

dimensionless)

Daae

Percentage

error Daae�DaamHacm�100%

1 Abu Dhabi 0.2794 0.2004 0.0790 0.0846 2.81

2 Accra 0.3039 0.2046 0.0993 0.0972 )1.003 Addis Ababa 0.3016 0.2208 0.0808 0.0797 )0.514 Adelaide 0.2575 0.1838 0.0738 0.0654 )4.555 Albuquerque 0.2537 0.1902 0.0636 0.0599 )1.906 Ankara 0.2395 0.1757 0.0637 0.0583 )3.087 Antalya 0.2486 0.1816 0.0671 0.0712 2.27

8 Archangel 0.1552 0.1125 0.0427 0.0350 )6.849 Arusha 0.3052 0.2198 0.0855 0.0961 4.83

10 Atlanta 0.2576 0.1825 0.0751 0.0704 )2.5211 Auckland 0.2517 0.1792 0.0725 0.0622 )5.7012 Bangkok 0.2968 0.1735 0.1233 0.0936 )17.1213 Bechar 0.2629 0.1886 0.0743 0.0772 1.55

14 Belfast 0.1899 0.1341 0.0558 0.0481 )5.7115 Bogota 0.3043 0.2166 0.0877 0.0815 )2.9016 Bombay 0.2892 0.2045 0.0847 0.0894 2.30

17 Bordeaux 0.2240 0.1587 0.0652 0.0615 )2.3518 Boston 0.2319 0.1665 0.0654 0.0620 )2.0319 Brisbane 0.2757 0.1978 0.0778 0.0757 )1.0820 Buenos Aires 0.2584 0.1804 0.0780 0.0663 )6.5121 Bulawayo 0.2895 0.2094 0.0801 0.0723 )3.7522 Buraimi 0.2798 0.2001 0.0797 0.0803 0.29

23 Cairo 0.2667 0.1933 0.0734 0.0775 2.11

24 Calcutta 0.2829 0.1970 0.0859 0.0864 0.24

25 Cambridge 0.1986 0.1369 0.0616 0.0526 )6.6326 Canberra 0.2565 0.1811 0.0754 0.0574 )9.9327 Caracas 0.3003 0.2064 0.0939 0.0845 )4.5728 Casablanca 0.2578 0.1882 0.0696 0.0747 2.70

29 Charleston 0.2596 0.1873 0.0722 0.0761 2.09

30 Chittagong 0.2835 0.1967 0.0868 0.0862 )0.2931 Cleveland 0.2350 0.1677 0.0673 0.0622 )3.0732 Colombo 0.3031 0.2172 0.0859 0.0976 5.37

33 Copenhagen 0.1863 0.1287 0.0576 0.0479 )7.5434 Dakar 0.2956 0.2008 0.0948 0.0924 )1.1735 Dallas 0.2596 0.1894 0.0702 0.0737 1.82

36 Dar Es Salaam 0.3039 0.2201 0.0838 0.0968 5.87

37 Dhaka 0.2804 0.1914 0.0890 0.0852 )1.9938 Djakarta 0.3042 0.2129 0.0913 0.0973 2.81

39 Edmonton 0.1946 0.1481 0.0465 0.0476 0.75

40 Eldoret 0.3055 0.2174 0.0881 0.0814 )3.1041 Entebbe 0.3055 0.2121 0.0934 0.0881 )2.50


Table 6 (continued)


daily extrater-

restrial insola-

tion (calcu-

lated, dimen-

sionless) H ao

Annual mean

daily clear sky

insolation

(measured,

dimensionless)

H acm

Annual mean

daily insola-

tion loss due

atmosphere

(measured,

dimensionless)

Daam

Annual mean

daily insola-

tion loss due

atmosphere

(estimated,

dimensionless)

Daae

Percentage

errorDaae�Daam

Hacm

�100%

42 Fairbanks 0.1544 0.1167 0.0377 0.0322 )4.7343 Freiburg 0.2133 0.1456 0.0677 0.0538 )9.5844 Genova 0.2252 0.1606 0.0646 0.0614 )2.0245 Goa 0.2946 0.2045 0.0901 0.0812 )4.3146 Goteborg 0.1789 0.1231 0.0558 0.0446 )9.1247 Hamburg 0.1936 0.1293 0.0642 0.0506 )10.5448 Hanoi 0.2859 0.1895 0.0965 0.0876 )4.6849 Harare 0.2930 0.2109 0.0822 0.0741 )3.8250 Havana 0.2819 0.2024 0.0795 0.0855 2.99

