Solar Enemy, Vol. 21, pp. 485-489 0038-092X/78/1201--0485/$02.00/0 © Pergamon Press Ltd., 1978. Printed in Great Britain

HOURLY VS DAILY METHOD OF COMPUTING INSOLATION ON INCLINED SURFACES

M. IQBAL

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

(Received 4 December 1977; revision accepted 21 Jane 1978)

Abaract--Insolation on south-facing inclined planes has been computed using hourly values of total and diffuse radiation, obtained from experimental data. Such a computation procedure is then compared with the widely used method by Liu and Jordan for obtaining daily insolation on surfaces tilted toward the equator. Very small differences are noted between the results obtained by the two methods. These differences are mainly due to three factors; (a) Liu and Jordan's formulation uses a theoretical day-length while the hourly method uses day-length as incicated by the radiation data; (b) hourly method takes into account the asymmetries of total and diffuse radiation around solar noon while the daily method implicitly assumes symmetry of the same, (c) the daily method assumes uniform atmospheric transmissivity to beam radiation throughout the day. On the other hand, the hourly method assumes constant atmospheric transmissivity for one hour only.

INTIIOIXJCTION

Design of a flat-plate solar collector requires detailed information on the hourly diffuse and beam radiation incident on its surface. On the other hand, design of a solar system, e.g. a heating system, also requires the total value of insolation on the collectors [1].

Data of insolation on inclined planes are extremely rare. The data generally available are those of daily total radiation on horizontal surfaces. Therefore, estimation methods are required to compute both hourly and daily values of insolation on sloped surfaces.

To compute daily insolation on an inclined surface, one first of all has to separate the diffuse component from the total horizontal radiation. A number of in- vestigators have presented such separation techniques [2, 3]. The most widely used method is the one by Liu and Jordan[4]. Liu and Jordan have presented two cor- relations between diffuse and total radiation on horizon- tal surfaces. One correlation is for daily values and the other for long-term monthly averages of daily values.

In another article, Liu and Jordan[5] have presented a method to compute daily insolation on surfaces tilted toward the equator. Klein[6] gave a review of [2, 4 and 5] and extended the method of calculation of insolation on inclined planes applicable to surfaces of a wide range of orientation.

A second approach to the computation of insolation on tilted surfaces is through the use of hourly values. That is, assuming hourly values of diffuse and total radiation on a horizontal surface are either known or can be estimated [3, 7,8], then these quantities can he trans- posed to give hourly insolation on inclined planes. Summation of such hourly values over a period of a day should give insolation on the sloped surface for the day.

Hottei and Woertz[9], Duffle and Beckman[10], Kondratyev [ l l, 12] and Heywood[13,14] have given various aspects of computing instantaneous insolation on inclined planes. By assuming an hour is a short enough period, methods in [9, 10] are widely used to obtain hourly insolation on sloped surfaces.

In this report, comparison has been carried out be- tween the two methods of computation, daily vs hourly. To avoid difficulties associated with estimation of hourly or daily diffuse radiation on horizontal surfaces, actual iongterm average data of three widely spread Canadian locations are employed (Table 1).

In the next section, mathematical formulations of the two methods are presented.

M A ~ T I C A L Ia~,MULATiONS

In order to place the two calculation methods, daily and hourly, in their true perspective, it is essential to repeat the mathematical formulations available in the literature cited earlier.

Monthly average daily insolation on inclined planes can be computed from the following expression given by Liu and Jordan [5],

f l B = ( f l - - l ~ l d ) R b + l ~ l d ~ +:Ipl-c°s2 ~ (I)

Equation (1) assumes that the sky diffuse radiation and the ground reflected diffuse radiation are both isotropic.

is the conversion factor for the beam radiation. It is the ratio of the beam radiation incident upon the tilted surface during a day to the beam radiation incident upon a horizontal surface during the same period. Equation (1) can be approximated by assuming J~, as the ratio of the

Table 1. Three Canadian stations used in this study with regular hourly measurements of diffuse and total solar radiation on a horizontal surface

Station Longitude W Latitude N Record Used

Montreal 73037 ' 45030 ' Oct. 1964-I~ec. 1965 Toronto, MRS 79033 ' 43o48 ' Aug. 1967-Dec. 1975 Goose Bay 60027 ' 53°18 ' May 1962-Dec. 1975

485

486 M. IQBAL

daily extraterrestrial irradiation on a tilted surface to that on a horizontal surface.

