The estimation of daily, clear sky, solar radiation intercepted by a tilted surface

Y, olar Energy Vol. 24. pp. 287 294 Persamon Press Ltd., 1980. Printed in Great Britain

THE ESTIMATION OF DAILY, CLEAR SKY, SOLAR RADIATION INTERCEPTED BY A TILTED SURFACEr

THOMAS A. WEISS

Dubois & King, Inc., Randolph, VT 05060, U.S.A.

and

GEORGE O. G. LOF

Solar Energy Applications Laboratory, Colorado State University, Fort Collins, CO 80523, U.S.A:

Abstract--The amount of solar energy that is intercepted by surfaces of any orientation is estimated from a new model of the clear sky, spatial distribution of solar radiation, The model was developed from measurements made during clear sky conditions and uses direct, isotropic reflected, and anisotropic diffuse radiation. The effects of azimuth, tilt, season, latitude, atmospheric turbidity, and reflectivity of the surroundings were computed using hourly measurements of normal beam and horizontal total radiation at four stations in the United States. A transformation of the co-ordinates of orientation produced a general relationship between orientation and intercepted energy. The general relationship was tested against measurements from six locations in the Northern Hemisphere and was found to be valid. The model is also a better estimator of energy intercepted by a tilted surface than are the more commonly used models.

I. INTRODUCTION

Designers of solar systems that use fiat-plate collectors need to know the amount of solar energy intercepted by a tilted surface. Estimates of the solar

energy intercepted by a tilted surface with an azimuth directly facing the equator are available from several sources [1-5]. However, designers often need to know how much energy can be intercepted by surfaces with other azimuths. A summary of an investigation into the amount of solar energy intercepted by surfaces of any orientation is presented. The investigation had four objectives.

1. Develop a more accurate model of the spatial distribution of solar radiation than had been used in other studies of collector orientation.

2. Determine the effects of orientation, latitude, reflectivity of the surroundings, time of year, and atmospheric turbidity on the amount of energy intercepted by a surface of unit area.

3. Compare the results of this model with measurements of radiation intercepted by non-horizontal surfaces.

4. Compare this model with some of the more commonly used, simpler models of solar radiation.

2. DESCRIPTION OF THE MODEL

Estimating the amount of solar energy that is intercepted by a surface having a given orientation

~" Presented at the 1977 Annual Meeting of the American Section of the International Solar Energy Society. Orlando. Florida. 6-10 June 1977.

requires knowing the amount of energy coming from each point of the sky and ground and the angle

between each point and the surface. The computer model developed for this study numerically integrates four components of solar radiation (beam radiation, reflected radiation, hemispherical diffuse radiation, and eircumsolar diffuse radiation) over time and the space visible to the surface. The energy distribution of each of these components was obtained from measurements reported in the literature.

The beam radiation is the radiation coming from the sun and a small region near the sun as measured by a normal incidence pyrheliometer aimed at the sun. The beam radiation is modeled as if it were all coming from a point at the center of the sun even though the measured beam radiation data in the computations come from a disc with a diameter of about 5.8 ° .

The radiation reflected from the surroundings is assumed to be isotropically distributed, so that the model will be more generally applicable than if the surroundings at a specific location were modeled. The mean reflectivity [6, 7] of the surroundings changes as the orientation of the surface and the objects in its field of view change. For many types ,of surroundings the change in mean reflectivity i s small, and the proper choice of a mean reflectivity should provide sufficient accuracy for engineering computations.

