Estimation of symmetric and asymmetric hourly global and diffuse radiation from daily values

Solar Energy Vol. 48. No. 1, pp. 7-14, 1992 0038-092X/92 $5,00 + .00 Prinled in the U.S.A. Copyright © 1991 Pergamon Press ptc

ESTIMATION OF SYMMETRIC AND ASYMMETRIC HOURLY GLOBAL AND DIFFUSE RADIATION

FROM DALLY VALUES

V. V. SATYAMURTY and P. K. LAHIRI Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur-721302, India

Abstract--The validity of the correlations[ 1-3 ] to estimate the hourly global and diffuse solar radiation components for an independent dataset of fourteen locations is examined in this article. The correlations for the diffuse component[2,3] are found to be in rather poor agreement with the data. An improved correlation for the diffuse component that includes a daily diffuse fraction as a parameter is developed. The influence of this improved correlation on estimating the beam radiation component is examined. A convenient form of describing asymmetry for these three components of solar radiation distribution is proposed and validated.

1. INTRODUCTION

Interest to estimate the monthly average hourly values of global, diffuse, and beam components of solar radiation from the corresponding daily values stems from a variety of reasons such as: (i) generation of synthetic data for use in the simulation methods such as TRNSYS [ 4 ] and; (ii) calculating monthly-average or even yearly-average tilt factor for beam radiation [ 5,6 ], utilizability [7,8 ], and transmittance-absorptance product [ 9,10 ] (the parameters of interest in the design of solar energy systems).

The correlation of Collares-Pereira and Rabl[l ] to predict the ratio of hourly global radiation to the daily global radiation, rt, is widely used. Similarly the ratio of hourly diffuse radiation to the daily diffuse radiation, ra, can be predicted from the Liu and Jordon[2] correlation. Both these correlations assume solar radiation distribution to be symmetric about solar noon. A con- sequence of employing the symmetric correlations is that the values of the monthly average, daily tilt factor are equal for equal positive and negative surface azi- muth angles which is contradicted by the numerical values obtained by numerical integration employing the hourly solar radiation data [ 5]. This has been at- tributed in [5] to asymmetry in the solar radiation distribution.

Recently correlations [ 3 ] were developed to predict rt and ra taking into account the asymmetry in the solar radiation distribution. This approach involves obtaining location dependent constants that can be evaluated when detailed hourly meteorological information is available. Also presented were location independent correlations valid when the solar radiation distribution is assumed to be symmetric. These correlations have been developed from the data of a limited number of locations (three French and one Ca- nadian). The relative accuracies of the correlations [ 1- 3] for a common independent dataset have not been established.

The objective of the present investigation is to test the correlations [ 1-3 ] available in the literature to pre-

diet rt and rd (which can be subsequently used to evaluate the hourly beam radiation component) against an independent dataset of fourteen Indian locations. From the numerical results, it has been found that the correlations reported in [ 2 ] and [ 3 ] predict rd in excess of 9.9% rms error. In the present article an improved correlation for rd that includes daily diffuse fraction as a parameter has been developed and validated. In order to develop correlations to predict rt and rd when the solar radiation distribution is not symmetric about solar noon, the methodology developed in the present article facilitates description of asymmetry through a parameter, the ratio of forenoon solar radiation (global, diffuse, or beam) to the corresponding afternoon solar radiation. Evaluated were the influence of the improved correlation for rd to estimate the beam radiation component and the performance of the correlations developed in the present article for r~ and ra that include asymmetry in the solar radiation distribution as well.

2. DATABASE

For establishing the relative performance of the correlations for r~ and rd[ 1-3 ] as well as the correlations developed in the present study, a dataset of fourteen Indian locations (except for Port Blair for which five years of hourly-global and three years of hourly- diffuse radiation data were available) with eight years hourly- and monthly-average hourly values was uti- lized [ 11,12 ]. The locations and their latitudes are given in Table 1. The performance of the correlations has been assessed against actual data as well as symmetric values generated from the data by taking the average of the values for corresponding hour pairs.

The data for the fourteen locations chosen are based on careful measurements [ 12 ] made over long periods (up to 21 years). All data are based on the World Ra- diometric Reference (WRR). The original data on the International Pyrheliometric Scale 1956 (IPS-56) have been modified to obtain the data on the WRR.

