Linear Solar Concentrator Nasa Contract Report by jfx10560

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THE LINEAR FRESNEL
LENS SOLAR CONCENTRATOR:
TRANSVERSETRACKINGERROR                              EFFECTS

Robert M . Cosby

Prepared by
BALL STATEUNIVERSITY
Muncie, Ind.
for GeorgeC.Marshall   SpaceFlightCenter

NATIONAL
      AERONAUTICS
                AND           SPACE ADMINISTRATION   9   WASHINGTON, D. 'C.   AUGUST 1977
1. REPORT NO.                                     2. GDVERNkENT ACCESSION NO.                                  3.    REClPlt.   -        OOb3673
"
.
      NASA .CR-2889 .
                  .            .~   ~~
                                             "             ."
                  SUBTITLE
                                                  ~~




4.    TITLE AND                                                                                                5. REPORT
                                                                                                                       DATE
      The Linear Fresnel Lens Solar Concentrator: Transverse                                                         August 1977
                                                                                                               6. PERFORMINGORGANIZATION           CODE
      Tracking Error Effects .
-~                       ~-    ~.        -   .~        ~    ~~~         ~. . ..   .   "_
7. AUTHOR(S)                                                                                                   8. PERFORMING ORGANIZATION REPOR'r         #

R o b e r t M. Cosby            ~        .                                __           .
                                                                                      . . .-
                                                                                      ~~       .   -.   "~ ~
                                                                                                         ~
                                                                                                                     "228
9. PERFORMING      ORGANIZATION      NAME AND~ADDRESS                                                          10.   WORK UNIT, NO.

      Department o Physics and Astronomy
                   f
                                                                                                               1 1.CONTRACT     OR GRANT NO.
      Ball State University
      Muncie , Indiana                                                                                               NCA8-00121.Mod            3
                                                                                                               13. TYPE     OF REPORT & PERIODCOVERED
-~
12.   SPONSORINGAGENCY       NAME AND ADDRESS
                                                                                                                      Contractor Report
      National Aeronautics and Space Administration
                      20546
      Washington, D. C.                                                                                        -
                                                                                                               1.
                                                                                                                .
                                                                                                                ;              AGENCY
                                                                                                                      SPONSORING    CODE




             Real sun-tracking linear solar concentrators imperfectly follow the solar disc, opera-
     tionally sustaining both transverse and axial misalignments. This report details an analysis of
     the solar concentration performance of a line-focusing, flat base Fresnel lens in the presence
     of small transverse tracking errors. Simple optics and ray-tracing techniques a r e used to
     evaluate the lens solar transmittance and focal plane imaging characteristics. Solar trans-
     mission losses by Fresnel reflection and material absorption are included and an analysis of
     groove edge losses is presented. Computer-generated example data a r e presented for lenses
     with parameters corresponding to two NASA test articles: a 0. 56 m wide, f l i . 0 lens and a
     1. 83 m wide, f/O. 9 lens. Results indicate less than a 1 percent transmittance degradation for
     transverse errors up to 2. 5 . In this range, solar image profiles shift laterally in the focal
                                  O
     plane, the peak concentration ratio drops, and profile asymmetry increases with tracking error.
     With profile shift as the primary factor, the ninety percent target intercept width increases
     rapidly for small misalignments, e. g. , almost threefold for a io error for the small test lens.
     The analytical model and computational results provide a design base for trackingand absorber
     systems for the Fresnel lens solar concentrator.




                                                                  '   *For sale by the National Technical Information Service, Springfield, Virginia 22161.
                                CONTENTS

                                                                        Page
ABSTRACT ...........................                                      i

NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . .             iv
LIST OF ILLUSTRATIONS . . . . . . . . . . . . . . . . . . . . .            vi

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . .          viii

I.    INTRODUCTION . . . . . . . . . . . . . . . . . . . . . .                 1

I1   .    THEORY..........................                                     2

          A . Transmission haracteristics . . . . . . . . . . . .
                         C                                                     2

          B . Concentrated lux istribution
                          F D               . . . . . . . . . . .          10

I11   .                     . . . . . . . .
          THEORETICAL RESULTS                   . . . . . . . . .   .      19

      A . Example Data . 0.56 m Lens . . .      . . . . . . . . .   .      19

          1. Lens Transmission . . . . .        . . . . . . . . .   .      19

          2 . Focal lane ntensity rofiles
                    P    I         P              . . . . . . . .   .      26

      B . Example Data . 1 . 8 3 m Lens . . .   . . . . . . . . .   .      32

IV.   S M A Y AND CONCLUSIONS . . . . . .
       U MR                                     . . . . . . . . .   .      37

v.    REFERENCES . . . . . . . . . . . . . .    .........                  38

APPENDIX . GROOVE BLOCKING LOSSES . . . .       . . . . . . . . .   .          39




                                     iii



                                           . . . . . .                   ...
                                                                        ...
                          NOMENCLATURE


Symbo1                                 Definition

A        totaltransmittedfraction                            of incident sunlight

         transmission for jth wavelength interval

FuJ FR   fractionsofincidentsunlightlosttogroove
         b l o c k i n gi nu p p e ra n dl o w e rl e n sh a l v e s ,
         respectively

f        l e n sf o c a ll e n g t h

         totalintensity                 at position Y

         j t h wavelengthintensity                       at Y

         serration index

         wavelengthindex

         beam s p r e a d f o r s u n l i g h t w i t h i n j t h w a v e l e n g t h i n t e r v a l
         refracted from ith serration

AR       d e f o c u sl e n g t hp a r a m e t e r

N        design index of refraction

n        index of refraction

9        i n c i d e n ts o l a ri n t e n s i t y

         r a y - l e n sg e o m e t r i c a lp a r a m e t e r s ;r e f e rt or a yd i a g r a m s

         b u l kt r a n s m i t t a n c ef a c t o r

         transmittanceofsurface                          1

         transmittanceofsurface                          2

         transmittancethroughithserrationforsunlightwithin
         j th wavelength interval

         sunlighttransmittanceofithserration

         g r o o v eb l o c k i n gt r a n s m i t t a n c ef a c t o r

         l e n st h i c k n e s s



                                         iV
I .
      W                  lenswidth

      Y                  serration position variable with respect                       to lens axis

      (AY1               serration width

      Y              ,   'position variable with respect               to l e n g t h a x i s o f l e n s
                          and i n a p l a n e p a r a l l e l t o and beneath the concentrator

                         extremerayintercepts

                         apparentangulardiameter                        o f thesun

      a* a,'
          'a             e x t r e m er a yr e f r a c t i o na n g l e sa t   first lenssurface
          1' 2   3

      '
      1    -'7
                         anglesbetweentheemergentrefractedextremeraysand
                         t h e normal t o t h e p l a n e o f t h e l e n s

      6                  transverseerrorangle

                         grooveangle            for i t h s e r r a t i o n

                         wavelength

                         angles of incidence

                         angles of r e f r a c t i o n

                         solar flux weighting factor




                                                 V
                          LIST OF ILLUSTRATIONS


FIGURE                                   TITLE                                                 PAGE

         Refraction f ays
                  or                         from c e n t e r f
                                                            o              sun   ..... .         4

         Groove e d g e o s s e s o r p p e r e n s a l f
                      l         f u         l h                                   . . .’ . .     6

         Groove e d g e o s s e s o l o w e l e n s a l f
                      l         f r         r h                                     .....        7

  4      Extreme r a y p a t h s i n u p p e r l e n s h a l f
         serrations; 6 > a             .   . ..............                                     12

 5       Extreme r a y p a t h s i n u p p e r l e n s h a l f
         serrations; 6 < a                 ....... .                       ........             13

 6       Y r extreme ray paths                  inlowerlenshalf
         s e r r a t i o n s ; 6 > CY      .. ... ....... .. . .                                15

  7      YI1 e x t r e m e r a y p a t h s i n l o w e r l e n s h a l f
         s e r r a t i o n s ; 6 2 c1      ... .............                                    17

         Yr e x t r e m e r a y p a t h i n l o w e r           lenshalf
         serrations; 6 < a . . . .                        ... . . .. . .. ..                    18

