Solar Energy, Vol. 20, pp. 101-105. Pergamon Press 1978. Printed in Great Britain

TECHNICAL NOTE

Estimation of the monthly average of the diffuse component of total insolation on a

horizontal surface

M. IQBAL Department of Mechanical Engineering, The University of British Columbia, Vancouver, B.C., Canada

(Received 21 February 1977; in revised form 4 June 1977)

INTRODUCTION Design of many solar energy utilizing devices requires knowledge of separate values of diffuse and beam components of total insolation on inclined surfaces. However, most measuring sta- tions generally record only total shortwave radiation data on horizontal surfaces. Therefore methods are needed to estimate the diffuse and beam components of total radiation on a horizon- tal surface. Once these two components are known, they can then be transposed to determine their value on an inclined plane.

Liu and Jordan[l] have presented two types of relations between diffuse and total radiation. For individual days, they have presented a relation between the ratio of the daily diffuse radiation to daily total radiation on a horizontal surface, D/H and the cloudiness index Kr = HIHo. This correlation was based on data from only one observatory, namely Blue Hill (Mas- sachusetts, U.S.A.) and is given in Fig. 7 of their paper.

Liu and Jordan have also presented a correlation based on long-term monthly averages of the daily values. This second correlation gives the ratio/)//~ as a function of £ 'r --- fllflo and was based on a statistical method with data from several stations in the U.S.A. This correlation was further verified by actual long-term diffuse data from 4 widely dispersed stations namely: Blue Hill (Massachusetts, U.S.A.), Nice (France), Helsingfors (Finland) and Kew (U.K.). This second correlation is given in Fig. 14 of their paper.

The above mentioned correlations have been generally treated with almost universal applicability. They have been extensively employed by designers and researchers. However, recently their applicability on some parts of North America has been questioned[2, 3]. Orgili and Hollands[3] have also presented, for northern latitudes, a correlation for the hourly diffuse radiation on a horizontal surface. Hay[4,5] has presented a revised hourly relationship which incorporates data on local cloud cover and regional albedos. It requires computation of the hourly values of diffuse radiation to obtain daily diffuse radiation. Although Hay's analysis, gives a clear understanding of the multiple reflection processes of the shortwave diffuse radiation between earth's surface and the atmosphere, his computational method is some- what complicated.

In some of the newer[6] design methods of solar heating of buildings, one needs only monthly averages of total insolation on inclined surfaces. Therefore, the need remains for development of simple methods to estimate monthly averages of the daily diffuse radiation. In the method described below, a very simple procedure of computing /9/H is proposed. It incorporates an aspect of local climatic data--the hours of bright sunshine.

PRESENT METHOD It seems obvious that, if a point on the earth's surface remains

covered by cloud the whole day, then all the shortwave radiation received at that point would be of diffuse nature. On such a day, a sunshine recorder would indicate zero number of hours of bright sunshine. On the other hand on an absolutely clear day,

very small amounts of solar radiation would be of diffuse nature and a sunshine indicator could record, almost the same number of hours of sunshine as the day-length. It therefore appears that, the percentage diffuse radiation could be linked with the percent possible sunshine hours. In this report, as we are interes!ed in long-term monthly averages, we may express monthly average daily diffuse radiation in terms of the ratio of monthly average hours of bright sunshine to the monthly average hours of day- length. That is

/5 ti -= = 1 - - = (1) H N "

In eqn (1), ti is the monthly average hours of bright sunshine per day given by weather records and the quantity N is given as

= 1 f"~ N dn • (2) n2- nl Jnt

The day-length N is given by

= ~ cos-t(-tan ¢k tan 8). N 13

(3)

Equation (2) is very simple to compute, although it requires sunshine data records. However, these records almost inv~triably exist wherever total solar radiation is measured. In fa~.!, !he sunshine records are much more abundant than the radtatlon records.

