- Email: [email protected]
A comparative study of optical designs and solar flux concentrating characteristics of a linear fresnel reflector solar concentrator with tubular absorber
~ t ~ k ELSEVIER
Solar Energy Materials and Solar Cells 32 (1994) 169-186
A comparative study of optical designs and solar flux concentrating characteristics of a linear fresnel reflector solar concentrator with tubular absorber G.D. Sootha
a,,,
B.S. Negi b
a Solar Thermal Division, Ministry of Non-Conventional Energy Sources, Block No. 14, CGO Complex, Lodi Road, New Delhi 110 003, India b D.N.E.S. Monitoring cell, Bhubaneswar, Orissa, India
(Received 9 July 1992; in revised form 12 August 1993)
Abstract This paper presents a comparative study of the optical designs ~ind solar flux concentrating characteristics of two designs of the linear Fresnel reflector solar concentrator (LFRSC) derived using two different approaches for the tubular absorber configuration. The concentrator uses long plane mirror elements suitably positioned on a flat base so that each mirror element reflects solar radiation on to an absorber placed in the focal zone along the length of the concentrator. In one approach the concentrator is generated from the prespecified diameter of the absorber, while in the second approach, using a prespecified equal width of the mirror elements, an appropriate diameter of the tubular absorber is determined for the desired size of the concentrator. For each concentrator design, the distribution of the local concentration ratio (LCR) on the surface of the tubular absorber has been determined using the ray trace technique, taking into consideration both the uniform intensity on the solar disc and the solar limb darkening effect (non-uniform intensity). The surface area factor aspect of each concentrator design, a measure of the cost of reflective surface used in the concentrator, has also been studied. Results of some numerical calculations carried out to illustrate the performance of the concentrators are graphically presented and compared.
1. Introduction T h e c o n c e p t o f c o n c e n t r a t i o n o f s o l a r e n e r g y offers a n effective w a y to h a r n e s s s o l a r e n e r g y for efficient p h o t o v o l t a i c a p p l i c a t i o n s [1] a n d for use in t h e r m a l
* Corresponding author. 0927-0248/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0927-0248(93)E0106-N
170
G.D. Sootha, B.S. Negi / Solar Materials and Solar Cells 32 (1994) 169-186
systems that require thermal energy at temperatures above 100°C [2,3]. Solar concentrators consist of either a refracting or reflecting optical surface which concentrates the incident solar radiation on to an absorber surface. Hence, higher temperatures are generated due to the increased solar flux on the absorber and the lower heat losses from the comparatively smaller absorber. The increased solar flux at the focus can also be effectively used for efficient photovoltaic applications, in which highly efficient solar cells are exposed to the concentrated solar flux. An increased electrical output is thereby obtained from a reduced solar cells area [3-14]. A wide variety of solar concentrators have been developed in recent years and have been tested for their suitability for various processes. A typical solar concentrator of this kind is a linear Fresne! reflector solar concentrator (LFRSC). Such a concentrator uses a series of long plane mirror elements suitably positioned on a flat base. The angle of inclination of each mirror element with respect to the base of the concentrator is so determined that the incident solar radiation, after reflection from the mirror elements, is intercepted by an absorber placed in the focal zone along the length of the concentrator [17-21]. An appropriate space is introduced between two consecutive mirror elements in order to avoid blocking of solar radiation reflected from the constituent mirror elements. From the fabrication point of view, the concentrator is very simple and inexpensive. However, the disadvantage of this concentrator is that because of the space introduced between the mirror elements, only a fraction of the base area of the concentrator covered by reflecting surfaces contributes to the concentration on the surface of the absorber. Nevertheless, when mounted on a simple framework, the existence of the gaps between the mirror elements offers an advantage, when considering that wind loading can be considerably reduced. Two different approaches have been proposed for designing linear Fresnel reflector solar concentrators [18-20]. In one approach, the concentrator is generated from the prespecified size of the absorber. In this approach, the angle of inclination of each mirror element with respect to the base of the concentrator and the width of the mirror element are determined so that incident solar radiation, after reflection, is intercepted within the absorber and, hence, illuminates the absorber from one end to the other. The approach allows a variation in the width of the constituent mirror elements. In the second approach, the concentrator is generated from a prespecified equal width of the mirror elements and an appropriate size of the absorber which will intercept all the solar energy reflected from the constituent mirror elements and which is determined for the desired aperture diameter of the concentrator. Obviously, the shape of the concentrated flux produced by a LFRSC in its focal zone will depend on the geometry of the absorber. Detailed designs and performance analysis of LFRSC designs with flat horizontal [18] and triangular absorber [20] configurations have shown that the concentrator can provide a considerable uniform flux concentration on the absorber surfaces. Hence, the system appeared suitable for photovoltaic applications. On the other hand, in the case of a LFRSC design with a flat vertical absorber, a poor performance in terms of uniform flux concentration and peak value of
G.D. Sootha, B.S. Negi / Solar Materials and Solar Cells 32 (1994) 169-186
171
concentration [19] was showed as compared to the LFRSC designs with flat horizontal and triangular absorbers. Recently, a concentrator design based on the Fresnel reflector concept has been developed and operated in conjunction with solar cells for electric power and water heating [12]. Some preliminary efforts made to study the optical and thermal performance of a LFRSC with tubular absorber employing ordinary black painted and selective coating indicated that the system can be effectively utilised for solar thermal applications [21]. However, for large scale practical utilization, it is important to carry out a detailed study on design and solar flux concentrating characteristics of LFRSC with tubular absorber to arrive at a suitable design approach for the desired performance. Furthermore, as the solar flux concentration is considerably affected by non-uniform intensity on the solar disc, known as the solar limb darkening effect [15,16], it is of considerable interest to study the effect of solar limb darkening on the local concentration ratio on the surface of the tubular absorber employed with a LFRSC. This paper