Thermodynamic Availability of Solar Radiation with Special Attention

19 Thermodynamic Availability of Solar Radiation with Special Attention to Atmospheric Rayleigh Scattering R. H. EDGERTON and J. A. PATTEN Downloaded by UNIV OF PITTSBURGH on May 4, 2017 | http://pubs.acs.org Publication Date: November 11, 1983 | doi: 10.1021/bk-1983-0235.ch019

School of Engineering, Oakland University, Rochester, MI 48063

This paper examines the available energy flux of scattered solar radiation in the atmosphere. It is shown that the available energy to energy flux ratio for Rayleigh scattered solar radiation is approximately 0.80 to 0.90. This is for clear sky conditions and with solar energy flux near the zenith and near the horizon. The available energy to energy flux of Rayleigh scattered radiation is shown to be nearly constant over the sky. From this analysis i t can be implied that the solar radiation has high thermodynamic potential even i f diffusely scattered over the sky. In the evaluation of second law efficiencies of solar energy converters, a determination must be made of the available work function of the incident radiation. The author has derived this function for arbitrary spectral distributions and spacial or s o l i d angle radiation inputs (1). The results have shown that the spectral distribution of typical solar radiation gives available energy inputs much higher than the inputs assuming a thermal equilibrium distribution with the same energy flux. The spacial effect on the available energy flux has also been examined theoretically with the available energy loss with scattering described. The spacial loss is apparent in the limited focusing capability of optical systems for diffuse radiation. Both these effects require knowledge of the spectral and spacial radiation distributions of the radiation flux on a surface. The determination of both these distributions at a given location is a d i f f i c u l t instrumentation problem. In this paper, the effect of scattering processes in the atmosphere on the available energy of solar radiation on a surface is examined. Both the spectral and spacial effects of Rayleigh scattering are demonstrated. In the establishment of a measure of the available energy of radiation energy inputs to processes, the question of the 0097-6156/ 83/0235-0395$06.00/0 © 1983 American Chemical Society Gaggioli; Efficiency and Costing ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

SECOND LAW

Downloaded by UNIV OF PITTSBURGH on May 4, 2017 | http://pubs.acs.org Publication Date: November 11, 1983 | doi: 10.1021/bk-1983-0235.ch019

396

ANALYSIS OF PROCESSES

angular d i s t r i b u t i o n of the r a d i a t i o n becomes c r i t i c a l . The a u t h o r O ) and others (2-8) have attempted to resolve t h i s problem by i n t r o d u c t i o n of the angular or s p a c i a l dependence by d i s c u s s i n g the f l u x dependence on the s o l i d angle of the r a d i a t i o n input. In t h i s approach the entropy and a v a i l a b l e energy of a r a d i a t i o n input are a l l expressed as f l u x terms r a t h e r than as thermodynamic p r o p e r t i e s of space. I f one examines the question of a v a i l a b l e energy of s o l a r r a d i a t i o n , two approaches appear a t t r a c t i v e . The f i r s t i s a quantum p e r s p e c t i v e that the electromagnetic energy of the photon stream from the sun i s a l l c o n v e r t i b l e to thermodynamic work. The second, based on the engineering p e r s p e c t i v e of d e a l i n g with s o l a r c o l l e c t o r s which u t i l i z e the s o l a r energy to heat substances i s that a thermodynamic l i m i t i s imposed by the sun surface temperature as a r a d i a t i n g object at approximately 6000K. For p r a c t i c a l purposes both these answers are s u f f i c i e n t i f one i s i n t e r e s t e d i n second law a n a l y s i s of energy conversion devices operated outside the atmosphere. A "heat" source of 6000K has an a v a i l a b l e energy f l u x which i s nearly equal to i t s radiant energy f l u x . The measurement of the s p e c t r a l d i s t r i b u t i o n of s o l a r r a d i a t i o n outside the atmosphere and the subsequent a s s o c i a t i o n of t h i s s p e c t r a l d i s t r i b u t i o n with the s p e c t r a l d i s t r i b u t i o n of r a d i a t i o n i n a blackbody c a v i t y has, I b e l i e v e , biased the attempts to c h a r a c t e r i z e the a c t u a l r a d i a t i o n i n the atmosphere to an undue extent. Figure 1 i n d i c a t e s t y p i c a l s p e c t r a l d i s t r i b u t i o n s of r a d i a t i o n i n the atmosphere as compared to that of s o l a r r a d i a t i o n outside the atmosphere. Outside the atmosphere m • 0 and i f the f l u x i s d i r e c t l y through m - 1. I f slanted at and angle from the z e n i t h angle 0 , then m i s approximately 1/cos 9. The processes of s c a t t e r i n g and absorption of radiation i n the atmosphere so s i g n i f i c a n t l y a l t e r the s p e c t r a l d i s t r i b u t i o n that any s i m i l a r i t y to extra t e r r e s t r i a l r a d i a t i o n i s almost coi n c i d e n t a l . Experiments with r a d i a t i o n between surfaces have shown that blackbody r a d i a t i o n theory can be extended successf u l l y to many r a d i a t i o n heat t r a n s f e r s i t u a t i o n s . In these s i t u a t i o n s the s t r i c t e q u i l i b r i u m requirements of the i n i t i a l model have so f a r not proved to be necessary f o r p r a c t i c a l designs. Most importantly the concept of temperature has proved u s e f u l i n non-equilibrium r a d i a t i o n f l u x s i t u a t i o n s ( 3 ) • The s p a c i a l d i s t r i b u t i o n of s o l a r r a d i a t i o n i s the other important c h a r a c t e r i s t i c of t e r r e s t r i a l r a d i a t i o n which a f f e c t s i t s a v a i l a b l e energy. When an earth-sun system i s examined i n a " u n i v e r s a l " p e r s p e c t i v e , the sun can be v i s u a l i z e d as a s p h e r i c a l r a d i a t i o n emitter with the energy emitted from the source expanding out i n t o the universe (a divergent f l u x ) . The r a d i a t i o n energy density of space decreasing with distance from the sun. The energy f l u x per u n i t area decreasing but the O