Sunshine records, to estimate terrestrial insolation, haste been employed by a number of researchers. At sites where acttml data of total insolation do not exist, Fritz[7], Page[8] and Nprris[9] have presented simple correlations to estimate the average total insolation from hours of bright sunshine (or cloudiness), day- length and "clear sky" or extraterrestrial radiation. Recently, Smith[10] has presented a statistical method to obtain hourly radiation values through data on hours of bright sunshine. I As can be seen, the use of sunshine records to estimate terrestrial insolation is not new. However, a distinct feature of the ipresent correlation is that, through a very simple expression, the monthly averages of the daily diffuse radiation are obtained when {he total terrestrial insolation is known.

In the next few sections, accuracy of the present correlation and the accuracy of results obtained from the use of this cor- relation in a solar heating system will be examined. An analysis of the solar heating system would require computatiola of in- solation on inclined planes. In this report, procedures la~id down by Liu and Jordan[ll, 12] to compute insolation on !inclined planes has been followed.

In the following section, insolation on inclined surface~s with/5 based on different sources is presented. These insolation values are then used as input in a solar heating system study! and the results compared. The final interest is to investigate wh&her the present method gives satisfactorily accurate results.

I01

102 Technical Note

RESULTS AND DISCUSSIONS

In this study, four sites in Canada, spread over a distance of 5000 km are considered. These stations are: Montreal (45°50'N), Winnipeg, (49°90'N), Edmonton (53°55'N) and Vancouver (49 ° 27'N). Records of sunshine hours of these cities published by the Atmospheric Environment Service Canada[13] are plotted in Fig. 1.

For all the 4 stations, Hay, using his revised method [4, 5], has presented[14] tables of total insolation on inclined surfaces. The measured values for zero inclination used by Hay (i.e. measured

~ ; t I I '1 I I I l I I . I

,~ a

~ . . .

co ~

>w // ..." o < ,~ /.:." .. .

I[ MONTREAL 45 ° 50" N ...............

i

I I I 1 I 1 i I I I JAN FEEl MAR API~ MAY JUN JUL AUG SEP OCT NO',,' DEC

MONTHS

Fig. 1. Bright sunshine hours per day--monthly averages.

same total value. As inclination increases, the differences, if any, begin to appear and they depend upon surface inclination and time of the year. In this report, insolation on a surface sloped at 500 toward the equator only is computed. This is the collector inclination at which a solar heating system would be analysed later in this report. Results of this computation are presented in graphical form in Fig. 2-5. These plots show that, in some cases there are substantial seasonal variations. Results from Hay's

'>.

o

o - -

q - -

I I I '1" I I I I I I MONTREAL QUEBEC 45 ° 50 ' N

SLOPE 5 0 . 0 0 DEGREES

: [

o ; :':' °o = ' ..__- o LIU AND JORDAN [1 ] "

• HAY [ 1 4 ] . . . . . . . . . . . . . . . .

PRESENT METHOD • Ira, m

I I t t I I 1 I I JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC

MONTHS

Fig. 2. Monthly average of the daily insolation on a surface sloped at 50 ° toward the equator. Comparison of three methods

for Montreal.

values) are given in Table I. Data in this table form a starting common position to obtain HT values by Liu and Jordan's correlation and the method proposed here (I). However, to obtain/'?T values, one requires data on ground albedos. In order to be able to make precise comparison withE14], albedo values employed by Hay were obtained from him[15] and used in the present analysis.

Total insolation on inclined planes can be computed for any desired inclination. At zero inclination, all methods start with the

Table 1. Monthly average daily total radiation on a horizontal surface,/-7, MJ m -2 day -1

Montreal Winnipeg Edmonton Vancouver Month 45 ° 50'N 490 90'N 530 55'N 49 ° 27'N

Jan. 5.32 5.25 3.73 2.89 Feb. 8.80 9.16 7.06 5.68 Mar, 12.57 14.08 12.50 10.00 Apr. 15.88 17.44 17.49 14.93 May 18.73 20.78 20.65 20.54 June 20.36 23.06 21.99 21.67 July 21.20 22.84 22.63 22.91 Aug. 16.97 19.32 18.34 18.84 Sept. 13.45 13.45 12.54 13.23 Oct. 8.38 8.20 7.79 7.34 Nov. 4.48 4.53 4.01 3.53 Dec. 3.77 3.83 2.69 2.26