         Transmittanceversusserrationposition
         f o r 6 = 1.5O;
                       0.56 m lens       .                ... ........                          25

10       Transverseorientation                      e f f e c t s on i n t e n s i t y
         p r o f i l e0 . 5 6
                      ;       m lens               .. . . . . .... . . ..                       27

11       Transverseorientation                      e f f e c t s on i n t e n s i t y
                      ;
         p r o f i l e0 . 5 6 m lens               . ... .. ... . .. . .                        28

12       Transverse orientation effects                            on p r o f i l e
               p
         p e a k o s i t i o n0 . 5 6
                              ;       m lens                  ...........                       29

13       Transverse orientation effects                            onpeak
         concentration;
                      0.56   m lens                       ............                          30

 14      Transverseorientationeffects                              on t a r g e t
         width;
              0.56  m lens . . . . .                              ..........                    31

 15      Transverseorientationeffects                              on i n t e n s i t y
         p r o f i l e1 . 8 3
                      ;       m lens . . . .                      . . .. . ... ..               33

16       Transverseorientationeffects                              on p r o f i l e
               p
         p e a k o s i t i o n1 . 8 3
                              ;       m lens .                    ..........                    34

 17      T r a n s v e r s e o r i e n t a t i o n e f f e c t s onpeak
         c o n c e n t r a t i o n1 . 8 3
                                  ;             m lens . . . . .                 .. ....        35




                                              vi
              LIST OF ILLUSTRATIONS (Cont.)


FIGURE                     TITLE                            PAGE

  18     Transverseorientationeffects      on t a r g e t
         width.   ....................                       36

  A1     Ray diagramsforgrooveedge      l o s s e si n
         upperlenshalf     . . . . . . . . . . . . . . . . 40
  A2     Ray diagrams for groove edge   l o s s e si n
         lowerlenshalf     . . . . . . . . . . . . . . . . 45




                          vii
                                             I.      INTRODUCTION


         The economics of tracking systems for solar concentrators depend

d i r e c t l y on t h e p r e c i s i o n        demanded i n f o l l o w i n g t h e s o l a r d i s c .                 The

required precision                 i s determinedbytheconcentrator'sperformance

sensitivity to tracking errors                            andby        theconcentrationrequirements

o f a p a r t i c u l a ra p p l i c a t i o n .S p e c i f i e de n e r g yc o l l e c t i o nc o n d i t i o n s

areachieved             by a p p r o p r i a t e d e s i g n o f       a primary con-centrator-absorber

system o r primary concentrator-secondary concentrator-absorber system.

Such d e s i g n i s p o s s i b l e o n l y           i f thesolarimaging                    and f l u x t r a n s f e r r a l

(transmission/reflection) p r o p e r t i e s o f t h e p r i m a r y c o n c e n t r a t o r a r e

known f o r a v a r i e t y o f c o n d i t i o n s , i n c l u d i n g i m p e r f e c t               sun t r a c k i n g .

Designand           optimum placement of                    a secondary concentrator and/or absorber

require an evaluation of the concentrator imaging sensitivity to defocusing.

          n
         A economicallyattractiveconcentratoroftherefractortype

f o r medium c o n c e n t r a t i o n a p p l i c a t i o n s          i s t h e l i n e a r f l a t baseFresnel

lens.        The s o l a rc o n c e n t r a t i o nc h a r a c t e r i s t i c s ,i n c l u d i n gd e f o c u s i n gs e n -

sitivities,of               a p e r f e c t l yt r a c k i n g ,l i n ef o c u s i n gF r e s n e ll e n sh a v eb e e n

a n a l y z e dd u r i n gt h i ss t u d y         afid r e p o r t e di nd e t a i li nR e f e r e n c e[ l ] .I n

actual c o n c e n t r a t o r o p e r a t i o n , b o t h a x i a l           and t r a n s v e r s e t r a c k i n g e r r o r s

will occur.             A g e n e r a lo b j e c t i v eo ft h ep r e s e n tp r o j e c t                i s t o examine t h e

e f f e c t s on performanceof                    a transverse tracking error for                             t h i s typeof

solarconcentratorusingsimpleopticalanalysis,raytracingtechniques,

a n dc o m p u t e rg e n e r a t i o no fe x a m p l ed a t a .Z e r oa x i a la l i g n m e n te r r o r                        is

assumedand            a l l incidentsolarraysareapproximated                                        as having no a x i a l

component.            S p e c i f i co b j e c t i v e sa r et o          compute t h el e n st r a n s m i t t a n c e

d e g r a d a t i o n and t h e image p r o f i l e s h i f t               and d i s t o r t i o n u n d e r        small

transversetrackingerrorconditions                                     (52.5')                   AA
                                                                                     f o r two N S t e s t a r t i c l e s :

     m
a 56 c wide,f/1.0                     and a 1.83 m e t e r w i d e , f / 0 . 9 l e n s .
                                                 11.      THEORY


         The s o l a r t r a n s m i s s i o n a n d c o n c e n t r a t i o n          characteristics o f a

F r e s n e ll e n sw i t h        a small t r a n s v e r s e t r a c k i n g e r r o r           (<2.5')        are

s t u d i e du s i n go p t i c a lr a yt r a c et e c h n i q u e ss i m i l a rt ot h o s ei n

previousanalysisfor                       a perfectlytrackingconcentrator                                 [ 1 , 2 , 3 ] . The major

c h a n g eo c c u r si nt h e           l o s s of      about
                                                  symmetry                           t h e l e n sa x i s .       The l e n s

i s assumed t o have a compressionmoldedgeometryand                                                 tobefreeof

             d
manufacturing efects,                                        thermal
                                            wind l o a d , and             effects.
                                                                   expansion      Other

assumptionsinclude:

         - Thec ahl ee ngght ht o. f
           fo
                       i                a s e r r a t i o n on t h e l e n s          i s much l e s s t h a n t h e


             D i f f r a c t i o n by grooveedges                 is negligible.

         *   Any anomalous d i s p e r s i o n e f f e c t s n e a r a b s o r p t i o n b a n d s i n t h e
             acrylic have negligible effect.

         - Thesun             i s a u n i f o r ms o u r c eo fr a d i a t i o n .

         *   The s o l a r f l u x r e f r a c t e d b y   a s i n g l e s e r r a t i o n i s uniformly
             distributedoverthe                   beam s p r e a d w i d t h i n t h e i n t e r c e p t p l a n e
             b e n e a t ht h el e n s .

             Lens o r i e n t a t i o ni nt h es e a s o n a l( l o n g i t u d i n a l )d i r e c t i o n is
             p e r f e c t ;s o l a rr a y sa r ea p p r o x i m a t e d       as having no a x i a l components.

A.     Transmission C h a r a c t e r i s t i c s

                Following a p r e v i o u s a n a l y s i s              [I], thetotaltransmission

       c o e f f i c i e n t i s w r i t t e n as a p r o d u c t :

                T = T1TaT2Ts,                                                                                           (1)

       w i t h Tl t h eF r e s n e lt r a n s m i t t a n c ef a c t o r              f o r t h e f i r s t lens s u r f a c e ,

      Ta a b u l k t r a n s m i t t a n c e f a c t o r ,         T2 t h e F r e s n e l f a c t o r f o r t h e

       secondsurface,
                    and                      Ts a " s h a d i n g "f a c t o r .W h i l et h ee m p i r i c a l

       t r e a t m e n to fa b s o r p t i o n         (Ta) i s unchanged, t h e F r e s n e l f a c t o r                    TITZ
       i s now e v a l u a t e d f o r r a y s           from t h e s u n ' s c e n t e r i n c i d e n t a t a n


                                                          2
angle 6, t h e t r a n s v e r s e e r r o r a n g l e , r a t h e r                  than forrays               normal

tothelenssurface.