presents a comparison of optical designs and solar flux concentrating characteristics of two designs of LFRSC derived using two different approaches discussed above, with tubular absorber. The analysis has been carried out taking into consideration the sun's rays coming in the form of cone with the apparent diameter of the sun subtending an angle of 32' on the earth. In one approach, the concentrator design is generated from a prespecified diameter of the tubular absorber. In the other approach, using prespecified equal width of the mirror elements, an appropriate diameter of the tubular absorber which would intercept all the solar radiation reflected from the mirror elements employed in the concentrator, is determined for the desired aperture diameter of the concentrator. The concentrator is characterised by design parameters associated with each mirror element, the location of the mirror element on the concentrator base with respect to the centre of the aperture of the concentrator, the angle of inclination of the mirror element with respect to the base of the concentrator, the width of the mirror element, and the shift associated with the mirror element in order for a mirror element not to block the solar radiation reflected from the adjacent mirror element. Generalised expressions for the design parameters for each design of LFRSC are described. The distribution of local concentration ratio (LCR) on the surface of tubular absorber has been determined for each LFRSC design using the ray trace technique. Both the uniform intensity on the solar disc and the solar limb darkening, i.e., the fact that the intensity on the solar disc is not uniform, and is maximum at the centre and decreases considerably with the distance from the centre of the sun, have been taken into consideration for the determination of LCR on the absorber surface. Solar limb darkening effectively indicates that if I 0 is the intensity at the centre of the solar disc and if the solar disc were of uniform intensity Io, uo, it would radiate the same energy as a nonuniform disc with its intensity at the centre Io.Nu o equal to 1.256 I0,~D as described in Refs. [15,16]. A comparison of the different correlations describing limb darkening is also available in literature [16]. Another important aspect of the present analysis is to study the surface area factor of the concentrator, since the area of the reflective surface
G.D. Sootha, B.S. Negi/ Solar Materials and Solar Cells 32 (1994) 169-186
172
would vary as the distance of the absorber varies from the base of the concentrator. The surface area factor defined as the ratio of reflective surface area to aperture area will indicate the cost of the reflective material as a multiple of the absolute minimum possible. The above analysis has been carried out assuming the mirror elements specularly reflecting and the solar concentrators perfectly tracking. The advantages and disadvantages of both LFRSC designs have been discussed. Results of the above analysis have been illustrated in graphic and tabular forms, and have been compared.
2. Analysis Figs. (1 and 2) show the geometry used for deriving the designs of LFRSC using two different approaches, discussed above, for tubular absorber configuration. In the case of the concentrator design generated from the prespecified diameter of the tubular absorber (Fig. 1), the width and the angle of inclination of each mirror element with respect to the base of the concentrator are so determined that the extreme outer rays of the incident cone after reflection from the extreme edges of the mirror elements, touch the tubular absorber tangentially. This ensures that all the rays of the cones incident on mirror elements, after reflection, are intercepted by the absorber. This approach varies the width of the mirror elements. In the case of the concentrator design generated from a prespecified equal width of the mirror
i
I i ,~,,
~
, -~--- , ~
i
1':1
\ rn
x,-
t-s.4
O~--L,.*t~2°'~1"~-~2
~s.4"
Fig. 1. Linear fresnel reflector solar concentrator (LFRSC) with a tubular absorber design generated from the prespeeified diameter of the absorber.
G.D. Sootha, B.S. Negi/Solar
Materials and Solar Cells 32 (1994) 169-186
173
Fig. 2. linear fresnel reflector solar concentrator (LFRSC) with tubular absorber design generated from the prespecified equal width of the mirror elements.
elements, the designing is carried out in such way that a ray coming from the centre of the sun and impinging normally on the mid point of each mirror element is reflected to a common point F in the focal zone of the concentrator (Fig. 2). Obviously, the size of the spreading produced by the solar radiation reflected from the mirror elements of equal width will be different for different mirror elements. Hence, an appropriate value of the diameter of a tubular absorber which would intercept all the solar radiation reflected from all the constituent mirror elements is calculated for the desired aperture diameter of the concentrator. In each concentrator design, an appropriate space is introduced between two constituent mirror elements so that the radiation reflected from a constituent mirror element is not blocked by the adjacent mirror element. Detailed design procedures for two different concentrator designs are given below.
3. LFR!K design derived from prespecified diameter of the absorber In this concentrator design, the angle of inclination of each mirror element with respect to the base of the concentrator (say nth mirror element) is determined so that the extreme outer ray P,Q, of the cone impinging on the lower edge of the mirror element, after reflection just touches the tubular absorber of prespecified diameter 2R, tangentially at point A,. The extreme inner ray of the cone after reflection meets the absorber at point B,.Using simple geometrical optics the
G.D. Sootha, RS. Negi~Solar Materials and Solar Cells 32 (1994) 169-186
174
generalised expression for the angle of inclination of the nth mirror element can be given by:
Xa 2
tan- 1
,
[L~f-R~(x~,~-RZ)+(Y.,~-f)
O"=3[
I
- ~o ,
z
(1)
where tan(90 + 2 0 n + G0)R Xa'n
=
~/1 + tan2(90 + 2On + G0)
R and Ya,n----f--
~/1 + tan2(90 + 2 0 n + G0)
are the ( x , y ) coordinates of the point A n on the circular cross-section of the absorber, R is the radius of the tubular absorber, f is the distance between the centre of the absorber and the concentrator base (i.e. point O), L n is the location of the nth mirror element on the concentrator plane with respect to the centre of the concentrator aperture, and G0 is the half of the angular subtense of the sun on the earth ( = 16'). L n is calculated from: L n •Ln_ 1 + W~_ 1 cos On_ 1 + S n,
(2)
with L 0 ffi W0 = S~ = 0 and O 0 = 0 as initial values. However, since the concentrator is perfectly tracked, the tubular absorber will east a shadow on the concentrator plane and, hence, the location of the first mirror element, L 1, is determined so that it is just beyond the edge of the shadow. If 2G0 ( = 32') represents the angular subtense of the sun at any point on the earth, then the location of the first mirror element on either half of the concentrator is defined as: L 1 = R + f tan G0.