0

Gaggioli; Efficiency and Costing ACS Symposium Series; American Chemical Society: Washington, DC, 1983.


EDGERTON AND PATTEN

Availability of Solar Radiation

397

F i g u r e 1. S p e c t r a l d i s t r i b u t i o n of d i r e c t s o l a r r a d i a t i o n through a c l e a r atmosphere of d i f f e r e n t a i r masses.


398

SECOND LAW


i n t e n s i t y or f l u x per u n i t area per s o l i d angle remaining constant. This constant i n t e n s i t y means that the temperature or s p e c t r a l d i s t r i b u t i o n of the r a d i a t i o n remains constant even though the f l u x per u n i t area i s decreasing. In an earth perspective however, the sun appears as a v i s i b l e disk with r a d i a t i o n confined w i t h i n a cone with convergence of r a d i a t i o n w i t h i n a s o l i d angle of approximately 7 x 10"* steradians. This s o l i d cone angle i s c r i t i c a l i n the development of focusing c o l l e c t o r s used to increase the f l u x per u n i t area or energy density of the electromagnetic r a d i a t i o n . This i n turn l i m i t s the maximum temperature obtainable with a passive c o l l e c t o r to the sun temperature of 6000K at a c o l l e c t o r concent r a t i o n r a t i o of approximately 40,000 to 1. Atmospheric s c a t t e r i n g f o r the sun near the horizon can increase t h i s s o l i d angle to 0.05 steradian. In r a d i a t i o n through the atmosphere, the electromagnetic energy i s s c a t t e r e d and absorbed so that part of the r a d i a t i o n i s observed to be d i s t r i b u t e d over the e n t i r e sky. The radi a t i o n w i t h i n the sun s o l i d angle i s u s u a l l y considered as d i r e c t r a d i a t i o n , but i n a d d i t i o n , contains smaller amounts of s c a t t e r e d r a d i a t i o n . The s c a t t e r i n g and absorption processes depend both s p a c i a l l y and s p e c t r a l l y upon the atmosphere comp o s i t i o n and cloud d i s t r i b u t i o n . I t i s u s e f u l to a r b i t r a r i l y separate the input r a d i a t i o n to a surface i n t o the d i r e c t and d i f f u s e or s c a t t e r e d r a d i a t i o n . Figure 2 i n d i c a t e s the r e l a t i v e s p e c t r a l magnitudes of each of these components. In a d d i t i o n , "sky r a d i a t i o n " i n d i c a t i n g r a d i a t i o n from the gases i n the atmosphere are sometimes included. In a r a d i a t i o n c o o l i n g s i t u a t i o n t h i s i s the r a d i a t i o n f l u x which governs the net f l u x from a surface at night. This sky r a d i a t i o n i s dependent on the s o l a r absorption heating of the atmosphere and has been estimated as the order of magnitude of 20% of the e x t r a t e r r e s t r i a l f l u x . The a v a i l a b l e energy of t h i s f l u x i s small because the temperature i s c l o s e to the environmental temperat u r e . The sky has an equivalent e m i s s i v i t y of about 0.75. For a sky energy f l u x of 300 W/m^ the temperature i s approximately 270 K. In energy f l u x terms the input r a d i a t i o n appears to be reasonably considered as a d i r e c t f l u x i n the cone subtended by the s o l a r d i s k . The s c a t t e r e d r a d i a t i o n i s d i s t r i b u t e d over the sky i n complicated geometrical ways depending on the p o s i t i o n of the sun. The two p r i n c i p a l s c a t t e r i n g mechanisms are molecular or Rayleigh s c a t t e r i n g , and Mie or p a r t i c l e s c a t t e r i n g . Figure 3 shows a representative sky d i s t r i b u t i o n of Rayleigh s c a t t e r e d light. To examine the a v a i l a b l e energy of s o l a r r a d i a t i o n , l e t us b r i e f l y examine the thermodynamic r e s u l t s f o r blackbody radi a t i o n w i t h i n an i n s u l a t e d c a v i t y . There i s no net f l u x of