         The transmissivity f o r n a t u r a l                        l i g h t i n c i d e n ta tt h eb o u n d a r y

between two o p t i c a l media is given by
                    s i n 2$i s i n 2$t 11 + s e c 2( $ i                                   - tJt)l
     T ( @ i > @ t=
                  )                                                                                               (2)
                         2 s i n 2 ( $ i + $t)
where $i and$t                    aretheanglesofincidence                             and r e f r a c t i o n ,

r e s p e c t i v e l y .R e f e r r i n gt oF i g u r el ( a )               and u s i n gS n e l l ' s         law of

refraction,theanglesforthefirstsurfaceare

          $i = 6
                        s
          $t = Arcsin (-),i n 6

wheren          is t h e a p p r o p r i a t e i n d e x          of r e f r a c t i o n .
          Fortheserratedsurface                             on t h e " u p p e r " h a l f o f t h e l e n s ,

theincidentangle                         I
                                       $ refractionangle
                                        and                                         ;
                                                                                    $       are(Figurel(a))

          4
          ;     =   e    +   $t

          :
          $     = Arcsin (n s i n $:),                                                                            ( 4)

and f o r t h e " l o w e r " h a l f ( F i g u r e l ( b ) )

          $
          ;     =   le       - $tl>                                                                               ( 5)

     ;
with $          asabove.

          Here      € is
                     I        thegrooveanglegiven                          by [I]
                                  r                                                     >
          8 = Arctan
                                      N[ y2 + ( f - t ) 2 ]
                                                              Y
                                                                     '   - (f-t)
withytheserrationdistance                                   from t h e l e n s c e n t e r l i n e ,         f thefocal

l e n g t h , t t h el e n sc e n t e rt h i c k n e s s ,               and N thedesignindexof

refraction.

          Thus t h e p r o d u c t          TIT2 can be evaluated from
Figure 1.   Refraction of r a y s from c e n t e r o f s u n
         Rays o f s u n l i g h t i n c i d e n t             on a s e r r a t i o n may b er e f r a c t e ds u c h

thattheyeitherstriketheserrationedgesurface                                                       or, afterpassing

t h r ut h el e n s ,        are o b s t r u c t e d by a na d j a c e n ts e r r a t i o n .                 In t h i s

a n a l y s i ss u c h    rays a r e assumed l o s t , i.e., do n o t c o n t r i b u t e t o
t h ei n t e n s i t yp r o f i l ei na n              imageplanebelowthelens.Sincethe

stepheightsarediminutive                               and t h e t r a n s v e r s e e r r o r        i s assumed small,

t h el o s s e sa r ea l s oe x p e c t e dm i n o r .

         As d e p i c t e d i n F i g u r e s 2 and 3 f o r t h e u p p e r                     andlower           lens
h a l v e s ,r e s p e c t i v e l y ,v a r i o u sp o s s i b l e" s h a d i n g "c a s e sf o ri n c i d e n t

                   be
s u n l i g h t must             considered.               The method o f a n a l y s i s f o r t h e f r a c t i o n s ,

Fu and FQ, o f i n c i d e n t l i g h t l o s t t h r u t h e v a r i o u s i l l u s t r a t e d c a s e s

for a givenserration(theith)                                  and f o r a p a r t i c u l a r w a v e l e n g t h          is out-

l i n e di nt h e        Appendix.            The r e s u l t s a r e                 below:
                                                                             summarized



CASE:              Z    ,   h
              Figure (a)upper alf.

                         6 tanei
               ,
              F'            n

              F = (6 + p)Z t a n e i ,
              ;                                                   6 < a.
                               4an


CASE:         F i g u r e Z(b)      , u p p e rh a l f .

              F;    = 0,




                                               f o r 6 < a and nei                5a - 6

              F;I = 0                   f o r 6 < a and n8i > a                    - 6.




                                                   5


                                         _.      "
Figure 2 .   Groove edge losses f o r upper lens h a l f .




                          6
Groove edge losses for lower l e n s half.
CASE:     Figure (c),
               2                  upper h a l f .




andwhere

        A = sin f i ;
                 J

        B = cos 9 .    ’                                                                (16)
                    I ’
                                                                   s i n Bi
        @io A r c s i n { n s i n
          =                               [ei -     Arcsin (
                                                                        n
                                                                              111   ;   (17)

                                                               t
p(x) :       (1   - n’A’)         +   (2n AB)x      - (B2)x2        ;




CASE:     Figure (a),
               3                  lower h a l f

          F Q = O ,                           6- u ;
                                               <
          ~i      (a - 6 ’ tangi,
                        )                           6 < a.
                           4 cm




                                          8
                                              LIST OF TABLES

Table                                                                             Page

1.              s
     T e slte n c h a r a c t e r i s t i c s .          . . .. .. . ... ..              20

2.   Large           t c
                t e sl e n s h a r a c t e r i s t i c s .    .. .......                 21

3.   Solar      and l e n s p e c t r ap a r a m e t e r s
                                         l                      ........                 22

4.   Computed s u n l i g h tt r a n s m i t t a n c e        of t e s tl e n s
                        ;
     s e r r a t i o n s0 . 5 6 m lens. .             .
                                                      .   ..........                     23

5.   Lens t r a n s m i t t a n c eo v e rt h es o l a rs p e c t r u m
     f o r a t r a n s v e r s e t r a c k i n g e r r o r o f 1.5';
     0.56mlens. . . . . . .                       .       .
                                                       .. . . .       . ....         ,   23
I

    CASE:            Figure 3(c),                   h
                                                lower alf.




                      f o r nei > 6 (24) and
                                     + a,



             FIIt =
                           tanei+.l
              R                          .'    nAB(nOi      -   @io) + 2(n
                                                                       A2     2   ei 2 -   $io2)
                                   2an




                       -   2n2A (AB2 - 1 ) -1
                                  2B2
                                                                o(nei)    ,       f o r 6 + a > nei.   (25)


    The l o s t f r a c t i o n o f i n c i d e n t l i g h t            is

             F       = F'          + F" + F"
                 u         u          u    u        '




    fortheupper                     andlowerserrations,respectively.

    Then

             Ts = 1            -    Fu        (upper)

             and

             Ts = 1 - F                       (lower)   ,
                                     R




                                                        9
                                                           (1)
          The transmission coefficient evaluated from Equation may
                                               Ti(y)
     be used to calculate the serration transmission as a function
     of serration position, the fraction Aj transmitted for one wavelength

     interval, and the total sunlight transmittance A:




           -
     The w 1 are spectral weighting factors, is the lens width,
                                          W
     (AY)~ the serration width, and the summations, and (I), are
                                                  (C)
                                          j        i
     over all lens serrations and all solar wavelengths[l]. In deriving
     the above equations, the decrease in incident flux caused by the
     small tracking error is assumed negligible since the cos6 factor
                     :O
     is essentially 1O    for all errors studied.
B.   Concentrated Flux Distribution
          The local concentration ratio in an image plane below the
     lens is given by
                    [l]




     with Lij the beam spread width. Now

          L=Yr-Yg,                                          (32)

     where Yr and YE are the extreme ray intercepts in the image plane
                                                of
     for light exiting a serration. Determination these intercepts
                                   of a
     for all serrations in the presence transverse tracking error



                               10
                      of
constitutes the balance the analysis. The study of refraction
of extreme rays is divided by considering separately the upper and

                                                     of the
lower lens halves and subdivided according to the magnitude
tracking error compared to the angle subtended by the solar
                                 (2a)

disc.
1. Upper Half of Lens
   a.    6- a
          >
         Extreme rays exiting at the serration edges are depicted in
   Figure 4. For this case,

                                (AY 1i
         ,y
          ,        =   yi +      -
                                7 (f +            AR    - t) tan y1 ,         (33)


         YRu       =   yi   -   --
                                2
                                         (f   +   AR - t - (Ay)itanei)tany2, (34)

   where AR is a defocus parameter and y1, are ray exit angles.
                                         y2

   Applying Snell's law at the two surfaces.

         y1 = Arcsin [n sin (a; +                  ,)
                                                   e]     - ei   ;


         y2 =      Arcsin [n sin (a; +             ei]   - Oi ;

   where
         a,;   =   Arcsin
   and
         ai = Arcsin



         For this case, the ray determining
                                       YR is identical to that
                                              3)
   in Figure 4. Hence YR is given by Equation( 4 .                                 ()
                                                                          Figures S a ,
     b,
    ( ) and (c) illustrate three possible refraction scenarios for
                     Yr
   the ray determining depending on the groove angle and the ray
   position with respect to the serration normal at the grooved surface.
                         11
   -p
                                                                                                                     Intercept
                                                                                                                       Plane




                                                                 \
                                                                                                                         "T
            C
            "         "




                                                                                                                                 Y,
       Yi
                                                                                                            \
                                                                                                                -T-
Optic                                                I
                                                                                                                 Y! -
                                                                                                                  1
Axis
                            <                                       t>
                                                                    Al
                                                                    f-




                Figure 4.   Extreme r a yp a t h s       i n u p p e rl e n sh a l fs e r r a t i o n s ;   6 > a.




