Pergamon Solar Energy, Vol. 52, No. 3, pp. 233-239, 1994

Copyright 0 1994 Elsevier Science Ltd Printed in the USA. All rights reserved

0038-092X/94 $6.00 + .OO

ESTIMATION OF MONTHLY MEAN HOURLY GLOBAL RADIATION FOR LATITUDES IN THE 1 “N-81 “N RANGE

ALFONSO SOLER and K. K. GOPINATHAN* Departamento de Fisica e Instalaciones, Escuela T&nica Superior de Arquitectura,

Avda. Juan de Herrera, 4,28040 Madrid, Spain

Abstract-A way to estimate monthly mean hourly global radiation (r) from monthly mean daily global radiation (I?) has been to use fits which give ? = i/I? as a function of $, the monthly mean daily maximum possible number of sunshine hours. In the present work it is shown that these plots are by no means of universal validity, but are clearly dependent on latitude for high latitudes. A new type of correlation is proposed, which gives Fas a function of Z, the solar elevation at mid-hour. The values of coefficients for the Tversus & correlations are given for each month of the year, for the hours centered at 0.5 h. 1.5 h, 2.5 h, etc. from the true solar noon, and stations with latitudes in the I “N-8 I “N range. The given coefficients are most likely valid for any latitude, north or south.

1. INTRODUCTION

When experimental or estimated values of the monthly mean daily global radiation (I?) are known for a certain location, it has become customary to estimate values of the monthly mean hourly global radiation (I) from plots of F = i/I? versus & given for the hours centered at 0.5 h, 1.5 h, etc., from the true solar noon, so being the monthly mean daily maximum possible number of sunshine hours [ 1-9 1. Values of ? are obtained as average values for hours symmetric with respect to the true solar noon. The dependence on turbidity and cloudiness of these plots has been studied for a specific location[ IO]. Similar plots have been obtained for global illuminances [ 111.

It has been stated that the rversus so plots are of “universal application”[ 12]in the sense that ifone fits the experimental rversus $ points of a few stations for each of the hours centered at 0.5 h, 1.5 h, etc., from solar noon, the curves obtained will be valid for any station. However, this statement was apparently based on data for stations with latitudes in the 30”N-50”N range.

In this work we present rversus & plots for stations with latitudes between 50”N and 8 1 “N, and we show that the “universal validity” of the rversus & plots is lost for high latitudes. A new type of correlation valid for latitudes in the 1 “N-8 I “N range is proposed.

2. DEPENDENCE ON LATITUDE OF THE CORRELATION _ _

BETWEEN i = I/H, AND &

Using the data given in [ 131, empirical values of r - - = I/H were obtained for the following locations: Uccle, Yakutsk, Wrangle, Dickson, Kotelny, Fedorov, and

* Permanent address: Department of Physics, The Na- tional University of Lesotho, Roma, Lesotho.

Krenkel. Latitudes and longitudes for these locations are given in Table 1. The data correspond to the four- year period 1986-l 989. For some months of the year, during the so-called polar night, no global radiation is measured for the mentioned stations, with the excep tion of Uccle and Yakutsk. For Krenkel, Fedorov, and Kotelny, polar night corresponds to January, Novem- ber, and December; for Dickson and Wrangle, to Jan- uary and December.

Astronomical daylength in hours So, is the com- puted time during which the center of the solar disk is above an altitude of zero degrees (without allowance

Table 1. Latitudes and longitudes for the 26 locations chosen for the present work

Station Latitude

Krenkel 80”37’ N Fedorov 77”43’ N Kotelny 76”OO’ N Dickson 73”3O’N Wrangle 70”58’ N Chetyrekhstolbovi lo”38 N Olenek 68”3O’N Verkhoyansk 67”33’ N Sodankyla 67”22’ N Turukhansk 65”47’ N Arkangelesk 64”35’ N Oimyalon 63”16’N Yakutsk 62”05’ N Helsinki 60”19’N Kaunas 54’43’ N Chita 52”Ol’ N Uccle 50”48’ N Kiev 50”24’ N Khabarovsk 48”31’ N Yuzhno-Sakhaliusk 46”55’ N Odessa 46”29’ N Vladivostok 43”07’ N Casablanca 23”lO N Le Raizet 16”16’N Cayenne-Rochambeau 4”50’ N Singapore l”22’N