(3)
The width of the mirror element under consideration is determined by drawing a line parallel to BnQ n (i.e., the reflecting ray associated with the extreme inner ray of the cone incident on the extreme lower edge of the first mirror element) such that it is tangential to the tubular absorber at point Cn and intersects the line making an angle On with the aperture plane of the concentrator, at point Tn. Hence, QnTn should be the required width of the mth mirror element and the line CnTn corresponds to the reflected ray associated with the extreme inner ray SnTn of the cone incident on the extreme upper edge of the first mirror element. This condition ensures that all the solar radiation reflected from the mirror element is intercepted by the tubular absorber (i.e., from point A n to point Cn). The width of the nth mirror element may be calculated from:
IV,, = ~(Xc,n --Xb.n) 2 + (Yc,n --Yb,n) 2 COS(2On -- Go - an),
(4)
where (Xo,n,yb, n) and (Xc,n,yc, n) are the (x,y) coordinates of the points B n and Cn on the circular cross section of the absorber, respectively, and a n is the angle of inclination of the line joining the points B n and C n. a n is calculated from:
1(,c an
-- tan
-
k
Xc,n _ X b , n
.
G.D. Sootha, B.S. Negi/ Solar Materials and Solar Cells 32 (1994) 169-186
175
The shift associated with the nth mirror element is expressed as:
=
(Ln_ 1 + Wn_ 1 cos O~_ 1 +x~.a)W~_ 1 sinO~_ 1 y.,, - W~_ 1 sin O,_x
(5)
The aperture diameter, D, of the concentrator for a desired number of mirror elements is defined by: D=2
1
Ll + E (W~ cos On + S~) ,
[
'
n=l
(6)
where n varies from 1 to k, and k is the total number of mirror elements employed on each half of the concentrator. The surface area factor of the concentrator defined as the ratio of reflective surface area to aperture area is calculated from: k
2Ew. SA
D
(7)
The surface area factor will indicate the cost of the reflective material as a multiple of the absolute minimum possible. From the geometry of the concentrator it is clear that for a certain given aperture diameter of the concentrator the reflective surface area will vary according to the distance the focal region is from the base of the concentrator.
4. LFRSC design derived from mirror elements of equal width In this design approach, the angle of inclination of mirror elements of equal width, W, with respect to the base of the concentrator, is chosen such that a ray incident normally on the concentrator aperture and striking the mid point of the mirror element, is reflected to a common point F in the focal zone of the concentrator (Fig. 2). In the present analysis, since the incident solar rays are being considered in form of cones, the above ray corresponds to the central ray of the cone emanating from the sun. A tubular absorber of an appropriate diameter is placed in the focal zone along the length of the concentrator in such a way that the centre of the absorber coincides with the point F, and hence captures all the rays reflected from the mirror elements. Since, in this case, the absorber size (i.e., the radius of the tubular absorber) is not given, the absorber may be determined from the size of the spreadings produced by the rays reflected from the mirror elements, and measured with respect to the point F in the focal zone of the concentrator of a desired aperture diameter, so that all rays reflected from the mirror elements are intercepted by the tubular absorber. Thus, the radius of the tubular absorber may be determined by drawing perpendiculars from the point F on the extreme outer reflecting rays associated with the extreme rays PnQn and SnT~ of the cones incident on the extreme upper and lower edges of the mirror elements. The largest
G.D. Sootha, B.S. Negi/ SolarMaterialsand Solar Cells32 (1994) 169-186
176
perpendicular distance from the point F is taken to be diameter of the tubular absorber. However, as a first approximation the designing may be initiated considering the radius of the absorber as equal to the half of the width of the mirror element. Similar to the case of the first design of LFRSC, an appropriate space is introduced between two consecutive mirror elements to avoid blocking of solar radiations reflected from the mirror elements. This effectively means that the extreme outer ray of the cone impinging on the lower edge of the nth mirror element, after reflection, just touches the extreme upper age of the ( n - 1)th mirror element and is finally intercepted by the absorber. Based on the simple geometrical considerations as for the first design, the following generalised expressions for the shift S~, location L~ and the angle of inclination with respect to the base of the concentrator parameters associated with the nth mirror element can be derived: S~ = W sin 0._ 1 tan(20~ + ~o),
(8)
L~ =L~_ 1 + W sin 0~_1 + S . ,
(9)
and
O.
I
=
~- tan-1
( L,, + W/2 cos O,, ) (10)
f - W/2 sin On
'
with O 0 = 0, L 0 -- So -- 0.0 and L i = W / 2 + f tan O 0 as initial values. Eqs. (9-11) are solved iteratively, n varies from 1,2. . . . . k, and k is the total number of mirror elements on each half of the concentrator. The radius of the tubular absorber, the maximum length of the perpendicular dropped from the point F on the extreme outer rays of the cone reflected from the extreme edges of the mirror element, which would intercept all the solar rays reflected from all the constituent mirror elements, is calculated from: [ L . + W cos On _ f ] sin(20, R = W sin O~ + [ ~ a n ( - 2 ~ - ~-~ , ¢0)
(11)
Since the above design equations have been derived presuming the diameter of the absorber equal to the width of the mirror element, the actual size of the shading caused by the absorber on the concentrator plane and hence the actual location of the first mirror element will be larger than assumed. Thus, the actual value of L has to be calculated so that no part of the mirror element is shaded by the absorber. This effectively means that the shading caused by the absorber of radius given by Eq. (11) would extend beyond the assumed value of L 1 used for initiating the designing of the concentrator. This necessitates the use of an iterative procedure in which the value of R obtained from Eq. (11) is used to calculate another value of L 1 a s : L 1 = r + f tan s%.
(12)
Using the value of L 1 given by Eq. (12) the design procedure is repeated (i.e., Eqs. (8-10)) to get another set of values for shift, angle of inclination and location
G.D. Sootha, B.S. Negi / Solar Materials and Solar Cells 32 (1994) 169-186
177
parameters for each constituent mirror element. This results in a new value of R. The new value of R is again used to calculate another value of L 1. The iterations are continued until the difference between the two consecutive values of L 1 is within a prespecifled convergence limit. The aperture diameter D of the concentrator is defined by: D=2
L~+
+S~) .
