The Interrelationship and of Direct, Diffuse and ... - Science Direct

The Interrelationship and Characteristic Distribution of Direct, Diffuse and Total Solar Radiation* By Benjamin Y. H. Liu$ and Richard C. Jordan~ Based upon the data now available, this paper presents relationships permitting the determination on a horizontal surface of the instantaneous i n t e n s i t y o f d i f f u s e r a d i a t i o n o n clear d a y s , t h e l o n g t e r m a v e r a g e h o u r l y a n d d a i l y s u m s o f diffuse radiation, and the daily sums of diffuse radiation for v a r i o u s c a t e g o r i e s o f d a y s o f differing d e g r e e s o f c l o u d i n e s s . F o r t h e s e d e t e r m i n a t i o n s , it is necessary to have, either from actual measurem e n t s or e s t i m a t e s , a k n o w l e d g e o f t h e t o t a l (direct p l u s diffuse) r a d i a t i o n o n a h o r i z o n t a l s u r f a c e - - i t s m e a s u r e m e n t is n o w r e g u l a r l y m a d e at 98 l o c a l i t i e s i n t h e U n i t e d S t a t e s a n d C a n a d a . For localities where only an estimate of the long t e r m a v e r a g e t o t a l r a d i a t i o n is a v a i l a b l e , r e l a t i o n s h i p s p r e s e n t e d i n t h i s p a p e r c a n be u t i l i z e d t o determine the statistical distribution of the daily t o t a l r a d i a t i o n at t h e s e l o c a l i t i e s . NOMENCLATURE D and/)

f

H and 17

Ho

I~., ID,~

= daily and m o n t h l y average daily diffuse radiation received on a horizontal surface, B t u / d a y - s q ft = fractional time during which the daily total radiation received on a horizontal surface is less t h a n or equal to a certain value, dimensionless = daily and m o n t h l y average daily total (direct plus diffuse) radiation received on a horizontal surface, B t u / d a y - s q ft = extraterrestrial daily insolation received on a horizontal surface, B t u / day-sq ft = intensity of direct radiation incident upon a horizontal surface, B t u / h r - s q ft = intensity of direct, radiation at normal incidence, B t u / h r - s q ft

* This paper is in part the result of researches sponsored bv a grant from the National Science Foundation, Washingtmi, 1). C. Portions of the material presented in this paper are drawn from the thesis of Benjamin Y. H. Liu prepared in partial fulfillment of the requirements of the degree of 1)octor of Philosophy. ? Assistant Professor, Department of Mechanical Engineering, University of Minnesota. :~ Professor and Head, Department of Mechanical Engineering, University of Minnesota.

I~h

= intensity of diffuse radiation on a horizontal surface, B t u / h r - s q ft ]ah = long term average of the hourly diffuse radiation received on a horizontal surface = long term hourly average of the intensity of diffuse radiation on a horizontal surface, B t u / h r - s q ft lob = intensity of solar radiation incident upon a horizontal surface outside the atmosphere of the earth, B t u / h r - s q ft Io,~ = intensity of solar radiation at normal incidence outside the atmosphere of the earth = r I . , , B t u / h r - s q ft I~ = solar c o n s t a n t = 442 B t u / h r - s q ft = 2 ly/min I rh = intensity of total (direct plus diffuse) radiation incident upon a horizontal surface, B t u / h r - s q ft = long term average of the hourly total radiation received on a horizontal surface = long term hourly average of the intensity of total radiation on a horizontal surface, B t u / h r - s q ft Kd a n d / ~ ' , = D / H o and f ) / H o , dimensionless K~ = ( H -- D ) / H , , , dimensionless K r a n d / ~ ' r = H / H o and f i l / H o , dimensionless L = l'aitude, degrees m = air mass = cscc~ except at low altitude, dimensionless = ratio of solar radiation intensity at normal incidence outside the atmosphere of the earth to solar constant, dimensionless = I,+h/D = ratio of hourly to daily difr~ fuse radiation, dimensionless = ratio of hourly to daily total radiation, rr dimensionless = solar altitude angle, degrees a = solar declination, degrees ~i rD = I v . / I o , , = l v h / I o , = transmission coefficient for direct solar radiation, dimensionless = l,~h/l,,~ = transmission coefficient for 7"d

TABLE 1.--Solar Declination, ~, and the ratio,

r,

of Solar Radiation Intensity at Normal Incidence Outside Earth's Atmosphere to Solar Constant Day of Month

Month

8

15

22 r

January . . . . . . . . . . . . . . . . . . February . . . . . . . . . . . . . . . . . March . . . . . . . . . . . . . . . . . . . . April . . . . . . . . . . . . . . . . . . . . . May . . . . . . . . . . . . . . . . . . . . . . June . . . . . . . . . . . . . . . . . . . . . . July . . . . . . . . . . . . . . . . . . . . . . August . . . . . . . . . . . . . . . . . . . September . . . . . . . . . . . . . . . . October . . . . . . . . . . . . . . . . . . November . . . . . . . . . . . . . . . December . . . . . . . . . . . . . . .

rr

¢0 ¢0~

--23°04 ' --17o19 ' --7053 ' 4o15 ' 14°51 ' 21°57 ' 23010 ' 18013 ' 8034 ' --2054 , _ 14Oll , -21°41 '

1.0335 1.0288 1.0173 1.0009 0.9841 0,9714 0,9666 0.9709 0.9828 0.9995 1.0164 1.0288