Hourly components of global and diffuse solar ra-

V. V. SATYAMURTY and P. K. LAHIRI

Table 1. Locations and their latitudes used for evaluating the performance of the correlations for rt and ra

Location number Location Latitude, °N

1 Ahmadabad 23.07 2 Bhavnagar 21.75 3 Bombay 19.12 4 Calcutta 22.65 5 Goa 15.48 6 Jodhpur 26.30 7 Madras 13.00 8 Nagpur 21. I 0 9 New Delhi 28.58

10 Poona 18.53 11 Port Blair 11.67 12 Shillong 25.57 13 Trivandrum 8.48 14 Visakhapatnam 17.72

the errors in rt and rd that are ratios can be expected to be similar. For the purpose of establishing the relative performance of the correlations for rt and rd, the present data base can be expected to yield reasonable conclu- sions.

3. PERFORMANCE OF THE EXISTING CORRELATIONS

FOR r~ AND rd FOR AN INDEPENDENT DATASET

Values o f r t d , rdd , and rbd have been calculated from the meteorological data [ l l, 12 ] for the set o f locations given in Table 1. From the symmetric data (the average

value of the hour pairs) ltd.,. , Fdds, and rbd., have also been calculated. Employing the correlations of [ 1 ] and [2], r~c and rdL have been calculated.

rbcL = [rtc - ( H d l H ) r u L ] / [ 1 - ( H a ~ H ) ] , ( 1 )

diation have been measured with a Moll-Gorczynski pyranometer and a Moll-Gorczynski pyranometer with shading ring. Shading ring correction factors, as- suming that the diffuse radiation from the sky is uni- form and the receiver is an ideal one, have been applied to the diffuse component . The Moll-Gorczynski pyr-

anometer has been classified [13 ] as a second class in- strument. Although second class instruments are likely to furnish data less reliable than first Class instruments,

Similarly, rtj and rdj have been evaluated using the correlations [ 3 ]. The values for the corresponding ratio for beam radiation rb, rhcL and rbjj have been obtained.

Based on the numerical results, the rms differences between the predicted values using the correlations and the actual values on a yearly basis are shown in Table 2 (for locations 1-7 listed in Table 1 ) and Table 3 (for

locations 8-14 listed in Table 1 ). The rms differences for the fourteen locations put together are also shown

Table 2. Relative performance of correlations for predicting r,, rd, and rb for all locations and locations 1-7 of Table 1

% rms Location number Row difference

number between All 1 2 3 4 5 6 7

1 rtc, rtas 3.0 2.5 3.8 2.6 2.1 2.6 3.0 3.6 2 rtc, rta 4.8 3.1 5.1 4.7 4.4 3.3 3.2 4.4

3 ra, rtds 2.9 2.6 4.2 2.5 1.6 2.8 3.2 3.6 4 ra, rtd 4.8 3.1 5.4 4.7 4.2 3.5 3.4 4.4

5 raL, raa~ 8.9 11.9 13.1 8.9 5.2 10.7 11.1 5.4 6 raL, raa 9.9 12.2 13.4 9.5 6.6 11.4 11.3 8.4

7 raj, raa, 8.9 11.9 13.2 9.0 5.2 10.7 11.1 5.3 8 raj, raa 9.9 12.2 13.5 9.6 6.6 11.4 11.3 8.3

9 raP, rdds 3.1 2.8 3.9 3.2 2.3 3.9 2.7 2.3 10 ray, rdd 5.3 3.9 4.9 4.7 4.7 5.5 3.4 6.8

1 1 fouL, rba~ 8.1 6.5 1 1.3 6.7 6.1 5.1 3.8 6.3 12 FOUL, rbd 12.1 9.2 12.5 1 1.4 8.5 8.0 4.6 9.3

13 r~jj, rbds 9.0 6.9 12.3 8.6 5.8 6.0 2.7 6.1 14 rbj.I, rbd 12.7 9.4 13.4 12.6 8.3 8.6 3.7 9.1

15 rOUp, rbas 7.2 4.9 8.8 4.6 4.8 5.0 4.5 7.3 16 rOUt,, rba 11.5 8.1 10.3 10.3 7.6 7.9 5.2 10.0