                                                            12
Figure 5.   Extreme I :ay pathIS i n upper l e n s h a l f s




                                  13
                                     5(a),
           For the situation in Figure




           y3 + Arcsin [n sin          (ei   - a;)] - Bi ,
     and
           a;   =   \a;\

           For the rays in Figure
                                5(b) and 5 c
                                          ()                  ,




     where

           Y4 = - Y3       .                                      (43)


2.   Lower Half of Lens

     a.     >
           6- a

           Figure 6(a) and 6(b) display possible ray paths determining
     the intercept Yr f o r serrations in the lower lens half. Nith

     the aid o f Figure 6(a),




     where


           y5 = Arcsin[n si.n(ei -                    .
                                             u’)l - e 1
                                              1           ’



           r.
            1
                = si   tan a; ,

     and




                                  14
                                   b )


Figure 6 .   Yr extreme ray p a t h s i n l o v e r l e n s h a l f s e r r a t i o n s ;   6 2 a.
       The case i n F i g u r e 6 ( b )                may o c c u r f o r s m a l l g r o o v e a n g l e s

and t h er e s u l t sa r ea l s od e s c r i b e db yE q u a t i o n s                       ( 4 4 ) t h r u( 4 7 ) .

       Rays determining Yll a r ed e p i c t e di nF i g u r e7 ( a ) ,( b ) ,                                 and

(c).         Using he ketch n
                  t s      i                        (a),




where

       y6 = A r c s i n [ ns i n ( e i            - a2) -
                                                     ']             8.
                                                                      1
                                                                          ,                               (49)

F o rt h ec a s e si nF i g u r e7 ( b )a n d( c ) ,


       Y
           llb
                 =   - yi     +
                                   (AY) i
                                  --
                                    2
                                                       cf    + AII - t        - (Ay)itan Bi+J               tan y
                                                                                                                     6'
                                                                                                                            (50)


b.     6 c a
       F o ra ne r r o ra n g l e              less t h a n t h e h a l f a n g l e s u b t e n d z d          by

t h es u na n df o rs e r r a t i o n si nt h el e n sl o w e rh a l f ,t h er a y

shown i n F i g u r e         S d e t e r m i n e st h ei n t e r c e p tp o s i t i o n            Yrb:


       'rb
                 -
                 -      Yi    -   "
                                      2
                                                       cf + All -         (Ay)itan      ei]       tan 7
                                                                                                     y


where


       y7 = A r c s snn n
                    ii[                    (€Ii+ a;)]            - ei     .                               (52)

       The i n t e r c e p t YQb i s d e f i n e dt h r uF i g u r e                      Equations
                                                                                      7 and                              (48)

t h r u (50).

       UsingEquations                 (32) t h r u ( 5 2 ) , t h e beam spreadwidth                            L

f o rs u n l i g h tw i t hw a v e l e n g t h              X . and r e f r a c t e d by a n y s e r r a t i o n
                                                             1
     computed.
may be                            Summing o v e r a l l v a v e l e n g t h sa n ds e r r a t i o n s

i nE q u a t i o n( 3 1 )y i e l d st h ei n t e n s i t yp r o f i l e                 i n t h ec h o s e n

intercept plane.




                                          16
                                                      '"t
                                                    (a t CY)

"7                                                    3" -
                                                        "




F i g u r e 7.   Yk e x t r e m er a yp a t h si nl o w e rl e n sh a l fs e r r a t i o n s ;   6 2 a.




                                                 17
Figure 8.   Y,   extreme ray p a t h in lower lens h a l f serrations; 6 < a.




                                   18
                                      111.     THEORETICAL RESULTS

          Basedon           the preceding theoretical                      model, a computerprogram                         was

developed t o p r o v i d e exampleperformancedata.Lensparameters                                                          were

s e l e c t e dt oc o r r e s p o n dw i t he x i s t i n ge x p e r i m e n t a lc o n c e n t r a t o r s( T a b l e s

1 and 2 ) t o f a c i l i t a t e c o m p a r i s o n s o f a n a l y t i c a l / e x p e r i m e n t a l r e s u l t s .

Performancedataforotherlens                                  sizes may be approximately determined by

using appropriate scaling factors.

          For t h e c o m p u t a t i o n s , t h e s o l a r s p e c t r u m p r o p o s e d          by Moon [4] as

a standardsolarradiationcurve                                 was incremented as i l l u s t r a t e d i n T a b l e

3 and a p p r o p r i a t ew e i g h t i n gf a c t o r sa s s i g n e d .              Bulk t r a n s m i t t a n c ef a c t o r s

for a c r y l i c were determined by t h e method o u t l i n e d i n                                 [ I ] and are a l s o
listed in Table                 3 alongwiththeindicesofrefractionobtained                                                  from

m a n u f a c t u r e r ' sd a t a[ S I .

          With t h e s ei n p u tp a r a m e t e r s          and d a t a , t h e l e n s t r a n s m i s s i o n           and f o c a l
plane solar             images were s t u d i e d as a f u n c t i o n o f t r a n s v e r s e t r a c k i n g e r r o r
up t o 2.5" f o r t h e 56 c l e n s and 0.75" f o r t h e 1 . 8
                            m                                                                    m lens.
A.      Example Data - 0.56 Meter Test Lens

        1.        Transmission
                Lens

                  Totallenssunlighttransmittance                                 was p r a c t i c a l l y u n a f f e c t e d     by

        t h ep r e s e n c eo f         a small t r a n s v e r s et r a c k i n ge r r o r .            The computed t r a n s -

        mittancedecreasedby                       less than 1%(from87.4                         to86.6%)         as t h e t r a c k i n g

        e r r o r was i n c r e a s e d from 0" t o 2.5'.                      The d e c r e a s e may b e a t t r i b u t e d

        togrooveedgelossesandtoincreasedreflectionlossesforupperhalf

        grooves.            A s i l l u s t r a t e di nT a b l e      4 , h i g h e rt r a n s m i s s i o nf o rl o w e rh a l f

        s e r r a t i o n s p a r t i a l l y compensates f o r t h e i n c r e a s e d u p p e r h a l f r e f l e c t i o n

        losses.             changes
                          The                   i nt r a n s m i s s i o nf o rt h e        two l e n sh a l v e sw i t hr e s p e c t

        t ot h ez e r ot r a c k i n ge r r o r            case are i l l u s t r a t e d b yt h ed a t a .I ng e n e r a l ,




                                                        19
               TABLE 1.     LENS
                          TEST CHARACTERISTICS

                                                                    . .~ .-~
                                                                       -.   ."
I
                                                                                      ~
                                                      ~~                              "~




I
    LensType                           C y l i n d r i c a lF r e s n e l ,           Grooves Down


    Material                           Rohm and Haas P l e x i g l a s VS
I
i
I
    Fabrication Technique              CompressionMolding


    Manufacturer                       OpticalSciencesGroup,Inc.


    f-number                           1.0


    Center Thickness                         m
                                       0.434 c ( 0 . 1 7 1i n . )


    Groove Density                     13.58/cm (34.                S/in.)


    DesignWavelength                   5893

                                                           -. .   ~i-~."..    -   i         ."