Yearly average 12.49 13.49 12.62 11.98

T

== co

Q

c) z

z <

z O

t -

WINNIPEG MANITOBA 49 ° 90 ' N

SLOPE 5 0 . 0 0 DEGREES o

,:/

o V'.....,,.~ 0~ - % . _ .

LIU AND JORDAN [1 ] o - - HAY [14] . . . . . . . . . . . . . . . . "4

PRESENT METHOD =m am Im m

o | I I I I I I I I I d - JAN FEB MAR APR M A y JUN JUL AUG SEP OCT NOV DEC

MONTHS

Fig. 3. Monthly average of the daffy insolation on a surface sloped at 50 ° toward the equator. Comparison of three methods

for Winnipeg.

Technical Note 103

C~

o3

7- <

~I I i I I I I I I I "I I EDMONTON ALBERTA 53 O 551 N

] SLOPE 50.00 DEGREES q

o . , - " " " . o

o

JAN FEB MAR APR MAY JUN JUt AUG SEP OCT NOV DEC MONTHS

Fig. 4. Monthly average of the daily insolation on a surface sloped at 50* toward the equator, Comparison of three methods

for Edmonton.

c~

c~

o I I I I I I I I I I I VANCOUVER B.C, 49 ° 27' N

SLOPE 50.00 DEGREES o

......................... ... -

o

o

LIU AND JORDAN [I ] o

HAY b41 . . . . . . . . . . . . . . . . . ~ "

PRESENT METHO0 '== ~ === ===

I I I I I I I I I I (5 JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC

MONTHS

Fig. 5. Monthly average of the daily insolation on a surface sloped at 50 ° toward the equator. Comparison of three methods

for Vancouver.

procedure and from the present method are in general quite close to each other, specially for Winnipeg and Edmonton. Yearly averages o f /~r from Figs. 2-5 are given in Table 2. It is evident that there is little difference between the yearly averages produced by different methods. Extent of the influence of seasonal variations would depend upon a particular application where the results are employed. In this study a solar heating system is examined with insolation inputs from various methods.

Modelling of solar heating of homes has been actively studied since the recent "energy crisis". Among many worthy attempts is that of Klein et al.[6]. In the present report, their design pro- cedure has been utilized. Primary reason for using Klein et al's design procedure is that it employs monthly average values of total insolation on inclined collectors--a quantity readily ~com- puled from different sources discussed above. In addition, Klein et al's procedure is also expected to gain widespread use in future. Therefore, its use now is only logical.

A house of 150m 2 floor area, with insulation conforming approximately to ASHRAE standards 90-75116] is considered. Monthly space heating loads are determined from the local degree-day distribution. A constant service hot water Mad of 43 MJ per day is assumed. A south facing collector system with the following specifications is employed

FR(~) = 0.674

FR UL = 11.28 kJ hr- t °C-1 m-2 F~/FR = 0.979

slope = 50 °.

Monthly fraction of load supplied by the solar syste m was obtained from/-charts in [61. Yearly fraction of load supl~lied by the solar system was computed from the monthly fr~ctions. Variation of the yearly fractions with collector area are given in Figs. 6-9. It is apparent from these plots that at sunnier locations (Edmonton and Winnipeg), the results from the percentage sun- shine relation (1) are identical to those from Hay's taltles[14]. For other stations, the present method gives results which are slightly on the conservative side. However, they are, in general, closer to Hay's results than those obtained from Liu and ~ordan's correlation. Maximum differences between the three i results depends upon the collector area. However, these differdnces do not appear to be large, for small collector areas.