17 ~ , rta 3.2 2.6 3.9 2.7 2.5 2.7 '3.0 3.7 18 ~ , rtd 3.2 2.6 4.3 2.7 2.1 2.9 3.2 3.7 19 ~aL, raa 9.1 12.0 13.2 9.0 5.3 10.9 11.2 6.0 20 ~ , raa 9.1 12.0 13.2 9.1 5.3 10.8 11.2 5.9 21 ~ , raa 3.8 3.2 4.1 3.7 2.6 4.4 3.1 3.7 22 r~CL, rbd 9.0 6.7 11.6 7.1 6.5 5.6 3.9 6.8 23 ?~JJ, rba 9.8 7.0 12.6 9.0 6.3 6.4 2.8 6.5 24 r~cp, rba 8.1 5.2 9.0 5.0 5.4 5.4 4.6 7.7

Symmetric and asymmetric radiation 9

Table 3. Relative performance of correlations for predicting rt, ra, and rb for all locations and locations 8-14 of Table 1

% rms Location number Row difference

number between All 8 9 10 11 12 13 14

1 rtc, r, as 3.0 2.6 2.3 1.9 4.0 3.7 2.6 2.7 2 rtc, r~a 4.8 4.3 2.6 2.9 5.5 10.4 4.1 3.1

3 ra, rtas 2.9 2.6 2.6 1.9 3.4 3.6 2.8 2.4 4 rtj, rta 4.8 4.3 2.9 2.9 5.1 10.3 4.2 2.9

5 ra~, raas 8.9 10.3 8.5 11.2 4.4 4.4 4.6 8.2 6 raL, raa 9.9 10.6 9.2 11.5 5.9 10.0 5.7 9.1

7 raj, raas 8.9 10.0 8.7 11.2 4.3 4.8 4.6 8.1 8 raj, raa 9.9 10.6 9.3 11.5 5.9 10.2 5.7 9.0

9 rap, raas 3.1 3.1 3.5 2.8 2.7 4.2 1.8 2.2 10 rap, raa 5.3 4.6 4.9 4.0 4.8 10.0 3.8 4.5

1 I rbcL, rbas 8.1 7.8 5.0 7.2 13.3 13.8 4.3 5.1 12 rb('L, rba 12.1 9.3 6.5 8.0 20.7 23.9 8.8 6.1

13 rbjj, reas 9.0 8.2 3.9 8.5 17.1 14.4 4.0 4.5 14 rbjj, rba 12.7 9.6 5.8 9.2 23.2 24.3 8.6 5.6

15 rbcp, rbas 7.2 5.9 5.2 4.4 13.9 11.4 5.1 4.9 16 rbcP, rba 11.5 7.7 6.7 5.6 20.9 22.6 9.2 5.9

17 ~ , rid 3.2 2.6 2.4 2.0 4.2 4.0 3.9 2.9 18 ~ , rta 3.2 2.6 2.7 1.9 3.7 3.9 4.0 2.6 19 ~aL, raa 9.1 10.4 9.0 11.3 5.1 5.0 5.4 8.3 20 ~ , raa 9.1 10.4 9.2 11.3 5.0 5.3 5.4 8.2 21 ~ap, raa 3.8 4.2 4.7 3.3 3.7 5.0 3.3 2.6 22 r~L, rba 9.0 8.0 6.0 7.3 16.1 14.3 7.8 5.3 23 r~jj, rba 9.8 8.5 5.2 8.6 19.7 14.9 7.7 4.8 24 r'k:p, rba 8.1 6.2 6.2 4.5 16.1 11.9 8.3 5.1

in the first column of both Tables 2 and 3. From the rms differences given in Tables 2 and 3, the following points emerge. 1. The correlations of Collares-Pereira and Rabl[1]

and Jain et al.[3] predict rt with comparable accuracy. The rms differences (see rows 1 and 3 of Tables 2 and 3) between rtc, rtds and rtj , rtas are comparable, equal to 3% and 2.9%, respectively. However, the rms differences compared to the actual data (see rows 2 and 4 of Tables 2 and 3 ), i.e., between rtc, rtd and r,j, rtd are comparable but larger, equal to 4.8%.

2. The accuracy in predicting rd using the correlations of Liu and Jordon [ 2 ] and Jain et aL [ 3 ] also is comparable. The rms differences (see rows 5 and 7 of Tables 2 and 3) between rdL, rdds and rdj, rdds are 8.9%. The rms differences between rdL, rdd and rdj,

tad are (see rows 6 and 8 of Tables 2 and 3) 9.9%. 3. rb is predicted using the correlations for r,[ 1 ] and

rd[ 2,3 ] with rms differences (between rbcL, rbds and rbjj, rbas; see rows i1 and 13 of Tables 2 and 3) of 8.1% and 9.0%, respectively. The rms differences between rbCL, rbd and rbjj, rba ( see rows 12 and 14 of Tables 2 and 3) have been obtained as 12.1% and 12.7%, respectively. Thus, it appears that rbCL is in slightly better agreement with the data than rbjj.