                                  20
            TABLE 2.            LENS
                        LARGE TEST CHARACTERISTICS



Lens Type                     C y l i n d r i c a lF r e s n e l ,G r o o v e s   Down


Material                      Rohm and Haas P l e x i g l a s V(811)


Fabrication Technique         Compression Molding


Manufacturer                  O p t i c a lS c i e n c e s     Group, I n c .


Width                                m
                              182.9 c (72 i n )A c t i v eA p e r t u r e
                                    m
                              186.7 c ( 7 3 . 5i n )T o t a lA p e r t u r e


FocalLength                   168.0 c ( 6 6 . 1 5i n )
                                    m
(for design wavelength)


Geometric F-Number            0.9


Center Thickness                     m
                              0.594 c ( 0 . 2 3 4 i n )


Groove Density                8 . 8 cm-l ( i n n e r 18 i n c hp a n e l )
                              1 3 . 2 cm-' ( o u t e r1 8i n c hp a n e l )


Design Wavelength             625nanometers




                                21
                                                  SPECTRAL
                            TABLE 3. SOLAR AND LENS      PARAMETERS


        Wavelength       Center        Weighting             AcrylicIndex          Acrylic Bulk
        Increment       Wavelength      Factors              of Refraction      TransmittanceFactor
           (ax) j           Aj                 w
                                                   j
                                                                   n
                                                                     j                (Tal j
         (microns)
        (microns)


          0.374
        0.295-0-40                                           1.5250
                                                             0.962
                                         2 . 6 7 ~ 1 0 - ~ (estimate)                           (0.675).
0.416   0.40-0-.43                                                                      1       (0.9951,
        0.43-0.45       0.441           2.44                 1.5018                     1
        0.45-0.47       0.460           2.91                 1.4999                     1
        0.47-0.49       0.480           3.20                 1.4982                     1
        0.49-0.51       0.500           3.27                 1.4968                     1
        0.51-0.53       0.520           3.23                 1.4954                     1
        0.53-0.55       0.540           3.22                 1.4942                     1
        0.55-0.57       0.560           3.19                 1.4930                     1
        0.57-0.60       0.585           4.73                 1.4918                     1
        0.50-0.63       0.615           4.73                 1.4906                     1
        0.63-0.66       0.645           4.75                 1.4895                     1
        0.66-0.69       0.675           4.56                 1.4886                     1
        0.69-0.73       0.709           5.37                 1.4876                     1
        0.73-0.78       0.753           5.91                 1.4865                     1
        0.78-0.83       0.804           5.62                 1.4854                     1
        0.83-0.89       0.857           6.23                 1.4845                     1
        0.89-0.99       0.953           6.06                 1.4832                     1
        0.99-1.06       1.024           5.65                 1.4826                     1
        1.06-1.21       1.129           6.21                 1.4818                    0.9 48
        1.21-1.52       1.274           6.49                 1.4812     (estimate)     0.912
        1.52-2.2        1.642           6.81                 1.4808     (estimate)     0.570



        ?Values i n parentheses used for 1.8           m lens computations.




                                                        22
     TABLE    4. COMPUTED SUNLIGHT                   OF TEST
                                           TRANSMITTANCE       LENS   SERRATIONS;
                0.56 M LENS


 Serration      Serration
                            I             Sunlight    Transmittance
                            i
  Number        Position            6 = 0"       1            6 = 2.5O
                  Y i/W         (Each Lens Half)             Lens Half
                                                       Upper              Lower
    0         6.494~10'~            -8878              .8878              .8878
   20         2.662~10-~            -8878              .8864              .8485
   40         5.260~10-~            .8878              .8850              .8878
   60         7.857~10-~            .8877              .8836              .8878
   80          .lo45                .8875              .8821              .8878
  100          .1305                .8873              .8805              .8876
  120          .1565                .8869              .8787              .8874
  140          .1825                .8862                                 .8869     .8766
  160          .2084                .8852                                 .8863     .8741
  180          ,2344                .8839                                 .8853     .8711
  200          .2604                .8820                                 .8840     .8675
  220          .2864                .8795              .8631              .8822
  240          .3123                -8763              .8578              .8800
  260          .3383                .8724              .8512              .8772
  280          .3643                .8675                                 .8739     .8434
  300          .3903                .8616              .8338              .8699
  320          .4162                .8546                                 .8653     .8224
  340          .4422                .8464                                 -8600     .8086
  360          .4682                -8369              .7921              .8540
  380          .4942                .8260                                 .8473     .7723




              TABLE 5 LENS TRANSMITTANCE
                     .                   OVER THE SOLAR SPECTRUM
                       FOR A TRANSVERSE TRACKINGERROR OF 1 5 ;
                                                          .'
                       0.56 M LENS


 Wavelength         Transmittance             Wavelength         Transmittance
 Increment                                    Increment
   (AA 1j                                       (MI j
 (microns)                                    (microns)


0.295-0.40            0.8592                  0.63-0.66               .9058
0.40-0.43              .9002                  0.66-0.69               .9061
0.40-0.45              .9016                  0.69-0.73               .9065
0.45-0.47              .9022                  0.73-0.78               .9068
0.47-0.49              .9028                  0.78-0.83               .9072
0.49-0.51              .9033                  0.83-0.89               .9075
0.51-0.53              .9038                  0.89-0.99               -9079
0.53-0.55              .9042                  0.99-1.06               .g081
0.55-0.57              .9046                  1.06-1.21               .8611
0.57-0.60              .9050                  1.21-1.52               .8286
0.60-0.63              .9054                  1.52-2.2

Total sunlight1 xmsmittance = 0.870


                                      23
transmittance decreases with increasing serration distance                                                       from t h e

 l e n sc e n t e r ,    as i l l u s t r a t e d i n F i g u r e       9 f o r a 1.5O t r a c k i n g e r r o r , d u e

tothelargergrooveangles                              a n dh e n c ei n c r e a s e da n g l e so fi n c i d e n c e .

A trackingerrorincreasestheanglesofincidence                                                    a t t h e grooved

surface for the upper lens half                             and d e c r e a s e s t h e s e a n g l e s          for the

lower half.

           Grooveedge           lossesresultin                   a reducedtransmit8ancebut                            are of

m i n o ri m p o r t a n c ef o rt h e         small t r a c k i n ge r r o r sc o n s i d e r e d .              For upper

halfserrations,edgelossesincreasemonotonically                                                    from n e a r z e r o f o r

t h ec e n t e rs e r r a t i o nt o ,t y p i c a l l y ,           1 t o 2% f o r t h e o u t e r m o s t s e r r a t i o n ;

e . g . , a t 1 . 5 't r a c k i n ge r r o rt h e            maximum l o s s i s 1.33%.Forthe

lowerhalfand                6 > a, b l o c k i n g l o s s e s o c c u r o n l y f o r s e r r a t i o n s w i t h

                 a          F           3
small g r o o v e n g l e s . i g u r e s ( a )                   and 3 ( b )i l l u s t r a t et h eb l o c k i n g

mechanisms r e s p o n s i b l e f o r t h e t r a n s m i t t a n c e " d i p " n e a r t h e l e n s c e n t e r

shown i nF i g u r e          9 f o r a t r a c k i n ge r r o ro f1 . 5 ' .T h e s eb l o c k i n g                     mecha-

n i s m sc e a s et of u n c t i o n         when t h e g r o o v e a n g l e         becomes s u f f i c i e n t l y

1a r g e   .
         Table 5 l i s t s t h e l e n s t r a n s m i t t a n c e f o r e a c h o f t h e                     22 i n t e r v a l s

o ft h es o l a rs p e c t r u mu s e di nt h ec o m p u t a t i o n s .A b s o r p t i o ni nt h el e n s

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

F o re x a m p l e ,h i g ha b s o r p t i o nd r o p st h et r a n s m i t t a n c ei nt h es p e c t r a l

range1.52-2.2      from
             microns   above                                     90% t o below52%.                 The r e f l e c t i o n

l o s s e s are seentodecreaseonlyveryslowlywithwavelength.