Further calculations carried out, but not presented here, at collector inclination of 70 ° substantiate the above mentioned pattern of results.

Assuming Hay's correlation[4,5] and his subsequently pre- pared tables[14] are accurate, it is evident that the simPle cor- relation proposed here gives fairly accurate (although sSmewhat conservative) results as far as their use in solar heating Of homes in northern climates is concerned.

It is known that solar terrestrial radiation estimation[ methods based on number of hours of bright sunshine generally g~ve crude results. In this light, it is somewhat puzzling to note that Hay's rather sophisticated procedure yields results very close to those obtained by the simple method proposed in this report.

Acknowledgemems--Financial support of the Nationall Research Council of Canada is gratefully acknowledged. Digital com- putations were carried out by Mr. Lance MacLeod. Thanks are due to Dr. J. E. Hay, for supplying albedo values fie used in computation of tables [14].

Table 2. Yearly average of the daily total radiation on a surface inclined at 50 ° toward the equator,/iv, MJ m -2 day -~

Montreal Winnipeg Edmonton Vancouver Method 45 ° 50'N 49 ° 90'N 53 ° 55'N 49 ° 27'N

Liu and Jordan [1] 14.26 17.23 16.83 13.24 Hay[14] 13.49 15.63 15.06 13.07 Present 12.87 15.34 14.97 12.19

104 Technical Note

>-

w Z 1.1.1

<

m

if)

<

< w

0 V---

e~ u .

¢.0 c:;

,q.

d

¢4 6

q o 0 .0

I I I I MONTREAL QUEBEC 45 e 50 " N

SLOPE 5 0 . 0 0 DEGREES

- - ~k ~ " HAY ['14] . . . . . . . . . . . . . . . . .

PRESENT M E T H O D === a m a m . m e

r I i ~ i 20 .0 40.0 60.0 80 .0 100.0

2 C O L L E C T O R A R E A ( M )

Fig. 6. Fraction of yearly load supplied by solar energy vs collector area, Comparison of insolation inputs from three

methods for Montreal.

> .

n-

Z

¢ r ,<

> - m

~3 w

o..

co

£3 <

O . . J

< t.u > .

o

d

6

c/

o ,O

0.0

i

I I I I EDMONTON A L B E R T A 53 ° 55 " N

SLOPE 50 .00 DEGREES

m

, I F - PRESENT M E T H O D . . . . . . . . . . . . . . . . .

' I I I I 20.0 40.0 6 0 . 0 8 0 . 0 100 .0

2 C O L L E C T O R AREA ( M ~,

Fig. 8. Fraction of yearly load supplied by solar energy vs collector area, Comparison of insolation inputs from three

methods for Edmonton.

" I I I I N I N N I P E G M A N I T O B A 4 9 ° 901 N

SLOPE 5 0 , 0 0 D E G R E E S C~ W LU

E <

en

W m

S d / . 0 . 0 o . _

w 6 °

- - ~ - HAY h 4 ] • . . . . . . . . . . . . . . . . . z c~ O I

I...- ~ P R E S E N T METHOD- ~ I = ~ i {.3 / <

o I I I I o

0.0 20 .0 4 0 . 0 6 0 . 0 8 0 . 0 1 0 0 . 0

C O L L E C T O R A R E A ( M 2 )

Fig. 7. Fraction of yearly load supplied by solar energy vs collector area, Comparison of insolation from three methods for

Winnipeg.

" I I I I >" VANCOUVER B.C. 4 9 e 2 7 ' N n- w SLOPE 5 0 . 0 0 DEGREES Z

)- m

n

if)

q - -

o

<

) -

,<

Z

O ,¢

o, o 0.0

i

....;;.#;;.;-;;;" -

- f H ~ [142 . . . . . . . . . . . . . . . .

P R E S E N T METHOD am= =m a m m m

r I I I I 20.0 40.0 50.0 80.0 1oo.o

2 C O L L E C T O R A R E A ( M )

Fig.9. Fraction of yearly load supplied by solar energy vs col- lector area. Comparison of insolation inputs from three methods

for Vancouver.