5



EDGERTON AND PATTEN

Wavelength 10

2

I


I II i—|—Mill

0 7

|i i — |

(I0~

6

m)

0-3

0 5

1

1

1

F i g u r e 2. S p e c t r a l d i s t r i b u t i o n of s c a t t e r e d s o l a r radi a t i o n compared to e x t r a t e r r e s t r i a l and d i r e c t s o l a r radiation. 180

0 F i g u r e 3. D i s t r i b u t i o n of Rayleigh s c a t t e r e d r a d i a t i o n over the sky f o r the sun at 45° from the z e n i t h as shown. I n t e n s i t y curves are normalized to 15 at the z e n i t h .


400

SECOND LAW ANALYSIS OF PROCESSES

energy at any point, and the r a d i a t i o n i s uniformly d i s t r i buted over a s p h e r i c a l angle of 4TT. The expected energy i n the volume V at a temperature T can then be shown t o be

E - 0.658 (2irkT/hc) kTV 3

In t h i s expression, h i s Planck's constant,k i s Boltzmann's constant and c i s the speed of l i g h t .


The entropy i s S = (4/3)E/T The G i b b s ' f r e e energy i s G » E + P V - T S = 0 The closed system a v a i l a b l e energy i s 4

A » E + P V - T S - E(l +(T /T) /3 -(4/3)T /T) Q

The open system A v a i l a b l e

Q

Q

Q

energy i s

B = H - T S = (4/3)(1 - T / T ) E - (1 - T / T ) H 0

Note: T

0

0

0

i s the environmental temperature and H i s the enthalpy.

In d e a l i n g with problems of s o l a r r a d i a t i o n , as opposed to blackbody r a d i a t i o n , the e f f e c t of the s o l i d angle i n which the r a d i a t i o n i s confined has been examined (2-4) by considering the volume density of photons to be reduced. Landsberg(6) considers d i l u t e r a d i a t i o n i n the sense that the s p e c t r a l d i s t r i b u t i o n i s retained but the r a d i a t i o n density i s reduced. This leads to d e f i n i n g the temperature of a s p e c t r a l component as T = hv/k l n ( l +

3

2

(hv G/c e ) v

Here ft i s the s o l i d angle of the r a d i a t i o n and e i s a s p e c t r a l energy f l u x per u n i t area per u n i t frequency. v

Press (3) makes a s i m i l a r approximation , comparing the closed system a v a i l a b l e energy of a volume of r a d i a t i o n i n a cone compared to that i n a f u l l s p h e r i c a l space. H i s r e s u l t s t r a n s l a t e d i n t o f l u x terms i n d i c a t e that approximately 38% of the a v a i l a b l e energy f l u x i n a s o l a r s o l i d angle i s retained i f the energy i s uniformly s c a t t e r e d over a f u l l 4irsolid angle.



19.