The hemispherical diffuse radiation comes from all of the sky and has an intensity equal to that measured in regions away from the sun. The beam and circumsolar diffuse radiation are superposed over the hemispherical diffuse radiation around the sun. The intensity of the hemispherical radiation depends on the

287

288 THOMAS A. WEISS and GEORGE O. G. L6F

i 20 3O 40 5O 6O 70 80 gO

A l t i t u d e Angle o f a Point in the Sky

Fig. 1. Spatial distribution of the hemispherical diffuse ' radiation.

altitude angle of the point from which the energy comes [8-10] but is assumed to be independent of the azimuth of the point. A comparison of the measured, clear-sky, anisotropic, hemispherical diffuse radiation [8-10] with the isotropic distribution is given in Figs. 1 and 2. Figure 1 shows the ratio of the hemispherical diffuse radiation at a point as measured under clear skies to the isotropic diffuse radiation. Figure 2 gives the ratio of hemispherical diffuse radiation intercepted by a tilted surface to that intercepted by a horizontal surface for the two distributions. Use of the isotropic distribution will result in an underestimation of the hemispherical diffuse radiation for surfaces at all slopes because of the greater intensity of the sky radiation near the horizon than near the zenith, as shown in Fig. 1.

The circumsolar diffuse radiation is the radiation from the region near the sun in excess of the hemispherical radiation but not included in the beam radiation. Measurements show that this radiation is con- centrated in a region around the sun with a radius of 16°[9--11]. Figure 3, based on measurements presented by Kondratyev [11], shows the relative intensity distributions for several turbidities. (The relative intensity is defined as one at 1 ° from the sun's center.) Tests of the model show that for a given amount of

~ 1.2

~ 1.C

i 0 .~

-~ 0.4 "C

I i I I I I I I

~ a n i s o ~ o p i ¢

. . . . . . i s o t r o p i c

I I I I J I I I 10 20 30 40 50 60 70 80

S u r f a c e Ti I t From Hor izonta l (degrees)

9 0

Fig. 2. Ratio of hemispherical diffuse radiation on a tilted surface to that on a horizontal surface.

1.o

8 = .B c

>E

m~- .2

I I I I ' 1 | I

- - T u r b i d i ~ = 7.6 . . . . . . . . T u r b i d i t y = 5.2

k . . . . . T u r b i d i t y = 3.3 , ~ . . . . T u r b i d i t y = 2.8

, 4 6 8 10 12 14 1 6

Angle F r ~ the Sun's Center

Fig. 3. Spatial distribution of the circumsolar diffuse radiation.

circumsolar radiation, the effect of the different distributions on the daily total intercepted energy was less than 0.5 per cent. The amount of circumsolar radiation increases with increasing turbidity as shown in Fig. 4 [12]. Clearness ratio is a measure of turbidity and is the ratio of the actual beam radiation to Moon's beam radiation [13] for the same airmass. Turbidities of 2 and 5 are about equal to clearness ratios of 1.2 and 0.6, respectively. The decrease in circumsolar diffuse radiation for low solar altitudes and low clearness ratios is caused by the increasing fraction of the circumsolar radiation that is below the horizon. None of the circumsolar radiation that is below the horizon or behind the surface is counted as part of the intercepted energy.

Input data for the computer model include daily sets of hourly total radiation on a horizontal surface, hourly normal beam radiation, and geographic and astronomic data. The solar radiation data are measurements made during clear days and hazy but cloudless days at the times of solstices and the vernal equinox. Symmetry about solar noon is imposed by using the mean of the values at the two hours equidis- tant from solar noon instead of the measured values. Circumsolar diffuse radiation is computed from Fig. 4 after computing the clearness ratio. Then the circumsolar diffuse and beam radiation are converted to values for a horizontal surface and are subtracted from the total horizontal radiation to obtain the hemispherical diffuse radiation on a horizontal surface. The energy intercepted by a surface of any orientation can then be calculated since all of the incidence angles are defined by the hour, the season, and the orientation and latitude of the surface.

Output data for each daily set of input include tables of daily solar radiation intercepted by surfaces

-41 I I I I I

°"

- . 6 .8 1.0 1.2

Clearness Ratio

Fig. 4. Amount of ¢ircumsolar diffuse radiation.