--22°21 ' -- 15o14' --5Oll ' 6055' 16053 ' 22°47 ' 22034 ' 16°22 ' 5°59 ' -5036 ' --16021 , -22°39 '

diffuse r a d i a t i o n on a h o r i z o n t a l surface, dimensionless = I rh/Io~,, = t r a n s m i s s i o n coefficient for t o t a l r a d i a t i o n on a h o r i z o n t a l surface, dimensionless = h o u r angle, degrees = s u n s e t h o u r angle, r a d i a n s INTRODUCTION

T h e increase in r e c e n t y e a r s of t h e p r o b l e m s w i t h w h i c h solar r a d i a t i o n is i n v o l v e d has m a d e solar r a d i a t i o n i n f o r m a t i o n f r e q u e n t l y n e e d e d b y w o r k e r s in m a n y fields. A l t h o u g h solar r a d i a t i o n d a t a are now a v a i l a b l e for m a n y localities, difficulties are s o m e t i m e s e n c o u n t e r e d in utilizing these d a t a since t h e y consist p r i m a r i l y of t o t a l ( d i r e c t plus diffuse) r a d i a t i o n o n l y a n d a k n o w l e d g e of t h e diffuse c o m p o n e n t is often required. Since t h e t h e o r e t i c a l c o m p u t a t i o n of diffuse r a d i a t i o n is e x t r e m e l y difficult if n o t i m p o s s i b l e a t t h e p r e s e n t t i m e , a n a t t e m p t is m a d e in t h i s s t u d y to investigate, from the limited data now available, the r e l a t i o n s h i p s b e t w e e n diffuse a n d t o t a l r a d i a t i o n in o r d e r t h a t t h e y m a y be utilized for t h e e s t i m a t i o n of diffuse r a d i a t i o n for localities w h e r e o n l y t h e t o t a l r a d i a t i o n is k n o w n . T h i s was first a t t e m p t e d b y P a r melee ~ for cloudless d a y s only, b u t no extension was m a d e to c l o u d y d a y s . I n t h i s i n v e s t i g a t i o n it h a s also b e e n f o u n d n e c e s s a r y to s t u d y t h e c h a r a c t e r i s t i c d i s t r i b u t i o n of t o t a l r a d i a tion. T h e results o b t a i n e d serve to i n d i c a t e t h e t y p e s of p a r a m e t e r s to be used in t h e i n v e s t i g a t i o n of o t h e r s t a t i s t i c a l c h a r a c t e r i s t i c s of solar r a d i a t i o n should t h e need for t h e s e c h a r a c t e r i s t i c s arise. T h e p r e s e n t s t u d y is r e s t r i c t e d to r a d i a t i o n on a h o r i z o n t a l surface only. Solar Constant

Since a definite v a l u e for t h e solar c o n s t a n t m u s t be used c o n s i s t e n t l y t h r o u g h o u t this s t u d y , t h e m o s t

1.0325 1.0263 1.0140 0.9963 0.9792 0.9692 0.9670 0.9727 0.9862 1.0042 1.0207 1.0305

--21°16 ' -- 12o56' --2026 , 9030 ' 18°41 ' 23°17 ' 21%9' 14°18 ' 3020 ' _8o14 , --18°18 ' --23°14 '

1.0315 1.0235 1.0103 0.9913 0.9757 0.9680 0.9680 0,9757 0.9898 1.0087 1.0238 1.0318

--19o51 ' --10028 ' 0o20 ' 11°56 ' 20°14 ' 23o27 ' 20026 ' 12°03 ' 0o37 ' --10047 ' --19058 , --23027 '

1.0300 1.0207 1.0057 0.9875 0.9727 0.9670 0.9692 0.9785 0.9945 1.0133 1.0267 1.0327

r e c e n t v a l u e of 2.00 l y / m i n , * or 442 B t u / h r - s q ft, given b y J o h n s o n , 2 in t e r m s of t h e S m i t h s o n i a n P y r h e l i o m e t r i c Scale of 1932t shall be a d o p t e d . T h e solar c o n s t a n t is t h e r a t e at, w h i c h solar e n e r g y is i m p i n g i n g u p o n a u n i t surface, n o r m a l to s u n ' s r a y s , in free space, a t t h e e a r t h ' s m e a n d i s t a n c e f r o m t h e sun. It, is in general s l i g h t l y different f r o m t h e solar r a d i a t i o n a t n o r m a l incidence a t t h e o u t e r l i m i t of t h e a t m o s p h e r e d u e to t h e v a r i a t i o n of t h e d i s t a n c e b e t w e e n t h e e a r t h a n d t h e sun. (See T a b l e 1.) Relationships and Diffuse

Between Radiation

the Intensities on Clear Days

of Direct

As solar r a d i a t i o n p e n e t r a t e s t h e a t m o s p h e r e it is d e p l e t e d b y a b s o r p t i o n a n d s c a t t e r i n g . N o t all of t h e s c a t t e r e d r a d i a t i o n is lost, since p a r t of it e v e n t u a l l y a r r i v e s a t t h e surface of t h e e a r t h in t h e f o r m of diffuse r a d i a t i o n . T h e t e r m , diffuse r a d i a t i o n , is used here in t h e c u s t o m a r y w a y to d e n o t e t h i s s h o r t w a v e l e n g t h r a d i a t i o n coming f r o m all p a r t s of t h e sky. I t should be d i s t i n g u i s h e d clearly f r o m t h e a t m o s p h e r i c t h e r m a l r a d i a t i o n which, a l t h o u g h also diffuse in n a t u r e , is of m u c h longer w a v e l e n g t h s . T o f a c i l i t a t e t h e discussion, t h e following d i m e n s i o n less t r a n s m i s s i o n coefficients shall b e defined: rD = t r a n s m i s s i o n coefficient for d i r e c t solar r a d i a t i o n =