                                                            24
.go,          1                  1         1         I         I        1          1         I




     I                                               I
                                                     I
                                                     I
                                                     1
                                                     I
         -                                           I
             < pe
              U
             - pr            Lens Half " +
                                        -            I   -Lower             Lens Half   #)
                                                                                         -




                                                     I
                                                     I
                                                     I
75             I        I         1        ,         I            I     I          I         I

   0.5       0.4       0.3      0.2       0.I      0         0.I        .
                                                                       02         0.3        0.4
                                        Serrution Posit ion (y/W)

                    Figure 9. Transmittance Versus serration position for 6 = 1.5';
                              0.56 m lens.
2.       Plane
     Focal           Profiles
             Intensity

              The l o c a l c o n c e n t r a t i o n r a t i o      as a f u n c t i o n o f f o c a l p l a n e p o s i t i o n

     hasbeen          computed f o r t h e           test lensfortransversetrackingerrors                                       in
                                                                                                                                       ~   .-
     therange           0'-     2.5'.       Figures 10and              1 d e p i c tt h e
                                                                        1                         computed i n t e n s i t y

     profiles.            The presenceof               a t r a c k i n ge r r o rm o d i f i e st h ed i s t r i b u t i o n

     of concentrated sunlight                      by (1) l a t e r a l l y s h i f t i n g t h e p r o f i l e , . . ( 2 )       gen-

     erally reducing the                  peak concentration, and (3Daltering the profile                                         symmetry

              The l a t e r a l s h i f t i n g o f t h e p r o f i l e            i s quantified in Figure                   1 2 by

     p l o t t i n g t h e peak p o s i t i o n s h i f t as a f u n c t i o n o f o r i e n t a t i o n e r r o r .

     The shift increases approximately linearly over the range                                                  examinedand

     may be compared w i t h t h e image displacement 6 x f expected for a mono-

     chromatic point source.

              The change i n t h e f o c a l p l a n e               peak concentration with increasing

     t r a c k i n ge r r o r    i s d e p i c t e di nF i g u r e . 1 3       .     For a smallmisalignmentangle

     larger than            a and f o r t h i s p a r t i c u l a r i n t e r c e p t p l a n e , t h e           computed peak

     concentration i s g r e a t e r t h a n t h e z e r o t r a c k i n g e r r o r c a s e , b u t t h e n

     monotonicallydecreases                      as t h ee r r o ri n c r e a s e s .            For 6     <   a , . t h e peak

     concentration remains nearly constant.

              P r o f i l e asymmetry becomes conspicuous f o r t h e l a r g e r t r a c k i n g e r r o r s ,

     with a long " t a i l T 1developing in a d i r e c t i o n away from t h e f o c a l l i n e .

     This redistribution of energy simultaneously sharpens the profile                                                         on t h e

     other side.

              Increasing profile shift                      andskewness             with tracking error result

     in large increases in the target                             widthdesigned              t o i n t e r c e p t an acceptable

     fractionoftheconcentratedenergy.                                      The t a r g e tw i d t h sr e q u i r e df o r            90%

     interception beneath concentrators with tracking systems                                                  whose design

     t o l e r a n c e sa r e   2 6 a r ei l l u s t r a t e di nF i g u r e            14   .    As       example,
                                                                                                          an                   for



                                                         26
6or
I
     1   I              I                                I          I




 t
                                                                                        0.I

50                                                                                  In




                                                (centimeters)



                                  FOCAL PLANE POSITION (Y/W)

         Figure 1 0 .       T r a n s v e r s eo r i e n t a t i o ne f f e c t s   on i n t e n s i t y p r o f i l e ;
                            0.56 m lens.
        60       I          1     I        I                    I           I             I          I           I                 I             I




           I




                                                                                                              7
           L
                                                                                                          .7f

        50 -


    0                                                                               17i
    k'




N
W




                                                                    (centimeters)
                                                                                                                                                          1
    -.h
                                                                                                                                       I

                      -.i. -ab -.oa -.of -.oh -.oi -.04
                                                                                                             I

                                                                                              -.oi
                                                                                                                        1                    I       I

               4'1.                                                                                      -.02 -.OI                 0       .OI   .02     .03
                                                                   POSITION
                                                         FOCAL PLANE                                      (Y/W)

                         Figure 11.   T r a n s v e r s eo r i e n t a t i o ne f f e c t s          on i n t e n s i t yp r o f i l e ;
                                      0.56 m l e n s .
                                         0

0       Computed Shift
        6 x f                        0   / '5
                                             ?.O
                                                    R
                                                    z
                                                    =!
                                                3
                                              .
                                              5
                                             1 m
                                                    -I
                                                    m
                                                    XI
                                                    v)
                                                    Y

                0

                                             I .o



                                             0.5



                                             0

             ORIENTATION (DEGREES)
    TRANSVERSE         ERROR
                                                                                                          0
3(



2!




2c




1:




IC



 e
 *




0         0
                  U




+5
                  1                      t                      I                   I                      I

                   .
                  05                    I .o                   I.5                20                     2.5
                       TRANSVERSE ORIENTATION ERROR (DEGREES)


     Figure 13.    T r a n s v e r s eo r i e n t a t i o ne f f e c t s   on p e a k. c o n c e n t r a t i o n ;
                   0.56 m l e n s .


                                                       30
       1                 I
                                                  I                       1                  I               I   O
                                                                                                                 I
                                                                                                         0
     I-
    .6
                                                                                                   0

    .I4-
                                                                                                         A
                                                                                             0
       I
       I                                                                                           A
    .121-                                                                            0
                                                                                            A
    .io -                                                                0
                                                                                     A
                                                             0
    .08-                                                                A
                                                 0

    .06-                              0
                                                             A

                                                 A
                         0                                         o TARGET               WIDTH
                                     A
                0
                                                                   A TARGET WIDTH INCREASE
                         A

                A        I                        I                       I                  I               I

I                        0.5                     I.o       1.5                              2 .o         2.5
                    TRANSVERSE                   ORIENTATION ERROR                          (DEGREES)

            Figure 14.       T r a n s v e r s eo r i e n t a t i o ne f f e c t s       on target -width;
                             0.56 m l e n s .
     6 = + l o ,a t h r e e - f o l d i n c r e a s e i n t a r g e t w i d t h o v e r t h e p e r f e c t

     alignment case is computed f o r t h e                                 t e s t l e n s( 4 . 1          c vs 1.4 cm).
                                                                                                            m

     P r o f i l e s h i f t i s r e s p o n s i b l ef o r             most o f t h i s i n c r e a s e .           If t h ez e r o

     misalignmentprofile is simply shifted to the 1" profile peak position,

     the change in the target width represents roughly 70% of the increase

     computed f o rt h e1 "p r o f i l e .F o rl a r g e rt r a c k i n ge r r o r s ,t h e

     importanceofprofileskewnessgrows.


B.   Example Data - 1.8Meter                            Test Lens

               The t r a n s m i t t e d f r a c t i o n o f s u n l i g h t s t r i k i n g t h e t o t a l l e n s

     a p e r t u r e was computed t o b e 0.842 f o r a f l a w l e s s l y t r a c k i n g l e n s .

     The t r a n s m i t t a n c e d r o p p e d          by less t h a n t w o - t e n t h s o f o n e p e r c e n t

     fortransversedeviations                              up t o 0.75".

               F o c a lp l a n ei m a g ep r o f i l e sw e r ed e t e r m i n e df o re r r o r s                     o f 0, 0.15,

     0.26, .52, nd .75" Figure 5). hese rofiles xhibit
         0    a 0     (       1 T      p      e                                                                         similar

     c h a r a c t e r i s t i c sw i t hr e s p e c tt op e a ks h i f t( F i g u r e1 6 ) ,p e a kc o n c e n -

     t r a t i o nr e d u c t i o n( F i g u r e1 7 ) ,a n ds k e w n e s s                   as i n t h e p r e v i o u s

     e x a m p l e .A g a i n ,s u b s t a n t i a li n c r e a s e sw i t ht r a c k i n ge r r o ra r e

     o b s e r v e di nt h et a r g e tw i d t hr e q u i r e dt oi n t e r c e p t                         90% o f t h e t r a n s -

     m i t t e df l u x( F i g u r e1 8 ) .               For example, f o r a l e n sc o n c e n t r a t o rs y s t e m

     d e s i g n e d t o t r a c k t h ec e n t e ro ft h es u nw i t h i n+ 0 . 2 5 " ,t h ea p e r t u r eo f

     thesecondaryconcentrator                              o r absorber must be increased in width

     by a p p r o x i m a t e l y o n e t h i r d o v e r t h a t r e q u i r e d f o r t h e                   flawless

     trackingcase.