4. In general, the errors in the correlations for r, are lower (symmetric 3.0%, actual 4.8% rms) whereas rd is predicted with relatively large (symmetric 8.9%, actual 9.9% rms) errors.

4. IMPROVED CORRELATION FOR rd

The correlations for ra[2,3] contain the seasonal dependence only through ~os, the sunset hour angle for a horizontal surface, but do not contain any other climatic information. Indeed rdL is the ratio of hourly to daily extraterrestrial radiation, I o / H o . It can be envis- aged that it is improbable that the hourly component of diffuse radiation is independent of all climatic parameters. Thus, it can be expected that the correlation for ra can be improved if a parameter to describe the climatic information is included. It has been found from the extensive database used in the present study that ra displays a systematic dependence on the daily diffuse fraction. A plot of values of raas at different hour pairs is shown in Fig. 1. The parameter in Fig. 1 is the monthly-average, daily-diffuse fraction. In obtaining this plot, different months and locations (listed in Table 1 ) have been chosen from the database that have comparable sunset hour angles (within 1 ° ). Figure 1 clearly shows that raas does depend on the monthly- average, daily-diffuse fraction.

l0 V. V. SATYAMURTY a n d P. K. LAHIRI

The same trend shown in Fig. 1 has been observed for several other fixed ws. In order to examine the functional form to describe the dependence of rda, on the diffuse fraction, values of rdd,/(Io/Ho) have been plotted against the hour angle I~01 , for fixed O~s and different diffuse fractions (Fig. 2). One functional form that satisfies the variation of rdd,/(lo/Ho) with w, is a ' + b'cos ~) where the values of a ' and b' will depend upon the diffuse fraction. Determining a ' and b' from the best fit curves of all months and the locations, plots of a ' and b' versus the daily diffuse fraction obtained are shown in Fig. 3. It is evident from Fig. 3 that there is a strong correlation for a ' and b' with the daily diffuse fraction. The correlation with H d / H has been found to be better than that with the daily clearness index, Kr. Even when H a / H is not available, the same can be obtained with a correlation, usually in terms ofKT, suitable for the particular location. Thus, by employing H d / H as a parameter the uncertainty in estimating H d / H from a correlation is avoided at the stage of developing the correlations. Best fit to these plots in Fig. 3 describe a ' and b' as

0.4922 + 0.27~(Ha~H)

0.1 _< (H~IH) <_ 0.7 a t ~

0.76 + 0 . 1 1 3 / ( H d / H )

0.7 < ( H d / H ) < 0.9 (2)

b' = 0.6369 - 0.3405~(Ha~H). (3)

Though the correlations for a ' and b' are given by eqns (2) and (3), b' can be obtained from a ' by im- posing that rda, when summed up for all the hours in a day, becomes unity. This condition is described by,

1.5 Curve Loc. No Month Hd//H

1 AOHt 0.30 l ~ - ~ / • ® , .o~ o.~g /

@ ll. August 0.62 1 3 -- ( ~ 10 Augusl 0.77 //

I 9~.0s* ~: COs ~ 94.s00 / /

1.2

0,9

0.8

0,7 -

0.6 7.5 22.5 37.5 52.5 67.5 82.5

Hour Angle , ICOI

Fig. 2. Dependence ofrdd~/(lo/Ho) o n hour angle and diffuse fraction.

fo ~' rddsdw = r / 2 4 . (4)

Noting that rd, l,. = ( a' + b'cos w )( lo/ Ho), from eqn (4), it follows,

b' = 2( 1 - a ' ) (s in o~, - ~,cos ws)/

(ws - 0.5 sin 2o~s). (5)

Though eqn ( 3 ) for b' has been obtained indepen-

0.1L,

0.~21

0.10

Loc. No Month Hd/H ] I April 0,30 I

1 (~ 7 May 0.39

14 August 0.62 J (~ 10 August 0.77 L 4.050 ~ COS ~ g4"600

0.08

g £o

0,0E

0.0 z

0.02

0.00 ~ 7.5 22.5 37,5 52.5 67.5

Hour Angle t I~1

Fig. 1. Dependence of rd on daily diffuse fraction.