(13)
The surface area fact or, S A, in this case is calculated from k
2~w SA ~
n=l
D
(14)
5. Distribution of local concentration ratio (LCR) The distribution of the local concentration ratio (LCR) on the surface of the tubular absorber has been determined using the ray trace technique. In the ray trace technique, equations for the incident rays and the corresponding reflected rays are derived and then the intersection points of the reflected rays with the absorber placed in the focal cone of the concentrator are calculated. The aperture diameter, D, of the concentrator is divided into a large number of devises (ray N ) of equal width. On each division a cone containing 33 rays with an angular interval of 1' has been considered. The coordinates of the point of intersection of the incident rays of each cone with the mirror elements are calculated. This is followed by deriving equations for the reflected rays associated with the incident rays for which the intersection points are calculated, and finally calculating the points of intersection of the reflected rays with the circular cross-section of the absorber. Obviously, the incident cones falling within the space introduced between mirror elements are assumed to be lost. Similarly, incident rays directly striking the tubular absorber have not been considered. The circular cross-section of the absorber is also divided into equal angular divisions. Thus, the ratio of the number of rays striking a particular division of the absorber of arc length equal to the width of the division on the concentrator aperture to the number of rays incident on a division of equal width on the concentrator plane gives the LCR for that particular division of the absorber. This provides a smooth LCR distribution curve with a peak at the point (f-R) on the circular cross section of the absorber. Calculations of the ray trace technique were done on a computer.
6. Consideration of limb darkening effect As explained earlier the limb darkening describes a variation in the brightness on the solar disc; the brightness is maximum at the centre and decreases consider-
178
G.D. Sootha, B.S. Negi / Solar Materials and Solar Cells 32 (1994) 169-186
ably with the distance from the Centre of the sun disc. This effectively indicates that ff I 0 is the intensity at the centre of the solar disc, and if the solar disc were of uniform intensity, it would radiate the same energy as a non-uniform disc with its intensity at the Centre, Io, NuO, equal to 1.256 Io,~o as Evans [15] has suggested the relationship between two as: 0.785 lo,uo = 0.625 IO,NuO.
(15)
Obviously, in this case each ray of the cone will carry a different intensity. The procedure provided in Ref. [16] for studying LCR distribution on a flat absorber employed with a cylindrical parabolic trough and a polygal trough, respectively, has been used to assign the appropriate value of weight factor to each of the rays emanating from the sun, and thus determine the LCR on the surface of the absorber. The ray trace equations derived for determining the LCR on the surface of the tubular absorber are described below. The distance of an incident point of a cone on any division on either half of the concentrator from the centre of the plane of the concentrator (xx) is given by O x, = 2
( m - 1) N '
(16)
where N is the total number of divisions of equal size on the aperture plane of the concentrator and m varies from 1 to N + 1. In the present work 1000 divisions were made on the concentrator aperture for an aperture diameter D equal to 1.0 m. As discussed above, the projection of a cone is simulated by 33 rays, each separated by an angular interval of 1'. The incident cone would strike a typical mirror element, say the nth, if x lies between L n and L n + Wn cos On, i.e., between the lower and the upper edge of the mirror element. Any typical ray of the incident cone under consideration may be represented by an equation as: Y =Yi,n + (X --X,,n) tan(90 + ~),
(17)
where (xi, n, Yi,n) are the ( x , y ) coordinates of the point of intersection of the incident cone with the nth mirror element, and is the angular deviation of a ray of the incident cone from the ray emanating from the centre of the sun (g varies from 0 to + 16'). The positive sign with ~ is taken for those rays of the cone which make an angle greater than 90 ° with the concentrator plane and the negative sign for those rays which make angle less than 90°. Y~,n is calculated from: Y,,n = ( xi,n - Ln) tan On.
(18)
Once the intersection points (xi,n,y~, n) of the incident cone with the mirror element are calculated, the equation of the reflected ray of a cone associated with the ray of the incident cone under consideration can be derived. The equation of the reflected ray is: Y =Y,,n +~v,n( x - x i , n ) , where /3v,n = tan[90 + (2On + g)]
(19)
179
G.D. Sootha, B.S. Negi / Solar Materials and Solar Cells 32 (1994) 169-186
is the slope of the reflected ray represented by Eq. (19). The positive sign before (20 n -t-~) corresponds to the rays falling on the mirror elements placed on the right hand-half of the concentrator and the negative sign corresponds to the reflected rays associated with the incident rays on the mirror elements on the left half of the concentrator. All the 33 rays of each cone reflected from all mirror elements have to be intercepted by an absorber placed in the focal zone and for which the concentrator has been designed. Thus the procedures is followed by the calculation of intersection points of the reflected rays with the tubular absorber cross-section. The circular cross-section of the tubular absorber can be represented by an equation as: x2 + ( Y _ f ) 2 = R 2.
(20)
The x-y coordinates of the point of intersection of a ray reflected from the nth mirror element represented by Eq. (19), with the circular cross-section of the absorber described by Eq. (21), are: Zn + ~ z2 -- Sec2(2On + ~){[ Xi,n + Yi,n tan(20~ ± g)]2 + f 2 _ R 2) Yip,n =
Sec2(20n ± ~)
, (21)
and Xia,n = xi,n - tan(20~ + g )( y,p,~ - Yi,~),
(22)
with Zn = f + x , , n
tan(20, +~:) +y,,, tan2(2On + ~ ) .
The above procedure is repeated for all the 33 rays of the cones incident on the different divisions made on the aperture diameter of the concentrator to obtain the LCR curve on the tubular absorber cross-section.
