7/26/2019 Solar Energy Volume 18 Issue 3 1976 [Doi 10.1016%2F0038-092x%2876%2990027-x] Ari Rabl -- Tower Reflector

1/3

Solar Energy,

Vol. 18, pp. 26%271. Pergamon Pr ess 19 76. printed in Great Britain

T E C H N I C A L N O T E

Tow er ref lector for so lar power plant t

A R I R A B L

S o l a r E n e r g y G r o u p , A r g o n n e N a t i o n a l L a b o r a t o r y , A r g o n n e , I L 6 0 43 9 , U . S . A .

(Received 15 August 1975; in revised orm 14January 1976)

S e v e r al d i f f e re n t p o w e r p l a n t c o n c e p t s h a v e b e e n p r o p o s e d f o r t h e

c o n v e r s i o n o f s o l a r e n e r g y t o e l e c t r i c i t y . O n e o f t h e m o s t

p r o m i s i n g s c h e m e s , t h e c e n t r a l r e c e i v e ~ o r p o w e r t o w e r / l / ,

emp loys a l a rge f ie ld o f he l io s ta t s which co l lec t s l igh t and

t r a n s m i t s i t t o a c e n t r a l a b s o r b e r l o c a t e d o n t o p o f a t o w e r . H i g h

co l lec t ion tempera tu re and e f f ic iency a re th e ou ts tand ing advan-

tages o f th i s app roach . How ever , s ince the towe r he ight li e s in the

range f rom 100 to 500 m, the re a re p roblem s a ssoc ia ted wi th the

h e a t t r a n s f e r f r o m t h e a b s o r b e r a t t h e t o p o f t h e p o w e r p l a n t a t t h e

b a s e o f t h e t o w e r . I t w o u l d b e d e s i r a b l e t o t r a n s p o r t t h e e n e r g y i n

the f o rm o f so la r r ad ia t ion al l the way to the pow er p lan t . In th i s

n o t e w e p r o p o s e a m e t h o d , c a ll e d t o w e r r e f l e c t o r , b y w h i c h t h is

c a n b e a c c o m p l i s h e d w i t h o u t e x c e s s i v e o p t i c a l l o ss e s . P r e l i m i n a r y

e s t i m a t e s a p p e a r f a v o r a b l e , a n d w e r e c o m m e n d t h a t a d e t a i l e d

s y s t e m s a n a l y s i s b e p e r f o r m e d c o m p a r i n g t h e t o w e r r e f l e ct o r w i th

t h e p o w e r t o w e r .

In p r inc ip le , a ho l low cyl ind r ica l p ipe could t r ansmi t the

c o n c e n t r a t e d s o l a r r a d i a t i o n f r o m t h e t o p o f t h e t o w e r t o t h e

bo t tom , but th i s wou ld en ta i l in to le rab le op t ica l lo sses . I f r ad ia t ion

is concen t ra ted on to a sma l l a rea a t the top o f the tower , i t w i l l

necessa r i ly have a l a rge angula r sp read , and h ence i t w i l l unde rgo

many re f lec t ions when t ransmi t ted through a cy l ind r ica l passage

to the base o f the tower . ( To tal ly d i f f use rad ia t ion makes on th e

ave rage ( n ) = l / r re f lec t ions [ 2] w hen pass ing thro ugh a cy l ind r ica l

re f lec to r o f l eng th l an d rad ius r ) . T he num ber o f re f lec t ions can

be kep t reasonably sma l l on ly i f the angula r d iv e rgence 0 o f the

r a d i a ti o n c o m i n g f r o m t h e t o p o f t h e t o w e r i s c o n f in e d t o v a l u e s

b e l o w a b o u t 5 But in tha t ca se i t i s ea sy to sho w tha t the ra t io o f

towe r rad ius to he l io s ta t f i eld rad ius mu s t be a t l ea s t s in 0 = 0 .1

e v e n i f p e r f e c t o p t i c s a r e u s e d . S u c h a f a t t o w e r s e e m s

imprac t ica l . Re f lec t ion lo sses could , o f course , be avo ided by

means o f a so l id l igh t p ipe , r e ly ing on to ta l in te rna l r e f lec t ion ;

un f o r tuna te ly , such a me thod would requi re too much ma te r ia l .