Longitude

58”03’ E 104”17’E 137”54’ E 80”24 E

178”32’ W 162”24’ E 112”26’ E 133”23’ E 26”39’E 87”57’E 40”30’ E

143”09’ E 129”45’ E 24”58’E 23”53’ E

113”2O’E 4”21’ E

30”27’ E 135”lO’E 142”44’E 30”38’ E

131”M’E 82”21’W 61”31’W 52”21’W

103”59 E

233

234 A. SOLER and K. K. GOPINATHAN

o KRENKEL 0 FEDOROV A KOTELNY x DICKSON f WRANGLE ??YAKUTSK

i-

0.25 r

04 ? /

0,20

O.OF

i

0.00

I i I I I t I

0 5 0 5

12 16 20 2L

o KENKEL 0 FEDOROV A KOTELNY x CICKSON + WRANGCE . YAKUTSK e UCCLE

o IKRENI<EL 0 FEDOROV A KOTELNY x DICKSON + WRANGLE ??YAKUTSl< e UCCLE

I ’ I 1 I I I I I I

IO 12 14 16 18 20 22 24 so

(b)

Fig. I. The ?versus SC, plots for seven locations with latitudes between 50.8”N and 80.6”N. (a) The hours centered at 0.5 h from the true solar noon. (b) The hours centered at 3.5 h from the true solar noon. (c)

The hours centered at 6.5 h from the true solar noon.

for atmospheric refraction). Monthly mean values are in the present work for each month of the year are calculated from: taken from[ 14 1.

so = l/7.5 cos-‘( -tg&g4) (I)

where 4 is the latitude, d is the solar declination, and cos-’ is to be taken in degrees. For stations located in the Arctic circle, if (-tgStg+) > 1, then ??a = 0 hours, and if (-tg6tgd) < -1, then $0 = 24 hours[l4]. For a location with 4 - 7S’N for example, & equals 0 hours for January, November, and December, and 24 hours for May, June, and July. The values of 6 used

In Fig. 1 (a-c) we have plotted for the seven stations Fversus s,, values for the hours centered at 0.5 h, 3.5 h, and 6.5 h, from solar noon, together with the best fits obtained using second degree polynomials. Values of correlation coefficients r, are very high. For the hours centered at 0.5 h, 0.980 5 r 5 0.998. For 3.5 h, 0.947 5 r 5 0.996, with the exception of Krenkel ( r = 0.829). For 6.5 h, 0.984 s r s 0.999.

Solid lines are drawn for each polynomial fit, for the ranges of experimental values off. Out of the ranges of experimental values, dashed lines are given, to em-

Monthly mean hourly global radiation 235

o 2.5h + 3.5h R 4.5h @ 5.5h A 65h

0.15

i

0.10

?? 0.5h o 1.5h x 2.5h ??3.5h 8 L.Sh

Fig. 2. Experimental values of rversus G, and best fits as given by eqn. (2), for locations in the I “N-8 1 “N latitude range, and the hours centered at 0.5 h, 1.5 h, etc. from true solar noon. (a) April; (b) October.

phasize the general conclusions drawn from the study The curve for 4 = 80.6”N is lying lower than all the of Fig. 1 (a-c): the F versus & plots show a clear de- others. However, for & < - 10 h, all the curves appear pendence with latitude for each of the hours. clearly differentiated: the higher the latitude, the lower

For the hours centered at 0.5 h from the true solar the curves. noon, & > -10 h, and 73.5”N 2 qS 2 50.8’N, the For the hours centered at 3.5 h from the true solar curves are practically coincident, and the same can be noon, and & -C -10 h, the lower the latitude, the iaid of the curves for C#I = 76.O”N and 4 = 77.7”N. lower lie the curves.

Table 2. Values of a, b, c, d, r and Q, and the ranges of values of 5 for the months of January, February, and March

0.5 1.5 2.5 3.5 4.5

January

: c d r 0

pdegrees)

February a b

2 r d

pd WW March

z C d r d

Gegrees)

0.309 0.171 0.066 0.025 0.006 -0.94 I 0.004 0.459 0.277 0.135

I.927 -0.472 -1.453 -0.174 1.285 -1.359 0.581 1.410 -0.576 -5.491

0.983 0.976 0.980 0.997 0.999 0.011 0.004 0.003 0.002 0.000

1.52-66.70 0.14-58.79 1.00-47.16 2.25-34.10 I. 12-20.44

0.226 0.170 0.089 0.036 0.015 -0.312 -0.122 0.207 0.255 -0.006

0.380 0.098 -0.656 -0.500 I.941 -0.154 -0.010 0.628 0.294 -0.653

0.992 0.990 0.925 0.989 0.994 0.003 0.002 0.003 0.002 0.001

1.07-73.96 2.48-63.53 0.41-50.17 1.22-34.96 1.94-21.57

0.126 0.114 0.092 0.069 0.045 0.133 0.137 0.133 0.028 -0.260

-0.290 -0.335 -0.399 -0.079 2.249 0.177 0.235 0.375 0.138 -5.113 0.925 0.93 I 0.943 0.853 0.92 I 0.001 0.001 0.001 0.001 0.001