51 Indianapolis 0.2401 0.1743 0.0658 0.0645 )0.7652 Irkutsk 0.1984 0.1443 0.0541 0.0459 )5.7053 Izmir 0.2441 0.1709 0.0732 0.0696 )2.0754 Jerusalem 0.2625 0.1897 0.0727 0.0682 )2.3755 Karachi 0.2785 0.1952 0.0833 0.0844 0.58

56 Kasama 0.3017 0.2168 0.0849 0.0817 )1.4857 King�s Park 0.2836 0.1832 0.1004 0.0861 )7.7858 Kinshasa 0.3049 0.2012 0.1037 0.0928 )5.4259 Kodaikanal 0.3005 0.2052 0.0954 0.0799 )7.5760 Kunming 0.2782 0.1715 0.1068 0.0782 )16.6461 Lahore 0.2631 0.1834 0.0796 0.0740 )3.0462 Lodwar 0.3049 0.2134 0.0915 0.0921 0.25

63 London 0.2011 0.1390 0.0620 0.0537 )6.0264 Los Angeles 0.2568 0.1931 0.0638 0.0733 4.92

65 Luanda 0.3026 0.2106 0.0921 0.0948 1.28

66 Lusaka City 0.2963 0.2130 0.0833 0.0776 )2.6867 Macon 0.2191 0.1526 0.0665 0.0596 )4.5168 Madison 0.2295 0.1690 0.0605 0.0600 )0.3069 Madras 0.2977 0.2083 0.0894 0.0938 2.10

70 Madrid 0.2379 0.1663 0.0717 0.0592 )7.5171 Mansa 0.3009 0.2154 0.0854 0.0819 )1.6372 Marseille 0.2203 0.1637 0.0566 0.0641 4.63

73 Melbourne 0.2495 0.1768 0.0727 0.0613 )6.4174 Mendoza 0.2630 0.1930 0.0700 0.0599 )5.2375 Milan 0.2220 0.1582 0.0638 0.0589 )3.0776 Mombasa 0.3050 0.2184 0.0867 0.0975 4.98

77 Mongu 0.2965 0.2116 0.0849 0.0790 )2.7978 Montreal 0.2218 0.1661 0.0557 0.0613 3.38

79 Moscow 0.1860 0.1297 0.0563 0.0450 )8.6580 Naha 0.2757 0.1999 0.0758 0.0825 3.35

81 Nairobi 0.3055 0.2203 0.0852 0.0855 0.11

82 Nancy 0.2110 0.1445 0.0665 0.0538 )8.8083 Ndola 0.2990 0.2146 0.0844 0.0799 )2.12



Table 6 (continued)


daily extrater-

restrial insola-

tion (calcu-

lated, dimen-

sionless) H ao

Annual mean

daily clear sky

insolation

(measured,

dimensionless)

H acm

Annual mean

daily insola-

tion loss due

atmosphere

(measured,

dimensionless)

Daam

Annual mean

daily insola-

tion loss due

atmosphere

(estimated,

dimensionless)

Daae

Percentage

errorDaae�Daam

Hacm

�100%

84 New Delhi 0.2703 0.1900 0.0803 0.0767 )1.8985 New York 0.2369 0.1671 0.0698 0.0655 )2.5786 Nice 0.2278 0.1634 0.0644 0.0638 )0.3387 Omsk 0.1889 0.1395 0.0494 0.0470 )1.7088 Ottawa 0.2223 0.1664 0.0559 0.0611 3.08

89 Paris 0.2105 0.1436 0.0669 0.0563 )7.4190 Passau 0.2113 0.1472 0.0641 0.0514 )8.6091 Perth 0.2652 0.1867 0.0785 0.0693 )4.9892 Phoenix 0.2568 0.1877 0.0691 0.0698 0.38

93 Quebec 0.2278 0.1685 0.0593 0.0635 2.48

94 Reno 0.2408 0.1801 0.0607 0.0561 )2.5395 Rome 0.2337 0.1682 0.0656 0.0653 )0.1896 Salt Lake City 0.2369 0.1816 0.0553 0.0548 )0.2397 Santiago 0.2876 0.1967 0.0909 0.0855 )2.7498 Santiago 0.2613 0.1940 0.0673 0.0611 )3.2299 Sapporo 0.2298 0.1676 0.0622 0.0643 1.29