82.5

'0.492 2 + 0. 2 7 / ( H d / H ) 0.1 ~< ( Hd /H ),~0.7

0.76 + 0 ,113/ (Hd/H ) 0.7 < ( H d / H }~< 0.9

• - %;°"

0 .6369- 0 . 3 4 0 5 / ( H d / H ) 0.1-% ( H d / H ) ~ 0 . 9

-z.,/ I ] 1 ] t L ] I 0.1 o.2 o.3 o.z, o.5 0.6 o.7 o .a o.9

H~/H

Fig. 3. Variation of the constants with diffuse fraction in the proposed correlation for rdd s.

Symmetric and asymmetric radiation

dently, the values of b' given by both the eqns (3) and (5) are found to be extremely close. Thus, it is rec- ommended that eqn (2) be used for a', and b' be obtained from eqn (5) that satisfies the normalization condition [given by eqn (4)] also. Thus the present correlation to predict ra is given by,

r d p ~ : (a' + b'cos w)(Io/Ho) : (a' + b'cos ¢o)rdL

= r(cos ~o - cos o~s)(a' + b'cos o~)/

[24(sin ws - ¢0scos ws)] (6)

where the constants a' and b' are given by eqns (2) and (5). From Fig. 3, it is interesting to note that a ' = 1 and b' = 0 when the daily diffuse fraction is approximately equal to 0.5. Thus the present form ofrap~ reduces to the correlation given by Liu and Jordon [ 2 ] for an "average" climate, with a daily diffuse fraction of 0.5.

The new form of the correlation given by eqn (6) (developed in the present study) has been tested against the values of raa and raas obtained from the data of the fourteen locations. The rms differences between rapt, the present correlation and raa is 5.3% (see row 10 of Tables 2 and 3) and between raps and rads is 3.1% (see row 9 of Tables 2 and 3). It is evident that the present form of the correlation for predicting rd that includes daily diffuse fraction as a parameter is in better agreement with the data than the correlations of Liu and Jordon [ 2 ] and Jain et al. [ 3 ].

The hourly beam component of solar radiation, predicted by rbcp calculated using rtc and raps in eqn ( 1 ) agrees with rbds and rbd given in rows 15 and 16 of Tables 2 and 3 with rms differences of 7.2% and 11.5%, respectively. Though the overall rms difference for all the locations is lower compared to rt,cL and rbjj, some of the locations (nos. 6, 7, 9, and 13) have a higher rms difference. Also, the improvement in predicting rb is not as significant as that in ra, through the present correlation. This is because improvement in rd influ- ences rb to a small extent, particularly at low diffuse fractions as evident from eqn ( 1 ). Further, the mean bias errors for rtc and rdPs have been found to be op- posite for these locations.

5. CORRELATIONS FOR ASYMMETRIC DISTRIBUTIONS

5.1 Characterization o f asymmetry In the present study it is proposed that the daily or

monthly-average-daily asymmetry for global, diffuse, or beam radiation components be described by the ratio of forenoon solar radiation to the afternoon solar radiation. Thus the asymmetry factors Ag, Ad, and At, for global, diffuse, and beam radiation are defined by

Ag = H f / t l ~ , (7)

Ad =ndf/Haa, ( 8 )

Ab =Hbf lHba , (9)

where H, Ha, and lib represent the daily- or monthly-

11

average-daily global, diffuse, and beam radiation. The quantities with subscript frefer to the forenoon and with subscript a refer to the afternoon. Maximum asymmetry found in the database of the locations given in Table 1 for global, diffuse, and beam components of radiation is shown in Table 4. Asymmetry factors in the range 0.85-1.15 have been found to be very common in the present database, particularly for the beam radiation. The factors A e, Aa, and Ab are related to each other by,

Aa = Ag(Hdf /Hf ) l (Hd . lHa) , (lO)

At, = Ag( 1 - HastHs)l( 1 - Ha.IHa). ( I 1 )

The factors Ag, Ad, and A b satisfy the integral relations

fO °~ rt,d, bd~ = 7r/[12( 1 + Ag,d,t,) ]. (12)