EDGERTON AND PATTEN


401

Scattered r a d i a t i o n has g e n e r a l l y been considered as l e s s a v a i l a b l e f o r energy conversion because of i t s d i f f u s e s p a c i a l d i s t r i b u t i o n . This d i f f u s e s p a c i a l d i s t r i b u t i o n imposes a c o l l e c t i o n l i m i t on c l a s s i c a l focusing systems. The c a p a b i l i t i e s of i n c r e a s i n g the energy density or f l u x per u n i t area are reduced by s c a t t e r i n g processes. The s p a c i a l f l u x charact e r i s t i c s are a l s o important f o r f l a t p l a t e c o l l e c t o r s since the energy a b s o r p t i v i t y of a surface i s dependent on the i n c i dent angle of the r a d i a t i o n . The a b s o r p t i v i t y f o r normal i n c i dence being high compared to that f o r low i n c i d e n t angle radiation. The r a d i a t i o n absorption and s c a t t e r i n g of d i r e c t r a d i a t i o n by the atmosphere decreases the d i r e c t r a d i a t i o n f l u x at a s u r f a c e . The s c a t t e r i n g however increases the d i f f u s e component of the energy f l u x ( 9 ) . The Rayleigh and Mie s c a t t e r i n g of l i g h t by the atmosphere are w e l l understood. S c a t t e r i n g by clouds however presents an unsolved problem because of computational and measurement d i f f i c u l t i e s associated with the l a r g e v a r i e t y of cloud c h a r a c t e r i s t i c s . Analysis In t h i s paper two basic conditions have been assumed. 1. In r a d i a t i o n problems, i t i s the a v a i l a b l e energy f l u x which i s of concern, not the a v a i l a b l e energy density. This means that d i r e c t i o n a l c h a r a c t e r i s t i c s become more important and the steady flow a v a i l a b l e energy, b = h - T s i s the more a p p r o p r i a t e measure of thermodynamic work. 2. The s p e c t r a l d i s t r i b u t i o n of the r a d i a t i o n at the earth surface i s only weakly r e l a t e d to the s p e c t r a l d i s t r i b u t i o n outs i d e the atmosphere. This means that a measurement of the spect r a l d i s t r i b u t i o n and s p a c i a l d i s t r i b u t i o n w i l l u s u a l l y be requ i r e d at a s i t e to determine the a v a i l a b l e energy f l u x . C a l c u l a t i o n r e s u l t s are presented f o r the a v a i l a b l e energy f l u x at a h o r i z o n t a l surface due to Rayleigh s c a t t e r e d r a d i a t i o n i n the atmosphere. Rayleigh s c a t t e r i n g was chosen because of the good p r e d i c t i o n of t h i s component of energy f l u x both spect r a l l y and s p a c i a l l y under c l e a r sky c o n d i t i o n s . I t a l s o provi d e d a non uniform s p a c i a l d i s t r i b u t i o n which could i l l u s t r a t e f l u x aspects of the a v a i l a b l e energy of s o l a r r a d i a t i o n . The R a y l e i g h s c a t t e r i n g model of Coulson (10) was used t o c a l c u l a t e the a v a i l a b l e energy f l u x from the energy f l u x . Determination of the a v a i l a b l e energy of s u n l i g h t i f d i f fused uniformly over the sky have been p r e v i o u s l y examined by the author (_1) with the r e s u l t s i n d i c a t e d i n Figures 4 and 5. The p r i n c i p a l r e s u l t being that the a v a i l a b l e energy to energy f l u x r a t i o f o r uniform s o l a r r a d i a t i o n i s from 0.5 t o 0.7. Q



402


WAVELENGTH (MICRON) F i g u r e 4. A v a i l a b l e e n e r g y o f s o l a r r a d i a t i o n i f s c a t t e r e d u n i f o r m l y over the sky as a f u n c t i o n o f the upper wavelength r e c e i v e d by a c o l l e c t o r . Reproduced w i t h p e r m i s s i o n from R e f . 1, C o p y r i g h t 1980, Pergamon P r e s s I n c .

0

1

2

3 WAVELENGTH

4

(MICRON)

F i g u r e 5. A v a i l a b l e energy to energy f l u x r a t i o f o r s o l a r r a d i a t i o n i f s c a t t e r e d uniformly over the sky. Reproduced w i t h permission from R e f . O ) Copyright 1980, Pergamon Press Inc. Gaggioli; Efficiency and Costing ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

19.

Assumptions

and Procedures

In the c a l c u l a t i o n s of the a v a i l a b l e energy f l u x reported i n t h i s paper f o r Rayleigh s c a t t e r e d s o l a r r a d i a t i o n , the f o l l o w i n g assumptions and procedures have been u t i l i z e d . The Coulson-Dave-Sekera Tables (10) of d i f f u s e r a d i a t i o n emerging from the bottom of a p l a n e - p a r a l l e l atmosphere due t o R a y l e i g h s c a t t e r i n g were used to c a l c u l a t e the s p e c t r a l energy i n t e n s i t y e . e i s the energy f l u x per u n i t area, per u n i t frequency v, per u n i t s o l i d angle w.This c a l c u l a t i o n procedure i s o u t l i n e d i n the appendix. I n t h i s model, m u l t i p l e s c a t t e r i n g and a ground r e f l e c t i v i t y of 0.25 i s assumed. The input s p e c t r a l r a d i a t i o n f l u x outside the atmosphere F i s taken from the data of Thomas and Thekaekara (11). The s p a c i a l geometry u t i l i z e d i n t h i s d e s c r i p t i o n i s shown i n Figure 6. The b a s i c equations used to evaluate the a v a i l a b l e energy f l u x are d e r i v e d i n r e f e r e n c e d ) . They are o u t l i n e d below. The steady flow a v a i l a b l e energy f l u x per u n i t area i s taken as vu)