I•,,/Io+

=

I.h/Ioh

rd = t r a n s m i s s i o n coefficient for diffuse r a d i a t i o n on a h o r i z o n t a l surface = I a h / I o h T h e s e t r a n s m i s s i o n coefficients a r e f u n c t i o n s of t h e solar a l t i t u d e , a t m o s p h e r i c w a t e r v a p o r c o n t e n t , d u s t c o n t e n t , o z o n e c o n t e n t , a n d a n y o t h e r r a d i a t i o n de* 1.0 langley/rain = 1,0 gm cal/min-sq cm ~ 3.687 Btu/minsq ft -- 69.7 milliwatts/sq cm. t The 1932 Pyrheliometric Scale of the Smithsonian Institution is 2.5% below the scale of 1913, 1.0% above the Angstrom scale and 0.5% below the recently proposed International Pyrheliometric Scale.a

pleting factors. However, in a nonindustrial locality where the atmosphere is relatively clean and the effect of dust small, the daily variation of these transmission coefficients for the sun at a fixed altitude, is primarily due to the variation of the atmospheric water vapor. Thus as the atmospheric water vapor content varies from d a y to day causing both rD and r~ to vary, a functional relationship between ro and r,~ is generated. The four upper curves of Fig. 1 show the theoretical relationships between ro ~nd rj for a cloudless and dust free atmosphere for four air masses, l, 2, 3 and 4 corresponding to solar altitude angles of 90, 30, 19.5 and 14.5 respectively. These relationships can be derived readily from the transmission coefficients computed by Kimball when the assumption is made that the diffuse radiation received on a horizontal surface is half of the solar radiation scattered by the atmospheric constituents. 4 However, due to the fact that Kimball's computations are based upon a zero air mass solar spectrum low in the ultraviolet, the theoretical values of rD and r,t in Fig. I have been reduced by 3 %. Since these theoretical relationships have been derived without considering the effect of dust and are based upon an assumption which is known to be only approximately correct, they should not be expected to represent the correct relationships between r . and rd under actual cloudless sky conditions and are derived here to serve merely as guides in the search of experimental relationships.

The experimental points of Fig. 1 are derived from the measurements made at H u m p Mountain, N o r t h Carolina, by Moore and Abbot 5 whose data appear to be the best available. In computing the experimental transmission coefficients, however, the direct and diffuse radiation intensities from the original data have been reduced by 2.5 % in order that the radiation intensities be expressed in the 1932 Smithsonian Pyrheliometric scale upon which the presently adopted solar constant of 2.00 ly/min is based. The fact that both the theoretical curves and experimental points of Fig. 1 show that the values of Td corresponding to a fixed value of ro depend only very moderately upon the air mass indicates that, for the degree of accuracy here sought, a relationship which is independent of the air mass is adequate. The following equation of a straight line, obtained b y the method of least squares, best fits the experimental points: ra = 0.2710 -- 0.2939ro

[1]

A total of 149 points representing the data of 28 clear days were used in obtaining this equation. The probable error computed by the equation, A[//

~12

Probable error - 0.6745 'l/ n ~

is 0.0052, where ~ is the difference between the experimental value of r,~ and the value of rj given by the straight line at the same value of r . , and n the total number of points used.

0.20

z 0 0

n,," ,.= 0.16 W

(/3

I--(

',

I.L

.

t:3

','

0.12

r~

0.08

,-2~it. TM ~ O

O N

12) 7 "I" 0 . 0 4 O

~z IE

7 ne

~-

o

0 0.40

0.44 0.48 0.52 TRANSMISSION COEFFICIENT

0,56 0,60 0,64 0.68 0.72 FOR DIRECT RADIATION, 'I"D= I o n / / I o n = I D h / I o h

0.76

I:]~;. l--Theoretical and experimental relations between the intensities of direct and diffuse radiation oil -~ horizontal surface for a cloudless atmosphere at 4800 ft elevation.

z 0 I-. -

-5

uJ Q.

~j 60C

D 2000

(.9 Z

~N m

Itn2: u.i n-uJ z 1-40

tin

I

G

1600

J

G400

"I-

1200800-

lX hl

200

400-

0

O_

0

I0

20

30 40 50 60 NORTH LATITUDE, L, DEGREES

70

80

90

FIG. 6 - - E x t r a t e r r e s t r i a l daily insolation received on a horizontal surface.

0

6

0.1

RAT I0 KT=

0.2

Ho

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 DAILY TOTAL RADIATION ON A HORIZONTAL SURFACE EXTRATERRESTRIAL DAILY INSOLATION ON A HORIZONTAL SURFACE

m

¢r

FIc. 7 - - T h e ratio of t h e daily diffuse r a d i a t i o n to t h e daily total r a d i a t i o n as a f u n c t i o n of t h e cloudiness index KT.

LI.

CO _1

N II

Ld..J

,~1--

1.0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 DAILY DIRECT RADIATION ON A HORIZONTAL SURFACE RATIO KD= H-D= HO EXTRATERRESTRIAL DAILY INSOLATION ON h HORIZONTAL SURFACE 0

0.1

0.2

I! FIG.

