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Optical designs and concentration characteristics of a linear Fresnel reflector solar concentrator with a triangular absorber
Solar Energy Materials 21 (1990) 237-251 North-Holland
237
Optical designs and concentration characteristics of a linear Fresnel reflector solar concentrator with a triangular absorber R.P. Goswami a, B.S. Negi
b,
H.K. Sehgal b and G.D. Sootha a,.
Solar Energy Centre, Department of Non-Conventional Energy Sources, Ministry of Energy, Government of India, Block 4, 9th Floor, C.G.O. Complex, Lodhi Road, New Delhi 11003, India b Department of Physics, Indian Institute of Technology, Hauz Khas, New Delhi 110016, India
Two different approaches for designing a linear Fresnel reflector solar concentrator (LFRSC) for a triangular absorber configuration are presented. The first approach allows a variation in the width of the mirror elements, while the second uses mirror elements of equal width. For each design, the distribution of local concentration ratio (LCR) on the surfaces of two sides of the absorber which intercept the solar radiation reflected from concentrator has been investigated using the ray trace technique. Results are plotted graphically and discussed.
I. Introduction H i g h - q u a l i t y energy f r o m the sun for m a n y processes r e q u i r i n g h e a t at t e m p e r a tures a b o v e 100 ° C can b e g e n e r a t e d b y c o n c e n t r a t i o n of solar r a d i a t i o n [1,2]. T h e c o n c e n t r a t i o n effect is a c c o m p l i s h e d b y the use of either reflecting or r e f r a c t i n g elements which are l o c a t e d so t h a t solar r a d i a t i o n is focused o n t o the receiver p l a c e d in the focal zone. T h e c o n c e n t r a t i o n a p p r o a c h c a n also b e effectively utilized for reducing the costs of p h o t o v o l t a i c p o w e r p l a n t s which is highly e x p e n s i v e at p r e s e n t [2-14]. In such a system highly efficient solar cells are e x p o s e d to the c o n c e n t r a t e d solar flux at the focus of a solar c o n c e n t r a t o r t h e r e b y r e d u c i n g the solar cells a r e a a n d increasing the electrical o u t p u t . T h e latter, however, requires u n i f o r m flux c o n c e n t r a t i o n on the solar cells for efficient o p e r a t i o n of the c o n c e n t r a t o r - p h o t o voltaic system. M o r e o v e r , in such a c o m b i n e d system, it is n e c e s s a r y to r e m o v e the intense heat g e n e r a t e d b y the focused r a d i a t i o n in o r d e r to keep the solar cells at an a d e q u a t e l y low t e m p e r a t u r e . It is, therefore, of interest to d e v e l o p a s u i t a b l e design of a solar c o n c e n t r a t o r - a b s o r b e r which can p r o v i d e sufficient u n i f o r m c o n c e n t r a tion in the focal zone of the c o n c e n t r a t o r . A n i m p o r t a n t c o n c e n t r a t o r which has received c o n s i d e r a b l e a t t e n t i o n in recent years for b o t h p h o t o t h e r m a l a n d p h o t o v o l t a i c c o n v e r s i o n of solar e n e r g y is a linear F r e s n e l reflector solar c o n c e n t r a t o r ( L F R S C ) [11-17]. A n L F R S C b a s i c a l l y consists * Advisor, Department of Non-Conventional Energy Sources, and to whom all correspondence should be addressed. 0165-1633/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
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R.P. Go~'wami et al. /" Linear h'esnel reflector ,solar concentrator
of long plane mirror elements suitably fastened to a flat base. The tilt of each mirror element is chosen such that normally incident solar radiation, after reflection from the mirror element, illuminates the absorber placed along the length in tile focal zone of the concentrator. The distribution of concentration at the focus, however. depends on the geometry of the absorber. This paper presents detailed designs of an L F R S C for a triangular absorber configuration. Such a concentrator-absorber system provides an improved performance in terms of uniformity in concentration on the absorber surfaces in contrast to the case of an L F R S C with a flat vertical absorber which showed a poor performance [14]. Two different approaches have been used for designing an L F R S C for a triangular absorber configuration. In the first approach the size of the mirror elements and hence the size of concentrator is determined from a prespecified size of the absorber. This approach allows a variation in the width of the constituent mirror elements [14]; whereas in the second approach using an equal width of the mirror elements, an appropriate size of the absorber is determined. For each design, the generalized expressions for the various design parameters have been derived. The angular subtense of the sun ( = 32') at any point on the earth has been taken into account while designing the concentrators. The concentration characteristics of each design have been studied in detail. The distribution of local concentration ratio (LCR) on the surfaces of two sides of the triangular absorber which intercept the solar radiation reflected from constituent mirror elements, for each design, has been investigated using the ray trace technique. The results of L C R distribution obtained from the ray trace technique have been compared with that obtained using an analytical technique [12-141.
2. Optical designs of LFRSC The designing of an L F R S C for a triangular absorber configuration has been carried out such that the solar radiation incident normally on the concentrator aperture, after reflection from the constituent mirror elements, illuminates two sides of the absorber. This effectively means that the mirror elements placed on the left half of the concentrator illuminates the left side of the absorber and the mirror elements placed on the right half illuminates the right side of the absorber. As described earlier, two different approaches have been used for designing an L F R S C for a triangular absorber configuration. In one approach the concentrator is designed from a prespecified width of the two sides of the absorber. This approach varies the width of constituent mirror elements. In the second approach using mirror elements of equal width the size of the absorber, i.e. the width of two sides of the absorber, is determined. The absorber size is determined such that it can intercept all the radiation reflected from constituent mirror elements. In each case an appropriate space is introduced between two consecutive mirror elements so as to avoid blocking of radiation reflected from mirror elements. Thus each constituent mirror element in both designs is characterised by four design parameters: shift ( S ) associated with it, its location ( L ) on the concentrator plane from its centre, its tilt
239
R.P. Goswami et al. / Linear Fresnel reflector solar concentrator
(0) with the concentrator plane and its width (W). In order to facilitate the designing of linear Fresnel reflector solar concentrators, the following simplifying assumptions have been made: (i) the concentrator is perfectly tracked so as to follow the apparent movement of the sun; (ii) the mirror elements are specularly reflecting; and (iii) the solar radiation is incident axially. The detailed design procedure for each LFRSC design is given below.
2.1. The first design of LFRSC The geometry used for deriving the design of an LFRSC from a prespecified size of the triangular absorber is shown in fig. 1. The absorber size is specified by the width, a, of two sides of the absorber which intercept the solar radiation reflected from constituent mirror elements, and the apex angle 2y between the two sides of the absorber. The tilt (0) and the width (W) of each mirror element is determined such that normally incident solar radiation, after reflection from the mirror element, produces an image on the plane of either sides of the absorber, of width equal to the width of the side of the absorber. Thus each mirror element placed on the left half of the concentrator would illuminate the complete surface of the left-hand side of the absorber and each mirror element placed on the right half would illuminate the complete surface of the right-hand side of the absorber. Further, since the concentrator is perfectly tracked, the absorber will cast a shadow on the concentrator plane and hence no mirror element is placed underneath the absorber. In other words, the location of the first mirror element on either half of the concentrator plane (XX') is determined such that it is just beyond the edge of the shadow. Taking into consideration the angular subtense of the sun, 240 ( = 32'), at any point
,/5 "
/
I \
~ rcx,,,A
~
k~ |lip n 2~°
\
\
~-Sn~IQn
Ln-1 L n Fig. 1. LFRSC employing mirror elements of varying width.