403


EDGERTON AND PATTEN

v a )

o v

b - e + pv - T s 0

T h i s i s found by i n t e g r a t i o n of the s p e c t r a l and s p a c i a l funct i o n s over the spectrum of the r a d i a t i o n and the s o l i d angle over the hemisphere above the s u r f a c e . The a v a i l a b l e energy intensity i s b

e

v o T vu> +

3

2

(2hv /c )(l-(ln(D)/ln(l-D))) 2

2

- k T ( (2 v / c ) l n ( l+( 1/D) )+( e /hv ) l n ( 1+D) 0

va)

3

2

Note: D - 2 h v / c e

v w

The d i r e c t i o n of the measurement of the energy f l u x i s g i v e n by the z e n i t h angle 0(or u^cosS) and the azimuth angle < | > of a v e r t i c a l plane through the d i r e c t i o n of observat i o n . The sun i s taken a t an azimuth angle of zero. The d i r e c t i o n of the sun i s then noted as U Q ^ C O S S Q and -0. The energy f l u x through a h o r i z o n t a l s u r f a c e e i s then found from the a c t u a l energy f l u x from a s o l i d angle as o

v a )

e

V(D

a!

(e o/ft)cos 6 pv

The reason f o r t h i s c o m p l i c a t i o n i n the computation i s that e Q i s the s p e c t r a l energy f l u x one would expect to measure with a s p e c t r a l radiometer with an aperture which accepts r a d i a t i o n i n a s o l i d angle Q. T h i s energy f l u x i s then measured on a s u r f a c e p e r p e n d i c u l a r to the d i r e c t i o n i n which i t i s pointed. The s u r f a c e f o r which the energy f l u x i s to be found w i l l be a t a f i x e d angle and only the cos0component w i l l be a f l u x through the s u r f a c e . p V



404


F i g u r e 6. Geometry used f o r c a l c u l a t i o n of a v a i l a b l e energy f l u x and energy f l u x with s p a c i a l l y v a r i a b l e i n t e n sity.


19.

405


EDGERTON AND PATTEN

Integrations over the spectrum and s p h e r i c a l angles were done n u m e r i c a l l y with azimuth angle increments A* 30°* 0.5236 radians, and z e n i t h angle increments Au A(cos9) =0.1. The a v a i l a b l e energy f l u x per u n i t frequency i s computed as 88

b

v

s

b

I I va> A(cos8)A cos 6 < ) >


The t o t a l a v a i l a b l e energy f l u x i s then b - £b v

v

Av

The c a l c u l a t i o n of b was s i m p l i f i e d by noting that f o r Rayleigh scattered solar radiation D »

1

Thus, l n ( l + 1/D) i s approximated by 1/D. f l u x on a h o r i z o n t a l s u r f a c e i s then b

V(D [e Q/ft+2hv /c (ls s

3

2

p V

- kT (e 0

p v Q

The a v a i l a b l e

energy

(ln(D)/(ln(l+D))))

/ h v f t ) ( l + l n ( l + D ) ) ] cos6

To the same l e v e l of approximation, the s p e c t r a l of a given s p e c t r a l energy f l u x component i s

temperature

T - (hv/k)ln(D) C a l c u l a t e d values of the separate components of the a v a i l a b l e energy f l u x are shown i n Table I I I . These are tabulated as A,B, and C as follows A

Energy Term :

e

B

Pressure Term

C

Entropy Term : - k T ( e p / h v f l ) [ l + l n ( l + D ) ]