8--Comparison of the relations between the daily direct radiation and the daily diffuse radiation on a horizontal surface on clear and cloudy days.

result would have been obtained had the "correct" average Ho been employed. This was not done in order that Ho for each month might have a definite value. To overcome this difficulty it is recommended that when KT > 0.75 the average ratio 0.16 of D / H for these points be used. The ratio D / H is shown as a function of K r in trig. 7. Since the diffuse and total radiation should be equal for completely overcast days, the ratio D / H should approach the limit one when KT approaches zero. An inconsistency in the data is seen for the points with values of K r near zero. The solid curve is so drawn that the ratio D / H is brought to unity when KT equals zero, and it is recommended that this curve be used for practical application. C o m p a r i s o n o f t h e C l e a r a n d C l o u d y Day R e l a t i o n s h i p s B e t w e e n t h e Daily D i r e c t a n d D i f f u s e Radiation

A comparison between the clear and cloudy day diffuse radiation is shown in Fig. 8 where Kd of Fig. 5 is plotted against KD with K~) = ( H -- D ) / H , , = KT -

gd

[7]

and represents the fraction of the extraterrestrial daily insolation transmitted through the atmosphere as direct radiation. Since Kd and KD are respectively the weighed averages of ~d and To, and since on clear days the relation between r~ and rz) is linear, when Kd and Kz) are substituted for rd and ~-, in Equation [1], an equation is obtained which represents the relationship between the daily direct and diffuse radiation received oil a horizontal surface on clear days. Thus, on clear days, Kd = 0.2710 -- 0.2939Kt~

[8]

which is also plotted in Fig. 8. The effects of clouds on diffuse radiation is clearly seen from Fig. 8 which shows that the value of Ku on cloudy days is higher than the corresponding value of Kd on clear days at the same values of K o . The higher diffuse radiation on cloudy days is obviously due to the additional scattering effects of clouds. Statistical Distribution of the Daily Total Radiation on a Horizontal Surface

In the majority of cases it is more important to have a knowledge of the monthly average of the daily dif-

I!

I!

"4

O v

a:

0

0.'2 0.4 0.6 0.8 FRACTIONAL TIME, f , DURING WHICH DALLY TOTAL RADIATION ~< H FIG. 9--Monthly Kr curves for/~T = 0.3.

fuse radiation, /5. However, in order to compute /3, with the use of the relationship of Fig. 7, the statistical distribution of the daily total radiation must be known. For reasons to be discussed it has been suspected that a correlation possibly exists among the statistical distribution curves of different localities. To test whether this indeed is the case, statistical distribution curves of daily total radiation for widely separated localities have been constructed and compared. Typi-

1.0

cal examples of these comparisons are shown in Figs. 9, 10 and II. Each of the curves of Fig. 9, 10 and 11 have been constructed using five year (primarily 1954-1958) data of daily total radiation from the Weather Bureau publication, Climatological Data, National Summary, and the curves have been so arranged that the values o f / ~ r for curves on the same graph are approximately equal where 10

fir

= I:I/Ho

[9]

A comparison of the different curves in the same figure shows that although the curves are not identical, the differences among them are not large and may be neglected for many practical purposes. The fact that such widely separated localities as Schenectady, New York, and Annette, Alaska, in Fig. 9 and localities with as much difference in elevation as Albuquerque, New Mexico, and Wake Island in Fig. 11 should have almost identical monthly K r curves indicates that such a

and /I, the monthly average of the daily total radiation for the particular month at the particular locality under consideration, and Ho, the extraterrestrial daily insolation, have been obtained from Fig. 6. The statistical distribution curves so obtained will be termed the "monthly K r curves" since they represent the statistical distribution of the quantity K r and are constructed on a monthly basis.

n

i! bhg

O n ¢r

FRACTIONAL TIME, f , DURING WHICH DAILY TOTAL RADIATION ~ H I~IG.

1 0 - - M o n t h l y KT ('IlFVeS f o r A T = 0 . 5 .

ll

1- I

O

II

o trr

~0

0.2 0.4 0.6 0.8 FRACTIONAL TIME, f , DURING WHICH DAILY

TOTAL

RADIATION

~

1.0

H

FIG. ll--Monthly K r curves for/~T = 0.7. correlation between the forms of the monthly K T curves and the values of /~ r need not be restricted to these localities. Thus if a set of monthly K r curves corresponding to different mean values of /~r is obtained, they m a y be used as a close approximation to the actual monthly K r curves and can be utilized to determine the statistical distribution of the daily total radiation when the monthly average daily total radiation is known. A set of such "generalized monthly KT curves" corresponding to /~T of 0.3, 0.4, 0.5, 0.6 and 0.7 has been obtained from the monthly K r curves

constructed with the data of the localities shown in Table 2. These generalized curves are shown in Fig. 12 and Table 3.

Relationship Between Monthly Average Daily Total and Monthly Average Daily Diffuse Radiation T h e generalized monthly K r curves of the preceding section m a y be used in conjunction with the curve in Fig. 7 to compute the monthly average of the daily diffuse radiation as follows. 12

TABLE 2 . - - S t a t i o n s S e l e c t e d for t h e C o n s t r u c t i o n of t h e G e n e r a l i z e d M o n t h l y KT C u r v e s Latitude (North)

Station

Elevation (ft.)

Month

1954-1958 1954-1958 1954-1958

298 310 434 429

•399 .400

404 357

.396 .406

1955-1958 1955--1958 1954-1958 1952, '54, '55, '57 '58 1950-1953 1954-1958

468 994 730 815 902 858

• 498 .500 .500 •498 •499 .502

1954-1958 1954-1958 1954-1958 1954-1958 1954-1958 1951, '52, '54, '55, '58

Aug. May Apr. Oct. Apr.

778 862 963 850 661 825

.6OO •599 •597 •598 .601 .600

1952, 1955-1958 1954-1958 1951, 1955-1958 1950-1953, 1957 1954-1958 1954-1958

July June July Sept. July Dec.

997 1019 998 797 997 635

• 703 .699 • 698 .700 • 700 • 698

1954-1958 1954, '55, '57, '58 1954-1958 1954-1958 1954-1958 1954-1958

110 217 721

Boston, Mass .......................... Cleveland, O ........................... Indianapolis, Ind ...................... Oak Ridge, Tenn .......................

42o22 ' 41030 ' 39o44 ' 36o01 '

15 787 793 905

Nov. Jan.