.I
240
R.P. Goswami et al. ,/ Linear Fresnel reflector solar concentrator
on the earth, the location of the first mirror element placed on either half of the concentrator plane is given by L 2 = a sin T + ( f + a cos y) tan ~0,
(1)
where a is the width of each of the two sides of the absorber, ~, is half the angle between the two sides of the absorber, f is the height of the absorber from the concentrator plane ( X X ' ) and ~0 is half the angular subtense of the sun ( = 16'). The tilt (02) of the first mirror element is chosen such that the extreme outer ray P~Q1 of the cone impinging on its lower edge, after reflection, meets the absorber at point F. The extreme inner ray PI'Q1 of this cone, after reflection, strikes the absorber at point R2. Now the width of the mirror element in question which has to illuminate the complete width of the absorber (i.e. FA = a) is determined by the inter section point T~ of a line drawn from the edge A of the absorber parallel to the line R2Q ! and a line passing through the point Q1 and making an angle 02 with the concentrator plane. This ensures that all the radiation reflected from the mirror element is intercepted by the one side (i.e. FA) of the absorber. In order to ensure that the solar radiation reflected from a constituent mirror element (excluding the first) is not blocked by the adjacent mirror element, appropriate values of the location and the tilt of the mirror element are determined. For example, the location (L2) and the tilt (02) of the second mirror element are chosen such that the solar radiation reflected from it is not blocked by the first mirror element and finally produces an image on the surface of one side of the absorber of width equal to the width of the absorber sides ( = a). This indicates that the incident ray PzQ2 of the cone impinging on the lower edge of the second mirror element, after reflection, just touches the upper edge of the first mirror element and finally meets the absorber at point F. This necessitates to introduce an appropriate space between the first and the second mirror element. Following the similar geometrical optics as described above, the following generalized expressions for the shift (S,), location (L,,), tilt (0,,) and the width (Wn ) parameters associated with the nth mirror element can be derived:
S"=
Wit 1 sin O n _ 2 ( L n .1 + Wn--I COS 0,, 2) f - W , , i sin0,, l '
L,=L,,
) + W,__ 2 cos 0,, i + S , ,
On = ½[tan-~( L , , / f ) -- ~o],
(2) (3) (4)
and
14:,,, =
a sin(20, + ~0) sin(20, + y - ~0) - L, sin 2~0 sin(20, + ~0) cos(0n - ~o)
(5)
with 00 = 0, W0 = $1 = 0 and L 2 = L 0 = a sin y + ( f + a cos y ) t a n ~0 as initial values. n varies from 1, 2, 3 . . . . . k; k being the total number of mirror elements placed on each half of the concentrator.
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R.P. Goswami et al. / Linear Fresnel reflector solar concentrator
The aperture diameter, D, of the LFRSC for a desired number of mirror elements may be expressed as D=2
L,+
~_, ( W . c o s O . + S .
(6)
.
n=l
2.2. The second design o f L F R S C
Fig. 2 shows the cross-sectional geometry of the LFRSC employing mirror elements of equal width W. Similar to the case of the first design, the tilt of each mirror element with the concentrator plane ( X X ' ) is chosen such that the extreme outer ray of the cones impinging on the lower edge of the mirror elements, after reflection, reaches the common point F on the triangular absorber. In the present case, since the absorber size is not known, it would be determined by the largest intercept produced on the plane originating from the point F and making an angle (90 - T) with the X-axis, by the solar radiation reflected from a constituent mirror element. 2T is the apex angle between two sides of the absorber which intercept the reflected radiation. It may clearly be noted from the geometry of the concentrator that the mirror elements of equal width will produce images of different widths on the absorber. However, the variation in the width of the images takes place in a different fashion for different values of T. For example, for values of T from 20 o to 60 °, the first mirror element placed near the centre of the concentrator aperture produces the largest image on the absorber plane and the width of the image therefore is taken to be the width of the two sides of the absorber. While for a value of T equal to 80 o, the last mirror element placed at the edge of the concentrator
xj Ln-ILn" ~
Fig. 2. LFRSC employingmirror elements of equal width.
,,~
R.P. Goswami et al. / l.inear Fresnel reflector so&r concentrator
242
produces the largest image on the absorber plane and the width of the image is taken to be the width of the two sides of the absorber in this case. However~ as a first approximation it may be assumed that the width of the two sides of the absorber, a, is equal to the width of the mirror elements. Similar to the case of the first design, the absorber will cast shadow on the concentrator plane, and thus the first mirror element is placed at a distance W sin 2/+ ( f + W cos "~) tan ~0, from file centre of the concentrator plane. Here W is the width of the mirror elements and ./ is the height of the absorber from the concentrator plane. Based on similar geometrical considerations as for the first design, the following generalized expressions for the shift (S,,). location ( L , ) and the tilt (0,,) parameters associated with the nth mirror element can be derived: Sn
Wsin0,,
=
l(L,, i + WcosO,, l) f-Wsin0,, 1
(7)
L,,=L,, 1+ WcosO~ 1+ S,,.
(8)
0, = ½[tan I ( L , / f ) - G ] .
(9)
with 00 = 0, $1 = 0 and L1 = L o = W sin 3' + ( f + W c o s ~,)tan ~o as initial values and n varies from 1, 2 . . . . . k: k being the total number of mirror elements on either half of the concentrator. The actual width, a, of each of the two sides of the triangular absorber which would intercept all the radiation reflected from the mirror elements is calculated from a =
f sin 2~ 0 + Wcos(20, + ~0) c o s ( 0 , - ~0) sin(Z0,, + ~, - ~o) cos(20, + ~0)
(10)
Obviously, the value of a obtained from eq. (10) would be greater than the width of the mirror elements as was considered earlier for determining the value of L~ (i.e. the location of the first mirror element). Thus the designing is carried out again using the actual value of a, i.e. the actual width of the two sides of the triangular absorber. An iterative procedure was used in which the value of a obtained from eq. (10) is used to calculate the new value of L~ given by L l=a
siny+(f+acosy)
tan ~0.
(11)
Using eq. (11), the design procedure is repeated to get another set of values for tilt. location and shift parameters for each constituent mirror element. The value of a is again calculated and is used to calculate another value of L~. The iterations are continued until the difference between the two values of L~ is within a prespecified convergence limit. The aperture diameter, D, of the LFRSC, for a desired number of mirror elements of equal width, W, may be expressed as D=2
F
LI+ Y~ (WcosO,+S,,). n=]
(12)
R.P. Goswami et al. / Linear Fresnel reflector solar concentrator
243
3. Distribution of local concentration ratio (LCR) The distribution of local concentration ratio (LCR) on the surface of the two sides of the absorber which intercept the solar radiation reflected from the constituent mirror elements has been investigated using the ray trace technique. For comparison, the L C R distribution has also been determined using an analytical technique [12-14].
3.1. Ray trace technique In the ray trace technique, intersection points of the incident rays with the mirror elements are calculated and then the equations for the reflected rays are derived. Using the equations representing the reflected rays and the absorber, intersection points of the reflected rays with the absorber are calculated. For this, the aperture diameter, D, of the concentrator is divided into a large number (say, N ) of divisions of equal width. On each division a cone subtending an angle equal to the angular subtense of the sun ( = 32') at any point on the earth, and containing 33 rays with an angular interval of 1' is considered. When the cones hit the mirrors, they are reflected back to the absorber. Those cones which fall on the shifts introduced between two consecutive mirror elements, are assumed to be lost. Once the points of intersection of the incident cones with the constituent mirror elements are known, the equations of the reflected rays can be derived using simple geometrical optics. This is followed by calculation of the points of intersection of the reflected rays with the surface of the absorber. The absorber surface is also divided into a large number of equal-sized divisions such that the width of each division is equal to the width of the divisions on the aperture of the concentrator. The L C R distribution for a particular division is calculated by dividing the number of rays striking a particular division on the absorber to the number of rays incident on a division of equal width on the concentrator aperture. The calculations of the ray trace analysis were done on a N E C computer. The ray trace equations used for determining the L C R distribution are described below. The distance of an incident point of the cone on any division on either half of the concentrator from the centre of the aperture of the concentrator may be specified as Xm
D 2
(I-1) N
'
(13)
where N is the total number of divisions of equal width made in the aperture diameter of the concentrator. For example, for D = 1.5 m, 1500 divisions were made in the present study. I varies from 1 to N + 1. The incident cone would strike a typical mirror element (say the nth) if X m lies between L, and L, + W, cos On (i.e. the edges of the mirror element). The equation of any typical ray of the incident cone may be given by
Y = Ym,n+
(X-Xm,n) tan(90 ° + ~ ) ,
(14)
where (Xm, n, Y,,,,) are the (X, Y) co-ordinates of the point of intersection of the incident cone with the n th mirror element and ~ is the angular deviation of a ray of
244
R.P. Goswarni et al. / Linear Fresnel re[h, ctor Sohlr concentrator
the incident cone from the ray emanating from the centre of the sun (~ varies from 0 to _+16'). The positive sign with ~ is applicable for those rays of the cone which make an angle greater than 90 o with the concentrator plane and vice-versa. }'~..... is given by t~.,, = ( X, .... - L,,) tan 0,,,
(15~
where L n and 0, are the location and the tilt of the nth mirror element. The equation of the reflected ray associated with the incident ray eq. (14) may be derived as Y = Y,..... + a ..... ( X - X,,,,,),
(16)
where a,,,, is the slope of the ray reflected from the nth mirror and is given by a ..... = tan[90 ° _+ (20,, +_~)].