When H/, Ha and Hat, Hda (or Hb/, lib.) are available, the factors Ag, Aa, and At, can be directly evaluated. If not, from the information expected to be more com- monly available, viz., Hf/Ha, the other two factors can be calculated from eqns (10) and (11) provided a method of estimating the diffuse fraction for the forenoon and afternoon is available. From the values of H/and Ha the corresponding half-day clearness indices Krcand Kr, can easily be calculated. It has been found that the correlations [ 1,14 ] for H a / H for Indian locations can be used with/£7. = KTr or KT, to predict the half-day diffuse fractions with the same accuracy as the correlations given in [1,14] yield for the full day. Alternatively, when only Ag is known, after calculating the asymmetric distribution for the global solar radiation I, la (and hence It,) can be directly obtained employing the correlations [ 15,16 ] for la/ I .

5.2 Description o f the asymmetric distribution o f solar radiation

The asymmetry factors for global, diffuse, and beam components on an hourly basis designated as ag, aa, and ab, respectively, are defined by

ag = rJrts, (13)

ad = rd/ra~, (14)

ab = rb/rt,s, (15)

Table 4. Maximum long-term average asymmetry found in the solar radiation distribution

Location Month Maximum asymmetry found

Bombay September 0.859 In global radiation Calcutta September 1.144 - - - Do - -

Poona March 0.918 In diffuse radiation Madras April 1.249 - - Do - -

Bombay September 0.690 In beam radiation Calcutta September 1.239 - - Do - -

12 V . V . S A T Y A M U R T Y a n d P . K . L A H I R I

where rt~, rd~, and rb~, respectively, are rt, rd, and rb when the distribution is symmetric.

A typical plot for ag, ad, and ab versus the hour angle w for Bombay is shown in Fig. 4. It is to be noted that these factors are calculated from the data and have nothing to do with the symmetric or asymmetric correlations available in the literature. The variation of the three factors can be described accurately by equa- tions of the form (P + Q~). From Fig. 4 it is evident that at noontime ag, ad, and ab are equal to unity lead- ing to P = 1. Referring to Fig. 4, Qg.d,b can be expressed in terms of the ordinates agm,dm,bm at w = wm,~ as

ag,d,b = ( agm,dm,bm -- l)/Wmax (16)

where OJma x is the hour angle at the midpoint corresponding to the last full hour of the day, reckoning from solar noon. Thus the proposed form to predict r,, rd, and rb when the solar radiation distribution is asymmetric in terms of the asymmetry factors [introduced in the present study defined by eqns ( 13 ), (14), and ( 15)] can be represented by

lation [ 1 ] for rt and the present correlation for ra are employed. Performing the integration in eqn (18), agm,amj,,n are given by

~ m a x ( 1 - - Ag,d,b ) agm,d,n,bm = 1 + ( 1 9 )

( I + Ag,d.b ) ( 60 s -- Rag, ad, ab )

where,

R"x = [ b ( - ~ + l - c ° S 2 ~ s ) + ( a - b c ° s w s ) 8

] a 2 X (1 - - COS ~os) - - ~ (~sCOS o~s)J/

(sin ~o~ -- w, cos w,); (20)

[ ( ~ l - c ° s 2 ~ ° ~ ) + ( a ' - b ' c o s c o 0 R~d = b' -~ 8

a' 1 X (1 - cos Ws) - " ~ (ws2COS ~o~) /

(sin w~ - to, cos w~); (21)

r*a,b = ag,d, brts,d~,bs = [ 1 + Qg,d, bO~] r~.,,d.~,b,

= 1 + ( a g m , d m , b m - - l ) rt~,d~,b,. ( 17 ) 6drnax

Using eqn ( 17 ) in eqn (12), agm,dm,bm can be evaluated from

E ' c: h ' o Rah = b" + + b"cos w~)

a" ] × (1 - COS ws) - -~ (w~COS w~) /

(sin w~ - ~o~cos w~). (22)

12( 1 + Ag,d,b ) (18)

To evaluate agm.dm.bm explicitly, the symmetric corre-

In eqn (22) a" and b" are given by

a" = [a - a ' ( H d / H ) ] / [ I - ( H a ~ H ) ] , (23)

b" = [b - b ' ( H d / H ) ] / [ 1 - ( H a ~ H ) ] . (24)

Using eqn ( 19 ) for agm,dm,bm in eqn ( 17 ), ag,d,b can be expressed as,

1.0 g

x ~

0.9

$

. /

/

0.8 / /

| / /

o.7 I I - 82.5 - 67.5 - 52.5

x / ~ / t ~ / i / /

- ~ ....e... t / ~ / /

/ /

Z / /

/ / /

/

Z / /

/ /

Location : Bombay • India

Month : August

• rtd / rids ] /

• rdd/ rdds ~* From data

I el rbd/rbdsJ

. . . . . Proposed linear form

[ I [ -37.5 -22.5 "7.5 0

I-lair Angle, to

Fig. 4. Variation of the hourly asymmetry factors with the hour angle.