Put-in-Bay, O ........................ State-College, Pa .....................

41o39 ' 40o48 '

575 1175

Jan.

Atlanta, Ga ........................... Blue Hill, Mass ....................... N e w p o r t , R. I . . . . . . . . . . . . . . . . . . . . . . . . Ottawa, Ont .......................... San Antonio, Tex ..................... Seattle, Wash

33o39 ' 42o13 ' 41029 ' 45020 ' 29°32 ' 47036 '

975 629 60

29o44' 46046 ' 41o30 ' 40o15 ' 32o01 ' 44o09 '

13 1650 787 8389 2854 3165

Mar.

35003 ' 39006 ' 36o04 ' 33o58 ' 34o56 ' 19o18 '

5310 4849 2162 1050 238 11

.

.

.

.

.

.

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.

.

.

.

.

.

.

.

.

.

.

.

Apalachicola, Fla B i s m a r c k , N. D . . . . . . . . . . . . . . . . . . . . . . . Cleveland, O .......................... G r a n d Lake, Colo . . . . . . . . . . . . . . . . . . . . . Midland, Tex ......................... R a p i d C i t y , S. D . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

A l b u q u e r q u e , N. M . . . . . . . . . . . . . . . . . . . G r a n d J u n c t i o n , Colo . . . . . . . . . . . . . . . . . Las Vegas, N e v . . . . . . . . . . . . . . . . . . . . . . . Riverside, Calif ....................... Santa Maria, Calif .................... W a k e I s l a n d s P.A . . . . . . . . . . . . . . . . . . . .

i

Nov.

Nov. Nov. Dec.

Dec.

NOV.

Jan.

July Sept. Apr. Apr. Aug.

792 14

I f,T = fI/Ho = ~

TABLE 3 - - T h e G e n e r a l i z e d M o n t h l y K r C u r v e s

Period of Data

•307 .295 .304

55o02 ' 42050 ' 46o28 '

.

KT

199 380 326

Annette, Alaska ....................... S c h e n e c t a d y , N. Y . . . . . . . . . . . . . . . . . . . . . S. Ste. M a r i e , M i c h . . . . . . . . . . . . . . . . . . . .

.

// (lg/day)

.401

•399

f

f=l

f=o

Hdf=

ff=lH

Value of f for KT =

= .5

.04 .08 .12 .16 .20 • 24 .28 .32 .36 .40 .44 .48 .52 • 56 .li0 • 1)4 • 68 .72 .76 .80 .84 • 88 .92 .96 1.00

073 162 245 299 395 496 513 579 628 • 687 748 • 793 •824 • 861 9O4 936 953 967 979 986 993 995 998

•015 .070 .129 • 190 •249 •298 .346 .379

•438 .493 .545 .601

• 654 .719 .760 •827 .888 • 931

.967 .981 .997

.999 •999 1.000

.o451 •082 .121 /

.160

194 • 234 • 277 •323 •358 •400 .460 .509 .614 • 703 • 792 •873 .945 .980 •993 1.000 •

'

[lOl

f=l

KT .3

df

f=o

•6

.7

.09O .008 •021 •039 •053 •076 • 101 .126 .152 .191 .235 .269 .310 .360 .410 .467 .538 .648 .758 .884 .945 .985 .996 .999 1.000

.000 .000 .007 .007 .007 .007 •013 .013 .027 .034 .047 .054 .081 .128 .161 .228 .295 .517 .678 .859 .940 .980 1.000

ff

KT, df

~0

However, since the ratio of D/H is a unique function of KT as shown in Fig. 7, when/~a is defined as

Kd = ]D/Ho

[11]

it is seen that

ff~= f/=x D d f = f=o H0

f f=IDKrdf f=0

[12]

Equation [12] states that when the ordinate, K T , of a monthly K r curve is multiplied by the ratio D/H from Fig. 7, the area under the curve so obtained is numerically equal to the value of/~d • The value o f / ) can then be obtained easily by multiplying /~a by the extraterrestrial daily insolation Ho from Fig. 6. When the coordinates K r of the generalized monthly K r curves of Fig. 12 are multiplied by the ratio D/H from Fig. 7, the curves of Fig. 13 are obtained. The areas under these curves, according to Equation [12], are numerically equal to the values of/~'a • A graphical integration produces the result shown in Table 4. The value of hTd corresponding to hT~ = 0.75 in Table 4 is taken from Fig. 5, since a locality w i t h / ~ = 0.75 should have almost constant clear weather from

It should be observed that the area under any monthly K r curve is numerically the same as the value of /~T, since from the definitions of /~r and "f" in Figs. 9, 10, l l or 12, 13

day to day and therefore Kd and K T should remain almost constant from day to day and be respectively equal to/~d a n d / ~ r • It should be noticed that the results of Table 4, in which /~d is a single valued function of -Kr, can be obtained only when a correlation between the monthly K r curves a n d / ~ r such as those shown in Figs. 9, 10 and 11 exists. Indeed this was the reason which initially led the authors to suspect such a correlation, since the relationship between Kd and K r of Fig. 5,

derived from the data of Blue Hill, strongly suggests the existence of a similar unique relationship between /~d a n d / ~ r . In examining Table 4, it should be noted that a locality may be considered to be extremely cloudy, if on a monthly average basis, the daily extraterrestrial solar radiation transmitted through the atmosphere is only 30% (/~r = 0.30). However, a locality may be considered to be very sunny i f / ~ r = 0.70. The results of Table 4 show that irrespective of the markedly dif-

- r : I :O m

g

b-0¢

v0

0.2

O.4

0.6

0.8

FRACTIONAL TIME, f , DURING WHICH DAILY TOTAL RADIATION -~ H FIG. 12--The generalized monthly Kv curves.