7. Results and discussion
In order to carry out a comparative study of optical design and performance characteristics of two LFRSC designs derived using two different approaches for tubular absorber configuration, some numerical calculations have been done, and the results are presented in this section. For the purpose of the analysis, being presented herein, the aperture diameter of each design of LFRSC has been taken as D -- 1.0 m, and 2R (for the first design)= W (for the second design). As the concentrator can use an integral number of mirror elements, the actual value of D may, however, not be the same as chosen. The actual value of D also varies as the height of the absorber varies with respect to the base of the concentrator. The variation in the actual value of D in the case of a LFRSC design generated from the prespecified field diameter of the absorber is found to be very small (i.e., about
180
G.D. Sootha, B.S. Negi / Solar Materials and Solar Cells 32 (1994) 169-186
Table 1 Variation in D, S n and SA with f for three different values of R, for the LFRSC design derived from the prespeeified diameter of the tubular absorber f (m)
0.3 0.5 0.7 0.9 1.1
2R = 0.03 m
2R = 0.04 m
2R = 0.05 m
D (m)
2S n (m)
SA
D (m)
2S n (m)
SA
D (m)
2S n (m)
SA
0.994 0.985 1.027 1.017 1.003
0.22 0.11 0.07 0.046 0.03
0.787 0.88 0.91 0.93 0.94
0.99 0.99 0.964 1.03 1.023
0.22 0.108 0.059 0.047 0.032
0.78 0.87 0.906 0.92 0.93
1.028 0.96 0.976 1.007 1.038
0.235 0.098 0.044 0.042 0.036
0.77 0.864 0.897 0.91 0.92
0.018 m). Whereas in the case of LFRSC using mirror elements of equal width the variation in the value of D is found to be more than for the former. The variation in the actual value of D and the total shift in the concentrator and the surface area factor SA for two designs of LFRSC, with the height of the absorber, f , is shown in Tables 1 and 2. As expected, in the case of the LFRSC design generated from a prespecified diameter of the absorber, the width of the mirror elements varied. However, in the case of the concentrator using a tubular absorber the variation in the width is very small, so small that the variation can be ignored, and an equal width of the mirror elements can be employed with almost negligible loss in flux concentration. For example, for 2R = 0.03 m, the width of the mirror elements vary from 0.0373 m (i.e., W1) to 0.0399 m (Wk = 14) for f - - 0 . 3 m, from 0.0254 m (W t) to 0.0255 m ( W k - - 1 7 ) for f = 0.5 m, from 0.0235 m (W 1) to 0.0231 (Wk = 20) for f - - 0 . 7 m from 0.0216 m (W 1) to 0.0211 m ( Wk = 22) for f = 0.9 m and from 0.0198 m (W 1) to 0.0192 m ( W k = 24) for f = 1.1 m. The angle of inclination of mirror elements with respect to the base of the concentrator, for each case, increases as one moves from the centre towards the rim of the concentrator aperture. The angle of inclination as well as the shift associated with each mirror element, however, decreases with an increase in the value of f. As a result, the number of mirror elements that can be employed in a concentrator of a given aperture, diameter, increases with an increase in f. The total shift 2ES n, however, first decreases drastically with increase in f and finally as can be seen in Table 1, the decrease is very small.
Table 2 Variation in D, 2S n, R and SA with f for three different values of W for the LFRSC design derived using mirror elements of equal width f (m) D (m)
W = 0.03 m 2S n (m) R (m) SA
D (m) 2S n (m) R (m) SA
D (m) 2S n (m) R (m) SA
0.3 0.5 0.7 0.9 1.1
0.227 0.101 0.064 0.039 0.032
0.9126 1.018 0.974 0.96 1.032
0.8914 0.9258 1.007 0.99 0.98
1.0016 0.96 0.99 0.97 1.02
0.016 0.017 0.018 0.019 0.02
W = 0.04 m
0.78 0.875 0.91 0.93 0.94
0.18 0.117 0.06 0.037 0.032
0.021 0.022 0.023 0.024 0.025
W = 0.05 m
0.79 0.865 0.904 0.92 0.93
0.17 0.089 0.064 0.0386 0.026
0.026 0.027 0.028 0.029 0.031
0.785 0.864 0.894 0.909 0.92
G.D. Sootha, B.S. Negi / Solar Materials and Solar Cells 32 (1994) 169-186
181
D:l.Om d : O.O3m
t~
50'
90°
~
UNIFORM INTENSITY SOLAR DISC
....
NON UNIFORM I N T E N S I T Y S O L A R
90°
DISC
0
#It ' ,,"l~1~ ~
~ " ~
F =O.'m
,=o.,°
#///'/20
jY
~
"
~
,.o.,.
10
////
/
160
120
80
40
ANGULARPOSITIONON
0
40
THE ABSORBER
80
120
i
160
(DEGREES)
Fig. 3. Distribution of local concentration ratio (LCR) on the surface of the tubular absorber for LFRSC design derived from the prespecified diameter (2R = 0.03 m) of the absorber for f = 0.3, 0.5 and 0.7 m.
Thus, the increase in the number of mirror elements slows down for f > 1.1 m. Consequently, the surface area factor S A increases with an increase in f . The increase in the value of S A is very small for values of f > 0.7 m. This is contrary to the case of the cylindrical parabolic trough in which for a fixed aperture area of the concentrator as the distance between the focal region and the concentrator base decreases the deepness and surface area increases, and hence SA increases. The reflector tends to a flat mirror when the focal region is moved far away from the base. In the case of a L F R S C design employing mirror elements of equal width, the actual aperture diameter of the concentrator varies considerably when f increases (Table 2). Nevertheless, the values of SA for each value of f and W are more or less the same as for the corresponding values of 2R and f in the case of LFRSC employing mirror elements of varying width. Hence, the cost aspects of both the L F R S C designs appear to be same even for smaller aperture diameters in the case of the concentrator design using equal width of the mirror elements. However, for later the diameter of the tubular absorber increases with increase in f .
182
G.D. Sootha, B.S. Negi I Solar Materiab and Solar Cells 32 (1994) 169-186 D : 1.Om d = O.O4m
40....
UNIFORM INTENSITY SOLAR DISC NON UNIFORM INTENSITY SOLAR DISC
30. F : O.7m
/,//..., , ~
10"
,'.f/.t"
E
/'/~
i
160
120
/
"~
I
80
F:O.5m --
40
0
40
80
120
160
ANGULAR POSITION ON THE ABSORBER (DEGREES) Fig. 4. Distribution of local concentration ratio (LCR) on the surface o f the tubular absorber for LFRSC design derived from the prespecfficd diameter (2R = 0.04 m) of the absorber for f = 0.3, 0.5 and 0.7 m. F=l.0m d : O.OSm
40-
----
30-
//
/'20"
s'*--
-',,xx ~
~
UNIFORM INTENSITY SOLAR DIsC NON UNIFORM INTENSITY SOLAR DISC
F= 0.7m F:O.5m
0.3m
!
160
__,,,m4r" ~"~.,,,dv
!
!
120 80 40 0 40 80 120 ANGULAR POSITION ON THE ABSORBER (DEGREES)
I
160
Fig. 5. Distribution of local concentration ratio (LCR) on the surface of the tubular absorber for LFRSC design derived from the prespecified diameter (2R ffi 0.04 m) of the absorber for f = 0.3, 0.5 and 0.7 m.