The estimation of daily, clear sky, solar radiation intercepted by a tilted surface

tilted from 0 ° to 90 ° and at azimuths from 0 ° (facing go the equator) to 90 ° (west), both by 5 ° intervals, for each reflectivity of the surroundings from zero to one ~ so at intervals of 0.2. Data for azimuths from - 9 0 ° to 0 ° are symmetric with those from 90 ° to 0 °. The model ~ 70 was tested with data from four stations which cover a ~ 6o range of latitudes, climates, and elevations found within the contiguous United States: Albuquerque, ~ 5̀0 New Mexico (35°N), Blue Hill, Massachusetts (42°N~ "= 4o

Lincoln, Nebraska (41°N), and Madison, Wisconsin (43ON).

2a

c' o

3. ANALYSIS OF THE RESULTS

Output from the model was plotted as dimension- less isolines of the ratio of daily intercepted energy at that orientation to the daily intercepted energy at the orientation intercepting the maximum energy (the other variables remaining constant). Figures 5-7 show three of these graphs. Haze and turbidity were found to have only a small effect on the locations of the isolines for a given station and could then be neg- lected from further analysis.

The isoline patterns for Blue Hill, Lincoln, and Madison were nearly identical but differed from the patterns for Albuquerque. A transformation of the slope-azimuth co-ordinate system brought the Albu- querque data into good agreement with the other data. The co-ordinate transformation describes the orientation of a given surface in terms of the orientation of a parallel surface at the transformed latitude, assuming that the surfaces are all on the same meridian. The equations for the co-ordinate transformation are:

cos s' = c o s s cos ( 4 ' - 4 )

- sin s sin (4' - 4) cos y (1) sin y' -- sin s sin y/sin s' (2)

where s is the slope and y is the azimuth at the origi- nal latitude 4, and s' is the slope and y' is the azimuth

Lati tude - 42°N Spring p=0.4

10 20 30 40 .50 60 70 8(3 90

289

Surface T i l t From Horizontal (degrees)

Fig. 6. Isolines of percent of maximum intercepted energy, vernal equinox.

at the transformed latitude 0'. For example, Fig. 5 gives s' and y' for 4 ' of 42 °. What surface at 42 ° corre- sponds to a surface with s = y = 40 ° at 4 = 35°? From the above equations, s' = 45 ° and y' = 35 °. For 4 = Y = 0°, these equations reduce to the familiar slope minus latitude transformation. The errors introduced into the isoline pattern are less than 5 per cent (not five percentage points) at all times and less than two per cent in the regions of maximum intercepted energy. The transformation would be exact except for the presence of the atmosphere and the horizon. The errors introduced by the transformation are acceptable for the differences in latitude (8 ° ) that were studied and probably acceptable for all latitudes covering the contiguous United States (25°N to 49°N). More

i solar radiation measurements, as yet unavailable, will be required to prove the accuracy of the transformation over the wider range of latitudes.

9c

u~ 8c == t.

7c "o

6c

5(:

4C 5 .~ 3C

~ 2c

N 1c

5

Latitude = 42°N Winter e=0.4

40

; 10 20 30 40 ,50 60 70 so 90 Surface T i l t From Horizontal (degrees)

Fig. 5. Isolines of percent of maximum intercepted energy, winter solstice.

Latitude = 42 ° Summer ,=0.4

0 10 20 30 40 ~0 60 70 80 90 Surface T i l t From Horizontal (degrees)

Fig. 7. [solines of percent of maximum intercepted energy, summer solstice.

290 THOMAS A. WEISS and GEORGE O. G. LIgF

100 =- w

9O

70 • ///// \k%%% kxN%%x

p -1 \

, / 2 . . . . . . . e~o \ zimuth = 0 . ~ \% -

I I I 1 I -40 - 2 0 0 20 40

T i l t of Surface-Latitude (degrees) 60

Fig. 8. Effect of surface tilt on intercepted energy.

Figures 5-7 show the isolines for a latitude of 42°N and a reflectivity of the surroundings of 0.4 for the two solstices and the vernal equinox. Changing the orientation by l0 ° from the orientation of maximum interception reduces the intercepted energy by one to two per cent; a 25 ° change reduces the intercepted energy by ten per cent. The maximum energy is intercepted when the azimuth of the surface is 0 ° while the

slope of the maximum interception depends on the other variables.