7.51-81.85 6.88-67.28 5.65-52.37 5.68-37.44 2.91-22.43

236 A. SOLER and K.K. GOPINATHAN

For the hours centered at 6.5 h from the true solar noon, and L?,, < - 16 h, the lower the latitude, the lower lie the curves. As similar observations are ob- tained using data for other hours and high latitude sta- tions, we must conclude that the J = IjI? versus $, curves are not of universal validity, but are clearly de- pendent on latitude for high values of 4. Relating the estimation of B, it has been recently shown [ 151 that for latitudes higher than about 60”N expressions of the coefficients in Angstrom’s type correlations, as used to estimate fl/fiO, from s/& differ from those pre- dicted using relations valid for lower latitudes.

3. PROPOSED CORRELATIONS FOR THE ESTIMATION -_

OF r= IfH

Due to the dependence on latitude of 7 versus & correlations demonstrated in the preceeding section, new correlations have been developed in the present work, from which r can be computed for 1 ON < C#I < 81°N.

The proposed correlations are of the following type:

- - ;= I/H = A + B& f Cc2 + DcU3 (2)

where A, B, C, and D are empirical coefficients, and solar elevation at mid-hour in degrees, (Y, is computed from:

& = sin-’ (sin fj sin d + Cos w cos I$ cos 6)

w = 15( 12 - t) degrees (3)

the value of I12 - t( being equal to 0.5 h, 1.5 h, etc., for the hours centered, respectively, at 0.5 h, 1.5 h, etc., from the true solar noon. Solar elevation LY, is the center of the solar disk’s angular height above the ob- server’s celestial horizon (without allowing for atmo- spheric refraction). Values of Z? in eqn. (2) have been obtained as usual from

do, da,, n.. ,,A

(4)

where nd is the number of days in the month and ny is the number of years of data utilized. In the present work, when G > 0 for an hour, the corresponding value off has been used to calculate fi from eq. (4). Values of fwhen Lu -C 0 are sometimes available but are not used. For small positive values of G, the sun will be obscured during part of the hour, so that (Y will be negative during part of the hour. As some predawn and postsunset radiation will most likely be recorded, the values of I? calculated in this way will be somewhat different from the real values, because to calculate the real values of 8, one should ideally measure the daily values of global radiation from the sunrise when cx = 0 to the sunset when 01 = 0 again Relating the values of Tfor small positive values of a! they are used together with values of a! obtained when the sun is above the horizon for the whole hour. It was checked for different

Monthly mean hourly global radiation

Table 4. Values of a, b, c, d, r, and (r, and the ranges of & for the months of June

237

June a b

0.5 1.5 2.5 3.5 4.5

0.062 0.056 0.376 0.013 0.272 -0.038 -0.142 -0.156 -2.465 -0.216 -3.110 1.031

0.649 0.787 6.294 2.111 13.219 -6.328 -0.464 -0.629 -4.974 -2.939 -17.496 14.338

0.950 0.967 0.910 0.930 0.920 0.990 0.006 0.004 0.005 0.003 0.002 0.001

32.38-83.10 31.70-69.93 30.38-55.61 28.56-42.00 26.37-3 1 .I9 7.45-24.19

6.5 7.5 8.5 9.5 10.5 11.5

5.5

June a b

d r D

(adegrees)

0.012 0.000 -0.002 0.001 0.002 0.001 -0.347 0.093 0.211 0.117 0.041 0.014

4.450 0.755 -0.748 -0.146 0.898 1.432 - 10.720 -2.050 2.980 2.328 -1.387 -3.540

0.994 0.996 0.994 0.996 0.999 0.997 0.001 0.001 0.001 0.001 0.000 0.00

10.38-21.53 0.62-19.21 3.62-17.16 0.27-15.51 0.51-14.35 0.64- 13.75

months and hours that suppressing the values of Fand

G for small values of a when using eqn. (2 ), did not change significantly the predicted values of Ffor values of rY for which the solar disk is visible during the whole hour.

The empirical coefficients in eqn. (2) are deter- mined for each of the months of the year, and each of the hours centered around the true solar noon. Third- order degree polynomials are used, as they give slightly better correlation coefficients than second-order degree polynomials, and because for some hours, visual in- spection of the Fversus (Y plots shows that a third order degree polynomial is more appropriate to fit the data.