For surfaces facing directly toward the equator,

Rb = cos (~k - p ) cos a sin a,, + (1r1180)co" sin (4' - p ) sin 8 cos ~b cos 8 sin ~o" + (¢r/180)oJ, sin ~b sin 8

(2a)

where ~o" is the sunset hour angle for the tilted surface which is given by

~o" = min{o,,, arcos [o,, - tan (4~ - P) tan a]}. (2b)

Equations (2a) and (2b) are in fact applicable to parti- cular days. When dealing with monthly averages, declina- tion could be chosen for an average day of the month. Klein[6] has given a table of recommended average days for each month at which declinations in eqns (2) should be computed•

Liu and Jordan[5] in their eqns (15) and (16) have pointed out that except for equinoxes, Rb, obtained through (2) is correct only when there is no hourly variation of the atmospheric transmissivity during the day.

Monthly average hourly insolation on sloped surfaces can be computed from the expression [11],

1+cos# -- 1 - cos p (3) fa = ( / - [a)rb +/a 2 +1 , , 2

Similar to eqn (1), eqn (3) is valid when sky diffuse radiation and the ground reflected diffuse radiation are both isotropic, rb is the conversion factor for hourly ,~ 4 beam radiation. Considering no change of atmospheric " transmission over an hour, rb is the ratio of the hourly extraterrestrial insolation on a tilted surface to that on a ~ o horizontal surface.

For surfaces facing directly towards the equator, Fig•

n~.JLTS

Insolation on inclined planes was computed using eqns (1) and (5). Only south facing surfaces were considered• Results of monthly total insolation were obtained for four slopes; /3 = 30", 50", 70* and 90*. All stations and slopes exhibited some difference between results obtained by the two methods. Differences were very minimal for slopes of 30*. For inclinations between 50* and 90", the differences were consistent, although they varied slightly with the station, slope and period of the year.

Figures i and 2 present results for 50* and 90 ° slopes for Montreal. For each slope, differences between the results are negligible. There does not appear to be any seasonal factor governing these differences. For some

- ~ 2 8

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20 - e,e - s

O 1 6

Z

Z 12

Z 0

8 Z g

I ! I l I I I I I I

M O N T R E A L 45030 , N SLOPE-50 .0 DEGeEES

DALLY BASIS . . . . HOURLY BASIS

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

M O N T H

1. Monthly variation of total radiation on an inclined plane.

cos (~ - p) cos a cos ~o + sin (~ - p) sin a rb = cos 4' cos 6 cos ~o + sin 4~ sin 8 (4)

2 8

'E

Equation (4) is in fact valid for any instant. However, considering one hour as a short enough period, rb is evaluated at mid-point of the hour angle ~o.

In dealing with the monthly average hourly values, there is some question whether the value of declination ..q in (4) should be the same as in (2)• This question can be z 16 resolved by determining the average day for each month and each hour applicable to the average hourly extrater- z restrial radiation. This means that in (4), the average day z la would be a function of the month as well as the hour in question. In this study, identical values of declination s were used in (2) and (4). z

From the hourly values I,, a daily summation can be obtained as, .,

Comparisons of results obtained from eqn (1) and (5) are presented in the next section.

I I 1 I I I 1 I I I

M O N T R E A L 4 s ~ o ' N SLOPE-90 .0 DEGREES

m a

DAILY BASIS . . . . . HOURLY BASIS

o o A N I , I I I I I I I , FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC

M O N T H

Fig. 2. Monthly variation of total radiation on an inclined plane.

Hourly vs daily method of computing insolation on inclined surfaces 487

months, the hourly basis value is higher than the daily basis value, and for other months, the reverse is true.

Figures 3 and 4 contain insolation values for Toronto for 50 ° and 900 slopes respectively. Comments made above with respect to Montreal also apply in this case.

Plots for Goose Bay are shown in Figs. 5 and 6. For this station, the differences between the two slope radia- tions are negligible when compared with the correspond- ing results for Montreal and Toronto. However, the general pattern of results remains the same as for the other two stations.

The correspondence of the results obtained by the two approaches is very good. The discrepancies whenever they arise are attributable to the following factors: (1) The daily method approach assumes a theoretical day- length obtained from the sunset hour angle co,. On the

-'-" 28

,., T O R O N T O 43"4B' N ~ : SLOPE - 50.0 0EGRets

24

16

12

8

_(2

5 4 DALLY BASIS . . . . . HOURLY BASIS

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

M O N T H Fig. 3. Monthly variation of total radiation on an inclined plane.