Changing the reflectivity of the surroundings (p) from zero to one increases the slope at which the maximum energy is intercepted by about 20 ° , as shown in Fig. 8. The maximum intercepted energy in this fgure is the greatest value obtained with a reflectivity of one for each season. For a reflectivity of zero, the percentages are related to the maximum at a reflectivity of one for the same season. Linear interpola- tion may be used for intermediate reflectivity.

4. COMPARISON OF THE MODEL WITH MEASUREMENTS

The accuracy of the model was tested against measurements at six locations in the Northern Hemi- sphere. Calculated and measured ratios of radiation on a sloping surface to that on a horizontal surface arc given in Table 1 and Fig. 9 for five of the six locations. (Leningrad is not included because it is too far north, 60°N, for the data based on latitude 42°N to bc applicable, even when using the co-ordinate transformation.) The calculated ratios are obtained from graphs like those of Figs. 5-7 for the appropriate season and reflectivity.

Table 1. Comparison of the calculated data with measured clear sky data

m eta crb c,i Location Y ~ (~) (*)

Blue Rill W .30 90 0 2.46 2.56 2.81 2.53 M a s s a c h u s e t t s W .30 90 -90 .78 .84 .84 .80 (42"N) W .30 90 -90 .81 .84 .84 .80

W .30 90 90 .84 .84 .84 .80 V .35 90 0 1.07 1.06 1.08 1.00 V .35 90 -90 .65 .74 .78 .68 V .35 90 -90 .66 .74 .78 .68 V .35 90 90 .67 .74 .78 .68 S .20 90 -90 .60 .54 .54 .52 S .20 90 90 .56 .54 .54 .52

Ashkabad S .20 90 0 .41 .37 .34 .34 Turkman SSR S .20 90 45 .60 .51 .51 .49 (38"N) S .20 90 90 .64 .53 .53 .51

Locarno-Montl W .70 25 0 2.10 2.13 2.34 2.13 Switzerland V .70 25 0 1.45 1.29 1.35 1.27 (46"N) S .20 25 0 1.10 1,06 1.07 1.05

Fort Collins W .30 45 0 2.20 2.28 2.50 2.30 Colorado V .35 45 0 1.23 1.32 1.37 1.28 (41ON) S .20 45 0 .87 .94 .94 .92

A .20 45 0 1.36 1.30 1.35 1.25

Crimea S .20 I0 0 1.04 1.04 1.05 1.03 Ukrainian SSR S .20 I0 -90 .98 1.01 1.02 1.01 (45"N) S .20 10 90 .99 1.01 1.02 1.01

S .20 20 0 1.04 1.06 1.07 1.05 S .20 20 -90 .92 .98 .98 .98 S .20 20 90 .96 .98 .98 .98 S .20 30 0 1.02 1.05 1.07 1.03 S .20 30 -90 .87 .94 .94 .93 S .20 30 90 .92 .94 .94 .93

p = The reflectivity of the surroundings. s = The surface's slope from horizontal. ), = The surface's azimuth: 0 is south, 90 is west. m = The measured and c is the calculated ratio of radiation on a tilted surface to

that on a horizontal surface (a is from the anisotropic, b the beam, and i the isotropic model).

Y = Season: W is winter, V is spring, S is summer, A is autumn. Sources: [14 18].

The estimation of daily, clear sky, solar radiatien intercepted by a tilted surface 291

s.

o ~ g -o

o

tn

3 .C

2.C

, , , , ,

f f l :

1.C

/° I I ! f i

1.0 2 .0 3 .0

Calculated Total Radiation (Tilted: Horizontal)

Fig. 9. Comparison of anisotropic model with measured data.

The reflectivities of the surroundings had to be estimated because they were not given with the radiation measurements. When the snow cover was less than 30ram, a mean reflectivity of 0.2 was used; for more than 30 mm of snow cover, 0.7 was used. Values of the reflectivity that were used for each calculation are also given in Table 1. At Blue Hill for the period when the measurements were made, there was less than 30 mm of snow on the ground for 80 per cent of the clear days near the winter solstice (4-10 days) and for 70 per cent of those near the vernal equinox. The weighted reflectivities for these seasons are given in Table 1. The same values are used for Fort Collins by similar arguments. There is no justification for any adjustments at Locarno~Monti for the winter and use of 0.2 or 0.7 makes no difference in spring.