(17)
The positive sign before (20, +_ ~) corresponds to the rays falling on the mirror elements placed on the right half of the concentrator and the negative sign corresponds to the rays falling on the mirror elements placed on the left half of the concentrator. Now the intersection point of the reflected ray represented by eq. (16) with the triangular absorber can be calculated. The two sides of the triangular absorber may be represented by
Y = f + _ X c o t y.
(18)
F r o m eqs. (16) and (18) the coordinates of the intersection point (X,, n, Y,,,) of reflected rays with the absorber may be determined. For example, for the right side of the absorber, the intersection points are given by
X,., = ( Y - f )
tan ~,,
Y . . = X.,,. + Ym.. tan(ZO. + ~) + f tan y ' tan , / + tan(20, _+ ~)
(19)
The above procedure is repeated for all 33 rays of the reflected cones incident on the different divisions made on the aperture diameter of the concentrator.
3.2. Analytical technique This technique includes the assumption that the solar radiation reflected from each constituent mirror element is distributed uniformly over the width of the image it produces on the absorber surface [12-14]. The concentration at any point on the surface of each side of the absorber is determined by summing up the contributions of the mirror elements contributing to the concentration at that point. Obviously, this technique would provide a uniform illumination, i.e. uniform flux concentration, on the surfaces of the absorber in the case of the first design of LFRSC. In the case of the second design, due to variation in the widths of the images, different concentration levels are obtained on the absorber surfaces. As mentioned earlier, the width of the images produced on the absorber surface by the mirror elements of
R.P. Goswami et al. / Linear Fresnel reflector solar concentrator
245
equal width vary in a different fashion for different values of 7. However, from the geometry of the concentrator in question it is clear that the shortest image produced on the absorber surfaces may be considered to be the region on the absorber which receives the radiation reflected from all constituent mirror elements of one half of the concentrator. Thus a uniform concentration is obtained within this region of the absorber. For the first design of LFRSC employing mirror elements of varying width, the contribution (Cn) of any constituent mirror element, say the nth, to the LCR at any point on the surface of either of the two sides of the triangular absorber may be calculated from ,
first
wo cos 0n a
(20)
The total concentration on the surface of either of the two sides of the absorber due to the mirrors of one half of the concentrator may be given by k C = E Cn,first" n=l
(21)
For the second design of LFRSC employing mirror elements of equal width, W, the contribution of any constituent mirror element (say, the n th) to the LCR at any point within the image produced on the surface of either of the two sides of the absorber may be given by Cn,seco,d --
W cos 0. Wn ,
(22)
where w~ is the width of the image produced on the surface of the absorber by the solar rays reflected from the nth mirror element and is calculated as eq. (10). The total concentration on the surface of each side of the absorber within the uniformly illuminated region of the absorber, i.e. the region on the absorber surface which receives the radiation reflected from all mirror elements placed in one half of the concentrator in the case of the second design of LFRSC, may be given by k
C = E C,,secood-
(23)
n=l
Due to variation in the width of the images on the absorber surface, in the case of the second design, the concentration beyond the uniformly illuminated region will be different, and it is of interest to determine the concentration levels at the entire surface of the absorber sides. The procedure described in refs. [13,14] is followed so as to determine the concentration on the entire surface of the absorber.
4. Results
and discussion
For illustrating the performance evaluation procedure, some numerical calculations have been made for each design of LFRSC, and the results obtained are
246
R, P. Goxwamt ('t aL / Linear t~'resnel r~/lector .solar com'entrator [3 : ] 5m
z'Oi
:: 00?m
f-O ?m / 05m
i-',
Y : 20°
"~' 30--
--
Roy trace techmque
O]m
/,j,
003
0102
001
10
00
001
PosJtion on the absorber
0.02
0.03
[metres)
Fig. 3. Distribution of local concentration ratio (LCR) on the surface of the absorber for first design of LFRSCwith7=20° a=0.03m. D=l.5m.
presented in this section. For each design of LFRSC, the aperture diameter, D, is taken to be 1.5 m. However, since the concentrator can use only an integral number of mirror elements, the actual aperture diameter may not necessarily be the same as chosen. In the present study the value of D varied not more than _+0.005 in from the design value. For the purpose of comparison of the performances of the two designs, the width of the two sides of the triangular absorber which intercept the solar radiation reflected from the mirror elements for the first design is taken to be the width of the mirror elements used in the second design of LFRSC (i.e., a = W = 0.03 m). However, it may be pointed out that the distribution of L C R obtained for one side of the absorber for each L F R S C design would be similar to that for the other side of the absorber. Therefore, for the purpose of the work, being presented herein, only the right-hand side of the absorber has been considered. Further, in order to study the effect of variation in the apex angle, 2y, of the absorber on the distribution of flux concentration on the surface of the absorber sides, four values of y (i.e., 20 o, 40 o, 60 o and 80 o) have been considered. Figs. 3 - 6 show the distribution of L C R on the surface of right-hand side of the absorber which intercept the solar radiation reflected from the right half of the concentrator obtained from the ray trace technique for the first design of L F R S C for four values of y. For each value of y, three values of f (i.e., 0.3, 0.5 and 0.7 m) have been considered. The L C R distribution, for one value of f , has also been compared with that obtained using an analytical technique. The analytical technique, as expected, provided a uniform flux concentration on the absorber surface. This is due to the fact that the analytical technique assumes that the solar radiation reflected from each constituent mirror element is distributed uniformly over the entire width of the image produced by the mirror element on the absorber surface. Thus, using the analytical technique, for each value of f , a uniform L C R distribu-
R.P. Goswami et al.
/ Linear Fresnel reflector solar concentrator
247
0 : 1Sin a = O03m ~" = ~0 ° 3t~0
........
Ray trace
--Anulyhcol
,/
]
0
~f=O
2
0
~
technique
technique
7m
03m
,(/
J ll
; o.~ /o'o~
oo
001
0~2
Position on the absorber [metres) Fig. 4. Distribution of local concentration ratio ( L C R ) o n the surface of the absorber for first design of L F R S C with 3' = 40 °, a = 0.03 m, D = 1.5 m.
tion will be obtained. On the other hand, the results of the ray trace technique show a relatively non-uniform concentration on the absorber surface. The width of the uniformly illuminated region obtained from the ray trace technique for 7 = 20 o and
El = 1 . 5 m a : 003m Y : 60° -Ray t r a c e technique --Analytical
30 -
/
'f-
.....
/f: ~¢'/
---~//
Fr
technique
07m 05m
O' m
~
I/ 0'02 0L0~ Pos'Ition
00
001
O!02--
on the absorber (metres)
Fig. 5. Distribution of local concentration ratio ( L C R ) on the surface of the absorber for first design of L F R S C w i t h I ' = 6 0 °, a = 0 . 0 3 m, D =1.5 m.
248
[~.P. O'o.s'wdmt dl ~ll.
L i n e a r ]:t'C~tIC[ Ff-~]e('[Ol" Wl[~II" ¢'Otl('¢'llfFUft)l" []-I Sm a:OO3m - 80 °
Roy troce fechmque
---30~ ~"-! .... ,
-Anolyhcol technique
/f:O ~/ j J/
7m OSm
-
002 Posphon
001
O0
001
002
on the absorber (metres)
Fig. 6. D i s t r i b u t i o n o f local c o n c e n t r a t i o n r a t i o ( L C R ) o n the s u r f a c e o f the a b s o r b e r f o r first d e s i g n of L F R S C w i t h y = 8 0 ° . a = 0.03 m, D = 1 . 5 m.
f = 0.3 m is approximately 0.013 m. The peak value of L C R obtained from the analytical technique, as can be seen in fig. 3 for a value of f = 0.3 m, is lower by about 17%. Clearly, the results of the ray trace technique appear to be more realistic in comparison to those obtained using the analytical technique. Further, it may be noted from these figures that the uniformity in concentration decreases when f is increased from 0.3 to 0.7 m. This is because when f increases the spreading of the reflected solar rays (i.e. reflected cones) increases due to the increase in path length of the reflected cones. As a result, the width of the region on the absorber which receives all the rays of the cones reflected from each mirror element decreases. The concentration becomes completely non-uniform for a value of f equal to 0.7 m in the case of the absorber with 1, = 20 ° (fig. 3). However, the peak value of LCR increases when f is increased from 0.3 to 0.7 m. This is because when f is increased the shift introduced between the two consecutive mirror elements decreases, and thus the number of mirror elements that can be used in a given aperture diameter. D, of the concentrator increases. Consequently, the concentration increases. The uniformity in concentration on the absorber surface increases when 7 is increased from 20 ° to 40°; thereafter, the width of the uniformly illuminated region decreases when 7 is increased to 80 o, i.e., when the absorber approaches to the horizontal position. Thus the best performance in terms of uniform concentration on the absorber surface is obtained for a value of 7 equal to 40 o. It may be pointed out that if the value of y is reduced to a minimum (i.e. when y = 0 ° ), uniformity in the concentration will decrease i.e. the case of the L F R S C with a flat vertical absorber. Figs. 7-10 show the distribution of L C R on the absorber surface for the L F R S C employing mirror elements of equal width. For this design, even the use of the
R.P. Goswami et aL
/
Linear Fresnel reflector solar concentrator D :I Sm W :0 03rn
-~30 f=O 7m
, I
~, 0 5 m \~ \
006
J
0 0z,
0 02
249
T =20 ° - RQy t . . . .