100 Seattle 0.2152 0.1612 0.0540 0.0591 3.14

101 Seoul 0.2466 0.1773 0.0693 0.0695 0.09

102 Seville 0.2470 0.1771 0.0699 0.0707 0.46

103 Shanghai 0.2640 0.1751 0.0889 0.0781 )6.16104 Strasbourg 0.2054 0.1418 0.0636 0.0549 )6.15105 Tashkent 0.2354 0.1719 0.0635 0.0599 )2.08106 Tucuman 0.2769 0.2035 0.0734 0.0703 )1.51107 Tunis 0.2487 0.1801 0.0686 0.0720 1.87

108 Turin 0.2217 0.1562 0.0656 0.0575 )5.18109 Ulan Bator 0.2135 0.1591 0.0545 0.0457 )5.49110 Valparaiso 0.2624 0.1954 0.0671 0.0682 0.57

111 Vancouver 0.2277 0.1699 0.0578 0.0639 3.57

112 Venice 0.2217 0.1423 0.0794 0.0616 )12.49113 Vienna 0.2124 0.1423 0.0701 0.0545 )10.98114 Visakhapatnam 0.2914 0.2087 0.0827 0.0906 3.78

115 Vladivostok 0.2295 0.1611 0.0684 0.0620 )3.99116 Warsaw 0.1984 0.1305 0.0678 0.0506 )13.17117 Wellington 0.2393 0.1691 0.0702 0.0567 )7.96118 Windhoek 0.2854 0.2094 0.0759 0.0680 )3.81119 Winnipeg 0.2067 0.1561 0.0507 0.0516 0.59

120 Zurich 0.2152 0.1512 0.0640 0.0513 )8.42

Means 0.2532 0.1796 0.0736 0.0696 4.00

The mean percentage error varies from )4.80% (82 locations) to +2.30% (38 locations).

Note: After the pertinent computer calculations, the data in columns 3–6 have been rounded to four digits after the



Fig. 2

surfac


H ao ¼1

12

X12j¼1

ðHoÞj ð23Þ

One should note that Eq. (12) gives the d needed to evaluate H ao. The sunset hour angle xs can becalculated from [17]:

cosxs ¼ � tan/ tan d ð24Þ
Using the data presented in Table 1 (which gives the dimensionless annual mean daily clear skyinsolation on a horizontal surface H acm using Eq. (2)) and Eqs. (12), (22), (23) and (24) (whichgive the dimensionless calculated annual mean daily extraterrestrial insolation on a horizontalsurface H ao), we have calculated the dimensionless annual mean daily insolation loss due to at-mospheric attenuation Daa for a given latitude / and altitude A. Table 6 presents the results for the120 locations under study. Also, the results have been plotted in Fig. 2 for five different altituderanges. From a study of this figure, we can deduce the following conclusions:
(a) Atmospheric attenuation is the smallest at the highest altitude, and it increases as the alti-tude decreases for given latitude /. This is an expected result, since as the altitude decreases,the solar beam travels further and especially in denser regions of the atmosphere containinggreater amounts of water vapor and aerosols, resulting in increased attenuation.

(b) For a given latitude / and altitude A, atmospheric attenuation is smaller in the southern hemi-sphere than in the northern hemisphere. This is caused by the greater pollution in the northernhemisphere atmosphere caused by a much greater number of polluting industries and cities inthe northern hemisphere as compared to the southern hemisphere.

. Dimensionless difference between annual mean daily extraterrestrial and clear sky insolations on a horizontal

e versus latitude for different altitude ranges.


(c) For a given altitude, A, atmospheric attenuation is largest in the equatorial regions, and it de-creases as one moves towards the higher latitudes both in the northern and the southern hemi-spheres. This is mainly caused by the cos hz effect (which decreases as the latitude increases)and the humidity effect (in equatorial regions, humidity is much greater than in regions withhigher latitudes).