p v

o/ Q 3

2

:(2hv /c )[i-(ln(D)/(ln(l+D)))] 0

pv


406

SECOND LAW


Discussion

of


Results*

The r e s u l t s of the computation of the a v a i l a b l e energy of Rayleigh s c a t t e r i n g are presented i n Tables I-V. In d i s c u s s i n g the r e s u l t s , i t should be kept i n mind that i f a l l the r a d i a t i o n could be converted to thermodynamic work then the r a t i o b/e would be equal to 4/3. In the steady flow a v a i l a b l e energy formulation the r a d i a t i o n pressure a l s o i s a v a i l a b l e f o r producing u s e f u l work. Let us begin by examining the o v e r a l l r e s u l t s and then d i s cuss the separate f a c t o r s which account f o r the r e s u l t s . Table I i n d i c a t e s that the r a t i o of a v a i l a b l e energy to energy f l u x on a h o r i z o n t a l surface i s 0.89 f o r the sun at the z e n i t h y » l and 0.82 f o r the sun at Uo^O.l or approximately 6° above the h o r i z o n . This i s a s u r p r i s i n g r e s u l t when compared to statements i n the l i t e r a t u r e concerning the low q u a l i t y of d i f f u s e s o l a r energy. In order to e x p l a i n t h i s d i f f e r e n c e l e t us f i r s t c o n s i d e r Table I which shows the v a r i a t i o n of the a v a i l a b l e energy to energy f l u x r a t i o as a f u n c t i o n of the wavelength of the s c a t t e r e d r a d i a t i o n . This table i l l u s t r a t e s the expected r e s u l t that the s h o r t e r wavelength r a d i a t i o n has a higher a v a i l a b l e energy than the long wavelength r a d i a t i o n as noted before i n the d i s c u s s i o n of uniform r a d i a t i o n f i e l d s . The a v a i l a b l e energy to energy r a t i o i s also higher f o r the sun at the z e n i t h than at the horizon. This d i f f e r e n c e i s not s i g n i f i c a n t and i s w i t h i n the computational e r r o r expected. The magnitude however requires some d i s c u s s i o n r e l a t i v e to the r e s u l t f o r d i r e c t s o l a r r a d i a t i o n . D i r e c t s o l a r r a d i a t i o n of energy f l u x 1.353 kW/m outside the atmosphere i n a s o l i d angle ft • 6.78xl0~5 would give an a v a i l a b l e energy to energy f l u x r a t i o of 1.26 thus the Rayleigh s c a t t e r e d r a d i a t i o n has reduced the a v a i l a b l e energy from the d i r e c t s o l a r energy. Computation of a v a i l a b l e energy to energy f l u x r a t i o s f o r a c t u a l d i r e c t s o l a r r a d i a t i o n , i f uniformly s c a t t e r e d over the f u l l sky hemisphere (as noted i n Figure 5 ), shows that the a v a i l a b l e energy to energy f l u x r a t i o i s smaller. The reason that the R a y l e i g h s c a t t e r e d r a d i a t i o n has a higher a v a i l a b l e energy than t h i s uniformly d i s t r i b u t e d s o l a r r a d i a t i o n , i s that Rayleigh s c a t t e r i n g i s predominantly a s c a t t e r i n g of short wavelength l i g h t as shown i n F i g u r e 7. Rayleigh s c a t t e r i n g decreases as X"". Thus the spectrum of s c a t t e r e d r a d i a t i o n i s s h i f t e d toward the short wavelengths compared to d i r e c t s o l a r r a d i a t i o n . T h i s s h i f t i n g i s a l s o shown i n Figure 2. The spectrum f o r the R a y l e i g h s c a t t e r e d l i g h t i s a high a v a i l a b l e energy spectrum. Examination of Table II f u r t h e r c l a r i f i e s the d i f f e r e n c e 0

2

4


407


EDGERTON AND PATTEN

Table I. A v a i l a b l e Energy to Energy Flux Ratio f o r Rayleigh S c a t t e r e d S o l a r R a d i a t i o n as a Function of Wavelength


Transmission

Wavelength

Sun Near Horizon u = U.l (b /e ) 0

X(y)

T

0.02 0.05 0.10 0.15 0.25 0.50 1.00

v

0.82 0.65 0.55 0.50 3.44 .37 .32

v

Sun At Zenith u = l.U (b /e ) 0

v

v

0.73 0.80 0.84 0.86 0.88 0.89 1.01

0.68 0.77 0.80 0.82 0.85 0.88 0.97

O v e r a l l A v a i l a b l e Energy to Energy Flux R a t i o s . Sun At Z e n i t h Sun Near Horizon

b/e =0.89 b/e =0.82

Table I I . A v a i l a b l e Energy to Energy Flux Ratio Per Unit Frequency (b^/e^) as a Function of Azimuth Angle

Z e n i t h Angle 6° 60°

30°

Azimuth Angle 60° 120° 90°

0.79

0.81

0.81

0.79

0.81

0.81

0.81

0.82

0.82

0.84

0.82

0.82

0.82

0.82

0O

150° 180°



408

Table I I I . A v a i l a b l e Energy Flux Components b ^ on a H o r i z o n t a l Surface f o r the Sun at the Z e n i t h . ©voT* A cos 6, pv = B cos 6, o vuT » M* » Transmission Parameter (See Table I ) . U n i t s are fW-s/m V(i)