--Anc;lytlcal
00
0 02
techn,Que technique
00z,
0106
Poslhon on the absorber Imetres)
F i g . 7. D i s t r i b u t i o n o f
local concentration ratio (LCR) on the surface of the absorber for the second design of LFRSC with y = 2 0 o, a = 0.03 m , D = 1.5 m.
analytical technique shows a non-uniform distribution of concentration on the absorber surface. This is clearly due to the reason that in this case mirror elements of equal width produce images of different widths on the absorber surface. As a result, the design yielded a larger width of the uniform flux concentration on the absorber surface in comparison to the case of the first design. For this design as well, the results of the ray trace technique show a considerable reduction in the width of uniformly illuminated region of the absorber. For example, for a value of 7 = 20 o and f = 0.3 m the width of the uniformly illuminated region in the case of the LCR distribution obtained from the analytical technique is approximately 0.032 m, while the results of the ray trace technique show a width of 0.019 m. The peak value of LCR obtained by using the analytical technique is lower by about 16%. It may be noted from these figures that the LFRSC employing mirror elements of equal width produce images of large widths on the absorber surface for "/= 20 °
O :]5m W : 0.0)m T : L,0° 3(
Roy froce technique
.......
Anolyhcol technique
- - -
/ f = O 2C j
/
7m 0Sin
03m
,}' // .- . . . . . . .
";@! _1 0.0z,
,/! 0.02
0.0
0.02
I
0 Ok
Position on the absorber (metres) F i g . 8.
Distribution of local concentration ratio (LCR) on the surface of the absorber for the second design of LFRSC with y = 4 0 o a = 0.03 m , D = 1.5 m.
250
R.P. Go~'wam~ et +,'1 , / L i n e a r fTesnel reflector solar concentrator
W : 0 03m T : 60 °
i
Roy i r a t e
~0 F.I
- -
I
techn que
A n a i y h ¢ o l t e ( h m q ue
i ,i
j- . f:
_Ao~. ,;:?.....
-I
+ - " -+ ....
0 9m
o s~
~.j- 03+
,,....... !-:?
0.0t~
002
0 0
0 02
Pos~h0n on t h e a b s o r b e r
O~t+
(metresl
Fig. 9. Distribution of local concentration ratio ( L C R ) on the surface of the absorber for the second d e s i g n o f L F R S C w i t h r = 6 0 ° ' a = 0.03 m, D = 1.5 m.
Thus a large absorber will be required in order to capture all the rays reflected from the constituent mirror elements. For example, the required width of the absorber in the case of T = 20 ° and f = 0.7 m is approximately 0.097 m. However, since most of the energy falls on the lower half of the absorber, the size of the absorber sides may be restricted to an appropriate size for the required performance of the concentrator-absorber system. For example, for photovoltaic application, the absorber width may be restricted to the uniformly illuminated region of the absorber side. Further, the absorber size decreases when y is increased from 0 to 60 o, and then further increase in 3' leads to an increase in the widths of the images and hence
D -ISm W - 0 0~rn T = 80 °
,oF I
......
-
-
Ray
trace
Analyhca{
fechmque
techmque
/ /
~___ 0 0L
,' /'I 0 02 Position
on t h e
0.0 absorber
002
0~:0L,
(metres)
Fig. 10, D i s t r i b u t i o n o f local c o n c e n t r a t i o n ratio ( L C R ) o n the surface o f the a b s o r b e r for the s e c o n d d e s i g n o f L F R S C w i t h T = 80 o, a = 0.03 m , D = 1.5 m.
R.P. Goswami et al. / Linear Fresnel reflector solar concentrator
251
the widths of the absorber. It m a y b e c o n c l u d e d f r o m these figures that the b e s t p e r f o r m a n c e in terms of r e d u c e d size of the a b s o r b e r a n d the p e a k value of c o n c e n t r a t i o n is o b t a i n e d for T = 60 o a n d f = 0.50 m. T h e w i d t h of the u n i f o r m l y i l l u m i n a t e d region does n o t v a r y m u c h in all cases in the p r e s e n t situation.
5. Conclusion The results of the a b o v e s t u d y have clearly s h o w n that an L F R S C with a t r i a n g u l a r a b s o r b e r p r o v i d e s a b e t t e r p e r f o r m a n c e in terms of u n i f o r m c o n c e n t r a tion on the a b s o r b e r surface t h a n an L F R S C with a flat vertical (i.e., the case when T = 0 o) absorber. The L F R S C design d e r i v e d f r o m the p r e s p e c i f i e d w i d t h of the sides of the t r i a n g u l a r a b s o r b e r offers a b e t t e r p e r f o r m a n c e in terms of p e a k value o f c o n c e n t r a t i o n on the a b s o r b e r surface. H o w e v e r , the b e s t p e r f o r m a n c e in terms of u n i f o r m c o n c e n t r a t i o n was o b t a i n e d for value of 3' = 40 °. T h e p e a k value of L C R o b t a i n e d for the s e c o n d design is c o n s i d e r a b l y less in c o m p a r i s o n with the first design. However, the design has o t h e r a d v a n t a g e s such as ease of m a n u f a c t u r e (as f a b r i c a t i o n of equal w i d t h m i r r o r e l e m e n t s is simple) a n d u n i f o r m c o n c e n t r a t i o n o n the a b s o r b e r surface. F o r the s e c o n d design, the b e s t p e r f o r m a n c e in terms of r e d u c e d a b s o r b e r size r e q u i r e d in o r d e r to c a p t u r e all the rays reflected f r o m the c o n s t i t u e n t m i r r o r elements a n d p e a k value o f c o n c e n t r a t i o n is o b t a i n e d for value o f T = 60 o. Thus, the objective of achieving a u n i f o r m c o n c e n t r a t i o n c a n be achieved b y an L F R S C with a t r i a n g u l a r a b s o r b e r . T h e results of the p r e s e n t s t u d y can be used for selection of an a p p r o p r i a t e size of the a b s o r b e r for the r e q u i r e d a p p l i c a t i o n .
References [1] J.A. Duffle and W.A. Beckman, Solar Engineering of Thermal Processes (Wiley, New York, 1980). [2] H.K. Charles, Jr., Solar Photovoltaic Energy Systems, in: Hand Book of Energy Technology and Economics, Ed. R.A. Meyers (Wiley, New York, (1983) ch. 16. [3] D.L. Marchi, Sandia Report SAND 77-0909 (Sandia Laboratories, Albuquerque, NM, USA, 1977). [4] D.L. Evans and L.W. Florschuetz, Sol. Energy 20 (1978) 37. [5] J. Sangrador and G. Sala, Sol. Energy 23 (1979) 53. [6] W.B. Ittner, Sol. Energy 24 (1980) 221. [7] U.H. Kurzweg, Sol. Energy 24 (1990) 44. [8] E.C. Boes, B.D. Shafer and D.G. Schueler, Sol. Cells 6 (1982) 3. [9] V.M. Andreev, V.R. Larionov, A.V. Maslov and E. Puron, Appl. Sol. Energy 24 (1988) 3. [10] M.I. Irshid and M.O. Othman, Sol. Cells 23 (1988) 159. [11] R.M. Cosby, NASA Report NASA-CR-120336 (NASA, Washington, USA, 1974). [12] R.N. Singh, S.S. Mathur and T.C. Kandpal, Int. J. Energy Res. 4 (1980) 59. [13] S.S. Mathur, B.S. Negi and T.C. Kandpal, Int. J. Energy Res. 14 (1990) 107. [14] B.S. Negi, T.C. Kandpal and S.S. Mathur, Sol. Wind Technol. 7 (1990) 379. [15] A.K. Singhal, M.S. Sharma, B.S. Negi and S.S. Mathur, Int. J. Energy Res. 10 (1986) 39. [16] B.S. Negi, S.S. Mathur and T.C. Kandpal, Sol. Wind Technol. 6 (1989) 589. [17] R.P. Goswami, B.S. Negi and G.D. Sootha, Paper Presented at the Indian National Solar Energy Convention, Udaipur, Rajashtan, 1-3 December 1989.