Using the information presented in Table 6 and Fig. 2 for the 120 locations under study, wehave correlated the data, deriving the following empirical relationship for the dimensionless an-nual mean daily insolation loss due to atmospheric attenuation:

Daa ¼ 0:1 ð1þ AÞ�0:1 � ½0:123þ 0:016ð�1Þi� /90

� �1:5ð25Þ

where A is the dimensionless altitude (non-dimensionalized by the altitude, 452 m, of the PetronasTowers) and i is the hemisphere power. The hemisphere power is given by:

i ¼ 1 for the northern hemisphere ð26Þ
and
i ¼ 2 for the southern hemisphere ð27Þ

In Table 6, we have also included the percentage error defined as:

Eaa ¼Daae � Daam

H acm

� 100% ð28Þ

It can be seen from Table 6 (last column) that our correlation gives a smaller attenuation (oroverestimates the insolation) for 82 locations with a mean percentage error of )4.8%, whereas itgives a greater atmospheric attenuation loss (or underestimates the insolation) for 38 locationswith a mean percentage error of +2.3%. Remembering that the measured insolation for the 120locations may actually include a mean cloudiness of about 5% (in other words, this means that thedata is about 5% lower than the actual), which translates into a )5% error, then our correlationwould give a mean error of )0.2% for 82 locations and a mean error of )7.3% for the remaining38 locations.

4.2. Extraterrestrial insolation

Extraterrestrial insolation on a horizontal surface for a given location depends on the coor-dinates of the location (latitude), day of the year n and time of the day. When we consider themonthly mean daily extraterrestrial insolation, then we take the daily extraterrestrial insolationfor the day specified for the mean insolation for the month in question. In this case, the monthlymean daily extraterrestrial insolation depends on the latitude of the location and the specified dayfor a given month. When we consider the annual mean daily extraterrestrial insolation for a givenlocation, it only depends on the latitude of the location /.Presently, the conventional way of calculating the annual mean daily extraterrestrial insolation

H ao is using Eqs. (12), (22), (23) and (24), which have the variables n, /, hz, d and xs for each


month, and then averaging the results for 12 months. These have been calculated in the lastsection and presented in column three of Table 6. Since the annual mean daily extraterrestrialinsolation is eventually a function of latitude only, we have obtained, by trial and error, the best fitand the shortest possible relationship to give, H ao. It is

Table

Dimen

Lat

(deg

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

Mea

Note:

period

H ao ¼ 0:305

0@ þ 0:051 90

/

� �2"� 1

#�1:51A cos/ ð29Þ

Table 7 presents the values for H ao calculated using the conventional method and estimations byEq. (29) for 18 different latitudes with 5� latitude intervals, together with the percentage error,defined as:

Eao ¼H aoe � H aom

H aom

� 100% ð30Þ

When we study the error column, we can see that from the equator 0� up to latitude 65�, which isthe most important region for solar energy, the percentage error varies from )0.43% to +0.13%with a mean absolute error of 0.1%. For the region between latitudes 70�–85�, the percentage errorvaries from )7.98% to +5.81% with a mean absolute error of 5.86%.

7

sionless annual mean daily extraterrestrial insolation, H ao

itude

rees) /Annual mean daily extraterrestrial

insolation (calculated, dimension-

less) H aom

Annual mean daily extraterrestrial

insolation (estimated, dimension-

less) H aoe

Percentage errorHaoe�Haom

Haom� 100%

0.3052 0.3050 )0.080.3037 0.3038 0.05

0.3003 0.3004 0.04

0.2946 0.2948 0.07

0.2872 0.2872 )0.010.2777 0.2775 )0.060.2662 0.2661 )0.040.2533 0.2530 )0.140.2388 0.2384 )0.140.2229 0.2226 )0.110.2058 0.2058 0.00

0.1882 0.1884 0.10

0.1705 0.1707 0.13

0.1541 0.1534 )0.430.1414 0.1374 )2.850.1333 0.1242 )6.830.1279 0.1177 )7.980.1248 0.1320 5.81

ns 0.2220 0.2210 1.38

After the pertinent computer calculations, the data in columns 2 and 3 have been rounded to four digits after the

, and to two digits in column 4.


4.3. Clear sky insolation

We are now ready to obtain the annual mean daily clear sky insolation for a given location(latitude and altitude) in the world using the relationships we have derived, viz.