14

1.0

bv

t:~-r II I!

v

0

0.1 0.2 0.5 0.4 0.5 0.6 0.7 0.8 0.9 FRACTIONAL TIME, f , DURING WHICH DALLY TOTAL RADIATION 4~ H

1.0

FIG. 13--Curves for the determination of ](,t. TAm.E 4--The Relation Between the Monthly Average ])ally Total and the Monthly Average Daily Diffuse Radiation /~T /~

0.3 0.4 0.179, 0.183

0.5 0.188

0.6 0.174

0.7 0.149

ther examination of the data of London shows that the reason that the experimental values of /~e are higher than those predicted is due, at least in part, to the fact that a "shading ring" correction of 1% to 0% had been added to the measured diffuse radiation. The agreement would have been entirely satisfactory if this correction had not been applied to the measured diffuse radiation as with the data of the other localities. The r a t i o / ) / / 7 plotted as a function of £ ' r is shown in Fig. 14. The experimental ratio for each month and for each of the four localities are also shown o~l the same graph for comparison.

(0.75) (0.125)

ferent atmospheric conditions associated with the different values of / ~ r , on a monthly average basis, the fractions of the extraterrestrial daily insolation transmitted through the atmosphere as diffuse radiation, i.e. the values of Kd, show only a very moderate variation--from a minimum of 0.125 to a maximum of 0.188. Furthermore an analysis of the data of daily total radiation published by the U. S. Weather Bureau shows that for a great majority (over 70%) of localities and months, the values o f / ~ r lie in the range of 0.3 to 0.6. Thus one should expect, from the results of Table 4 alone, that for many localities Kd is within the range from 0.174 to 0.188. From the limited diffuse radiation data available, the twelve months average of /~d is 0.178 at Blue Hill, Massachusetts, (10 year average); s 0.185 at Nice, France, (3 year average);9 0.189 at Helsingfors, Finland, (4 year average);10 and 0.205 at London, England, (5 year average). H A fur-

Relationship Between Hourly and Daily Diffuse Radiation A knowledge of the average intensity of diffuse radiation at different times of the day is needed in many problems dealing with solar radiation. Since solar radiation data are not presented for intervals shorter than one hour, the nearest approach to the true average intensity at an instant obtainable from the solar radiation data commonly available is the hourly average intensity. Again it must be emphasized that extremely variable cloudiness precludes the possi15

I, ll'= II

II

0

0.1

0.2

0

0.3

RATIO

I..-

0.4

0.5

0.6

0.7

0.8

0.9

1.0

KT='-'~o=

MONTHLY AVERAGE DAILY TOTAL RADIATION EXTRATERRESTRIAL DALLY INSOLATION

FIG. 14--The ratio of the monthly average daily diffuse radiation to the monthly average d:~ily total radiation as a function of the cloudiness index K T. Therefore

bility of obtaining a true instantaneous radiation intensity during cloudy days except from direct experimentation. By Equation [11]

rd =

If the assumption is made t h a t the same fraction, lid, also represents the ratio of the average intensity of diffuse radiation to the extraterrestrial radiation intensity, i.e.

r~ -

where _Tdj,is the average intensity of diffuse radiation received on a horizontal surface, then the ratio rd of the average intensity of diffuse radiation to the daily diffuse radiation is rd = T-dh/D = Io,,/Ho

[15]

The correctness of this assumption can be tested only when the ratio computed by means of Equation [15] is compared with the ratio derived from the experimental data. An expression for Ho is given in Equation [5], and the instantaneous radiation intensity can be shown to be Ioh= rI~c(cosLcos~cos~

~- s i n L s i n S )

C O S ¢0 - -

C O S COs

24 sin o~ -- ~ cps ~

[18]

Using the ten year data for Blue Hill, Massachusetts, s and the four year data for Helsingfors, Finland, ~2 the experimental ratio of the monthly average hourly to daily diffuse radiation for the hours: 11:00-12:00 a.m. and 12:00-1:00 p.m., 10:00-11:00 a.m. and 1:00-2:00 p.m., etc., are plotted as shown in Fig. 15 against the sunset hour angle ~s computed b y means of Equation [6] with the use of the mean solar declination for each month. The data for the morning and afternoon hours symmetrical with respect to solar noon have been combined in obtaining these ratios. If these average

[14]

id~ = f ~ I o h

[17]

When the sunset hour angle, o~,, of Equation [6], is substituted into Equation [17], the expression is obtained,*

[13]

D = ff~Ho

~r cos L cos/I cos ~ -t- sin L sin 24 cos L cos ~ sin ~8 q- o~ sin L sin

* An equation which is slightly different, but practically the same as Equation [18] was first derived by Whillier in his investigation of the relation between the hourly and daily total radiation on a horizontal surface to be discussed in the following section.

[16] 16

Thus both Equation [14] and Fig. 15 may be utilized for the determination of the average intensity of diffuse radiation when the value of /(d, which can be determined from Table 3 and Fig. 14, is known.

hourly diffuse radiation are considered as the average intensities of diffuse radiation at the mid-point of these hours, a comparison with the theoretical ratio rd of Equation [18] can be made. The ratio r~ computed by means of Equation [18] using the hour angle at ½, 1½, 2~, etc., hours from solar noon is shown plotted in Fig. 15 as the solid curves. The excellent agreement between the experimental ratios and those computed by means of Equation [18] substantiates the assumption made in Equation [14].