237
Optical designs and concentration characteristics of a linear Fresnel reflector solar concentrator with a triangular absorber R.P. Goswami a, B.S. Negi
b,
H.K. Sehgal b and G.D. Sootha a,.
Solar Energy Centre, Department of Non-Conventional Energy Sources, Ministry of Energy, Government of India, Block 4, 9th Floor, C.G.O. Complex, Lodhi Road, New Delhi 11003, India b Department of Physics, Indian Institute of Technology, Hauz Khas, New Delhi 110016, India
Two different approaches for designing a linear Fresnel reflector solar concentrator (LFRSC) for a triangular absorber configuration are presented. The first approach allows a variation in the width of the mirror elements, while the second uses mirror elements of equal width. For each design, the distribution of local concentration ratio (LCR) on the surfaces of two sides of the absorber which intercept the solar radiation reflected from concentrator has been investigated using the ray trace technique. Results are plotted graphically and discussed.
I. Introduction H i g h - q u a l i t y energy f r o m the sun for m a n y processes r e q u i r i n g h e a t at t e m p e r a tures a b o v e 100 ° C can b e g e n e r a t e d b y c o n c e n t r a t i o n of solar r a d i a t i o n [1,2]. T h e c o n c e n t r a t i o n effect is a c c o m p l i s h e d b y the use of either reflecting or r e f r a c t i n g elements which are l o c a t e d so t h a t solar r a d i a t i o n is focused o n t o the receiver p l a c e d in the focal zone. T h e c o n c e n t r a t i o n a p p r o a c h c a n also b e effectively utilized for reducing the costs of p h o t o v o l t a i c p o w e r p l a n t s which is highly e x p e n s i v e at p r e s e n t [2-14]. In such a system highly efficient solar cells are e x p o s e d to the c o n c e n t r a t e d solar flux at the focus of a solar c o n c e n t r a t o r t h e r e b y r e d u c i n g the solar cells a r e a a n d increasing the electrical o u t p u t . T h e latter, however, requires u n i f o r m flux c o n c e n t r a t i o n on the solar cells for efficient o p e r a t i o n of the c o n c e n t r a t o r - p h o t o voltaic system. M o r e o v e r , in such a c o m b i n e d system, it is n e c e s s a r y to r e m o v e the intense heat g e n e r a t e d b y the focused r a d i a t i o n in o r d e r to keep the solar cells at an a d e q u a t e l y low t e m p e r a t u r e . It is, therefore, of interest to d e v e l o p a s u i t a b l e design of a solar c o n c e n t r a t o r - a b s o r b e r which can p r o v i d e sufficient u n i f o r m c o n c e n t r a tion in the focal zone of the c o n c e n t r a t o r . A n i m p o r t a n t c o n c e n t r a t o r which has received c o n s i d e r a b l e a t t e n t i o n in recent years for b o t h p h o t o t h e r m a l a n d p h o t o v o l t a i c c o n v e r s i o n of solar e n e r g y is a linear F r e s n e l reflector solar c o n c e n t r a t o r ( L F R S C ) [11-17]. A n L F R S C b a s i c a l l y consists * Advisor, Department of Non-Conventional Energy Sources, and to whom all correspondence should be addressed. 0165-1633/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
238
R.P. Go~'wami et al. /" Linear h'esnel reflector ,solar concentrator
of long plane mirror elements suitably fastened to a flat base. The tilt of each mirror element is chosen such that normally incident solar radiation, after reflection from the mirror element, illuminates the absorber placed along the length in tile focal zone of the concentrator. The distribution of concentration at the focus, however. depends on the geometry of the absorber. This paper presents detailed designs of an L F R S C for a triangular absorber configuration. Such a concentrator-absorber system provides an improved performance in terms of uniformity in concentration on the absorber surfaces in contrast to the case of an L F R S C with a flat vertical absorber which showed a poor performance [14]. Two different approaches have been used for designing an L F R S C for a triangular absorber configuration. In the first approach the size of the mirror elements and hence the size of concentrator is determined from a prespecified size of the absorber. This approach allows a variation in the width of the constituent mirror elements [14]; whereas in the second approach using an equal width of the mirror elements, an appropriate size of the absorber is determined. For each design, the generalized expressions for the various design parameters have been derived. The angular subtense of the sun ( = 32') at any point on the earth has been taken into account while designing the concentrators. The concentration characteristics of each design have been studied in detail. The distribution of local concentration ratio (LCR) on the surfaces of two sides of the triangular absorber which intercept the solar radiation reflected from constituent mirror elements, for each design, has been investigated using the ray trace technique. The results of L C R distribution obtained from the ray trace technique have been compared with that obtained using an analytical technique [12-141.
2. Optical designs of LFRSC The designing of an L F R S C for a triangular absorber configuration has been carried out such that the solar radiation incident normally on the concentrator aperture, after reflection from the constituent mirror elements, illuminates two sides of the absorber. This effectively means that the mirror elements placed on the left half of the concentrator illuminates the left side of the absorber and the mirror elements placed on the right half illuminates the right side of the absorber. As described earlier, two different approaches have been used for designing an L F R S C for a triangular absorber configuration. In one approach the concentrator is designed from a prespecified width of the two sides of the absorber. This approach varies the width of constituent mirror elements. In the second approach using mirror elements of equal width the size of the absorber, i.e. the width of two sides of the absorber, is determined. The absorber size is determined such that it can intercept all the radiation reflected from constituent mirror elements. In each case an appropriate space is introduced between two consecutive mirror elements so as to avoid blocking of radiation reflected from mirror elements. Thus each constituent mirror element in both designs is characterised by four design parameters: shift ( S ) associated with it, its location ( L ) on the concentrator plane from its centre, its tilt
239
R.P. Goswami et al. / Linear Fresnel reflector solar concentrator
(0) with the concentrator plane and its width (W). In order to facilitate the designing of linear Fresnel reflector solar concentrators, the following simplifying assumptions have been made: (i) the concentrator is perfectly tracked so as to follow the apparent movement of the sun; (ii) the mirror elements are specularly reflecting; and (iii) the solar radiation is incident axially. The detailed design procedure for each LFRSC design is given below.
2.1. The first design of LFRSC The geometry used for deriving the design of an LFRSC from a prespecified size of the triangular absorber is shown in fig. 1. The absorber size is specified by the width, a, of two sides of the absorber which intercept the solar radiation reflected from constituent mirror elements, and the apex angle 2y between the two sides of the absorber. The tilt (0) and the width (W) of each mirror element is determined such that normally incident solar radiation, after reflection from the mirror element, produces an image on the plane of either sides of the absorber, of width equal to the width of the side of the absorber. Thus each mirror element placed on the left half of the concentrator would illuminate the complete surface of the left-hand side of the absorber and each mirror element placed on the right half would illuminate the complete surface of the right-hand side of the absorber. Further, since the concentrator is perfectly tracked, the absorber will cast a shadow on the concentrator plane and hence no mirror element is placed underneath the absorber. In other words, the location of the first mirror element on either half of the concentrator plane (XX') is determined such that it is just beyond the edge of the shadow. Taking into consideration the angular subtense of the sun, 240 ( = 32'), at any point
,/5 "
/
I \
~ rcx,,,A
~
k~ |lip n 2~°
\
\
~-Sn~IQn
Ln-1 L n Fig. 1. LFRSC employing mirror elements of varying width.