A be t te r and s imp le r so lu t ion i s based on the tower re f lec to r

show n in F ig. 1 . Her e the incoming sun ligh t i s r e f lec ted f rom a

f ie ld o f he l io s ta t s R , toward s a com mo n f oca l po in t F~ in the sky .

Bef o re reaching F , , the l igh t i s in te rcep ted by a second re f lec to r

R2 which d i rec t s a l l r ays to the f oca l po in t F2 on the g round . As

p rac t ica l va lues f o r the two r im ang le s , we sugges t d ' , - ~45 and

if2 = 5 . Due to mir ro r e r ro rs and f in i te s ize o f the sun , the l ight

wi l l be concen t ra ted on to a f oca l zone a round F2 , cove r ing

app roxim a te ly 1 / 100 to 1 / 50 o f the to ta l ape r ture ; th i s imp l ie s a

ra the r low va lue , 50-100 f o r the co ncen t ra t ion C~2 achieve d by the

f i r s t two s tages R , and R2 . However , the angula r sp read o f the

rad ia t ion reaching the f oca l zo ne i s sma l l , app roxima te ly ~2 = 5,

a n d t h u s a c o m p o u n d p a r a b o l ic c o n c e n t r a t o r ( = CPC)[3] c o n e c a n

bo ost the con cent rat io n by an addit ional facto r of ( l / s in t /~z) = 100

to va lues be twee n 5000 and I0 ,000 ( f ur the r reduc ed in p rac t ice to

2500-5000 becau se o f shad ing and b locking) .

In o rd e r to s imp l i f y the op t ica l ana lys i s , we take the sun l igh t to

h e i n c i d e n t a l on g t h e v e r t i c a l a x is a n d w e a s s u m e t h e s e c o n d s t a g e

to be a f la t ho r izon ta l Fresne l mi r ro r .~ Cons ide r a pa r t icu la r

inc iden t r ay which h i t s R~ a t P , a d i s tance r~ away f ro m the ax is .

t W o r k s u p p o r t e d b y U . S . E n e r g y R e s e a r c h & D e v e l o p m e n t

Admin is t ra t ion .

In p rac t ice , the op t ica l ax is wi l l be t i l t ed in the d i rec t ion tow ards

the su n a t noon , bu t the conc lus ion s wi l l r ema in e ssen t ia l ly

u n c h a n g e d .

t

I

I

/ \ \

= R r s t s ta g e ~ - -- 6 . . . . /~ lm ox

F r e s n e l m i r r o r s )

Optical axis

Fig . 1 . Op t ics o f tow er re f lec to r . F i r s t s tage = R , , cons i s t ing o f

Fresne l mi r ro rs ( hel io s ta t s) . Secon d s tage = R2 , cons i s t ing o f

Fresne l r e f lec to rs which d i rec t l igh t towards common f ocus F2 .

Thi rd s tage , cons i s t ing o f an a r ray o f com pound pa rabo l ic

c o n c e n t r a t o r ( = C P C ) cones a round F2 , i s no t shown in th i s

figure.

Af te r r e f lec t ion o f f R ~, the ray s t r ikes R2 a t P 2 a d i s tance r2 away

f rom the ax is be f o re be ing re f lec ted to F~. The s lope ( measured

f rom ho r izon ta l ) o f the re f lec to r Rz a t P _, i s g iven by

1

a(r2 ) = ~ (0, + 02). (1)

whe re 0~ and 05 a re the ang le s ind ica ted in F ig . 1 ( no te tha t they

have oppos i te s igns) . I f the f oca l l eng th o f RI i s f and i f R2 i s a t a

he ight h above the g round , then 0 , and 05 sa t i s f y the f o l lowing

re la t ions

a n d

r l - r2 r j

tan 0, h [ (2)

t an 0 2 = h = ( 1 - h ' l ~ (3 )

f J h

In o rde r to f ind the concen t ra t ion , one has to t ake mir ro r e r ro rs

and f in ite s ize o f the sun in to accou n t and ca lcu la te the re su l t ing

rad ius o f the f oca l zone . Fo r th i s purpose , we cons ide r a ray

c o m i n g f r o m P , t o w a r d s R 2 b u t d e v i a t i n g f r o m t h e c o r r e c t

d i rec t ion 01 by a sma l l amoun t d0~. Wh en h i t t ing the g round , th i s

ray wi l l miss the f oca l po in t F2 by a d i s tance

dr = dr2 + ~ d02. (4)