A 28

~ 24

i 20

Z 1 6

, ( 4 og

I I I I I l I I I I T O R O N T O 43°48 ' N S L O P E - 9 0 . 0 oeoeees

DALLY BASIS . . . . . HOURLY BASIS

o 2. ML, J . ' , L i JUL AUG see oc; NOV Oe¢

M O N T H Fig. 4. Monthly variation of total radiation on an inclined plane.

m , t

Z m

U

Z

Z <

Z O

Z _o

O

Fig.

28 I I I I I I I I I I

GOOSE BAY s3°m ' N SLOPE- 50.0 oeoeees

24

20

12

8 --

4 - - ~ DAILY BASIS . . . . . HOURLY BASIS

o I [ I I I I t I I t JAN FEB MAR APR MAY JUN JUt AUG $EP OCT NOV DEC

M O N T H

5. Monthly variation of total radiation on an inclined plane.

A

v

Z w , i r a

U

Z w

Z <

Z O

Z o_

Fig.

28

24

20

16

I I I f I I I I I I G O O S E BAY 53018 . N

SLOPE - 9O.O oeoeees

4 DALLY BASIS . . . . . HOURLY BASIS ~ ~ I

O N F~R l A~" "Lv . L , ! . , ! ~ ~ ! l ~ I , I J JAN EB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC

M O N T H

6. Monthly variation of total radiation on an inclined plane.

other hand, the day-length used in (5) is obtained from the actual hourly data during which solar radiation is recorded. The recorded day-length is slightly longer than the theoretical day-length due to pre-dawn and post- sunset diffuse radiation. This difference is one to two hours in the case of the three stations examined. The energy collected during very early morning and very late evening sunshine is, however, very small and cannot be the major reason for the small discrepancies. (2) The daily method implicitly assumes symmetry of diffuse and total radiation around solar noon as represented by equal sunrise and sunset hour.angles. On the other hand, examination of the actual hourly data showed asym- metries around solar noon. The hourly method of computing slope radiation takes into account these

488

asymmetries through eqns (3)--(5). Therefore asym- metries of the actual data appears to be a more important reason for the differences between the two results. (3) In calculating slope radiation through the daily method, the conversion factor J~ in eqn (1) is for the surface of the earth. As the hourly variation of the atmospheric trans- missivity to beam radiation during a day is not known, this factor is approximated by evaluating it in the ab- sence of the earth's atmosphere. However, as shown by Lin and Jordan[5], at equinoxes (8--0) , J~ obtained through eqn (2a) is exact.

In calculating slope radiation through the hourly method, the conversion factor rb should also be for the surface of the earth. However, as the atmospheric transmissivity may be considered constant over the period of an hour, rb expressed through eqn (4) is a very good approximation.

From the foregoing it may be concluded that if the day length differences and the asymmetries referred to in points I and 2 above are not strong reasons for the discrepancies between the two methods, then at equinoxes the two methods should give almost identical results. This can be verified by an examination of all the six diagrams, Figs. 1-6. For the months of March-April and September-October, the correspondence between the daily and the hourly methods is indeed very close.

Therefore, it may be concluded that variation of the atmospheric transmissivity during a day is the most important reason for the small discrepancies between the results obtained by the two methods.

The parameter rb, eqn (4), was computed at mid-hour. It was thought that the results might improve if the numerator and the denominator in eqn (4) were in- tegrated over one hour giving a new factor ?b defined as,

f '2[cos (4, - /~) cos 8 cos o, + sin (4' - /3) sin 8] do, /.b = I

f~ 2[cos ¢~ cos cos o, ¢~ 8] 8 + sin sin do,

(5)

The factors rb and ?b were computed and a negligible change was noticed in the results.

From the above it may be concluded that the slight differences between insolation on inclined planes computed by the daily and hourly methods are due to three reasons; (a) the differences between the theoretical day-length and the one recorded by radiation data; (b) the asymmetries around solar noon of the hourly diffuse and total radiation. (c) the variation of the atmospheric transmissivity during the day.

Finally, the hourly method of computing insolation on inclined planes should give slightly more accurate results than those obtained by the daily method. An experimen- tal verification of this study would be useful.

It has to be borne in mind, however, that the hourly data, particularly the diffuse one, is extremely rare. On the other hand, average daily total radiation on horizon- tal surfaces is widely available. In such a situation, the daily method developed by Lin and Jordan[5] producing

M. loe~

results very close to those obtained by the hourly method gives confidence in the continuous use of their formulation.