G.D. Sootha, B.S. Negi / Solar Materials and Solar Cells 32 (1994) 169-186
t 40
W : O.03m O:l.Om UNIFORMINTENSITYSOLAR DISC . . . . NONUNIFORMINTENSITYSOLAR DISC
N
-
160
120
80
40
183
0
40
80
F = O.3m
120
160
ANGULAR POSITION ON THE ABSORBER (DEGREES) Fig. 6. Distribution of local concentration ratio (LCR) on the surface of the tubular absorber for LFRSC design derived from the prespecified equal width (W= 0.03 m) of the mirror elements for f = 0.3, 0.5 and 0.7 m.
The second aspect of the present study is the comparison of the flux concentration on the absorber surface for the two designs of concentrator under consideration. The distribution of the local concentration ratio (LCR) on the circular cross-section of the absorber has been determined using the ray trace technique. Both uniform intensity and non-uniform intensity on the solar disc have been considered while determining the LCR distribution. Figs. (3-5) illustrates for the L F R S C design using mirror elements of varying width for three values of d, and Figs. (6-8) for the L F R S C design using mirror elements of equal width for three values of W, the distribution of LCR on the surface of the tubular absorber, for three values of f , e.g., 0.3, 0.5 and 0.7 m. It may be noted that in each case the LCR is maximum at the centre of the circular cross-section, i.e., at the point (f-R) on the absorber and the LCR decreases with an increase in the arc length on both sides with respect to the distance f-R on the absorber. Further, the consideration of the solar limb darkening, as expected, resulted in a different LCR, showing a considerable increase in the peak value of LCR near the centre of the circular cross-section (i.e., at the point (f-R) on the absorber) in comparison to the LCR obtained considering the uniform intensity on the solar disc. The peak value of LCR and the total concentrated flux density on the absorber surface, in each ease, increases with an increase in f. The increase in the peak value of LCR as well as
184
G.D. Sootha, B.S. Negi /Solar Materials and Solar Cells 32 (1994) 169-186 40-
D = 1.Ore W : O.O4m
30-
-
UNIFORM INTENSITY SOLAR DISC NON UNIFORM INTENSITY SOLAR DISC
....
- ~ ' ' 2 ~ . . . - ~ " F : 0.7m ~
F:O.Sm
0.3m
I
I
160
120
80
40
ANGULAR
0
I
I
20
80
-" - " ~
I
120
160
P O S I T I O N ON THE A B S O R B E R ( D E G R E E S )
Fig. 7. Distribution of local concentration ratio (LCR) on the surface of the tubular absorber for LFRSC design derived from the prespecLfied equal width (W = 0.04 m) of the mirror elements for f = 0.3, 0.5 and 0.7 m.
40.
D = 1.Ore W = O.05m
30"
UNIFORM INTENSITY SOLAR DISC NON UNIFORM INTENSITY SOLAR DISC
~
....
20: p/
~ F : O . , m
..
@ ~
~
~ . ~ - F : O.5m
~
O
.
~
3
- -
I
160
,11I~ ~ ' r
120
I
I
80
40
m _
_
_
i
0
40
80
120
160
ANGULAR POSITION ON THE ABSORBER (DEGREES) Fig. 8. Distribution of local concentration ratio (LCR) on the surface of the tubular absorber for LFRSC design derived from the prespecified equal width (W= 0.05 m) of the mirror elements for f = 0.3, 0.5 and 0.7 m.
G.D. Sootha, B.S. Negi / Solar Materials and Solar Cells 32 (1994) 169-186
185
total concentrated flux density decreases when f increases from 0.5 to 0.7 m and for values of f > 0.7 m the increase in concentration is very small and hence the same has not been plotted. This is due to the fact that when f increases from 0.3 to 0.7 m, the total shift decreases considerably, as explained above. As a result the number of mirror elements that can be employed in a given aperture diameter increases, hence the flux concentration on the absorber increases. For f > 0.7 m, since the decrease in the total shift becomes slow, and hence results in a small increase in the number of mirror elements. As a result of which the increase in flux concentration on the circular cross-section of the absorber is also not much. The concentration decreases when the diameter of the absorber (in the case of the first design) or the width of the mirror elements (in the case of the second design) increases. As can be seen in Figs. 3-8, due to plane geometrical configuration of the LFRSC, a considerable portion of the circular cross-section is not illuminated from the solar rays reflected from the constituent mirror elements. This portion of the absorber which does not contribute to the absorption of solar energy reflected from the mirror elements has to be insulated to prevent heat losses. A comparison of the results of the LCR distribution indicates that the peak value as well as the total flux density of the concentrated flux on the surface of tubular absorber for the first design of LFRSC is more than that for the second LFRSC design. The surface area factor aspect, however, does not show much variation for the two designs of concentrator. This effectively means the cost of the reflector and, hence, the total cost of the concentrator may be the same for two designs. The fabrication of mirror elements of varying width, in the case of the first design, may be cumbersome and hence may add upto the cost of the system. However, as indicated earlier that variation in the width of the mirror elements in the first LFRSC design is so small that the variation can be ignored and a common width of the mirror elements can be taken with a negligible loss in concentration. The LFRSC employing mirror elements of equal widths, on the other hand, yield an increase in the diameter of the absorber with an increase in the value of f, for a given value of IV. Thus, the increased diameter would result in an increase in the heat losses as well as the cost of the system compared to the first design of LFRSC. It may be concluded that the LFRSC generated from the prespecified diameter of the absorber is a suitable design for solar thermal applications.
8. References [1] IC Harry and J.R. Charles, Solar photovoltaic energy systems, Handbook of Energy Technology and Economics, Ch. 6 (Wiley, NY, 1983). [2] J.A. Duffle and W.A. Beckman, Solar Engineering of Thermal Process (Wiley, NY, 1980). [3] J.C. Duran and R.O. Nicolas, Comparative optical analysis of cylindrical parabolic solar concentrators, Appl. Optics, 26(3) (1987) 578. [4] D.L. Evans and L.W. Florschuetz, Terrestrial concentrating photovoltaic power system studies, Solar Energy, 20 (1978) 37. [5] Marshall E. Alper, Photovoltaics and solar thermal conversion to electricity status and prospects, J. Energy, 3(5) (1979) 263.