T

s

c

c

o

s

e

c

o

s

e

T

i

s

a

2


y/x

0.02

0.1

A B C

0.3

A B C

0.05

0.50

1.00

11.75 0.698 2.41 1.00

13.5 0.800 2.53 1.18

12.1 0.694 2.05 1.07

6.3 0.343 0.96 0.57

2.9 0.151 0.40 0.27

1.6 0.089 0.50 vu> 0.35

3.6 0.210 0.91 0.87

5.5 0.315 1.18 1.40

7.2 0.408 1.39 1.86

7.8 0.436 1.35 2.06

5.4 0.290 0.83 1.46

3.1 0.162 0.42 0.85

A B C

1.0 0.058 0.34 0.38

2.5 0.137 0.63 0.98

3.8 0.214 0.83 1.61

5.1 0.287 1.01 2.21

5.9 0.328 1.04 2.60

4.6 0.245 0.71 2.06

3.0 0.155 0.42 1.39

0.8 0.045 0.27 vu> 0.42

2.0 0.110 0.51 1.10

3.2 0.176 0.65 1.88

4.3 0.238 0.85 2.57

5.0 0.275 0.89 3.09

4.1 0.219 0.65 2.60

3.0 0.151 0.41 2.70

0.7 0.040 0.24 vu> 0.53

1.7 0.096 0.45 1.38

2.9 0.153 0.62 2.33

3.9 0.214 0.77 3.31

4.6 0.250 0.77 4.06

3.9 0.208 0.61 3.50

2.9 0.048 0.40 2.68

VU)

A B C b

1.0

0.25

8.8 0.530 2.11 0.72

b

0.7

0.15

4.2 0.252 1.27 0.35

b

0.5

0.10

A B C b



19. EDGERTON AND PATTEN

409

Table IV. Sample A v a i l a b l e Energy Flux Per Unit S o l i d Angle and Frequency D i s t r i b u t e d Over the Sky f o r the Sun Near the Horizon (u= 0.1). U n i t s are fW-s/m 2

WAVELENGTH


0 0.1 0.3 0.5 0.7 1.0

0.023 0.093 0.18 0.22 0.21

0

y/+ 0.1 0.3 0.5 0.7 1.0

0.70 0.80 0.76 0.69 0.43

30

60

0.022 0.086 0.17 0.20 0.21

30

0.821 MICRONS 120

90 0.021 0.073 0.15 0.17 0.21

0.020 0.066 0.14 0.16 0.21

0.021 0.071 0.15 0.17 0.21

180

150

WAVELENGTH 0.395 MICRONS 150 120 60 90

0.62 0.71 0.68 0.64 0.43

0.46 0.53 0.53 0.53 0.43

0.37 0.44 0.45 0.44 0.43

0.45 0.50 0.49 0.47 0.43

0.022 0.089 0.17 0.19 0.21

0.021 0.082 0.16 0.18 0.21

180

0.61 0.66 0.64 0.55 0.43

0.69 0.78 0.70 0.61 0.43

Table V. Equivalent Temperature of Rayleigh Scattered S o l a r R a d i a t i o n f o r C l e a r Sky Conditions Sun P o s i t i o n cos8 0

Sky D i r e c t i o n cos9

Wavelength microns

Temper< K

1.0 1.0 1.0 1.0

0.1 1.0 0.1 1.0

0.316 0.316 0.821 0.821

2290 2290 1050 950

0.6 0.6 0.6 0.6

0.6 1.0 0.6 1.0

0.316 0.316 0.821 0.821

2230 2160 1035 905

0.1 0.1 0.1 0.1

0.1 1.0 0.1 1.0

0.316 0.316 0.821 0.821

1870 1830 930 810


410



RAYLEIGH SCATTERING

FRACTION

.5

.6 .7 WAVELENGTH ( MICRONS )

F i g u r e 7. F r a c t i o n of d i r e c t s o l a r r a d i a t i o n s c a t t e r e d by molecular c o n s t i t u e n t s of the atmosphere (Rayleigh s c a t t e r i n g ) as a f u n c t i o n of wavelength.


19.

EDGERTON AND PATTEN


411

i n the a v a i l a b l e energy as a f u n c t i o n of wavelength. In direct s o l a r r a d i a t i o n outside the atmosphere the pv term i s a p p r o x i mately 0.33 of the energy f l u x and the T s term approximately 0.07 of the energy f l u x . I n the R a y l e i g h s c a t t e r e d r a d i a t i o n the pv term i s approximately 0.05 of the energy f l u x and i s e s s e n t i a l l y constant over wavelength and sky d i r e c t i o n . The pv term reduction has a small p r a c t i c a l s i g n i f i c a n c e since t e c h n i c a l methods to u t i l i z e t h i s e f f e c t have so f a r not been developed. The T s term however ranges from 0.35 a t long wavelengths to 0.13 at short wavelengths and i s important. The entropy e f f e c t i s however n e a r l y constant over the sky so that h o r i z o n s c a t t e r i n g does not have a strong e f f e c t on the a v a i l a b l e energy f l u x . I t i s of i n t e r e s t f o r a s s i g n i n g an equivalent temperature to s c a t t e r e d r a d i a t i o n to examine Table I I I . Equivalent tempe r a t u r e s c a l c u l a t e d using Wein's d i s t r i b u t i o n 0