A regression analysis, using the calculated (c) and measured (m) ratios listed in Table 1 with the method of least squares, is given in Table 2. The regression equation is not significantly different from m = c at the 90 per cent confidence level using Student's t distribution since the t values are less than 1.70.

5. C O M P A R I S O N W I T H

O T H E R M O D E L S

The anisotropic model presented above was compared with two commonly used models of solar radi-

._3a 2~I

v

w ~

I | ; i' i i ! i !

- - 7 - - -

aS J " ~

'J . ' ~ - - 7 . . . . . i s o t r o p i c J ~ 1 2 1 2 2 1 6 1 a z i m u t h : O"

~1/~ I I I ! I I I J - 0 10 2 0 3 0 4 0 ~ 0 6 0 "K) 80 9 0

Surface T i l t From Horizontal (degrees)

Fig. 10. Comparison of models (Blue Hill).

ation. The isotropic model assumes the diffuse radiation to be isotropicaUy distributed over the sky. The beam model treats the diffuse radiation as if it all came from the center of the sun. The treatment of diffuse radiation is the only difference among the three models. The beam and isotropic models are the two extremes for modeling the diffuse radiation; the anisotropic model is somewhere between. The three models are compared in Fig. 10 using data for Blue Hill. The isotropic model underestimates the intercepted energy; the beam model overestimates it by as much as 40 per cent during the winter heating season. Regression analyses were also performed for the beam and isotropic models using the data in Table 1; the results are given in Table 2. The regression equations for the beam and isotropic models are significantly different from m = c at the 90 per cent confidence level. Thus, the anisotropic model presented in this paper is the most accurate of the three models.

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

(I) A standard set of graphs showing patterns of intercepted energy as a function of slope and azimuth of a surface, season, and reflectivity of the surroundings can be established for bands of latitude at least 15 ° but less than 35 ° in width by using a transformation of the slope-azimuth co-ordinates and by plotting intercepted energy as a ratio to the maximum intercepted under given conditions.

(2) The maximum intercepted energy for the symmetric radiation conditions studied here occurs at an azimuth of 0 ° (lacing the equator) at a slope that depends on the other variables.

Table 2. Comparison of the radiation models ii

½odel R e | r e s o t o n

( . - a + b e )

• ~ a i g o t r o p t c m = . 0 3 6 ÷ . 9 5 3 ¢ . 9 9 I s o t r o p i c m = . 0 7 1 + . 9 4 0 c . 9 g Beam m = . 1 1 6 + . 8 4 7 c . 9 9

t s t a t i s t i c s r a b

1 . 1 0 1 . 6 6 2 . 4 9 2 . 3 8 4 . 3 7 6 . 9 8

m and c are the same as in Table I. r is the correlation co-efficient.

292 THOMAS A. WEISS and GEORGE O. G. LGF

(3) Intercepted energy is reduced no more than 10% ¢L for orientations as much as 25 ° from the orientation 4,t that maximizes intercepted energy. During the sum- 4,2 mer, intercepted energy is virtually independent of the azimuth of the surface for a given slope. ~,

(4) The beam model should not be used to estimate ~,~ the energy intercepted by a tilted surface, especially

during the winter, because of its large overestimation ~" of intercepted energy. The isotropic model is conser- O3

vative in estimating intercepted energy and is accept- o~L able for use when a model simpler than this anisotropic model is desired. ~o~

(5) Of the three models studied, the anisotropic

model provides the most accurate estimation of the total solar energy intercepted by surfaces of any

orientation.

