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REFERENCEIC/85/263
INTERNAL REPORT
International Atomic Energy Agency
and
Educational Scientific and Cultural Organization
INTERNATIONAL CESTHE FOR THEORETICAL PHYSICS
ACCURATE COMPUTATIONS
OF MONTHLY AVERAGE DAILY EXTRATERRESTRIAL IRRADIATION
AHD THE MAXIMUM POSSIBLE SUIJSHIHE DURATION *
ABSTRACT
The monthly average daily values of the extraterrtstrial irradiation on
a horizontal plane (H ) and the maximum posBitile sunshine duration (5 ) areo o
two important parameters that are frequently needed in various solar energy
applications. These are generally calculated Ijy solar Bcientists and engineers
each time they are needed and often by using the approximate short-cut methods.
Using the accurate analytical expressions developed by Spencer [5] for the
declination and the eccentricity correction factor, computations for these
parameters have been made for all the latitude values from 90 H to 9Q°S at
intervale of 1 and are presented in a convenient tabular form. Monthly
average daily values of the maximum possible sunshine duration as recorded on a
Campbell Stoke's sunshine recorder s' are also computed and presented. These
tables would avoid the need for repetitive and approximate calculations and serve
as a useful ready reference for providing the accurate values of K , S and
5* to the solar energy scientists and engineers.
P.C. Jain ••+
International Centre for Theoretical Physics, Trieste, Italy,
MIRAMARE - TRIESTE
December 1985
•*
To be submitted for publication.
Permanent address: Physics Department, The University Of ZambiaP.O. Box 32379, Lusaka, Zambia.
Member of the International Solar Energy Society.
i. INTRODUCTION
The monthly average daily values of the extraterrestrial irradiation on
a horizontal plane H and the maximum possible sunshine duration (MPSD) So o
are tvo Important parameters that are frequently needed in solar energy
applications. For instance, any attempts at estimating the global irradiation
and the diffuse irradiation at a plaoe invariably require the knowledge of one or
both of these parameters at that place. The values of H have been tabulatedo
by Duffie and Beckman [l] and Iqbal [2] for latitude intervals of 5°. Duffie and
Beclnnan [1] also present graphs for the values of H for recommended days of
each month and tiomogram for calculating the values of S . However these areo
often not convenient enough and it is observed that most of the solar energyscientists do their own calculations for the values of H and S at their
o o
location. For reducing the amount of calculations, short-cut methods of using
the middle day of each month or a single recommended day for each month [3] have
often been employed. Realizing the need to provide these values once for all in a
convenient form so as to avoid the repetitive and approximate calculations,
recently Jain [It] compiled a table for these values. However, as explained in
Kec. 3, these values are sometimes not sufficiently accurate mainly due to the
inaccuracy involved in the expression for the declination 6. The aim of this
note is to provide accurate values of H , S and S' in a convenient tabularo o o
form.2. COMPUTATIONS
All the computations were done on the ICTP computer Gould 32/87- The
values of H for a given day were computed using the equation
2Ho = Isc * Eo * ( c o s * cos S sin
The following accurate expressions for 6 (in radians) and E o,
developed by Spencer [5]i were used
i - 0.006918 - 0.399912 cos r + 0.07257 sin r
- 0.006758 cos sr + 0.000907 sin r - 0.O02697 cos 3r
+ 0.0011*8 sin 3r,
(2)
EQ = 1.00011 + 0.03^221 cos V + 0.001280 sin r
+ 0.000719 cos £T + O.0OOO77 sin 2V, (3)
where
(M
eqn (2) estimates 6 with a maximum error of 3' and eqn (3) estimates E with
a maximum error of 0.0001. The monthly averages of H for each month wereo
computed for a l l the ^-values from 90°N to 9Q°S at 1° interval. The number
of days in February were taken to be 28. The best value of the solar constant
available at present is 1367 W/m [6]. This value has been used In the
computations.
The values of S for a given day and latitude value were computed using
the relation
E = ~ cos'1 ( - tan + tan <S)o 15 (5)
and the monthly averages were taken for all the -values.
The values of S1 were computed using the expression
o • 2 -if cos 85 - sin <P Kin 6S = — cos °—' ™ — x ^o 15 cos <P cos 0
(6)
The North latitude values were taken as positive and the South latitude
values as negative.