This equa t ion shows two con t r ibu t ions : the f i r s t i s the d i sp lace -

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2 7 0 T e c h n i c a l N o t e

m e n t d r2 b y w h i c h t h e r a y m i s s e s P2 o n t h e s e c o n d s t a g e , a n d t h e

s e c o n d t e r m i s d u e t o t h e a n g u l a r d e v i a ti o n f r o m 0 ~,

dO2 = 2d a - dO,, (5)

r a y AOz = 2A a - A 0~, t he t e r m 2 A a = 80 dom i na t e s . S i m i la r ly , t he

c on t r i bu t i on o f A 02 i n eqn (4) i s m u c h l a r ge r t han t ha t o f ~r ~ , and

t h u s o n e c a n t a k e t h e r a d i u s A r o f t h e f o c a l z o n e s u r r o u n d i n g F ~

t o b e

o f t he r ay a f t e r i t i s r e f l ec t ed f r om R2 . The quan t i t y d02 a r i se s

f r o m t h e d e v i a t i o n d 0 , o f t h e i n c o m i n g r a y a n d f r o m 2 d a d u e t o

t h e w r o n g c u r v a t u r e a t t h e p o i n t o f i m p a c t . C o n t o u r e r r o r s d / 3 o f

t h e s e c o n d s t a g e c a n b e i n c l u d e d b y a d d i n g a f u r t h e r t e r m 2 d / 3 to

t he r i gh t hand s i de o f eqn (5) .

I f t he a ngul a r ha l f w i d t h A 0~ o f t he i nc o m i ng r ad i a t i on i s

suf f i c i en t l y sm a l l , t hen eqn s (4) and (5) fo r t he i n f i n i te s i m a l

q u a n t i t i e s c a n b e u s e d t o d e t e r m i n e t h e c o n c e n t r a t i o n . C o n c e n t r a -

t i on , o f c our se , i s de f ined a s r a ti o o f ape r t ur e t o abso r be r a r ea , and

h e n c e t h e c o n c e n t r a t i o n o f t h e f i r s t t w o s t a g e s i s ( a p a r t f r o m

b l o c k i n g a n d s h a d i n g )

w h er e r E . ,.~ is t he r ad i us o f t he f i r s t s tage and A r m ~ = r ad i us o f

foc a l z one i s t he l a r ges t va l ue o f t he l e f t hand s i de o f eqn (4) ,

cor res po ndi ng to 0~ = qS, and 02 = d~z . Inc lu ding a fac to r ~b =

e f f e c ti v e g r o u n d c o v e r t o a c c o u n t f o r s h a d i n g a n d b l o c k in g , o n e

f i nds t ha t

[ c o ~ 2 2 s i n ~ 2 ] - 2

C,~ = ~0 -~ cos26~ ( tan ~, + tan q~) s i n ~ C . . . . . . (6)

w i t h

C _ ~ 1 (7)

'~ ' - sin 2 he , - (hO,) ~

A C P C c o n e a s t h i r d s t a g e c a n b o o s t t h i s v a l u e b y a n a d d i t i o n a l

f a c t o r

C 3 = 'o ( l / s i n~h2 )2 , w i t h 0 ~0 .9 ,

(8)

r e su l t i ng i n a t o t a l c onc en t r a t i on

1 2 s i n 6 = ] -2

C,23=~r0 ~ c os= t h , ( t an4~,+ t an4 ~2) ] C . . . . . ( 9)