186
G.D. Sootha, RS. Negi / Solar Materials and Solar Cells 32 (1994) 169-186
[6] J. Sangrador and G. Sala, Static concentrator for two sided photovoltaic solar cells, Solar Energy, 23 (1979) 53. [7] W.B. Ittner, An array of directable mirrors as a photovoltalc solar concentrator, Solar Energy, 24 (1980) 221. [8] U.Z. KurzWeg, Characteristics of axic concentrators for use in photovoltaic energy conversion, Solar Energy, 24 (1980) 411. [9] F.A. Akhmedov, Sh.Z. Mirtursunov and R.A. Mumnov, An investigation of the possibility of using inexpensive concentrating systems in photovoltaic power modules, Appl. Solar Energy, 17 (1981) 10. [10] O.I. Chesta and V.A. Grilikhes, Analysis of characteristics of flat film concentrators for solar photoelectric units, Appl. Solar Energy, 23(6) (1987) 20. [11] R.Z. Zakhidov, Sh.I. Klycber, I.A. Ogneva and M.N. Shulman, Optical and energy characteristics of fresnel mirrors, Appl. Solar Energy, 26(4) (1990) 47. [12] I.I. Tkachenko, N.N. Kllmova, Yu. I. Meshchanov and P.P. Lavrov, Experimental solar unit for power supply to autonomous agricultural consumers, Appl. Solar Energy, 26(6) (1990) 17. [13] E.V. Tveryanovich, Optical concentrating systems for solar electric stations, Appl. Solar Energy, 25(3) (1989) 18. [14] A.V. Vartanyan and L.A. Gagiyan, Linear concentrator of solar radiation with uniform energy distribution in plane of generatrix's optical axix, Appl. Solar Energy, 26(4) (1990) 50. [15] D.L. Evans, On the performance of cylindrical parabolic solar concentrators with flat absorber, Solar Energy, 19 (1977) 379. [16] B.S. Negi, N.C. Bhowmik, S.S. Mathur and T.C. Kandpal, Ray trace evaluation of solar concentrators including limb darkening effects, Solar Energy, 36(3) (1986) 293. [17] R.M. Cosby, Concentration characteristics of fresnel solar strip reflection concentrator, NASA Report NASA-CR-120336, 1974. [18] S.S. Mathur, B.S. Negi and T.C. Kandpal, Geometrical designs and performance analysis of a linear Fresnel reflector solar concentrator with flat horizontal absorber, Int. J. Energy Res., 14(1) (1990) 107. [19] B.S. Negi, T.C. Kandpal and S.S. Mathur, Designs and performance characteristics of a linear Fresnel reflector solar concentrator with a flat vertical absorber, Solar Wind Techn., 7(4) (1990) 379. [20] R.P. Goswami, B.S. Negi, H.IC Sehgal and G.D. Sootha, Optical design and concentration characteristics of a linear Fresnel reflector solar concentrator with a triangular absorber, Sol. Energy Mater., 21(2,3) (1990) 237. [21] B.S. Negi and H.K. Sehgal, Performance characteristics of spray pyrolysed selective cobalt oxide coated tubular absorber operated with a linear solar concentrator, Int. J. Energy Res., 15(9) (1991) 715.
Solar Energy Materials and Solar Cells 32 (1994) 169-186
A comparative study of optical designs and solar flux concentrating characteristics of a linear fresnel reflector solar concentrator with tubular absorber G.D. Sootha
a,,,
B.S. Negi b
a Solar Thermal Division, Ministry of Non-Conventional Energy Sources, Block No. 14, CGO Complex, Lodi Road, New Delhi 110 003, India b D.N.E.S. Monitoring cell, Bhubaneswar, Orissa, India
(Received 9 July 1992; in revised form 12 August 1993)
Abstract This paper presents a comparative study of the optical designs ~ind solar flux concentrating characteristics of two designs of the linear Fresnel reflector solar concentrator (LFRSC) derived using two different approaches for the tubular absorber configuration. The concentrator uses long plane mirror elements suitably positioned on a flat base so that each mirror element reflects solar radiation on to an absorber placed in the focal zone along the length of the concentrator. In one approach the concentrator is generated from the prespecified diameter of the absorber, while in the second approach, using a prespecified equal width of the mirror elements, an appropriate diameter of the tubular absorber is determined for the desired size of the concentrator. For each concentrator design, the distribution of the local concentration ratio (LCR) on the surface of the tubular absorber has been determined using the ray trace technique, taking into consideration both the uniform intensity on the solar disc and the solar limb darkening effect (non-uniform intensity). The surface area factor aspect of each concentrator design, a measure of the cost of reflective surface used in the concentrator, has also been studied. Results of some numerical calculations carried out to illustrate the performance of the concentrators are graphically presented and compared.