Relating the dependence of 7 on (Y, it was noted long ago[ I-31 that the correspondence between f/H values and the values of &/I& (where & and fl,, are

the extraterrestrial values corresponding to jand H), although not perfect, is quite close [ 121. However, in [ l- 31, as in [4-IO], high latitudes were not considered. It has been shown [ 141, that &/I!& can be written as a sine function of solar altitude, and as a consequence - - from[ 1-3 1, I/H should be a sine function of solar al- titude. In a more recent work [ 16 1, using data for Rer- gen (60”24’N, 5”19’E) and Montreal (45”30’N, 73’37’W), it was found appropriate to write I as a function of sin 5 and (sin Cu ) I/*, but the validity of the “universal” Fversus so plots was not addressed.

Latitudes of the locations used to compute the em- pirical coefficients in eqn. (2) are in the 1 “N-8 1 “N range, and the data, obtained from[ 13lcorrespond to the four-year period 1986-l 989. Latitudes and longi- tudes of the 26 locations chosen are given in Table 1.

Table 5. Values of a, 6, c, d, r, and C, and the ranges of Cu for the months of July

0.5 1.5 2.5 3.5 4.5 5.5

July a -0.003 -0.072 -0.111 0.009 -0.080 0.044 b 0.250 0.710 1.015 -0.129 0.470 1.173

: -0.079 -0.036 -1.008 0.567 -1.793 1.193 -3.096 1.983 -5.085 1.495 -7.673 18.669 r 0.972 0.962 0.939 0.949 0.967 0.992 Ll 0.005 0.005 0.005 0.002 0.001 0.001 i&ees~ 30.50-82.77 29.79-69.09 28.49-50.13 26.67-41.39 2 1.44-30.50 7.50-22.39

6.5 7.5 8.5 9.5 10.5 11.5

July b” -0.056 1.195 0.001 0.061 0.115 0.001 0.09 0.002 1 0.078 0.001 0.003

-0.057 c -6.654 0.992 0.130 0.114 0.336 2.876 d 15.357 -1.913 1.725 3.128 3.043 -9.982 r 0.985 0.998 0.996 0.998 0.999 0.999 LT 0.002 0.001 0.001 0.0000 0.000 0.000 &grees) 1.73-19.65 0.92-17.33 1.86-15.27 0.50-13.61 0.12- 12.44 1.81-I 1.84

238 A. SOLER and K. K. GOPINATHAN

Table 6. Values of a, b, c, d, r, and m, and the ranges of a for the months of August and September

0.5 1.5 2.5 3.5 4.5 5.5 6.5

August a b

fi r IJ

zegrees)

0.019 -0.006 -0.088 -0.153 0.02 1 0.050 -0.020 0.357 0.543 1.264 2.130 -0.156 -1.127 0.927

-0.367 -0.842 -2.929 -6.548 4.165 9.306 -10.520 0.125 0.482 2.323 6.638 -12.270 - 16.866 53.420 0.977 0.974 0.98 1 0.900 0.920 0.810 0.989 0.004 0.003 0.002 0.002 0.002 0.006 0.001

22.97-78.47 22.30-68.12 21.02-53.73 19.23-39.55 17.07-25.77 7.62-15.54 3.7-2.25

7.5 8.5 9.5 10.5

August

z

t; r cl

Gegrees)

0.002 0.004 0.002 0.003 0.222 -0.278 0.191 0.149

-3.295 13.195 0.106 2.129 37.234 -88.722 7.301 -28.610

0.974 0.964 0.989 0.989 0.002 0.002 0.000 0.000

1.20-9.9 I 0.16-7.83 0.42-6.14 0.68-4.94

0.5 1.5 2.5 3.5 4.5 5.5 6.5

September a

b 2

r 0 Gegrees,

0.092 0.077 0.073 0.063 0.064 0.163 0.004 0.241 0.323 0.223 0.129 -0.270 -5.652 0.963

-0.411 0.23 1 -0.694 0.483 -0.533 0.414 -0.477 0.606 -1.857 1.178 -359.142 75.677 -662.862 6.367

0.925 0.977 0.952 0.840 0.865 0.973 0.999 0.003 0.001 0.001 0.001 0.002 0.001 0.001

12.21-82.36 11.56-67.47 10.32-52.54 8.57-37.59 6.44-22.65 4.07-8.03 0.23-1.63

Table 7. Values of a, b, c, d, r, and 0, and the ranges of G for the months of October, November, and December