T h e f a c t o r ~ = 0 . 9 i s t o a c c o u n t f o r t h e f a c t t h a t a t h r e e

d i m ens i on a l C P C (c one) f a ll s sho r t o f t he i dea l l i m it [4] by a bou t

1 0 p e r c e n t . F o r r e a s o n a b l e v a l u e s o f ~b, a n d ~ 2 t h e e x p r e s s i o n i n

t h e p a r a n t h e s i s o f e q n ( 9 ) i s n o t m u c h s m a l l e r t h a n o n e , a n d s i n c e

C ~, , = 40 ,000 fo r A 0, = t$~ = 5 m r a d (= ha l f ang l e o f s un ) , v e r y

h i g h c o n c e n t r a t i o n s c a n b e a c h i e v e d b y t h i s a r r a n g e m e n t , a t le a s t

in pr inc iple .

In p r ac t i c e t he angul a r dev i a t i on A 0, w i l l be som ew ha t l a r ge r

t h a n & b e c a u s e o f m i r r o r e r ro r s , a n d v a l u e s o f h 0 ~ = 7 -1 0 m r a d

c a n b e e x p e c t e d . E v e n t h o u g h t h e r e s u l t i n g d i s p l a c e m e n t s A r 2 a n d

h r ( s e e e q n 4 ) a r e n o t s m a l l e n o u g h t o b e c o n s i d e r e d i n f i n i te s i m a l ,

t h e a b o v e c o n c l u s i o n s r e m a i n s u b s t a n t i a ll y u n c h a n g e d . F o r a n

i l l us t r a t i on , l e t us c ons i de r a spec i f i c c ase w i t h bo t h t he t ow er

he i gh t and t h e he l i o s t a t fi e ld r ad i us R t aken t o be 100 m . Fo r r a ys

w i t h 0~ = ~ = 4 5 , w e f ind a d i sp l ac em en t

h

A r 2 = ~ A O , = 2 m f o r h O , = l O m r a d . ( 10 )

cos t~,

f r o m t h e c o r r e c t i m p a c t p o i n t P 2 o n R 2 . T h e r a d i u s r 2 ~ ,x o f t h e

s e c o n d s t a g e o u g h t n o t t o b e m u c h l a r g e r t h a n 1 0 m , i m p l y i n g t h a t

IArz/rzl = (1 /5) w hi c h i s no t r ea l l y i n f i n i t e s im a l . H enc e i t seem s

m or e app r op r i a t e t o app r oxi m a t e t he f i n i t e c hange i n r e f l ec t o r

s l o p e A a b y

Ar2 a . . . . w i t h a m . . = ~ O , - ~ h 2 ) = 2 0 . 1 1 )

Aa = r max

Si nc e t he l a r ges t va r i a t ion o f a oc c ur s ne a r t he op t i c a l ax i s , t h i s

e s t i m a t e i s a n u p p e r l i m i t o n A a . I n t h e d e v i a t i o n o f t h e r e f l e c te d

A r m x = c o~0 2 A 02 lm ax= 1 4 m .

(12)

W i t h t h e a s s u m e d r a d i u s r , ~ . . o f t h e h e l i o s t a t fi el d o f 1 00 m t h i s

i m p l i e s a c o nc en t r a t i on o f C ,2 = tP(r,m,dhr~ax)2= 5 0 0 f o r t h e

f i r s t t w o s t ages , if A 0, = 10 m r ad . Wi t h be t t e r m i r r o r s A 0~ m i ght

bec o m e a s sm a l l a s 7 m r ad , and C ,2 w oul d r eac h 100t p.

A C P C c o n e a s t h i r d s t a g e c a n b o o s t t h i s c o n c e n t r a t i o n b y a

fur t he r f ac t o r o f (1 / s i n ~2 )2= 100 t o a t o t a l o f C ,23= 2500 fo r

A 0, = 100 m r ad and C ,23 = 5 ,000 fo r A 0I = 7 m r ad , a ssu m i n g an

e f f ec t i ve g r ound c over f ac t o r ~ = 0 .5 .