NOMENCLATURE

A1 a constant to equate CN with the integrated C~(2) B n beam radiation on a horizontal surface B= Moon's normal beam radiation B N beam radiation on a surface normal to the sun B~ beam radiation on a surface of slope s Cu circumsolar diffuse radiation on a horizontal sur-

face C~(2) relative intensity distribution of the circumsolar

diffuse radiation CN circumsolar diffuse radiation on a surface normal

to the sun C~ circumsolar diffuse radiation on a surface of slope

s E equation of time Hn hemispherical diffuse radiation on a horizontal

surface H~(/l) intensity distribution of the hemispherical diffuse

radiation on a surface normal to a small area of the sky dome

H~ hemispherical diffuse radiation on a surface of slope s

In total radiation on a horizontal surface I~ total radiation on a surface of slope s LST local standard time M airmass based on zenith airmass equal to 1.0 m airmass based on zenith airmass of P/Po p mean air pressure at the elevation of the station Po mean air pressure at sea level R~ reflected radiation on a surface of slope s s slope of the surface from the horizontal

Greek characters

solar altitude angle fl altitude angle of a portion of the sky dome y azimuth angle of the outward normal of the sur-

face (south is 0 ° and east is - 9 0 °) y, relative azimuth between the sun and the outward

normal of the surface (equals y - o) 6 solar declination (north is positive) 0~ incidence angle between the sun and the outward

normal of the surface Oz solar zenith angle 2 angular distance from the center of the sun p reflectance of the surroundings of the surface ~b solar hour angle (south is 0 ° and east is - 9 i f )

latitude of the surface 4,c limit of integration for ~bl based on the surface

being the obscuring component in zone 4 4,u limit of integration for 4,1 based on the horizon

being the obscuring component in zone 4

limit of integration for 4,~ in zones 2 and 3 angular distance around a ring around the sun angle of incidence of a differential element of the circumsolar diffuse radiation an angle defined only for ease of computation longitude of the surface (west is positive) angle of incidence of a differential element of the hemispherical diffuse radiation longitude of the meridian determining standard time solar azimuth angle (south is 0 and east is - 9 i f ) azimuth of the intersection of the surface with the sky dome at a given fl azimuth of a differential element of the sky dome

REFERENCES

1. C. F. Becker and J. S. Boyd, Solar radiation avail- ability on surfaces in the United States as affected by season, orientation, latitude, altitude, and cloudiness. Solar Energy 1 (1), 13-21 (1957).

2. H. C. Hottel, Performance of fiat-plate solar energy collectors. Space Heating with Solar Energy, pp. 58 71. M.I.T. Press, Cambridge, Massachusetts (1954).

3. G. O. G. LGf and R. A. Tybout, A model for optimiz- ing solar heating design. Trans ASME, 72-WA/Sol 8 (1972).

4. R. N. Morse and J. T. Czarnecki, Flat plate solar absorbers, the effect on incident radiation of inclina- tion and orientation. CSIRO, Australia, Eng Section Rep. E.D.6, (1958).

5. J. L Threlkeld and R. C. Jordan, Solar radiation during cloudless days. Heating, Piping and Air Condition- ing 26(2), 117-122 (1955).

6. B. Y. H. Liu and R. C. Jordan, Daily insolation on surfaces tilted toward the equator. ASHRAE J. 3 (10), 53-59 (1961).

7. B. D. Hunn and D. O. Calafell II, Determination of average ground reflectivity for solar collectors. Solar Energy 19 (1), 87-89 (1977).

8. A. R. Moore and L. H. Abbot, The brightness of the sky. Smith. Misc. Coll. 71 (4), 1-36 (1920).

9. C. G. Abbot et al., Annals of the Astrophysical Observa- tory of the Smithsonian Institution, Vol. 2, Part 2, Chap. 3, pp. 146--168 (1908); and Vol. 3, Chaps. 6 and 7, pp. 141-165 (1913).

10. C. W. Morris and L. H. Lawrence, Anisotropy of clear sky diffuse solar radiation. Trans. ASHRAE 77, Part 2, 136-142 (1971).