OS 1.5 2.5 3.5 4.5 5.5

October i

C

d r Cr Gegrees)

-0.073 0.185 -0.02 0.152 1 0.099 0.07 1 0.258 0.043 0.016 0.157 -0.160 0.014

-0.086 -0.119 -0.110 -0.750 -0.183 -1.010 -0.135 0.145 -0.008 0.829 -0.052 33.491

0.953 0.96 1 0.837 0.963 0.964 0.999 0.005 0.002 0.002 0.003 0.002 0.003

0.60-77.42 2.66-65.40 1.08-5 1.28 1.46-36.72 0.23-2 1.99 3.38-7.20

November

z

2 r 0 (Qdegrees)

-0.706 0.279 0.177 0.104 0.076 0.348 0.026 0.315 0.008 0.112 -0.09 0.005 1

-0.948 1.377 -0.024 0.101 -1.076 1.017 -0.65 0.505 1 -3.455 0.937 2.118 9.27 1 0.99 1 0.985 0.957 0.996 0.997 0.999 0.006 0.003 0.003 0.002 0.001 0.000

0.48-68.90 1.61-60.32 1.62-48.16 0.70-34.74 0.25-20.82 1.75-6.61

Decmeber i

C

d r Q Gegrees)

-0.903 0.306 -0.032 0.177 0.059 0.552 0.016 0.423 -0.443 0.030 0.089 0.002

1.875 -0.350 -1.742 -0.852 5.581 0.112 -1.361 0.462 I .670 0.386 -15.929 4.670

0.94 1 0.971 0.973 0.997 0.999 0.999 0.02 1 0.004 0.003 0.002 0.008 0.001

1.52-64.54 0.03-57.2 1 1.28-46.10 0.22-33.42 9.80-20.04 0.34-6.35

Monthly mean hourly global radiation 239

Figure 2 (a,b) shows values of 7 versus G for the months of April and October. After looking at the data in Fig. 2, one can predict high correlations coefficients for the corresponding best fits.

Values of A, B, C, D were obtained using eqn. (2). In Tables 2 to 7 we give the values of a, 6, c, d; being a=A;b=BX102;c=CX104,andd=DX106. Best fits for April and October have been plotted in Fig. 2. The correlation coefficients r, the standard de- viations (I, and the ranges of values of G for the different months of the year, and each of the hours centered at 0.5 h, 1.5 h, etc., from the true solar noon are also given. Values of r are rather high as expected. Values of fl are rather low, indicating small dispersions or spreads of the data from the mean values.

Values of coefficients a, b, c, d (and, thus, ofA, B, C, D) in eqn. (2) are given in detail for the following reasons: (a) the availability of personal computers does not make it difficult to compute r form Cr using the values of a. b, c, d, given in the Tables. (b) Simple equations relating the different values of A, B, C, D to other variables, as for example, the months of the year ( I to 12) have not been found, but it is not ex- pected that (in the case these equations could be found) values of A, B, C, D from these equations could be used to predict values of fwith the same accuracy that the values given in Tables 2 to 7; (c) precise fits for all hours can be useful for testing a theory which justifies the dependence on 5.

4. CONCLUSIONS

Plots of f/H versus S,, are commonly used to esti- mate values of fwhen values of Z? are available or can be estimated. Up to now it was thought that these plots, as drawn for the hours centered at 0.5 h, 1.5 h, etc., from the true solar noon are of universal validity. Using data for stations with latitudes in the range 1 “N-8 I “N, new polynomial correlations are proposed to estimate values of F, which relate f to 6, the solar elevation at mid-hour, for each month and each of the hours cen- tered at 0.5 h, 1.5 h, etc., from solar noon. Very high correlation coefficients and low values of standard de- viations indicate that the new correlations are highly reliable to obtain values off. Although rversus Cu cor- relations have been developed for stations in the 1 ON- 8 I “N latitude range, they are likely to be valid for any latitude, north or south.

One can speculate that if fcan be obtained from (Y (or So for low latitudes) with very high reliability, this means that it depends little on climatic parameters such as the number of real sunshine hours. This may not be the case for the & versus S0 plots, r;l being the ratio of monthly average hourly to daily diffuse radia-

tion, because as shown for Uccle [ 171 fd versus $0 plots can show a dependence on the values of S/So, the daily sunshine Ii-action and, thus, they may be expected to be somewhat different, for example, for a location with mostly clear skies and for a location with mostly cloudy skies.

Acknowledgment-The authors are greatful to the Direccibn General de Investigacibn Cientifica y Ttcnica (DGICYT) for financial support for K. K. Gopinathan during his sabbatical.

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