Derivation of solar angles using vector algebra

Solar Energy Vol. 37. No. 6, pp. 429-430, 1986

0038-092X/86 $3.00 + .00 © 1986 Pergamon Journals Ltd.

Printed in the U.S.A.

D E R I V A T I O N OF SOLAR A N G L E S USING VECTOR ALGEBRA P. G. Jot.LV Solar Energy Research Centre, University of Queensland, St. Lucia, Australia 4067

I. I N T R O D U C T I O N

A f u n d a m e n t a l r e q u i r e m e n t in the design o f a n y solar e n e r g y s y s t e m is the ability to c a l c u l a t e the a m o u n t o f s o l a r r a d i a t i o n i n c i d e n t on the collector. T h e c o m p u t a t i o n o f the direct c o m p o n e n t of solar r a d i a t i o n r e q u i r e s an e q u a t i o n for the cosine o f the angle b e t w e e n the s u n ' s direct rays and the collector n o r m a l [ I ] . M a n y e x p r e s s i o n s h a v e b e e n published in v a r i o u s s o l a r d e s i g n b o o k s (e.g. [1] and [2]) for a v a r i e t y o f c o l l e c t o r regimes. H o w e v e r , difficulties c a n arise in using t h e s e e x p r e s s i o n s as different angle sign c o n v e n t i o n s are often u s e d for e a c h e x p r e s s i o n . T h i s m a k e s it t e d i o u s and difficult to c h e c k the validity of the desired e x p r e s s i o n s . T h i s p a p e r p r e s e n t s a m e t h o d , b a s e d on v e c t o r a l g e b r a , for c o m p u t i n g t h e d e s i r e d sun angle relat i o n s h i p s . T h e m e t h o d is simple to use and c a n be readily u s e d to d e r i v e a n y o f the sun angle relat i o n s h i p s p u b l i s h e d in the literature[4].

angle b e t w e e n a single axis E - W t r a c k i n g s u r f a c e a n d the s u n ' s unit v e c t o r will be d e r i v e d . C o n s i d e r an E - W t r a c k i n g s u r f a c e r o t a t i n g a b o u t a N-S axis inclined at an angle S. L e t r be the unit n o r m a l to the surface and I, m a n d n, be a s t a t i o n a r y set of o r t h o g o n a l a x e s as s h o w n in Fig. 1. m a l w a y s r e m a i n s parallel to unit v e c t o r i. Angle r is the angular d i s p l a c e m e n t b e t w e e n r a n d I. From trigonometry. r = c o s - r l + sin "rm. where I = cos Sk -

sin

Sj,

m=i. .'. r = sin -ri - cos "r sin Sj + c o s "r cos Sk.

(3)

T h e angle, 07, b e t w e e n the s u n ' s unit v e c t o r and the p l a n e ' s unit v e c t o r is f o u n d b y taking the dot p r o d u c t o f r a n d R, viz.

2. A N A L Y S I S

C o n s i d e r a n o r t h o g o n a l set o f a x e s at a point o n the e a r t h ' s s u r f a c e aligned in the east, n o r t h a n d vertical d i r e c t i o n s . T h e unit v e c t o r s i, j and k define t h e s e d i r e c t i o n s r e s p e c t i v e l y (see Fig. !). T h e s u n ' s unit v e c t o r c a n b e defined in t e r m s o f the a z i m u t h angle a n d the altitude angle: R = c o s 13 sin ",/i + cos 13 cos ~'j + sin 13k.

(1)

R c a n b e e x p r e s s e d in t e r m s of the f u n d a m e n t a l e a r t h - s u n angles by s u b s t i t u t i n g the following relationships[3]: sin y = - s e c

c o s 07 = r . R

x (cos d) sin 8 - c o s 8 sin 6 cos to) + cos'r cos S × (cos + cos to c o s 8 + sin + sin 8).

cos Or = sin v c o s 8 sin to + c o s to cos T c o s 8 c o s ( S - 0 ) -

sin 13 = c o s + c o s ~o cos 8 + sin + sin 8,

(4)

E q u a t i o n (4) c a n be simplified by g r o u p i n g t e r m s a n d u s i n g the a p p r o p r i a t e t r i g o n o m e t r i c identities:

13 cos 8 sin to,

c o s ' r sin 8 sin(S -

+).

(5)

F o r a c o n s t a n t s p e e d of r o t a t i o n of 15° p e r h o u r , "r to.

cos -/ = sec 13 (cos + sin 8 - c o s 8 sin + cos to).

.'. c o s Or = c o s 8 sin-" to + c o s 8 c o s z to

. ' . R = cos 8 s i n t o i

× cos(S -

+ (cos + sin 8 - cos 8 sin + cos o~)j + (cos ~b c o s to cos 8 + sin ~b sin 8)k

= sin "r cos 8 sin to - cos "r sin S

(2)

~b) -

cos to sin 8 sin(S - cb).

(6)

F o r a c a s e o f a p o l a r m o u n t , S = 6 , a n d e q n (6) simplifies to the k n o w n r e l a t i o n s h i p [ l ]

3. APPLICATION--SINGLE AXIS E-W TRACKING

cos 0 r = c o s 8.

TO d e m o n s t r a t e the a p p l i c a t i o n o f the v e c t o r alg e b r a m e t h o d a n e x p r e s s i o n for the c o s i n e o f the

(7)

F o r t h e c a s e o f a fixed plane, "r = 0, e q n (5) r e d u c e s 429

P. G. JOLLY

430

VERTICAL k

yj

R

..,.....~ WEST

SOUTH I

i

/

--

"-- .....

N

"" ------

EAST

~

I

"1

"---

/

~l

~

/

n

NORTH

/"

Fig. 1. Vector diagram.

to the following known relationship[I] cos 0T = COS tO COS 5 cos(S - +) -

cussions. This work was performed while at the Mechanical Engineering Department, University of the West Indies, Trinidad.

sin 5 sin(S - ~b). (8) NOMENCLATURE

4. RESULTS AND DISCUSSIONS

The method presented in this paper can be used to derive an analytical expression for the cosine of the angle between the sun's direct rays and the unit normal of any desired inclined surface. The method requires a vector equation to be derived linking the unit normal vector of the desired surface and the unit vectors of the stationary orthogonal axes imbedded in the earth's surface. The vector dot product of this equation and the vector equation for the s u n ' s unit vector, gives the required relationship. It is hoped that the method presented here will assist solar designers in deriving solar angle equations and allow published equations to be readily c h e c k e d for p a r t i c u l a r a p p l i c a t i o n s .

13 3' + 13 to

altitude angle azimuth angle measured from north clockwise latitude (north positive) declination angle hour angle (zero at noon, mornings positive, afternoon negative) "r angular displacement from surface normal r to unit vector I. S inclination of rotation axis from horizontal

REFERENCES

1. J. A. Duffle and W. A. Beckman, Solar Engineering of Thermal PROCESSES, p. 16. Wiley-lnterscience, New York (1980). 2. F. Kreith and J. Kreider, Principles of Solar Engineering, p. 58. Hemisphere, Washington, D.C. (1978). 3. J. L. Threlkeld, Thermal Environmental Engineering, 2nd edn. pp. 288-291. Prentice-Hall. Englewood Cliffs. NJ (1970). 4. W. B. Stine and R. W. Harrigan. Solar Energy Fun-

Acknowledgements--The author wishes to thank Prof. S.

damentals and Design, with Computer Applications.

Satcunanathan of Mechanical Engineering for helpful dis-

Wiley, New York (1985).