Technical Note 105

NOMENCLATURE

D daily diffuse radiation on a horizontal surface, kJm -2 day-I

/5 monthly average of the daily diffuse radiation on a horizontal surface, kJm -2 day -1

FR collector heat removal factor F~ collector-heat exchanger efficiency factor /-/ daily total radiation on a horizontal surface, kJm-2 day -1

monthly average of the daily total radiation on a horizontal surface, kJm -2 day -~

Ho daily extraterrestrial radiation on a horizontal surface, kJm -2 day-I

/~0 monthly average of the daily extraterrestrial radiation on a horizontal surface, kJm -2 day -~

Hr monthly average of the daily total radiation on an in- clined plane, kJm -2 day -l

KT HIH o, ratio of the daily total radiation on a horizontal surface to the extraterrestrial radiation

/~r ratio of the monthly average of the daily total radiation on a horizontal surface to the extraterrestrial radiation

~i monthly average of the hours of bright sunshine per day, hr

N day-length, hr /Q monthly average of the day-length, hr

UL collector overall energy loss coefficient, kJ hr -~ m-2*C -I {Td~ weighted monthly average of the transmittance-absorp-

tance product.

~ N C E S

1. B. Y. H. Liu and R. C. Jordan, The inter-relationship and characteristic distribution of direct, diffuse and total solar radiation. Solar Energy 4(3), l (1960).

2. D. W. Ruth and R. C. Chan, The relationship of diffuse radiation to total radiation in Canada. Solar Energy 18(2), 153 (1976).

3. J. E. Orgill and K. G. T. Hollands, Correlation equation for the hourly diffuse radiation on a horizontal surface. Sharing the Sun, Joint Con[. of the ISES and SESC 1, pp. 298. Winnipeg (1976).

4. J. E. Hay, A revised method for determining the direct and diffuse components of the total shortwave radiation. Atmosphere, Qu. Publication of the Canadian Meteorological Society 14(4), 278 (1976).

5. J. E. Hay, Climatologically related problems in determining the potential for solar energy utilization in Canada. Proc. WMOIUNESCO Symposium on Solar Energy. Geneva (1976).

6. S. A. Klein, M.A. Beckman and J. A. Duffie, A design procedure for solar heating systems. Solar Energy 18(2), 113 (1976).

7. S. Fritz, Solar radiation energy and its modification by ~he earth and its atmosphere. Compendium of Meteorology, American Meteorological Society (I 951 ).

8. J. K. Page, The estimation of monthly mean values of daily total shortwave radiation on vertical and inclined surfaces from sunshine records for latitudes 40"N--40°S. Proc. UN Con[. in New Sources of Energy 4, 378 (1964).

9. D. J. Norris, Correlation of solar radiation with clouds. Solar Energy 12, 107 (1%8).

10. G. E. Smith, Solar radiation data base development based on bright sunshine data. Sharing the Sun Joint Con[erence of the ISES and SESC. 1,226. Winnipeg (1976).

11. B. Y. H. Liu and R. C. Jordan, Availability of solar energy for flat-plate heat collectors, In Low Temperature Engineer- ing Applications of Solar Energy, ASHRAE. New YOrk, N.Y. (1%7).

12. B. Y. H. Liu and R. C. Jordan, Daily insolation on surfaces tilted toward the equator, Trans. ASHRAE, pp. 526-541 (1%2).

13. Atmospheric Environment Service: Climate Normals, Sun- shine, Cloud, Pressure and Thunderstorms 3, 24. Toronto (1%8).

14. Canada's Renewable Energy Resources--An assessmenit of Potential, p. 514. Middleton Associates, Toronto (1976).

15. J. E. Hay, Private communication (1976). 16. American Society of Heating, Refrigeration and Air Con-

ditioning Engineers, Standards 90-75, Energy Conservation in New Building Design, New York, N.Y. (1975).