H aclv ¼ H ao � Daa ð31Þ
Substituting Eqs. (25) and (29) in the above equation we obtain
H aclv ¼ ½0:123þ 0:016ð�1Þi� /90

� �1:5þ 0:305

0@ þ 0:051 90

/

� �2"� 1

#�1:51A cos/

� 0:1 ð1þ AÞ�0:1 ð32Þ

where i ¼ 1 for the northern hemisphere and i ¼ 2 for the southern hemisphere. We shall call thisderived equation the Clear Sky Insolation Lingamgunta–Veziroglu Correlation, H aclv.Table 8 presents the annual mean daily clear sky insolation measured in dimensionless form for

the 120 locations selected and the estimated dimensionless insolation using Eq. (32), as well as thepercentage error defined as:

Elv ¼H aclv � H aom

H aom

� 100% ð33Þ

It can be seen from Table 8 that our universal relationship gives much better estimates than thosecalculated using the conventional method (See Table 4). The Lingamgunta–Veziroglu correlationerror varies from )6.34 for Dar Es Salaam to +16.90% for Bangkok with a mean absolute error of3.70%, whereas it was 8.81% in the case of the conventional method. Themean absolute error for theuniversal relationship is 2.4 times less than that with the conventional method. As can be seen fromTable 8, the mean percentage error varies from )2.20% for 46 locations to 4.63% for 74 locations.This means that for most of the locations, our relationship gives a higher estimate (4.63% on ave-rage) than the measured data. When we remember that the measured data has, on the average,about 5% cloudiness, then we can say that for 74 locations, our relationship gives a mean error of)0.37%, and for the remaining 46 locations, it would give a mean error of )7.2%.It must be remembered that both the conventional method and the Lingamgunta–Veziroglu

correlation gives insolation estimates for normal atmospheric conditions. For locations wherethere is severe pollution, severe sandstorms and the like, the attenuation caused by insolationreducing events must be taken into account. Of course, the cloudiness factor also must be con-sidered.

5. Conclusion

The annual mean daily clear sky insolation can be estimated as a function of latitude and al-titude only. The newly derived Lingamgunta–Veziroglu Correlation not only estimates the clearsky insolation with fewer calculation steps and with one elegant relationship but also with betteraccuracy than the conventional method of calculation. Also, simple and accurate correlations

Table 8

Comparison of clear sky insolation data with Lingamgunta–Veziroglu correlation

No. Location Annual mean daily clear

sky insolation (measured,

dimensionless) H acm

Annual mean daily clear

sky insolation (estimated,

dimensionless) H aclv

Percentage errorHaclv�Hacm

Hacm� 100%

1 Abu Dhabi 0.2004 0.1941 )3.142 Accra 0.2046 0.2063 0.84

3 Addis Ababa 0.2208 0.2216 0.37

4 Adelaide 0.1838 0.1878 2.21

5 Albuquerque 0.1902 0.1929 1.42

6 Ankara 0.1757 0.1802 2.56

7 Antalya 0.1816 0.1765 )2.778 Archangel 0.1125 0.1198 6.51

9 Arusha 0.2198 0.2084 )5.1510 Atlanta 0.1825 0.1862 2.03

11 Auckland 0.1792 0.1850 3.21

12 Bangkok 0.1735 0.2029 16.90

13 Bechar 0.1886 0.1848 )2.0214 Belfast 0.1341 0.1415 5.49

15 Bogota 0.2166 0.2225 2.74

16 Bombay 0.2045 0.1993 )2.5517 Bordeaux 0.1587 0.1617 1.85

18 Boston 0.1665 0.1690 1.51

19 Brisbane 0.1978 0.1965 )0.6820 Buenos Aires 0.1804 0.1878 4.15

21 Bulawayo 0.2094 0.2146 2.51

22 Buraimi 0.2001 0.1988 )0.6323 Cairo 0.1933 0.1884 )2.5424 Calcutta 0.1970 0.1959 )0.5625 Cambridge 0.1369 0.1456 6.32

26 Canberra 0.1811 0.1947 7.54

27 Caracas 0.2064 0.2155 4.41

28 Casablanca 0.1882 0.1822 )3.1729 Charleston 0.1873 0.1825 )2.5630 Chittagong 0.1967 0.1967 )0.0331 Cleveland 0.1677 0.1719 2.54

32 Colombo 0.2172 0.2052 )5.5233 Copenhagen 0.1287 0.1382 7.37

34 Dakar 0.2008 0.2028 0.97

35 Dallas 0.1894 0.1850 )2.2936 Dar Es Salaam 0.2201 0.2061 )6.3437 Dhaka 0.1914 0.1945 1.64

38 Djakarta 0.2128 0.2059 )3.2639 Edmonton 0.1481 0.1466 )0.9940 Eldoret 0.2174 0.2236 2.88

41 Entebbe 0.2121 0.2169 2.26

42 Fairbanks 0.1167 0.1218 4.38

43 Freiburg 0.1456 0.1589 9.13

44 Genova 0.1606 0.1618 0.74

45 Goa 0.2045 0.2130 4.11



Table 8 (continued)