3. RESULTS ADD DISCUSSION
The computed values of H , S and S' are presented in Table 1 in thee 0 0 o
first, second and the third rows, respectively, against each $-v&lue. The
essential difference between thin table and the similar table in Ref. [h]
is the use of different expressions for the calculation of 6 and E in
the two cases. The table in Ref. [It] was compiled by using the expression
for 6 as used by Cooper [7]
S = 23.U5 Bin 36028U+H
and for E as used by Duffie and BecKnan [1]o
(7)
0.033 cos [3| (8)
-3-
While the use of eqn(3) instead of eqn (8) does not create an appreciable
difference, the use of eqn (2) instead of eqn (7) can indeed lead to
substantial differences in the values of H - upto 10? or even more. This
would not he obvious as eqn (8) is generally considered to tie good enough for
most of the practical purposes. But it has been shown [3] that even a small
error in the value of S can lead to relatively large errors in the value of
H Q particularly for higher values of the latitude. In fact, it can be directly
seen by a comparison of Table 1 in this paper and Table 1 of Eef. [I*]
that differences of upto 5-10/t are not very uncommon and even larger differences
are occasionally encountered. As the correlations and the models developed
these days often claim to predict the solar radiation to an accuracy of about
+ 5-10JS, it is necessary to use the values of H , S and S" to a higher
accuracy of about 1-2%. Table 1 in this paper is meant to meet this need.
It is highly recommended that in future work this table should be used for
obtaining the values of H , S and S' at any location. For intermediate
it-values, simply a linear interpolation between the two successive listed
^-values would provide the values of H , S and 5' without introducing any
appreciable error at all.
The chart on the Campbell-Stoke's sunshine recorder does not burn when
the solar elevation is less than 5 [9]. Therefore the value of the MPSD
recorded on a sunshine recorder is actually less than S ,
Some authors [10, 11] prefer to use the value of S' instead of So o
in their correlation. For this reason the values of £', as computed using
eqn (6), are also provided in Table 1.
The computation of H values in this paper has been done using the
recently suggested value of the solar constant [6], viz 136? W/m . The values
of the solar constant have undergone revisions several times in the past. An
account of them is given by Iqbal [2], It is not unlikely that no such
revisions would take place in future. The values of i5 corresponding to any
other value of the solar constant can be easily obtained from those presented
here simply by multiplying the tabulated values herein by the ratio of the new
value of the solar constant to 1367 W/m .
The use of eqn {2) instead of the simpler eqn (T) for the value of 6
does provide more accurate values of H . However, there still remain
uncertainties in the value of 6 due to the leap-year cycle, the use of a
single value of 6 for a day, etc. These uncertainties are relatively small
and one can expect the values of H provided in Table 1 to be accurate within
about +1%.-h-
AKNOWLEDGMENTS
The author would like to thank Professor Abdus Salarn, the International
Atomic Energy Agency and the UNESCO for the hospitality at the International
Centre for Theoretical Physics, Trieste. Thanks are also due to Professor
G. Furlan for his continued interest and encouragement.
NOMEMCLATURE
monthly average daily extraterrestrial irradiation on a horizontal
surface.
monthly average daily maximum possible sunshine duration.
monthly average daily maximum possible sunshine duration recorded
by a Campbell-Stoke'e sunshine recorder.
latitude.
declination,
sunset hour angle.
day of the year with 1 on January 1.
eccentricity correction factor,
solar constant.
REFEREHCES
1 .
2 .
3 .
k,
5 .
6.
7.
9.
1 0 .
J.A. Duffie and W.A, Beckman, Solar Engineering of Thermal Processes,John Wiley (1980).
M. Iqbal, An Introduction to Solar Radiation, Academic Press,
Canada (1963).
S.A. Klein, Calculation of monthly average insolation on t i l ted
surfaces!. Solar Energy 19, 325-329 (1977).
P.C. Jain, Table for monthly average daily extraterrestrial
irradiation on horizontal surface and the maximum possible sunshine
durationt ICTP, Trieste, Internal Heport IC/S^/?.
J.W. Spencer, Fourier series representation of the position of the
sunj Search 2(5), 172 (1971).
C. Frohllch and R.W. Brusa, Solar radiation and i t s variation in
time, Sol. Phys. jb, 209-215 (I98l).
P.I . Cooper, The absorption of solar radiation in solar s t i l l s^
Solar Energy 12, 3 (1969).
P.C. Jain, Study of the errors in the calculation of extraterrestrial
radiation on horizontal surface (H ) and recommended days for the
calculation of i! , ICTP, Trieste, Internal Report IC/83/193.
C.F, Brooks and E.S. Brooks, Sunshine recorders - A comparative
study of the turning glass and thermometric system. J. Meteo. I4,
105-115 (191*?)-
J.E. Hay, Calculation of monthly mean solar radiation for horizontal
and inclined surfacesj Solar Energy 23, 301-307 (1979).
A, Mani and S. Rangarajan, Techniques for the precise estimation of
hourly values of global, diffuse and direct solar radiation. Solar
Energy 31, Ho. 6, 577-595 (19S3).
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