.I
240
R.P. Goswami et al. ,/ Linear Fresnel reflector solar concentrator
on the earth, the location of the first mirror element placed on either half of the concentrator plane is given by L 2 = a sin T + ( f + a cos y) tan ~0,
(1)
where a is the width of each of the two sides of the absorber, ~, is half the angle between the two sides of the absorber, f is the height of the absorber from the concentrator plane ( X X ' ) and ~0 is half the angular subtense of the sun ( = 16'). The tilt (02) of the first mirror element is chosen such that the extreme outer ray P~Q1 of the cone impinging on its lower edge, after reflection, meets the absorber at point F. The extreme inner ray PI'Q1 of this cone, after reflection, strikes the absorber at point R2. Now the width of the mirror element in question which has to illuminate the complete width of the absorber (i.e. FA = a) is determined by the inter section point T~ of a line drawn from the edge A of the absorber parallel to the line R2Q ! and a line passing through the point Q1 and making an angle 02 with the concentrator plane. This ensures that all the radiation reflected from the mirror element is intercepted by the one side (i.e. FA) of the absorber. In order to ensure that the solar radiation reflected from a constituent mirror element (excluding the first) is not blocked by the adjacent mirror element, appropriate values of the location and the tilt of the mirror element are determined. For example, the location (L2) and the tilt (02) of the second mirror element are chosen such that the solar radiation reflected from it is not blocked by the first mirror element and finally produces an image on the surface of one side of the absorber of width equal to the width of the absorber sides ( = a). This indicates that the incident ray PzQ2 of the cone impinging on the lower edge of the second mirror element, after reflection, just touches the upper edge of the first mirror element and finally meets the absorber at point F. This necessitates to introduce an appropriate space between the first and the second mirror element. Following the similar geometrical optics as described above, the following generalized expressions for the shift (S,), location (L,,), tilt (0,,) and the width (Wn ) parameters associated with the nth mirror element can be derived:
S"=
Wit 1 sin O n _ 2 ( L n .1 + Wn--I COS 0,, 2) f - W , , i sin0,, l '
L,=L,,
) + W,__ 2 cos 0,, i + S , ,
On = ½[tan-~( L , , / f ) -- ~o],
(2) (3) (4)
and
14:,,, =
a sin(20, + ~0) sin(20, + y - ~0) - L, sin 2~0 sin(20, + ~0) cos(0n - ~o)
(5)
with 00 = 0, W0 = $1 = 0 and L 2 = L 0 = a sin y + ( f + a cos y ) t a n ~0 as initial values. n varies from 1, 2, 3 . . . . . k; k being the total number of mirror elements placed on each half of the concentrator.
241
R.P. Goswami et al. / Linear Fresnel reflector solar concentrator
The aperture diameter, D, of the LFRSC for a desired number of mirror elements may be expressed as D=2
L,+
~_, ( W . c o s O . + S .
(6)
.
n=l
2.2. The second design o f L F R S C
Fig. 2 shows the cross-sectional geometry of the LFRSC employing mirror elements of equal width W. Similar to the case of the first design, the tilt of each mirror element with the concentrator plane ( X X ' ) is chosen such that the extreme outer ray of the cones impinging on the lower edge of the mirror elements, after reflection, reaches the common point F on the triangular absorber. In the present case, since the absorber size is not known, it would be determined by the largest intercept produced on the plane originating from the point F and making an angle (90 - T) with the X-axis, by the solar radiation reflected from a constituent mirror element. 2T is the apex angle between two sides of the absorber which intercept the reflected radiation. It may clearly be noted from the geometry of the concentrator that the mirror elements of equal width will produce images of different widths on the absorber. However, the variation in the width of the images takes place in a different fashion for different values of T. For example, for values of T from 20 o to 60 °, the first mirror element placed near the centre of the concentrator aperture produces the largest image on the absorber plane and the width of the image therefore is taken to be the width of the two sides of the absorber. While for a value of T equal to 80 o, the last mirror element placed at the edge of the concentrator
xj Ln-ILn" ~
Fig. 2. LFRSC employingmirror elements of equal width.
,,~
R.P. Goswami et al. / l.inear Fresnel reflector so&r concentrator
242
produces the largest image on the absorber plane and the width of the image is taken to be the width of the two sides of the absorber in this case. However~ as a first approximation it may be assumed that the width of the two sides of the absorber, a, is equal to the width of the mirror elements. Similar to the case of the first design, the absorber will cast shadow on the concentrator plane, and thus the first mirror element is placed at a distance W sin 2/+ ( f + W cos "~) tan ~0, from file centre of the concentrator plane. Here W is the width of the mirror elements and ./ is the height of the absorber from the concentrator plane. Based on similar geometrical considerations as for the first design, the following generalized expressions for the shift (S,,). location ( L , ) and the tilt (0,,) parameters associated with the nth mirror element can be derived: Sn
Wsin0,,
=
l(L,, i + WcosO,, l) f-Wsin0,, 1
(7)
L,,=L,, 1+ WcosO~ 1+ S,,.
(8)
0, = ½[tan I ( L , / f ) - G ] .
(9)
with 00 = 0, $1 = 0 and L1 = L o = W sin 3' + ( f + W c o s ~,)tan ~o as initial values and n varies from 1, 2 . . . . . k: k being the total number of mirror elements on either half of the concentrator. The actual width, a, of each of the two sides of the triangular absorber which would intercept all the radiation reflected from the mirror elements is calculated from a =
f sin 2~ 0 + Wcos(20, + ~0) c o s ( 0 , - ~0) sin(Z0,, + ~, - ~o) cos(20, + ~0)
(10)
Obviously, the value of a obtained from eq. (10) would be greater than the width of the mirror elements as was considered earlier for determining the value of L~ (i.e. the location of the first mirror element). Thus the designing is carried out again using the actual value of a, i.e. the actual width of the two sides of the triangular absorber. An iterative procedure was used in which the value of a obtained from eq. (10) is used to calculate the new value of L~ given by L l=a
siny+(f+acosy)
tan ~0.
(11)
Using eq. (11), the design procedure is repeated to get another set of values for tilt. location and shift parameters for each constituent mirror element. The value of a is again calculated and is used to calculate another value of L~. The iterations are continued until the difference between the two values of L~ is within a prespecified convergence limit. The aperture diameter, D, of the LFRSC, for a desired number of mirror elements of equal width, W, may be expressed as D=2
F
LI+ Y~ (WcosO,+S,,). n=]
(12)
R.P. Goswami et al. / Linear Fresnel reflector solar concentrator
243
3. Distribution of local concentration ratio (LCR) The distribution of local concentration ratio (LCR) on the surface of the two sides of the absorber which intercept the solar radiation reflected from the constituent mirror elements has been investigated using the ray trace technique. For comparison, the L C R distribution has also been determined using an analytical technique [12-14].
3.1. Ray trace technique In the ray trace technique, intersection points of the incident rays with the mirror elements are calculated and then the equations for the reflected rays are derived. Using the equations representing the reflected rays and the absorber, intersection points of the reflected rays with the absorber are calculated. For this, the aperture diameter, D, of the concentrator is divided into a large number (say, N ) of divisions of equal width. On each division a cone subtending an angle equal to the angular subtense of the sun ( = 32') at any point on the earth, and containing 33 rays with an angular interval of 1' is considered. When the cones hit the mirrors, they are reflected back to the absorber. Those cones which fall on the shifts introduced between two consecutive mirror elements, are assumed to be lost. Once the points of intersection of the incident cones with the constituent mirror elements are known, the equations of the reflected rays can be derived using simple geometrical optics. This is followed by calculation of the points of intersection of the reflected rays with the surface of the absorber. The absorber surface is also divided into a large number of equal-sized divisions such that the width of each division is equal to the width of the divisions on the aperture of the concentrator. The L C R distribution for a particular division is calculated by dividing the number of rays striking a particular division on the absorber to the number of rays incident on a division of equal width on the concentrator aperture. The calculations of the ray trace analysis were done on a N E C computer. The ray trace equations used for determining the L C R distribution are described below. The distance of an incident point of the cone on any division on either half of the concentrator from the centre of the aperture of the concentrator may be specified as Xm
D 2
(I-1) N
'
(13)
where N is the total number of divisions of equal width made in the aperture diameter of the concentrator. For example, for D = 1.5 m, 1500 divisions were made in the present study. I varies from 1 to N + 1. The incident cone would strike a typical mirror element (say the nth) if X m lies between L, and L, + W, cos On (i.e. the edges of the mirror element). The equation of any typical ray of the incident cone may be given by
Y = Ym,n+
(X-Xm,n) tan(90 ° + ~ ) ,
(14)
where (Xm, n, Y,,,,) are the (X, Y) co-ordinates of the point of intersection of the incident cone with the n th mirror element and ~ is the angular deviation of a ray of
244
R.P. Goswarni et al. / Linear Fresnel re[h, ctor Sohlr concentrator
the incident cone from the ray emanating from the centre of the sun (~ varies from 0 to _+16'). The positive sign with ~ is applicable for those rays of the cone which make an angle greater than 90 o with the concentrator plane and vice-versa. }'~..... is given by t~.,, = ( X, .... - L,,) tan 0,,,
(15~
where L n and 0, are the location and the tilt of the nth mirror element. The equation of the reflected ray associated with the incident ray eq. (14) may be derived as Y = Y,..... + a ..... ( X - X,,,,,),
(16)
where a,,,, is the slope of the ray reflected from the nth mirror and is given by a ..... = tan[90 ° _+ (20,, +_~)].