( 1 - A g , d , b ) ag,d,b = 1 + w. (25)

( 1 "k- Ag,d,b)(60 s -- Rag, ad, ab )

Thus, using eqn (25) in eqn ( 17 ) Ft*,,d,b is given by,

* I (1 - Ag,d,b) ] rt,a,h = 1 + ( 1 + A g , d , b ) ( O d s - - Rag.ad, ab) W rtsAs, b s.

(26)

The form of the correlations to predict r,, rd and rb given by eqn (26) with the asymmetry factors (subject to validation of the asymmetry factors) are general enough to be used with any reliable correlation for predicting the symmetric hourly components of solar radiation. Equation (26) also reduces to the symmetric form when A g , d , b equal unity.

6. TEST OF THE ASYMMETRIC CORRELATIONS

The rms differences between rtd, rdd , and rbd and the values predicted with the present form of the cor-

Symmetric and asymmetric radiation

relations [eqn (26)] when the distribution is asymmetric are also shown in Tables 2 and 3. Asymmetric predictions are indicated with an as te r i sk '* ' in Tables 2 and 3. In all cases, rms differences are lower than the corresponding differences when the symmetric correlations have been employed. It is also interesting to note that the rms differences between symmetric correlations and symmetric data is exactly of the same order as the rms differences between the asymmetric correlations and the actual data. This indicates that irrespective of the absolute agreement in predicting the symmetric distribution, asymmetry is predicted accurately by the present asymmetry factors. To further establish this, a plot of ag calculated from eqn (25) v e r s u s rtd/rtds is shown in Fig. 5. The scatter appears to be large, due to a highly enlarged scale. In view of the detailed database, the asymmetry factor ag can be considered to be predicted reasonably accurately by the present expressions. Similar agreement has been found for the asymmetry factors aa and ab as well. The global asymmetry parameter, Ag,d,b introduced in the present study can be evaluated with ease if Hj and H~ are known, i.e., essentially at least long-term, average- hourly solar radiation data is needed• When such information is not available, measurements for a few typical days can be made and from such measurements, at least an upper and lower bound estimate of asymmetry can be obtained for a parametric study with the present form of the asymmetric correlations.

7 . C O N C L U S I O N S

Performance tests conducted on the correlations for rr[l,3] and ra[2,3] with an independent dataset of fourteen Indian locations reveal that these correlations are of comparable accuracy. However, ra is predicted with a rather large rms error of 9.9%. An examination of the extensive database revealed that a correlation for ra can be improved by including the daily diffuse

t .10

i.05 • : . - C : ~ . . . : ~ : 4 . - .

• . ..'. : . .~ . : . :

. .. _~ ~:." - 1 0 0 . " ~ ' * . .

"" . . . .

0.95 "'" " ~'::":: "

0.00 I I I 0.90 0.95 1.00 1.05 1.10

og ( Oa'~a ) : r i d / r i d s

Fig. 5. Comparison of actual and predicted values of the asymmetry factor ag for global radiation.

13

fraction as a parameter. Though the correlations for the location dependent constants have been developed for Indian locations, the idea of including diffuse fraction as a parameter in the correlation for rd can be explored for other locations as well. It has been found that inclusion of any asymmetry factor for all three components of solar radiation reduces the rms differences in predicting rt, rd, and rb by approximately 1.6%, 1%, and 3%, respectively. However, this improvement depends on the magnitude of the asymmetry itself. If the data is symmetric for a location, the present forms of the asymmetric correlations reduce to the symmetric forms. The two constants required to predict the asymmetry factors described as linear functions of the hour angle have been expressed in terms of global information such as the ratio of forenoon radiation to the afternoon radiation• The method of evaluating the location dependent constants remains valid for other locations also as long as the asymmetry factors can be described in a linear form, at least approximately.