                                                                   32
    60




    50




c
z
W
0
6 20
0



    IC



     0
         -a            -6            -4          -2           0           2           4         6
                                   IMAGE PLANE POSITION (cm)
              Figure 15.    Transverse orientation effects on intensity profile; 1.83 m lens.
                                                      0




                                     0




               0




      0




          I                 I              I               I
          02             On4             0.6              088
          TRANSVERSE                 ERROR          (DEG)
Figure 1 6 .       T r a n s v e r s eo r i e n t a t i o ne f f e c t s on
                   p r o f i l e p e a kp o s i t i o n ;       1.83 m lens.



                                34
                                                                        0




      0

                  0


                                0




     d                0.i              de4             6.6                  6.8
                   TRANSVERSE                  ERROR (DEG)
F i g u r e1 7 .T r a n s v e r s eo r i e n t a t i o ne f f e c t s         on peak
                      c o n c e n t r a t i o n ;1 . 8 3 m l e n s .




                                  35
                                 0

                       0




         I                 I                  I                 I               I

       0               0.2     04
                                .                       0.6                 0.8
                       TRANSVERSE                    ERROR              (DEG)

F i g u r e1 8 .T r a n s v e r s eo r i e n t a t i o ne f f e c t s           on t a r g e t
                      w i d t h ;1 . 8 3 m l e n s .


                                          36
                                      IV.       S M A Y AND CONCLUSIONS
                                                 U MR

 1.    A n o p t i c a lr a y       t r a c e a n a l y s i sf o ra s s e s s i n g         small (<5") t r a n s v e r s e

       trackingerroreffectsonthesolartransmissionandimagingproper-

       t i e s o f a l i n e a rF r e s n e ll e n s             was d e v e l o p e d .I na d d i t i o nt o

       t r a c k i n ge r r o r ,v a r i a b l ep a r a m e t e r si n c l u d ei n t e r c e p tp l a n ep o s i t i o n

       a n ds u c hl e n sc h a r a c t e r i s t i c s          as f-number,groovedensity,and

       design index of refraction.

 2.    Transmittance and image profile computations                                           wereperformed                for

       a 56 c wide,f/1.0and
             m                                        a 1 . 8 3 m, f / 0 . 9 t e s t l e n s t o p r o v i d e              a

       databaseforcomparisonwithexperimentaldata                                                    from NASA t e s t

       programs.

 3.    Lens t r a n s m i t t a n c e i s o n l ys l i g h t l yd e g r a d e d( < 1 % )f o rm i s a l i g n m e n t s

       up t o 2 . 5 " .

 4.    S o l a r image d e g r a d a t i o n w i t h t r a c k i n g e r r o r i n c l u d e s a n a p p r o x i m a t e l y

       linearlydependentprofileshift,                                    a p e a kc o n c e n t r a t i o nr e d u c t i o n ,

       andincreasedprofileskewness.

 5.    The 90% t a r g e t i n t e r c e p t w i d t h i n c r e a s e s r a p i d l y f o r                    small t r a n s v e r s e

       t r a c k i n g e r r o r s , a l m o s tt h r e e f o l df o r           a 1 "e r r o ro v e rt h ep e r f e c t

       trackingcaseforthe                         small t e s t a r t i c l e , and a t a similar r a t e f o r

       t h el a r g el e n s .

6.    The p r i m a r y c a u s e f o r t a r g e t w i d t h i n c r e a s e s i n t h e p r e s e n c e o f

       transversetrackingerrors                             is p r o f i l e s h i f t .

 7.    The t h e o r y a n d r e s u l t s i n t h i s s t u d y p r o v i d e                an a n a l y t i c a l b a s e ,

       albeitapproximate,forthedesignoftheinterrelatedtracking

       and t a r g e t a b s o r b e r s y s t e m s f o r           a flat linearFresnellenssolar

        concentrator.




                                                              37
V.   REFERENCES

     1.    L. H a s t i n g s , S. Allums,
                                         and   R. Cosby, "An A n a l y t i c a l and
           ExperimentalEvaluationof           a F r e s n e l Lens S o l a r C o n c e n t r a t o r " ,
          .NASA T e c h n i c a l Memorandum TM X-73333, August,1976.

     2.   L. H a s t i n g s , S. Allums, and R. Cosby, "An A n a l y t i c a l and
          Experimental Evaluation o f t h e P l a n o - C y l i n d r i c a l F r e s n e l Lens
          SolarConcentrator",Proceedingsofthe"Sharingthe                                Sun"
          JointConferenceofthe            American, Canadian,andInternational
               E           S
          Solar nergy ociety,                   Canada.
                                        Winnepeg,     Vol.                2 , (August      1976),
          p . 275.
     3.   R. Cosby, " C o n c e n t r a t i o nC h a r a c t e r i s t i c so ft h e. C y l i n d r i c a l
          F r e s n e l Lens S o l a r C o n c e n t r a t o r " , i n   1975 NASA-ASEE          Summer
          FacultyFellowResearchReports,                              BER Report No. 202-94,
          (George C. M a r s h a l l S p a c e F l i g h t C e n t e r / U n i v e r s i t y o f         Alabama/
          Auburn University),        September,        1975,     p.           111-1.

     4.   P. Moon, "ProposedStandardSolar-RadiationCurvesforEngineering
          Use", J. F r a n k l i n I n s t i t u t e , - 583 (1940).
                                                       23a,

     5.   Rohm and
                 Haas   Company, P h i l a d e l p h i a , PA, " P l e x i g l a sI n j e c t i o n
          and ExtrusionMoldingPowders,"Publication                   PL 1 6 5 f , December
          1968.




                                                          38
                                                       APPENDIX

                                              GROOVE BLOCKING LOSSES


         Assuming a l l s o l a r r a y s s t r i k i n g              a grooveedge             are l o s t , t h e p r o b l e m

is tocalculate,for                        a given small t r a c k i n g e r r o r , t h e a v e r a g e f r a c t i o n o f

i n c i d e n t r a y s on a s e r r a t i o n which are b l o c k e d f r o m r e a c h i n g t h e s o l a r                  image.

Notingourapproximationofzerorayaxialcomponents,each                                                          case i l l u s t r a t e d

i nF i g u r e s2 ( a ) ,( b ) ,( c )             and Figures 3 ( a ) , ( b ) , ( c ) must evaluated
                                                                                         be         sep-

arately.


CASE :       Figure Al'(a), u p p e r h a l f .

         Redrawing t h er a yd i a g r a m                of F i g u r e 2 ( a ) t o p r o v i d e      more d e t a i l , it i s

observedthatthelostfraction                                 o f r a y si n c i d e n t     a t anangle         $i o nt h e      ser-

r a t i o n o f width by i s


         F = - S
              AY

 o
Nw

         s = A t tan$t                ,
where


                y
         A t = A tan Oi.

Then

           s = A t a n O i t a n$ t
                y                                  .
Using S n e l l ' s law o f r e f r a c t i o n ,

                          sin$,
          tan$t =
                    ,      /      n       T


Combining Equations (A-1) t h r u (A-5).

                                          sin$i
          F ( I $ ~ )= t a n 0 -



                                                                  39
                                      f




                                                                 y
                                                                * +                  I   .   .
                                                                                             4
                                                                                             "




/                                                                7-"




                                                                               \
Figure A l .   Ray diagrams f o r 'grooveedge   losses i n upper l e n s h a l f .




                                          40
For a t r a c k i n g e r r o r       6, t h e a v e r a g e f r a c t i o n l o s t      is




where $o = ' 6        -    a i f 6.> .a and O0 = 0 i f 6 < a.