(17)
The positive sign before (20, +_ ~) corresponds to the rays falling on the mirror elements placed on the right half of the concentrator and the negative sign corresponds to the rays falling on the mirror elements placed on the left half of the concentrator. Now the intersection point of the reflected ray represented by eq. (16) with the triangular absorber can be calculated. The two sides of the triangular absorber may be represented by
Y = f + _ X c o t y.
(18)
F r o m eqs. (16) and (18) the coordinates of the intersection point (X,, n, Y,,,) of reflected rays with the absorber may be determined. For example, for the right side of the absorber, the intersection points are given by
X,., = ( Y - f )
tan ~,,
Y . . = X.,,. + Ym.. tan(ZO. + ~) + f tan y ' tan , / + tan(20, _+ ~)
(19)
The above procedure is repeated for all 33 rays of the reflected cones incident on the different divisions made on the aperture diameter of the concentrator.
3.2. Analytical technique This technique includes the assumption that the solar radiation reflected from each constituent mirror element is distributed uniformly over the width of the image it produces on the absorber surface [12-14]. The concentration at any point on the surface of each side of the absorber is determined by summing up the contributions of the mirror elements contributing to the concentration at that point. Obviously, this technique would provide a uniform illumination, i.e. uniform flux concentration, on the surfaces of the absorber in the case of the first design of LFRSC. In the case of the second design, due to variation in the widths of the images, different concentration levels are obtained on the absorber surfaces. As mentioned earlier, the width of the images produced on the absorber surface by the mirror elements of
R.P. Goswami et al. / Linear Fresnel reflector solar concentrator
245
equal width vary in a different fashion for different values of 7. However, from the geometry of the concentrator in question it is clear that the shortest image produced on the absorber surfaces may be considered to be the region on the absorber which receives the radiation reflected from all constituent mirror elements of one half of the concentrator. Thus a uniform concentration is obtained within this region of the absorber. For the first design of LFRSC employing mirror elements of varying width, the contribution (Cn) of any constituent mirror element, say the nth, to the LCR at any point on the surface of either of the two sides of the triangular absorber may be calculated from ,
first
wo cos 0n a
(20)
The total concentration on the surface of either of the two sides of the absorber due to the mirrors of one half of the concentrator may be given by k C = E Cn,first" n=l
(21)
For the second design of LFRSC employing mirror elements of equal width, W, the contribution of any constituent mirror element (say, the n th) to the LCR at any point within the image produced on the surface of either of the two sides of the absorber may be given by Cn,seco,d --
W cos 0. Wn ,
(22)
where w~ is the width of the image produced on the surface of the absorber by the solar rays reflected from the nth mirror element and is calculated as eq. (10). The total concentration on the surface of each side of the absorber within the uniformly illuminated region of the absorber, i.e. the region on the absorber surface which receives the radiation reflected from all mirror elements placed in one half of the concentrator in the case of the second design of LFRSC, may be given by k
C = E C,,secood-
(23)
n=l
Due to variation in the width of the images on the absorber surface, in the case of the second design, the concentration beyond the uniformly illuminated region will be different, and it is of interest to determine the concentration levels at the entire surface of the absorber sides. The procedure described in refs. [13,14] is followed so as to determine the concentration on the entire surface of the absorber.
4. Results
and discussion
For illustrating the performance evaluation procedure, some numerical calculations have been made for each design of LFRSC, and the results obtained are
246
R, P. Goxwamt ('t aL / Linear t~'resnel r~/lector .solar com'entrator [3 : ] 5m
z'Oi
:: 00?m
f-O ?m / 05m
i-',
Y : 20°
"~' 30--
--
Roy trace techmque
O]m
/,j,
003
0102
001
10
00
001
PosJtion on the absorber
0.02
0.03
[metres)
Fig. 3. Distribution of local concentration ratio (LCR) on the surface of the absorber for first design of LFRSCwith7=20° a=0.03m. D=l.5m.
presented in this section. For each design of LFRSC, the aperture diameter, D, is taken to be 1.5 m. However, since the concentrator can use only an integral number of mirror elements, the actual aperture diameter may not necessarily be the same as chosen. In the present study the value of D varied not more than _+0.005 in from the design value. For the purpose of comparison of the performances of the two designs, the width of the two sides of the triangular absorber which intercept the solar radiation reflected from the mirror elements for the first design is taken to be the width of the mirror elements used in the second design of LFRSC (i.e., a = W = 0.03 m). However, it may be pointed out that the distribution of L C R obtained for one side of the absorber for each L F R S C design would be similar to that for the other side of the absorber. Therefore, for the purpose of the work, being presented herein, only the right-hand side of the absorber has been considered. Further, in order to study the effect of variation in the apex angle, 2y, of the absorber on the distribution of flux concentration on the surface of the absorber sides, four values of y (i.e., 20 o, 40 o, 60 o and 80 o) have been considered. Figs. 3 - 6 show the distribution of L C R on the surface of right-hand side of the absorber which intercept the solar radiation reflected from the right half of the concentrator obtained from the ray trace technique for the first design of L F R S C for four values of y. For each value of y, three values of f (i.e., 0.3, 0.5 and 0.7 m) have been considered. The L C R distribution, for one value of f , has also been compared with that obtained using an analytical technique. The analytical technique, as expected, provided a uniform flux concentration on the absorber surface. This is due to the fact that the analytical technique assumes that the solar radiation reflected from each constituent mirror element is distributed uniformly over the entire width of the image produced by the mirror element on the absorber surface. Thus, using the analytical technique, for each value of f , a uniform L C R distribu-
R.P. Goswami et al.
/ Linear Fresnel reflector solar concentrator
247
0 : 1Sin a = O03m ~" = ~0 ° 3t~0
........
Ray trace
--Anulyhcol
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]
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~f=O
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0
~
technique
technique
7m
03m
,(/
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; o.~ /o'o~
oo
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Position on the absorber [metres) Fig. 4. Distribution of local concentration ratio ( L C R ) o n the surface of the absorber for first design of L F R S C with 3' = 40 °, a = 0.03 m, D = 1.5 m.
tion will be obtained. On the other hand, the results of the ray trace technique show a relatively non-uniform concentration on the absorber surface. The width of the uniformly illuminated region obtained from the ray trace technique for 7 = 20 o and
El = 1 . 5 m a : 003m Y : 60° -Ray t r a c e technique --Analytical
30 -
/
'f-
.....
/f: ~¢'/
---~//
Fr
technique
07m 05m
O' m
~
I/ 0'02 0L0~ Pos'Ition
00
001
O!02--
on the absorber (metres)
Fig. 5. Distribution of local concentration ratio ( L C R ) on the surface of the absorber for first design of L F R S C w i t h I ' = 6 0 °, a = 0 . 0 3 m, D =1.5 m.
248
[~.P. O'o.s'wdmt dl ~ll.
L i n e a r ]:t'C~tIC[ Ff-~]e('[Ol" Wl[~II" ¢'Otl('¢'llfFUft)l" []-I Sm a:OO3m - 80 °
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-Anolyhcol technique
/f:O ~/ j J/
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-
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001
O0
001
002
on the absorber (metres)
Fig. 6. D i s t r i b u t i o n o f local c o n c e n t r a t i o n r a t i o ( L C R ) o n the s u r f a c e o f the a b s o r b e r f o r first d e s i g n of L F R S C w i t h y = 8 0 ° . a = 0.03 m, D = 1 . 5 m.
f = 0.3 m is approximately 0.013 m. The peak value of L C R obtained from the analytical technique, as can be seen in fig. 3 for a value of f = 0.3 m, is lower by about 17%. Clearly, the results of the ray trace technique appear to be more realistic in comparison to those obtained using the analytical technique. Further, it may be noted from these figures that the uniformity in concentration decreases when f is increased from 0.3 to 0.7 m. This is because when f increases the spreading of the reflected solar rays (i.e. reflected cones) increases due to the increase in path length of the reflected cones. As a result, the width of the region on the absorber which receives all the rays of the cones reflected from each mirror element decreases. The concentration becomes completely non-uniform for a value of f equal to 0.7 m in the case of the absorber with 1, = 20 ° (fig. 3). However, the peak value of LCR increases when f is increased from 0.3 to 0.7 m. This is because when f is increased the shift introduced between the two consecutive mirror elements decreases, and thus the number of mirror elements that can be used in a given aperture diameter. D, of the concentrator increases. Consequently, the concentration increases. The uniformity in concentration on the absorber surface increases when 7 is increased from 20 ° to 40°; thereafter, the width of the uniformly illuminated region decreases when 7 is increased to 80 o, i.e., when the absorber approaches to the horizontal position. Thus the best performance in terms of uniform concentration on the absorber surface is obtained for a value of 7 equal to 40 o. It may be pointed out that if the value of y is reduced to a minimum (i.e. when y = 0 ° ), uniformity in the concentration will decrease i.e. the case of the L F R S C with a flat vertical absorber. Figs. 7-10 show the distribution of L C R on the absorber surface for the L F R S C employing mirror elements of equal width. For this design, even the use of the
R.P. Goswami et aL
/
Linear Fresnel reflector solar concentrator D :I Sm W :0 03rn
-~30 f=O 7m
, I
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006
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0 0z,
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249
T =20 ° - RQy t . . . .