11. K. Ya. Kondratyev, Radiation in the Atmosphere, pp. 272-277, 342-346, and 363-377. Academic Press, New York (1969).

12. A. G. Loudon, The interpretation of solar radiation measurements for building problems. Proc. of the 1965 C.I.E. Inter. Conf., Sunlight in Buildings, University of Newcastle-upon-Tyne, pp. 111-118 (1968).

13. P. Moon, Proposed standard solar radiation curves for engineering use. J. Franklin Inst., V. 230, No. 5, pp. 583-617, 1940.

14. C. V. Cunniff, Solar radiation on walls facing east and west. Air Conditioning, Heating, and Ventilating 55, (10}, 82-88 (1958}.

15. H. Flohn (editor), World Survey of Climatology, Vol. 2, pp. 109-116. Elsevier Publishing Company. Amster- dam (1969).

16. I. F. Hand, Preliminary measurements of solar energy received on vertical surfaces. Trans., Am. Geoph),s. Union 28 (5), 705-712 (1947).

17. Kh. Sh. Khadzhiev, The ratio of direct and diffuse solar radiation. Geliotekhnika (English trans.} 9 (6), 81-84 (1973).

The estimation of daily, clear sky. solar radiation intercepted by a tilted surface 293

18. J. C. Thams, Die globalstrahlung eines Siidhanges yon 25 Neigung. Archly f i i r Meteorolof/le, Geophysik, und Bioklimatolooie, Serie B, 7, 190-196 (1956).

19. T. A. Weiss. Collector orientation and its effect on intercepted energy. M.S. Thesis, Department of Civil Engineering, Colorado State University (1976).

APPENDIX

This appendix presents the equations that were used in the computer model that generated the data shown in Figs. 5-7.

Beam radiation

B . = BN COS O~ cos 0~ = sin s sin 0~ cos ),, + cos 0~ cos s cos 0~ = sin 6 sin ~ + cos t$ cos ~ cos sin co = sin z cos 6/sin 0~

= L S T + E + (d/, - ~)

BN is from measurements of the National Weather Service.

Hemispherical diffuse radiation

f .... f , / 2 H , /Hn = 2 . . . . = - ,/2~ # = o it- 1Hi(fl) cos ~'s cos fl dco, dfl

cos ~ = cos s sin fl - sin s cos fl sin co,

coL=n~2 for f l > s ; or sin coL -- tan fl/tan s for ~ < s H~(fl) is from Fig. 1

Hu = In - Bn - Cn

In is from measurements by the National Weather Service.

Reflected radiation

R, = In p (l-cos s)/2.

Circumsolar diffuse radiation

f~h :2~p, C, AI CI1~.) sin ). cos ~2 dR d~j

CN/B, is from Fig. 4. Bm is Moon's beam radiation

log B m - 3.12027 - 0.18503 m + 0.04304 m 2

- 0.00850 m a + 0.00083 m 4 - 0.0000296 m s

m = Mp/po.

The equation for Cs/CN given above must only be integrated over the region of circumsolar diffuse radiation that is visible to the surface. There are six zones for computing C,/CN based on the cause of some of the circumsolar diffuse radiation being obscured. These zones aie: (1) All of the circumsolar radiation is visible to the surface; (2) All of the circumsolar radiation is above the horizon but part of the radiation is behind the surface; (3) Part of the circumsolar radiation is below the horizon but none of the radiation that is above the horizon is behind the surface; (4) Part of the circumsolar radiation is behind the surface and an additional part is below the horizon; (5) Part of the circumsolar radiation is behind the collector while some of the obscured part is below the horizon: (6) None of the circumsolar radiation is visible to the surface. The limits of integration for C, are presented in Table 3.

The defining relationship for A1 is:

CN A 1Ct(2) sin 2 dR d~bl "l @l=@8° it=i.A

where all is known except Al. The limits of integration are as for zone 3. C~(:.) is from Fig. 3.

Total radiation

I, = B, + Hs + Cs + R~.

294 THOMAS A. WEISS and GEOgGE O. G. L6E

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Documents

The estimation of daily, clear sky, solar radiation intercepted by a tilted surface