Relationship Between the Hourly and Daily Total Radiation If the subscript "d" for diffuse radiation in Equations [13] and [141 is replaced by the subscript, " T " for total radiation, an expression which is identical to r~

0.2("

0.11

0.1(

z Iz _o

0.1'

,,

O.lt

°'°'

0

O.Oq

n.. 0,0,

0.0:

L

8 I

6O

9

I0 II 12 15 14 HOURS FROM SUNRISE TO SUNSET I 75 9'0 i 105 SUNSET HOUR ANGLE, ~ S ' DEGREES

15

16 I

120

FIG. 15--Theoretical and experimental ratio of the hourly diffuse radiation to the daily diffuse radiation.

17

C.__

5~ I~n,-

>

A

~o_

C

(l:

C

v

8 I

60

9

I0 II 12 13 14 HOURS FROM SUNRISE TO SUNSET I

I

15

16

t

75 90 105 SUNSET HOUR ANGLE,u) S , DEGREES

FIG. 16--Experimental ratio of the hourly total radiation to the daily total radiation. of Equation [18], is obtained for r r , the ratio of the average intensity of total radiation incident upon a horizontal surface to the daily total radiation received on the horizontal surface. I t was shown by Whillier L~ and Hottel and Whillier ~4 that the experimentally determined ratios r r are different from those computed b y means of Equation [18]. However, it was also shown t h a t when the experimental ratios r r derived from the d a t a of widely separated localities are plotted against the sunset hour angle, a mean curve for each hour is obtained such t h a t the deviation of a n y individual

point from the mean curve is no more than =t=5% for all hours between 9:00 a.m. and 3:00 p.m. sun time. The experimental curves in Fig. 16 in the range of 9 to 15 hours from sunrise to sunset are those presented by Hottel and Whillier) 4 T h e extension beyond this range has been made with data from Winnipeg, Canada. 1~ The following two examples illustrate some of the possible applications of the relations derived in the preceding sections. Example 1: Estimate the intensities of diffuse and 18

t o t a l r a d i a t i o n on a h o r i z o n t a l surface a t 12:00 noon on J u n e 23 a t a l o c a l i t y on 36°N l a t i t u d e . T h e d i r e c t r a d i a t i o n i n t e n s i t y , I ~ , , , a t n o r m a l incidence, a c c o r d ing to T h r e l k e l d a n d J o r d a n , ~6 is 280 B t u / h r - s q ft for the a s s u m e d basic a t m o s p h e r e . T h e solar a l t i t u d e is 77.5 ° . Solution: F r o m T a b l e l , on J u n e 23, r -- 0.9670. Therefore, Ion = r l ~ = ( 0 . 9 6 7 0 ) ( 4 4 2 ) = 428 B t u / h r sq ft a n d rD = ID,/lo,, = 280/442 = 0.655. B y m e a n s of E q u a t i o n [1], Td = 0.2710 -- (0.2939)(0.655) = 0.079. T h u s Iah = r~Io~ = (0.079) ( 4 2 8 ) ( s i n 77.5 °) = 33 B t u / h r - s q ft a n d Irl, = i,~, + Id~ = ( 2 8 0 ) ( s i n 77.5 °) + 33 = 307 B t u / h r - s q ft. H a d t h e i n t e n s i t y of t o t a l r a d i a t i o n of 307 B t u / h r - s q ft been given, Iah a n d I , , , can be c o m p u t e d as follows: Since r r = Irh/Iot, = 3 0 7 / ( 4 2 8 ) ( s i n 77.5 °) = 0.734, a n d b y m e a n s of E q u a t i o n [2] r~ = 0.3840 -- (0.4160) (0.734) = 0.079, I a h = ~',doh = 33 B t u / h r - s q ft as before. T h e r e f o r e , ID,, -- (I~h, -- I d h ) / s i n a = (307 -3 3 ) / s i n 77.5 ° = 280 B t u / h r - s q ft. E x a m p l e 2: T h e five y e a r (1954-1958) a v e r a g e of the d a i l y t o t a l r a d i a t i o n , / t , on a h o r i z o n t a l surface for J a n u a r y in I n d i a n a p o l i s , i n d i a n a , ( L a t . 39°44'), is 553 B t u / d a y - s q ft a c c o r d i n g to d a t a p u b l i s h e d b y t h e U. S. W e a t h e r B u r e a u in C l i m a t o l o g i c a l D a t a , N a t i o n a l S u m m a r y . T h e a v e r a g e of t h e d a i l y diffuse r a d i a tion a n d t h e a v e r a g e i n t e n s i t i e s of t o t a l a n d diffuse radiation during the hour l l :00-12:00 and 12:00-1:00 are to be e s t i m a t e d . Solution: B y m e a n s of Fig. 6 or E q u a t i o n [5] Ho = 1370 B t u / d a y - s q ft. T h e r e f o r e / ~ r = t t / H o = 5 5 3 / 1370 = 0.403, a n d b y m e a n s of Fig. 1 4 , / ) / / t = 0.454. Hence 1)= (D/H)(f-t) = (0.454)(533) = 242 B t u / d a y - s q ft. Since t h e s u n s e t h o u r angle, o~, is given b y E q u a tion [6], a n d on J a n u a r y 16, the d e c l i n a t i o n is - 2 1 °, cos ~.~ = - ( t a n 3 9 ° 4 4 ' ) ( - t a n 2 l ° ) = 0.323, ~o~ = 71 ° (or x~ 7~ × 2 = 9.45 hours b e t w e e n sunrise a n d suns e t ) . F r o m Figs. 16 a n d 15, r r = 0.172 a n d ra = 0.161. Therefore, t h e a v e r a g e i n t e n s i t i e s of t o t a l a n d diffuse r a d i a t i o n d u r i n g t h e hours 1 1 : 0 0 - 1 2 : 0 0 a n d 1 2 : 0 0 l : 0 0 are r e s p e c t i v e l y , irp~ = r r H = ( 0 . 1 7 2 ) ( 5 5 3 ) = 95 B t u / h r - s q ft a n d 1,,, = ref) = ( 0 . 1 6 1 ) ( 2 4 2 ) = 39 B t u / h r - s q ft. Using t h e generalized m o n t h l y K r curves of Fig. 12 or T a b l e 3, it is possible to d e t e r m i n e a p p r o x i m a t e l y , from t h e given v a l u e of H , t h e f r a c t i o n a l t i m e s d u r i n g which t h e d a i l y t o t a l r a d i a t i o n is less t h a n or equal to c e r t a i n values. F o r e x a m p l e , a c c o r d i n g to T a b l e 3 or Fig. 12, w h e n K r = 0.20, f = 0.249. T h u s for 25 % of t h e t i m e (or 7½ d a y s d u r i n g t h e m o n t h ) , t h e d a i l y t o t a l r a d i a t i o n is less t h a n or equal to K T H o = (0.20) (1370) = 274 B t u / d a y - s ( l ft d u r i n g J a n u a r y in I n -