Assuming small t r a c k i n g e r r o r s , s i n              $i = $i, and t h e i n t e g r a l s i m p l i f i e s t o




Evaluationofthissimpleintegralyields




and


                                                                                                           (A- 10)



CASE :     F i g u r e A 1 (b) , u p p e r h a l f .

        R e f e r r i n gt ot h ed i a g r a m ,       it is clearthatinthiscase,blocking

lossescanoccuronly                     i f 6 < a and when t h e a n g l e o f r e f r a c t i o n                at the

first s u r f a c e is g r e a t e r t h a n         or e q u a l t o      Oi, r e q u i r i n g

                           $i > Ai r c s i n ( (A-11) Oi).
                                s n            n

(Here O i i s t h e g r o o v e a n g l e f o r t h e i t h s e r r a t i o n w h i l e              qi.    is the .angle

 i
of ncidence.                         s
                           Thus t h e u b s c r i p t s a v e i f f e r e n t e a n i n g s . ) i n c e o r
                                                       h     d              m                 S        f

t h i s case $i h a s a maximum v a l u e o f                   a (when 6 = 0 ) , t h e s e c o n d i t i o n s          are

possibleonly              for small O i , i . e . , o n l y f o r            a few o f t h e g r o o v e s ,         i f any,


                                                             41
near t h e l e n s c e n t e r .                Hence t h e l o s s e s               are e x p e c t e dt ob ei n c o n s e q u e n t i a l .

          From Figure Al@), twice u s i n g t h e t r i a n g l e                                     law o f s i n e s ,      and recogniz-

ing that

                             A t = ~y tan ~                     i     ,
                                                                      +         ~                                        (A-12)

the lost fraction for                       rays w i t h a n g l e s              of incidence$i is




                                                                                                                         (A- 13)

Applying S n e l l ' s law o f r e f r a c t i o n                         a t t h e two l e n s s u r f a c e s            and u s i n g t h e

f a c t t h a t #i and Oi are v e r y small,

                             9 t = -9 i          9                                                                       (A- 14)
                                       n

and

                             $i        $i       - noi .                                                                  (A- 15)

Further

                             cos(+t         -    Oi) = cos                'pi =   cos $ t = 1
                                                                                       I                 .               (A- 16)

Then

                             F($i) = tan Oi+l                       [@i- (n - 1)                Oil      .               (A- 17)

I n t e g r a t i n g as i nE q u a t i o n( A - 7 ) ,t h ea v e r a g ef r a c t i o no fi n c i d e n ts u n l i g h t

lost thru this blocking                          mechanism i s

                                       tan O i + l
                             :
                             F     =                                      [ $ i - (n - 1)          Oil       dOi    .    (A-IS)


To d e t e r m i n e t h e a p p r o p r i a t e             limits o f i n t e g r a t i o n ,              we n o t e t h a t t h e    maxi-

mum $ i i s a - 6 and t h a t t h i s t y p e o f b l o c k i n g                                does n o t o c c u r         i f $< i n

                 zero.
FigureAl(b),becomes                                      Thus

                             $f= a - 6               ;   $    I $ ~   =   nOi     ,                                      (A- 19)

and p r o c e e d i n g w i t h t h e s i m p l e i n t e g r a t i o n i n ( A - 1 8 ) ,
and obviously

                         FE = 0                    f o r noi > a       -   6   .                   (A-21)


CASE:       Figure A 1 (c) , upper h a l f .

         Applyingthetriangle                  law o f s i n e s      t o determine s as i n t h e p r e v i o u s

c a s e ,t h er a yd i a g r a my i e l d s


                           F($i) = %- =       --
                                                   tm     gi+l    sin(gi       - 9)
                                                                                  ;   c o s ( @ t - ei)   .
                                   AY                         cos 4;       cos

                                                                                                   (A-22)

Again

                           $t". @i                                                                 (A- 2 3)
                                n

A 1s o

                           sin          = n sin(Oi        -   @i
                                                              -
                                                              )                                    (A- 2 4)
                                                              n

                                        = n sin      Oi   -   $i c o s . ei




where

                                                                                                  (A-27)

NOW

                                                                                                  (A- 28)



                                                           43
where t h e limits o f i n t e g r a t i o n f o r t h i s                  case are

                           @ f = a - 6                                                                        (A-29)
                                                          sin O i
                           @o = noi        -    n arcsin (
                                                            n     1                  .                        (A- 3 0)

E v a l u a t i o no ft h ei n t e g r a li n           (A-28) u s i n g (A-25) t h r u (A-30) y i e l d s

Equation (13) i n t h e t e x t                 of t h e ' r e p o r t . F o r t r a c k i n g e r r o r s         6 > a,
FG' = 0.


CASE:             A    l    h ol
            Figure 2(a), ower alf f ens.

        Assuming 6 < a and r e f e r r i n g t o t h e r a y d i a g r a m ,

                           F ( @ ~= 2 = t a n
                                   )
                                         AY
                                                           oi    t a n @t        .                            (A-31)

Using S n e l l ' s law,
                                                                                                              (A-32)


and the assumption of                  small 6,


                                                                                                              (A-33)



Then




                                                                                                              (A-34)


Using ( a - 6) << n,

                                                                                                              (A- 35)


         For 6 > a         , Fi     = 0.


CASE:       F i g u r e A 2 (b)           h
                                      lower alf.

         This blocking            mechanismshouldbe                      nontrivial only for serrations

very     near t h e l e n sc e n t e ra n df o r                t h e l a r g e rt r a c k i n ge r r o r s .U s i n g



                                                                44
                                                         ""_
                                                               T
                                                               bY



                                                  _"""         1
                                          (a>




"T"'




\                                                                   I




       Figure A 2 .                                                   half.
                      Ray diagrams for groove edge losses in lower lens




                                          45
t h e t r i a n g l e law o f s i n e s ,




Againusingthe                 f a c t t h a t a l l a n g l e s o f i n c i d e n c e Gi are v e r y small

f o r small 6 and u s i n g S n e l l ’ s                laws,


                                                                                                                [A- 3 7)



Then

                            F(Oi)       y    tan             [@i - (n - l)OiJ                   .              (A- 3 8)

Integratingoverpossibleanglesofincidence                                                and u s i n g t h e      limits

(for serrations which have                         n Oi < 6 + a ) ,

                            @o=nOi ; @ f = 6 + a                                 ,                             (A-39)
                            FII   - t a n 0‘-+I- ( + a - nei) [6 + a
                                            -     6                                         + (2       -   n ) ~ i ~ .
                                            4a
                                                                                                               (A-40)


CASE:             A2(c),
             Figure        half.
                       lower

         Observing the ray diagram                         and u s i n g t h e t r i a n g l e         law o f s i n e s ,

                                            tan Oi+l         COS($^ - Oi)sin(Oi                     - @<)
                            F(@i) =
                                                              cos       Ot       cos @;
                                                                                                               (A-41)
N o t i n gt h a t    (A-41) i s i d e n t i c a lw i t h               (A-22)       , Equations (A-23) t h r u
(A-28) are a l s of o u n dt oa p p l yt ot h ep r e s e n t                            case.       To d e t e r m i n et h e

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

ofrays,         it i s n o t e d t h a t          Go i s t h e a n g l e o f i n c i d e n c e f o r          which t h e

emergentray             is horizontal:
                                                                        sin Oi
                            $o    2:   n Oi      - :nArcsin         (                1 ,                       (A-42)
                                                                             n




                                                                 46
1-
                and

                                           Qf=nOi               ,     if       n O i < 6 + a           ;                  (A-43)

                                           Q f = 6 + a            ,       if     n O i > 6 + a .                          (A-44)

                E v a l u a t i o n of t h e i n t e g r a l i n E q u a t i o n           (A-28)    u s i n gt h e   limits i n

                (A-42;      4 3 ,4 4 )     y i e l d sE q u a t i o n s        (24)     and (25) i n t h e t e x t o f t h e

                report.




     NASA-Langley, 1977

                                                                                   47

								
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