T h i s d e r i v a t i o n s h o w s t h a t t h e c o n c e n t r a t i o n i s v e r y s e n s i t i v e to

t h e s u r f a c e a n d t r a c k i n g a c c u r a c y o f t h e h e l i o s t a t s o f R , , b u t

r a t h e r i n s e n s i t i v e t o e r r o r s i n s l o p e o r a l i g n m e n t o f t h e s e c o n d

s t age R2 ( e r ro r s A /3 i n R2 c oul d be a s l a r ge a s 1 o r 2 w i t h o u t a n y

s e r i o u s e ff ec t) . T h e a c c u r a c y r e q u i r e m e n t f o r t h e C P C t h i rd s t a g e

i s no t c r it i c a l e it he r , p r ov i ded i t s angu l a r c on t o ur e r r o r s a r e s m a l l

c om par ed t o ~ ,2 = 5.

A s f o r t h e d e s i g n o f t h e t h i r d s t a g e , a s i n g le c o n e w o u l d b e

r i d i c u l ous l y b i g. B ut e ssen t i a l l y t he s am e op t i ca l pe r fo r m anc e i s

o b t a i n e d b y a n a r r a y o f m a n y s m a l l C P C c o n e s , c l o s e l y s p a c e d

w i t h o u t g a p s , e a c h c o n e h a v i n g a n a p e r t u r e r a d i u s o f a b o u t

5 - 2 0 c m a n d a h e i g h t o f a b o u t 1 5 -6 0 c m .

O p t i c a l t r a n s m i s s i o n l o s s e s in a s y s t e m o f t h i s k i n d c a n b e f a i r ly

l a r ge s i nc e a l l r ays have t o unde r go t w o r e f l ec t i ons be fo r e

r e a c h i n g t h e C P C a n d a p p r o x i m a t e l y o n e a d d i ti o n a l r e fl e c ti o n , o n

t h e a v e r a g e , w h e n p a s s i n g th r o u g h t h e C P C . S e v e r a l m e t h o d s f o r

m i n i m i z i n g t h e s e l o s s e s a r e m e n t i o n e d .

T h e e n e r g y a b s o r b e d i n t h e C P C n e e d n o t b e w a s t e d ; i t c a n w e l l

s e r v e t o p r e h e a t t h e w o r k i n g fl ui d. O n e m i g h t , f o r e x a m p l e , p u m p

a g a s d o w n w a r d t h r o u g h t h e C P C c o n e s . T h e g a s s t r e a m w i ll n o t

o n l y e x t r a c t t h e h e a t f r o m t h e C P C a n d f r o m t h e a b s o r b e r , b u t i t

c a n s u p p r e s s c o n v e c t i v e l o s s e s i f a t r a n s p a r e n t c o v e r i s p la c e d o n

t o p o f t h e c o n e s . T h e c o v e r s h o u l d b e m a d e o f n o n g l a r e ( e tc h e d )

g l a ss [5] ( t ransm i ss i v i t y ->96 pe r c en t ) . Suc h a c ove r a l so o f f e r s t he

p o s s i b i li t y o f u s i n g a n i n e r t g a s s u c h a s H e t o p r e s e r v e t h e h i g h

r e f l ec t i v i t y o f a f i r s t sur f ac e s i l ve r c oa t i ng w hi c h w oul d be

d e s i r a b l e f o r t h e C P C c o n e s .

H i g h r e f l e c ti v i ty f o r t h e s e c o n d s t a g e i s n e c e s s a r y n o t o n l y f o r

t he sake o f go od op t i c a l e ff ic ienc y bu t a l so t o avo i d ove r hea t i ng .

W i t h t h e p r o p o r t i o n s s u g g e s t e d a b o v e , t h e s e c o n d s t a g e i s

e x p o s e d t o I 0 0 s u n s . E v e n i f o n l y 1 0 p e r c e n t o f t h i s i s a b s o r b e d ,

c a r e fu l des i gn w i l l be n eed ed t o avo i d ove r hea t i ng . (Refl ect iv i ti e s

a r o u n d 9 0 p e r c e n t a r e f e a s i b le w i t h s e c o n d s u r f a c e A g m i r r o r s ).