Hacm� 100%

46 Goteborg 0.1231 0.1343 9.07

47 Hamburg 0.1293 0.1426 10.27

48 Hanoi 0.1895 0.1977 4.37

49 Harare 0.2109 0.2166 2.73

50 Havana 0.2024 0.1957 )3.3151 Indianapolis 0.1743 0.1747 0.25

52 Irkutsk 0.1443 0.1521 5.41

53 Izmir 0.1709 0.1735 1.53

54 Jerusalem 0.1897 0.1934 1.91

55 Karachi 0.1952 0.1934 )0.9456 Kasama 0.2167 0.2186 0.85

57 King�s Park 0.1832 0.1968 7.44

58 Kinshasa 0.2012 0.2113 5.02

59 Kodaikanal 0.2052 0.2204 7.42

60 Kunming 0.1715 0.1993 16.23

61 Lahore 0.1834 0.1881 2.57

62 Lodwar 0.2134 0.2125 )0.4363 London 0.1390 0.1469 5.68

64 Los Angeles 0.1931 0.1827 )5.3965 Luanda 0.2106 0.2067 )1.8666 Lusaka City 0.2131 0.2167 1.75

67 Macon 0.1526 0.1587 4.02

68 Madison 0.1690 0.1686 )0.2069 Madras 0.2083 0.2035 )2.2870 Madrid 0.1663 0.1779 6.97

71 Mansa 0.2155 0.2175 0.95

72 Marseille 0.1637 0.1635 )0.1373 Melbourne 0.1768 0.1836 3.84

74 Mendoza 0.1929 0.1990 3.11

75 Milan 0.1582 0.1623 2.58

76 Mombasa 0.2182 0.2067 )5.3477 Mongu 0.2115 0.2155 1.86

78 Montreal 0.1661 0.1598 )3.8479 Moscow 0.1297 0.1407 8.47

80 Naha 0.1999 0.1925 )3.7281 Nairobi 0.2204 0.2195 )0.3782 Nancy 0.1445 0.1566 8.37

83 Ndola 0.2146 0.2175 1.33

84 New Delhi 0.1900 0.1928 1.46

85 New York 0.1671 0.1705 2.04

86 Nice 0.1634 0.1631 )0.1787 Omsk 0.1395 0.1416 1.51

88 Ottawa 0.1664 0.1605 )3.5489 Paris 0.1436 0.1536 6.98

90 Passau 0.1472 0.1592 8.17

91 Perth 0.1865 0.1920 2.83


Table 8 (continued)








Hacm� 100%

92 Phoenix 0.1877 0.1861 )0.8593 Quebec 0.1685 0.1635 )2.9794 Reno 0.1801 0.1838 2.05

95 Rome 0.1682 0.1676 )0.3496 Salt Lake City 0.1816 0.1812 )0.2597 Santiago 0.1967 0.2016 2.47

98 Santiago 0.1942 0.1961 1.10

99 Sapporo 0.1676 0.1646 )1.80100 Seattle 0.1612 0.1554 )3.57101 Seoul 0.1773 0.1762 )0.62102 Seville 0.1771 0.1754 )0.98103 Shanghai 0.1751 0.1850 5.66

104 Strasbourg 0.1418 0.1559 9.96

105 Tashkent 0.1719 0.1746 1.56

106 Tucuman 0.2036 0.2032 )0.17107 Tunis 0.1801 0.1758 )2.38108 Turin 0.1562 0.1635 4.68

109 Ulan Bator 0.1591 0.1671 5.07

110 Valparaiso 0.1954 0.1902 )2.67111 Vancouver 0.1699 0.1630 )4.04112 Venice 0.1423 0.1594 11.95

113 Vienna 0.1423 0.1573 10.53

114 Visakhapatnam 0.2087 0.2003 )4.01115 Vladivostok 0.1611 0.1667 3.47

116 Warsaw 0.1305 0.1473 12.84

117 Wellington 0.1692 0.1778 5.14

118 Windhoek 0.2094 0.2145 2.43

119 Winnipeg 0.1561 0.1546 )0.94120 Zurich 0.1512 0.1631 7.89

Means 0.1826 0.1796 3.70

The mean percentage error varies from )2.20% (46 locations) to +4.63% (74 locations).