--Anc;lytlcal
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0 02
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Poslhon on the absorber Imetres)
F i g . 7. D i s t r i b u t i o n o f
local concentration ratio (LCR) on the surface of the absorber for the second design of LFRSC with y = 2 0 o, a = 0.03 m , D = 1.5 m.
analytical technique shows a non-uniform distribution of concentration on the absorber surface. This is clearly due to the reason that in this case mirror elements of equal width produce images of different widths on the absorber surface. As a result, the design yielded a larger width of the uniform flux concentration on the absorber surface in comparison to the case of the first design. For this design as well, the results of the ray trace technique show a considerable reduction in the width of uniformly illuminated region of the absorber. For example, for a value of 7 = 20 o and f = 0.3 m the width of the uniformly illuminated region in the case of the LCR distribution obtained from the analytical technique is approximately 0.032 m, while the results of the ray trace technique show a width of 0.019 m. The peak value of LCR obtained by using the analytical technique is lower by about 16%. It may be noted from these figures that the LFRSC employing mirror elements of equal width produce images of large widths on the absorber surface for "/= 20 °
O :]5m W : 0.0)m T : L,0° 3(
Roy froce technique
.......
Anolyhcol technique
- - -
/ f = O 2C j
/
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03m
,}' // .- . . . . . . .
";@! _1 0.0z,
,/! 0.02
0.0
0.02
I
0 Ok
Position on the absorber (metres) F i g . 8.
Distribution of local concentration ratio (LCR) on the surface of the absorber for the second design of LFRSC with y = 4 0 o a = 0.03 m , D = 1.5 m.
250
R.P. Go~'wam~ et +,'1 , / L i n e a r fTesnel reflector solar concentrator
W : 0 03m T : 60 °
i
Roy i r a t e
~0 F.I
- -
I
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A n a i y h ¢ o l t e ( h m q ue
i ,i
j- . f:
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-I
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0 9m
o s~
~.j- 03+
,,....... !-:?
0.0t~
002
0 0
0 02
Pos~h0n on t h e a b s o r b e r
O~t+
(metresl
Fig. 9. Distribution of local concentration ratio ( L C R ) on the surface of the absorber for the second d e s i g n o f L F R S C w i t h r = 6 0 ° ' a = 0.03 m, D = 1.5 m.
Thus a large absorber will be required in order to capture all the rays reflected from the constituent mirror elements. For example, the required width of the absorber in the case of T = 20 ° and f = 0.7 m is approximately 0.097 m. However, since most of the energy falls on the lower half of the absorber, the size of the absorber sides may be restricted to an appropriate size for the required performance of the concentrator-absorber system. For example, for photovoltaic application, the absorber width may be restricted to the uniformly illuminated region of the absorber side. Further, the absorber size decreases when y is increased from 0 to 60 o, and then further increase in 3' leads to an increase in the widths of the images and hence
D -ISm W - 0 0~rn T = 80 °
,oF I
......
-
-
Ray
trace
Analyhca{
fechmque
techmque
/ /
~___ 0 0L
,' /'I 0 02 Position
on t h e
0.0 absorber
002
0~:0L,
(metres)
Fig. 10, D i s t r i b u t i o n o f local c o n c e n t r a t i o n ratio ( L C R ) o n the surface o f the a b s o r b e r for the s e c o n d d e s i g n o f L F R S C w i t h T = 80 o, a = 0.03 m , D = 1.5 m.
R.P. Goswami et al. / Linear Fresnel reflector solar concentrator
251
the widths of the absorber. It m a y b e c o n c l u d e d f r o m these figures that the b e s t p e r f o r m a n c e in terms of r e d u c e d size of the a b s o r b e r a n d the p e a k value of c o n c e n t r a t i o n is o b t a i n e d for T = 60 o a n d f = 0.50 m. T h e w i d t h of the u n i f o r m l y i l l u m i n a t e d region does n o t v a r y m u c h in all cases in the p r e s e n t situation.
5. Conclusion The results of the a b o v e s t u d y have clearly s h o w n that an L F R S C with a t r i a n g u l a r a b s o r b e r p r o v i d e s a b e t t e r p e r f o r m a n c e in terms of u n i f o r m c o n c e n t r a tion on the a b s o r b e r surface t h a n an L F R S C with a flat vertical (i.e., the case when T = 0 o) absorber. The L F R S C design d e r i v e d f r o m the p r e s p e c i f i e d w i d t h of the sides of the t r i a n g u l a r a b s o r b e r offers a b e t t e r p e r f o r m a n c e in terms of p e a k value o f c o n c e n t r a t i o n on the a b s o r b e r surface. H o w e v e r , the b e s t p e r f o r m a n c e in terms of u n i f o r m c o n c e n t r a t i o n was o b t a i n e d for value of 3' = 40 °. T h e p e a k value of L C R o b t a i n e d for the s e c o n d design is c o n s i d e r a b l y less in c o m p a r i s o n with the first design. However, the design has o t h e r a d v a n t a g e s such as ease of m a n u f a c t u r e (as f a b r i c a t i o n of equal w i d t h m i r r o r e l e m e n t s is simple) a n d u n i f o r m c o n c e n t r a t i o n o n the a b s o r b e r surface. F o r the s e c o n d design, the b e s t p e r f o r m a n c e in terms of r e d u c e d a b s o r b e r size r e q u i r e d in o r d e r to c a p t u r e all the rays reflected f r o m the c o n s t i t u e n t m i r r o r elements a n d p e a k value o f c o n c e n t r a t i o n is o b t a i n e d for value o f T = 60 o. Thus, the objective of achieving a u n i f o r m c o n c e n t r a t i o n c a n be achieved b y an L F R S C with a t r i a n g u l a r a b s o r b e r . T h e results of the p r e s e n t s t u d y can be used for selection of an a p p r o p r i a t e size of the a b s o r b e r for the r e q u i r e d a p p l i c a t i o n .
References [1] J.A. Duffle and W.A. Beckman, Solar Engineering of Thermal Processes (Wiley, New York, 1980). [2] H.K. Charles, Jr., Solar Photovoltaic Energy Systems, in: Hand Book of Energy Technology and Economics, Ed. R.A. Meyers (Wiley, New York, (1983) ch. 16. [3] D.L. Marchi, Sandia Report SAND 77-0909 (Sandia Laboratories, Albuquerque, NM, USA, 1977). [4] D.L. Evans and L.W. Florschuetz, Sol. Energy 20 (1978) 37. [5] J. Sangrador and G. Sala, Sol. Energy 23 (1979) 53. [6] W.B. Ittner, Sol. Energy 24 (1980) 221. [7] U.H. Kurzweg, Sol. Energy 24 (1990) 44. [8] E.C. Boes, B.D. Shafer and D.G. Schueler, Sol. Cells 6 (1982) 3. [9] V.M. Andreev, V.R. Larionov, A.V. Maslov and E. Puron, Appl. Sol. Energy 24 (1988) 3. [10] M.I. Irshid and M.O. Othman, Sol. Cells 23 (1988) 159. [11] R.M. Cosby, NASA Report NASA-CR-120336 (NASA, Washington, USA, 1974). [12] R.N. Singh, S.S. Mathur and T.C. Kandpal, Int. J. Energy Res. 4 (1980) 59. [13] S.S. Mathur, B.S. Negi and T.C. Kandpal, Int. J. Energy Res. 14 (1990) 107. [14] B.S. Negi, T.C. Kandpal and S.S. Mathur, Sol. Wind Technol. 7 (1990) 379. [15] A.K. Singhal, M.S. Sharma, B.S. Negi and S.S. Mathur, Int. J. Energy Res. 10 (1986) 39. [16] B.S. Negi, S.S. Mathur and T.C. Kandpal, Sol. Wind Technol. 6 (1989) 589. [17] R.P. Goswami, B.S. Negi and G.D. Sootha, Paper Presented at the Indian National Solar Energy Convention, Udaipur, Rajashtan, 1-3 December 1989.