NOMENCLATURE

r, The ratio of the hourly component of global radiation l, to daily global radiation, H

rd The ratio of the hourly component of diffuse radiation, la, to daily diffuse radiation, Ha

rh The ratio of the hourly component of beam radiation, lb, to daily beam radiation, Hb

r~a The ratio of the hourly component of global radiation to daily global radiation, obtained from the data

r,d~ The value of the ratio ofthe hourly component ofglobal radiation to the daily value, obtained from data made symmetric by taking tt{e average value of the hour pairs

rdd The value of the ratio of the hourly component of diffuse radiation to the daily value, obtained from the data

rda, The value of the ratio of the hourly component of diffuse radiation to the daily value, obtained from data made symmetric

rba The value of the ratio of the hourly component of beam radiation to the daily value, obtained from the data

rbd~ The value of the ratio of the hourly component of beam radiation to the daily value, obtained from the data made symmetric

r,c The predicted value of r, using the correlation of Col- lares-Pereira and Rabl [ 1 ]

rn The predicted value of rt using the correlation of Jain et al.[3]

rdL The predicted value of rd using the correlation of Liu and Jordon [ 2 ]

rdj The predicted value of rd using the correlation of Jain et al.[3]

rbCL The predicted value of rb using the combination of the correlations of Collares--Pereira and Rabl[1] and Liu and Jordon [ 2 ]

rbjj The predicted value ofrb using the combination of correlations for r, and rd of Jain et a/.[3] that can be obtained from ( 1 ) by replacing hc, raL with ra, rdj.

R E F E R E N C E S

1. M. Collares-Pereira and A. Rabl, The average distribution of solar radiation--correlations between diffuse and hemispherical and between daily and hourly insolation values, Solar Energy 22, 155 (1979).

2. B.Y.H. Liu and R. C. Jordon, The interrelationship and characteristic distribution of direct, diffuse and total solar radiation, Solar Energy 4, 3 (1960).

4 V.W. SATYAMURTY and P. K. LAHIRI

3. P. C. Jain, S. Jain, and C. F. Ratto, A new model for obtaining horizontal instantaneous global and diffuse radiation from daily values, Solar Energy 41,397 (1988).

4. TRNSYS, A Transient System Simulation Program, Ver- sion 12.1, Users'Manual, Solar Energy Laboratory, Uni- versity of Wisconsin, Madison ( 1983 ).

5. S. A. Klein and J. C. Theilacker, An algorithm for calculating monthly average radiation on inclined surfaces, ASME J. Solar Energy Eng. 103, 29 ( 1981 ).

6. C. Gueymard, Mean daily averages of beam radiation received by tilted surfaces as affected by the atmosphere, Solar Energy 37, 261 (1986).

7. S. A. Klein, Calculation of fiat plate collector utilizability, Solar Energy 21,393 (1978).

8. M. Collares-Pereira and A. Rabl, Simple procedure for predicting long term average performance of non-concentrating and of concentrating solar collectors, Solar Energy 23, 235 (1979).

9. A. A. Sfeir, Monthly average optical efficiency of flat plate collectors, Solar Energy 30, 397 ( 1983 ). Also, the letter of R. Beato, Solar Energy 35, 207 ( 1985 ).

10. L. M. Acharya, Studies on monthly average parameters in the design methods for solar energy heating systems, Ph.D. Thesis, Indian Institute of Technology, Kharagpur, India (1987).

11. Aerological data of lndia, Part I11. Radiation Data, India Meteorological Department, Government of India ( 1971- 1978).

12. A. Mani, Handbook of solar radiation data for India, Allied Publishers Private Limited, New Delhi ( 1981 ).

13. A. A. M. Sayigh (ed.), Solar energy engineering, Academic Press, New York (1977).

14. V. Modi and S. P. Sukhatme, Estimation of daily, total and diffuse insolation in India from weather data, Solar Energy 22, 407 (1979).

15. J. F. Orgill and K. G. T. Hollands, Correlation equation for hourly diffuse radiation on a horizontal surface, Solar Energy 19, 357 (1977).

16. D. G. Erbs, S. A. Klein, and J. A. Duffle, Estimation of the diffuse radiation fraction for hourly, daily and monthly-average global radiation, Solar Energy 28, 293 (1982).

Documents

Estimation of symmetric and asymmetric hourly global and diffuse radiation from daily values