Note: After the pertinent computer calculations, the data in columns 3 and 4 have been rounded to four digits after the



have been obtained for the extraterrestrial insolation and for atmospheric attenuation. These willbe very useful tools for solar energy systems designers and engineers to use.

Acknowledgements

The authors gratefully acknowledge the US–Egypt Joint Science & Technology Fund, the USNational Science Foundation and the N.S.F. Senior Program Director Dr. Osman Shinaishin forsupport of this study under the Research Project entitled ‘‘Investigation and Optimization of theDesign Parameters for Photocatalytic Solar Hydrogen Production’’.


Appendix A. Simple clear sky correlations

Hourwitz correlation (H) [8,9]:

Gc ¼ 1098 cos hz expð�0:057= cos hzÞ ðA:1Þ
Daneshyar–Paltridge–Proctor correlation (DPP) [10,11]:
Gc ¼ 950:2 1f � exp ½ � 0:075ðp=2� hzÞ�g þ 14:29þ 21:04ðp=2� hzÞ ðA:2Þ
Berger–Duffie correlation (BD) [12]:
Gc ¼ 1350 0:70 cos hz ðA:3Þ
Adnot–Bourges–Campana–Gicquel correlation (ABCG) [13]:
Gc ¼ 951:39 cos1:15 hz ðA:4Þ
Kasten–Czeplak correlation (KC) [14]:
Gc ¼ 910 cos hz � 30 ðA:5Þ

References

[1] Hottel HC. A simple model for estimating the transmittance of direct solar radiation through clear atmospheres.

Solar Energy 1976;18:129.

[2] Liu BYH, Jordan RC. The interrelationship and characteristic distribution of direct, diffuse and total solar

radiation. Solar Energy 1960;4:3.

[3] May BR, Collingbourne RH, McKay DC. Catalogue of estimating methods. In: Handbook of methods of

estimating solar radiation. Stockholm, Sweden: Swedish Council for Building Research; 1984. p. 4–32.

[4] Bener P. Survey and comments on various methods to compute the components of solar irradiance on horizontal

and inclined surfaces. In: Handbook of methods of estimating solar radiation. Stockholm, Sweden: Swedish

Council for Building Research; 1984. p. 47–77.

[5] Davies JA, McKay DC, Luciani G, Abdel-Wahab M. Validation of models for estimating solar radiation on

horizontal surfaces, vol. 1. IEA Task IX, Final report 1988. Atmospheric Environment Service of Canada,

Downsview, Ontario, Canada.

[6] Festa R, Ratto CF. Solar radiation statistical properties. Technical report for IEA Task IX 1993, University of

Genova.

[7] Viorel B. Verification of some very simple clear and cloudy sky models to evaluate global solar irradiance. Solar

Energy 1997;61:251–64.

[8] Hourwitz B. Insolation in relation to cloudiness and cloud density. J Met 1945;2:154–6.

[9] Hourwitz B. Insolation in relation to cloud type. J Met 1946;3:123–4.

[10] Daneshyar M. Solar radiation statistics for Iran. Solar Energy 1978;21:345–9.

[11] Paltridge GW, Proctor D. Monthly mean solar radiation statistics for Australia. Solar Energy 1976;18:235–43.

[12] Berger X. Etude du Climat en Region Nicoise en vue d�Applications a l�Habitat Solaire. Paris: CNRS; 1979.[13] Adnot J, Bourges B, Campana D, Gicquel R. Utilisation des courbes de frequence cumulees pour le calcul des

installation solaires. In: Lestienne R, editor. Analise Statistique des Processus Meteorological Appliquee a l� energieSolaire. Paris: CNRS; 1979. p. 9–40.

[14] Kasten F, Czeplak G. Solar and terrestrial radiation dependent on the amount and type of clouds. Solar Energy

1980;20:143–50.

[15] Available from: http://eosweb.larc.nasa.gov/sse. Surface Meteorology and Solar Energy Data Set (Release 3).

A renewable energy resource website sponsored by NASA�s Earth Resource Enterprise Program.[16] Cooper PI. The absorption of solar radiation in solar stills. Solar Energy 1969;12:3.

[17] Duffie JA, Beckman WA. Solar engineering of thermal processes. 2nd ed. New York: John Wiley and Sons; 1991.

http://eosweb.larc.nasa.gov/sse

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A universal relationship for estimating clear sky insolation