Ref l ec t i v i t y c l o se t o 100 pe r c en t c an be ac h i eved w i t h p r i sm s

us i n g to t a l i n t e r na l r e f l ect i on . F i gur e 2 sh ow s t he o r i en t a t i on o f

sm a l l sym m e t r i c r ec t angu l a r p r i sm t o be use d a t po i n t P ~ i n F ig . 1 .

T h e b r o a d s i d e o f th e p r i s m f a c e s d o w n w a r d i n s u c h a w a y t h a t t h e

Roy from ~ \

Rays to F

Fi g . 2 . O r i en t a t i on o f r ec t angul a r r e f l ec t i ng p r i sm on sec on d s t age .

P , , P 2 and F2 r e f e r t o po i n t s i n F i g . 1 . Rad i a t i on w hi c h em er ges

a f t e r m or e t han t w o i n t e r na l r e f l ec t i ons i s no t i nd i c a t ed ; i t w i l l

a l w ay s be pa ra l le l t o t he r ay s sho w n t o be go i ng t o F~.

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T e c h n i c a l N o t e 2 71

r a d i a t io n w h i c h i s r e f le c t e d f r o m t h e f r o n t r e a c h e s t h e f o c a l p o i n t

F : . T h e 9 0 e d g e f a c e s u p w a r d a n d r u n s a l o n g t h e r a d ia l d i re c t i o n

a w a y f r o m t h e o p t i ca l a x i s o f t h e s y s t e m . T h e r a y s w h i c h u n d e r g o

t o t a l i n t e r na l r e f l ec t i on i n s i de t he p r i sm w i l l em er ge w i t h a sm a l l

l a t er a l d i s p l a c e m e n t ; t h e a b e r r a t i o n s d u e t o t h i s d i s p l a c e m e n t a r e

neg li g ib l e s i nc e eac h p r i sm w i ll m ea sur e on l y a f ew m m i n w i d t h .

O b v i o u s l y s t r a i g h t ( i. e. n o n t a p e r e d ) p r i s m s c a n n o t b e p l a c e d s i d e

by s i de i n a r ad i a l a r r ay w i t hout gaps . A p r ac t i c a l so l u t i on w oul d

u s e a p a t c h w o r k o f f a ir l y s m a l l g l a s s p l a t e s (r o u g h l y 3 0 x 3 0 c m ,

e a s i l y m a s s p r o d u c e d ) e a c h o f w h i c h i s c o v e r e d o n t h e u p p e r s i d e

b y p a r a l l e l r e c t a n g u l a r p r i s m s . T h i s i m p l i e s t h a t s o m e o f t h e

p r i s m s w i l l n o t h a v e t h e c o r r e c t o r i e n t a t i o n , b u t t h i s h a s n o e f f e c t

o n t h e o p t i c a l p e r f o r m a n c e . t O f c o u r s e , t h e g la s s s h o u l d b e h i g h ly

t r a n s p a r e n t t o m i n i m i z e a b s o r p t i o n l o s s e s ; s o - c a l l e d w a t e r - w h i te

g l a s s w o u l d b e a c c e p t a b le .

T h e t o w e r r e f l e c t o r s c h e m e c a n r e a c h v e r y h i g h t e m p e r a t u r e s

and c o l l ec t i on e ff ic ienc y. Fo r a c on se r va t i ve e s t i m a t e , c ons i de r

t h e f o l l o w i n g s y s t e m :

C onc en t r a t i on C = 2500, c o r r e spond i ng t o f a i r l y l a r ge m i r r o r

a n d t r a c k i n g e r r o rs A 0 , = 1 0 m r a d ( o n e s i d e d d e v i a t i o n f ro m

per fec t ) , and g r ound c over f ac t o r ~ = 0 .5 .

Ref l ec t i v i t y o f f i r s t s t age p , = 0 .85 .

Ref l ec t i v i t y o f sec o nd s t age p2 = 0 .98 (g l a ss p r i sm s) .

T r a n s m i s s i v i t y o f n o n - g l a r e g l a s s c o v e r o n C P C = 0 . 96 .