d i a n a p o l i s . S i m i l a r l y for 76 % of t h e t i m e (or 23 d a y s in t h e m o n t h ) t h e d a i l y t o t a l r a d i a t i o n is less t h a n or equal to ( 0 . 6 0 ) ( 1 3 7 0 ) = 820 B t u / d a y - s q ft. T h e a c t u a l a v e r a g e (1954-1958) n u m b e r of d a y s in J a n u a r y w i t h r a d i a t i o n below t h e a b o v e v a l u e s are res p e c t i v e l y 6½ a n d 23 d a y s in I n d i a n a p o l i s . T h e generalized m o n t h l y K T curves can also be utilized to d e t e r m i n e a p p r o x i m a t e l y t h e s t a t i s t i c a l d i s t r i b u t i o n of t h e d a i l y t o t a l r a d i a t i o n for localities where o n l y a n e s t i m a t e of t h e m o n t h l y a v e r a g e of t h e d a i l y t o t a l r a d i a t i o n is k n o w n ~7 since t h e o n l y i n f o r m a tion n e e d e d is the v a l u e of H.

REFERENCES 1. Parmelee, (L V., "Irradiation of Vertical and Horizontal Surfaces by Diffuse Solar Radiation from Cloudless Skies," A S H V E Transaction 60: 341-358, 1954. 2. Johnson, F. S., "The Solar Constant," Journal of Meteorology, 11: 431-439, December, 1954. 3. Drummond, A. J. and Greer, H. W. : "Fundamental Pyrheliometry," The Sun at Work 3(2) : 3-5, 11, June, 1958. 4. Klein, W. H., "Calculation of Solar Radiation and the Solar Heat Load on Man," Journal of Meteorology, 5: 119-129, August, 1948. 5. Moore, A. F. and Abbot, L. H., "The Brightness of the Sky," Smithsonian Miscellaneous Collection, 71(4): 1-36, February. 1920. 6. Fritz, S., "Solar Radiation During Cloudless Days," Heating and Ventilating, 46: 69-74, January, 1949. 7. Hand, I. F., "Methods of Calculating Solar Radiation Values at Blue Hill Observatory, Milton, Massachusetts," Monthly Weathev Review, 82: 43-49, February, 1954. 8. Diffuse and total radiation data for Blue Hill, Massachusetts, were obtained from Mr. C. V. Cuniff, U. S. Weather Bureau, Blue Hill Observatory, Milton, Massachusetts. For instrumentation and methods of measurement see: Hand, I. F. and Wollaston, F. A., "Measurements of Diffuse Solar Radiation at Blue Hill Observatory," Technical Paper No. 18, U. S. l)epartment of Commerce, Weather Bureau, Washington, I). C., May, 1952. 9. Gorcynski, W., "Enregistrements due Rayonnement Solaire au Moyan des Solarigraphy et des Pyrheliographes," Nice. O~ce Meteorologique, Annalcs, Tome !1: 117-165, 1933. 10. Imnelund, H., "Contribution to the Knowledge of Solar Radiation in Finland," Finska Vetenskaps-Societeten, Helsingfors, Commentationes Physico-Mathematicae, 7(11): 1-58, February, 1934. l l . Blackwell, M. J., "Five Years Continuous Recording of Total :tnd Diffuse Radiation at Kew Observatory," Paper of the Meteorological Research Committee (London), No. 895, 1954. 12. Lunehmd, H., "Records of Solar Radiation in Helsingfors," Finska Vetenskaps-Societeten, Helsingfors, Com,mentationes Physico-Mathematicae 7(1) : 1-28, 1933. 13. Whillier, A., "The l)etermination of Hourly Values of Total Solar Radiation from Daily Summations," Archly fl~r Meteorologie, Geophysik i~nd Bioklimatologie, Vienna, Series B., 7: 197-244, 1956. 14. Hottel, H. C. and Whillier, A., "Evahmtion of Flat-Plate Solar Collector Performance," Transaction of the Conference on the (:se of Solar Energy: The Scientific Basis, Vol. II, Part I, Section A, pp. 74-104, 1955. 15. Data of total radiation for Winnipeg, Canada, are obtained from Mr. ]). M. Robertson, Regional Meteorologist, District Aviation Forecast Office, Winnipeg, Canada. 16. Threlkeld, J. L. and Jordan, R. C., "Direct Solar Radiation Available on Clear Days," Heating, Piping and Air Conditioning, 29:(12): 135-145, l)ecember, 1957. 17. Fritz, S. and MacDonald, T. H., "Average Solar Radiation in the United States," Heating and Ventilating, 46(7): 6164, July, 1949.

19