In this report, monthly average values of the insolation were used. However, the procedure is equally applicable to particular days.

Acknowledgements--Financial support of the National Research Council of Canada is gratefully acknowledged. Numerical computations were performed by Mr. C. Y. K. Lau.

~ C L A T U R l g /7 monthly average daily total radiation received on a

horizontal surface /~d monthly average daily diffuse radiation received on a

horizontal surface Ho monthly average daily total radiation received on a surface

inclined at/3 deg. to the ground /40 extraterrestrial monthly average daily insolation on a

horizontal surface. [ monthly average hourly total radiation received on a

horizontal surface [d monthly average hourly diffuse radiation received on a

horizontal surface [# monthly average hourly total radiation received on an in-

clined surface rb hourly ratio of the extraterrestrial radiation on a tilted

surface to that on a horizontal surface for the month. Rb daily ratio of the extraterrestrial radiation on a tilted sur-

face to that on a horizontal surface for the month

Greek symbols /3 surface tilt from the ground, deg. 8 declination, deg.

latitude, deg. ,o hour angle, deg.

~o, sunset hour angle for a horizontal surface, deg. co', sunset hour angle for a tilted surface, deg. p albedo

~ c g s

I. s. A. Klein, W. A. Beckman and J. A. Duttie, A design procedure for solar heating system analysis. Solar Energy t8(2), 113--127 (1975).

2. J. K. Page, The estimation of monthly mean values of daily total short-wave radiation on vertical and inclined surfaces from sunshine records for latitudes 400N-40°S. Proc. UN Conf. on New Sources of Energy, Paper No. S/98 (1961).

3. M. lqbal, Estimation of the monthly average of the diffuse component of total insolation on a horizontal surface. Solar Energy 20(1), 101-105 (1978).

4. B. Y. H. Liu and R. C. Jordan, The interrelationship and characteristic distribution of direct, diffuse, and total solar radiation. Solar Energy 4(3), !-19 (1960).

5. B. Y. H. Liu and R. C. Jordan, Daily insolation on surfaces tilted toward the equator. Trans. ASHRAE 526-541 (1962).

6. S. A. Klein, Calculation of monthly average insolation on tilted surfaces. Solar Energy 19(4), 325-329 (1977).

7. J. F. Orglll and K. (3. T. Hoilands, Correlation equation for hourly diffuse radiation on a horizontal surface. Solar Energy 19(4), 357-359 (1977).

8. J. E. Hay, A revised method for determining the direct and diffuse components of the total short-wave radiation. Atmos. 14(4), 278-287 (1976).

9. H. C. Hottel and B. B. Woertz, performance of flat-plate solar-heat collectors. Trans. ASME 64, 91 (1962).

10. J. A. Duffle and W. A. Beckman, Solar Energy Thermal Processes. Wiley, New York (1974).

Hourly vs daily method of computing insolation on inclined surfaces 489

1 I. K. Y. Kondratyev, Radiation in the Atmosphere. Academic Press, New York (1969).

12. K. Y. Kondratyev and M, P. Manolova, The radiation balance of slopes. Solar Energy 4(I), 14-19 (1960).

13. H. Heywood, The computation of solar radiation intensities,

Part 2, Solar radiation on inclined planes. Solar Energy 1O(I), 46-52 (1966).

14. H. Heywood, A general equation for calculating total radia- tion on inclined surfaces. Proc. ISES Cony. Paper No. 3/21 (1970).

Rc~um~-La pr~sente m~thode de calcul de l'insolafion sur des surfaces planes, inclines vers le sud, a pour fondement les donn~es horaires du rayonnement global et diffus, donn~es bas~s sur I'observation exp~rimentale. Nous raisons la comparalson de cette m~thode de calcul avec celle bien connue de Liu et Jordan pour obtenir I'insolation quotidienne sur des surfaces pench~es dans la direction de r~quateur. Les r~sultats obtenus selon les deux m~thodes varient & peine et cette variation provient avant tout de trois facteurs: I. La formulation de Liu et Jordan utilise une longueur de jour th~orique tandis que la m~thode de calcul horaire d~termine la Iongneur du jour en se basant sur l'actuel rayonnement. 2. La m~thode de calcul horaire tient compte de I'asym~trie du rayonnement aux environs de midi, heure solaire, alors que la m~thode de Liu et Jordan presume sa sym~trie. 3. La m~thode de Liu et Jordan suppose, ~ la difference de la n6tre, une valeur constante de la transmissivit~ atmosph6rique pour le rayonnement direct durant toute la journ~e.