0

T

3

2

»(hv/k)ln(2hv ft/c e

pvfl

)

a

are tabulated f o r the cases of the sun near the h o r i z o n y 0.1 and at the z e n i t h U o ^ l . O v e r a l l , these r e s u l t s i n d i c a t e t h a t the equivalent temperature f o r R a y l e i g h s c a t t e r e d s o l a r r a d i a t i o n ranges from around 2300K to 800K. For the sun at the z e n i t h the horizon b r i g h t e n i n g i s c l e a r w i t h equivalent temperatures there higher than d i r e c t l y overhead. The wavelength i s the dominant f a c t o r however with short wavelength temperatures of 2000K compared to long wavelenth temperatures of 1000K. I t should be noted that these temperatures are i n the range of temperatures that Bosnjakovic (12) assumes f o r h i s approximation f o r d i f f u s e r a d i a t i o n i n c a l c u l a t i n g a v a i l a b l e energy f l u x e s . 0

Conclusions The a v a i l a b l e energy f l u x of R a y l e i g h s c a t t e r e d s o l a r r a d i a t i o n i n the atmosphere was examined i n t h i s paper. I t was demons t r a t e d that the a v a i l a b l e energy to energy f l u x r a t i o on at the earth surface i s approximately 80% f o r m o l e c u l a r l y s c a t t e r e d s o l a r energy. T h i s c l a r i f i e s f u r t h e r the r o l e of s p e c t r a l and s p e c i a l d i s t r i b u t i o n e f f e c t s on the a v a i l a b l e energy of s o l a r energy i n the atmosphere. I t r e i n f o r c e s the p r i n c i p a l importance of the s p e c t r a l d i s t r i b u t i o n of the r a d i a t i o n i n determining the q u a l i t y of the r a d i a t i o n . Scattering




412

processes in the atmosphere reduce the energy flux on surfaces but do not significantly reduce the available energy to energy flux ratio* The results indicate that efforts to u t i l i z e scattered radiation should be directed toward research in spectral selective devices with less concern for the spacial distribution* The collection of scattered radiation remains an engineering challenge* One need be less concerned about the quality but more at producing a high energy flux* One outcome of this research may be more effort being placed on the u t i l i z a t i o n of emitted radiation from collective surfaces which have so far been thought to have low quality because of their wide spacial distribution. It is hoped that this work w i l l further efforts directed to using combined energy conversion systems where the transmitted, reflected and emitted radiation from a collection system are u t i l i z e d . Literature Cited 1. 2. 3. 4. 5. 6. 7. 8.

Edgerton, R. H. Energy, Int. J. 1980, 5, 693-707. Press, W. H. Nature 1976, 264, 734-735. Landsberg, P . T . ; Tonge, G. J.Phys. A . , 1979, 12, 551-562. Leontovich, N. A. Sov. Phys-USP 1975, 6, 963-964. Parrott, J. E . Solar Energy 1978, 21, 227. Landsberg, P. J.; Tonge, G. J. Appl. Phys. 1980, 51, R1-R20. Jeter, S. M. Solar Energy 1981, 26, 231-236. Landsberg, P. T. "Thermodynamics and S t a t i s t i c a l Mechanics"; Oxford University Press: Oxford, 1978. 9. Coulson, K. L . "Solar and Terrestrial Radiation"; Academic Press: New York, 1975. 10. Coulson, K . L . ; Dave, J. V . ; Sekera, Z., "Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering"; University of California, Berkeley, California, 1960. 11. Thomas, A. P.; Thekaekara, M. P . , "Experimental and Theoretical Studies of Solar Energy for Energy Conversion"; NASA/Goddard Space Flight Center, 1973. 12. Bosnjakovic, F . M . , "Studies in Heat Transfer"; Hartnett, J. et al., E d . ; Hemisphere Publishing Co: New York, 1979.

Appendix The computational procedure for the calculation of energy flux per unit frequency per unit solid angle e due to Rayleigh scattered solar radiation is outlined in this section. The geometry of Figure 6 is used to describe the calculations. The total energy flux through a surface due to a spectral radiation intensity I is given by integrating the intensity over the hemispherical space and the frequency as vw

v


19.

413


EDGERTON AND PATTEN

e * J 7 / l c o s e s i n e d6ddv v

f

If I i s i s o t r o p i c designated as I t h e n the energy would be v

v

flux

e'= i r / V d v The energy f l u x can then be found r e l a t i v e t o t h i s i s o t r o p i c energy f l u x as e = (e /Tr)///(I /l )cosesin6d8d

Thermodynamic Availability of Solar Radiation with Special Attention

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