Ref l ec t i v i t y o f t h i r d s t age (C P C , app r oxi m a t e l y one r e f l ec t i on

on a ve r age ) p3 = 0 .95 (A g i n H e a t m o sphe r e ) .

A b so r p t i v i t y o f c av i t y r ec e i ve r E = 0 .95.

T h e s e n u m b e r s i m p l y a n o p t i c a l e f fi c ie n c y

~ 0) = p~p2zp~ = 72%.

t N o t e t h a t r e c t a n g u l a r p r i s m s a r e e f f e c t i v e r e f le c t o r s f o r a 1 8 &

f i e l d o f v i ew i n t he d i r ec t i on a l ong t he p r i sm edge , w hi l e i n t he

t r a n s v e r s e d i r e c t i o n , t h e y a r e l i m i t e d b y t h e i n d e x o f r e f r a c t io n n .

Fo r n = 1 .5 , t he t r ans ve r s e f ie l d o f v i ew i s - --5 , l a r ge enou gh t o

a c c o m o d a t e a l l r a y s i m p i n g i n g u p o n t h e s e c o n d s t a g e .

A s s u m i n g in s o l a t io n o f S = 0 . 1 W c m 2 a t t e n u a t e d b y a f a c t o r o f

0 .6 t o a l low fo r c o s i ne f ac t o r an d fo r l o ss o f d i f fuse r ad i a t i on , t he

a v e r a g e e n e r g y r e a c h i n g t h e a b s o r b e r i s

Q~n = 0.6 C S r /(0) = 108 W cm 2. (13)

A t a n o p e r a t i n g t e m p e r a t u r e o f T = 1 3 0 0 K ( a p p r o x i m a t e l y

1000C ) , t he r ad i a t i ve hea t l o ss pe r un i t abso r be r a r ea i s

Q ra a = t r ( T ' - T 4 . . . . . ) 2 1 6 W / c m 2, (14)

i ndeed a sm a l l f r ac t i on o f t h e i nc om i ng fl ux .

A c k n o w l e d g e m e n t s - - I s h o u l d l ik e t o t h a n k P r o f . R . W i n s t o n f o r

h e l p f u l d i s c u s s i o n s .

REFEREN ES

I . G . Fr anc i a ,

So lar Energy

12, 51 (1968); V. A. Baum, R. R.

A p a r e s e a n d B . A . G a r f ,

So lar Energy

I , 2 (1957); A. F.

H i l d e r b r a n d t , e t a l. , EO S Trans . o .[ Amer ic an G eophys ic s

U n ion 53 (7) , 684--692 (1972); T. D. B rum lev e et al., A h i g h

t e m p e r a t u r e s o l a r e n e r g y s y s t e m . S a n d i a L i v e r m o r e L a b o r a t -

o r y , LSS-73-0059 ; H oneyw el l , Inc . and Bl ac k & Vea t c h ,

D y n a m i c c o n v e r s i o n o f s o l a r g e n e r a t e d h e a t t o e l e c t r i c i t y .

N A S A r e p o r t , N A S A C R - 13 4 72 6 ( 1 97 4 ).

2 . A . R a b l , R a d i a t i o n t r a n s f e r t h r o u g h s p e c u l a r p a s s a g e s . A r -

g o n n e N a t i o n a l L a b o r a t o r y R e p o r t , S O L 7 5 - 0 3 ( 19 7 5) , Int. J.

H ea t Mass T rans fer , t o b e p u b l i s h e d .

3 . H . H i n t e r b e r g e r a n d R . W i n s t o n , Rev. Sc i . lns tr . 37, 1094 (1966);

Rev. Sc i . Ins tr . 39, 121 7 (1968) .

4 . R . W i n s t o n , L i g h t c o l le c t io n w i t h i n t h e f r a m e w o r k o f

geom et r i c a l op t i c s . J .

Opt . Soc . Am. 60 ,

245 (1970).

5 . R . E . P e t e r s o n a n d J . W . R a m s e y , T h i n f il m c o a t in g s i n

s o l a r - th e r m a l p o w e r s y s t e m s . J . Vac . Sc i . Technol . 12, 0471

(1975).