1. Introduction T h e c o n c e p t o f c o n c e n t r a t i o n o f s o l a r e n e r g y offers a n effective w a y to h a r n e s s s o l a r e n e r g y for efficient p h o t o v o l t a i c a p p l i c a t i o n s [1] a n d for use in t h e r m a l
* Corresponding author. 0927-0248/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0927-0248(93)E0106-N
170
G.D. Sootha, B.S. Negi / Solar Materials and Solar Cells 32 (1994) 169-186
systems that require thermal energy at temperatures above 100°C [2,3]. Solar concentrators consist of either a refracting or reflecting optical surface which concentrates the incident solar radiation on to an absorber surface. Hence, higher temperatures are generated due to the increased solar flux on the absorber and the lower heat losses from the comparatively smaller absorber. The increased solar flux at the focus can also be effectively used for efficient photovoltaic applications, in which highly efficient solar cells are exposed to the concentrated solar flux. An increased electrical output is thereby obtained from a reduced solar cells area [3-14]. A wide variety of solar concentrators have been developed in recent years and have been tested for their suitability for various processes. A typical solar concentrator of this kind is a linear Fresne! reflector solar concentrator (LFRSC). Such a concentrator uses a series of long plane mirror elements suitably positioned on a flat base. The angle of inclination of each mirror element with respect to the base of the concentrator is so determined that the incident solar radiation, after reflection from the mirror elements, is intercepted by an absorber placed in the focal zone along the length of the concentrator [17-21]. An appropriate space is introduced between two consecutive mirror elements in order to avoid blocking of solar radiation reflected from the constituent mirror elements. From the fabrication point of view, the concentrator is very simple and inexpensive. However, the disadvantage of this concentrator is that because of the space introduced between the mirror elements, only a fraction of the base area of the concentrator covered by reflecting surfaces contributes to the concentration on the surface of the absorber. Nevertheless, when mounted on a simple framework, the existence of the gaps between the mirror elements offers an advantage, when considering that wind loading can be considerably reduced. Two different approaches have been proposed for designing linear Fresnel reflector solar concentrators [18-20]. In one approach, the concentrator is generated from the prespecified size of the absorber. In this approach, the angle of inclination of each mirror element with respect to the base of the concentrator and the width of the mirror element are determined so that incident solar radiation, after reflection, is intercepted within the absorber and, hence, illuminates the absorber from one end to the other. The approach allows a variation in the width of the constituent mirror elements. In the second approach, the concentrator is generated from a prespecified equal width of the mirror elements and an appropriate size of the absorber which will intercept all the solar energy reflected from the constituent mirror elements and which is determined for the desired aperture diameter of the concentrator. Obviously, the shape of the concentrated flux produced by a LFRSC in its focal zone will depend on the geometry of the absorber. Detailed designs and performance analysis of LFRSC designs with flat horizontal [18] and triangular absorber [20] configurations have shown that the concentrator can provide a considerable uniform flux concentration on the absorber surfaces. Hence, the system appeared suitable for photovoltaic applications. On the other hand, in the case of a LFRSC design with a flat vertical absorber, a poor performance in terms of uniform flux concentration and peak value of
G.D. Sootha, B.S. Negi / Solar Materials and Solar Cells 32 (1994) 169-186
171
concentration [19] was showed as compared to the LFRSC designs with flat horizontal and triangular absorbers. Recently, a concentrator design based on the Fresnel reflector concept has been developed and operated in conjunction with solar cells for electric power and water heating [12]. Some preliminary efforts made to study the optical and thermal performance of a LFRSC with tubular absorber employing ordinary black painted and selective coating indicated that the system can be effectively utilised for solar thermal applications [21]. However, for large scale practical utilization, it is important to carry out a detailed study on design and solar flux concentrating characteristics of LFRSC with tubular absorber to arrive at a suitable design approach for the desired performance. Furthermore, as the solar flux concentration is considerably affected by non-uniform intensity on the solar disc, known as the solar limb darkening effect [15,16], it is of considerable interest to study the effect of solar limb darkening on the local concentration ratio on the surface of the tubular absorber employed with a LFRSC. This paper presents a comparison of optical designs and solar flux concentrating characteristics of two designs of LFRSC derived using two different approaches discussed above, with tubular absorber. The analysis has been carried out taking into consideration the sun's rays coming in the form of cone with the apparent diameter of the sun subtending an angle of 32' on the earth. In one approach, the concentrator design is generated from a prespecified diameter of the tubular absorber. In the other approach, using prespecified equal width of the mirror elements, an appropriate diameter of the tubular absorber which would intercept all the solar radiation reflected from the mirror elements employed in the concentrator, is determined for the desired aperture diameter of the concentrator. The concentrator is characterised by design parameters associated with each mirror element, the location of the mirror element on the concentrator base with respect to the centre of the aperture of the concentrator, the angle of inclination of the mirror element with respect to the base of the concentrator, the width of the mirror element, and the shift associated with the mirror element in order for a mirror element not to block the solar radiation reflected from the adjacent mirror element. Generalised expressions for the design parameters for each design of LFRSC are described. The distribution of local concentration ratio (LCR) on the surface of tubular absorber has been determined for each LFRSC design using the ray trace technique. Both the uniform intensity on the solar disc and the solar limb darkening, i.e., the fact that the intensity on the solar disc is not uniform, and is maximum at the centre and decreases considerably with the distance from the centre of the sun, have been taken into consideration for the determination of LCR on the absorber surface. Solar limb darkening effectively indicates that if I 0 is the intensity at the centre of the solar disc and if the solar disc were of uniform intensity Io, uo, it would radiate the same energy as a nonuniform disc with its intensity at the centre Io.Nu o equal to 1.256 I0,~D as described in Refs. [15,16]. A comparison of the different correlations describing limb darkening is also available in literature [16]. Another important aspect of the present analysis is to study the surface area factor of the concentrator, since the area of the reflective surface
G.D. Sootha, B.S. Negi/ Solar Materials and Solar Cells 32 (1994) 169-186
172
would vary as the distance of the absorber varies from the base of the concentrator. The surface area factor defined as the ratio of reflective surface area to aperture area will indicate the cost of the reflective material as a multiple of the absolute minimum possible. The above analysis has been carried out assuming the mirror elements specularly reflecting and the solar concentrators perfectly tracking. The advantages and disadvantages of both LFRSC designs have been discussed. Results of the above analysis have been illustrated in graphic and tabular forms, and have been compared.
2. Analysis Figs. (1 and 2) show the geometry used for deriving the designs of LFRSC using two different approaches, discussed above, for tubular absorber configuration. In the case of the concentrator design generated from the prespecified diameter of the tubular absorber (Fig. 1), the width and the angle of inclination of each mirror element with respect to the base of the concentrator are so determined that the extreme outer rays of the incident cone after reflection from the extreme edges of the mirror elements, touch the tubular absorber tangentially. This ensures that all the rays of the cones incident on mirror elements, after reflection, are intercepted by the absorber. This approach varies the width of the mirror elements. In the case of the concentrator design generated from a prespecified equal width of the mirror
i
I i ,~,,
~
, -~--- , ~
i
1':1
\ rn
x,-
t-s.4
O~--L,.*t~2°'~1"~-~2
~s.4"
Fig. 1. Linear fresnel reflector solar concentrator (LFRSC) with a tubular absorber design generated from the prespeeified diameter of the absorber.
G.D. Sootha, B.S. Negi/Solar
Materials and Solar Cells 32 (1994) 169-186
173
Fig. 2. linear fresnel reflector solar concentrator (LFRSC) with tubular absorber design generated from the prespecified equal width of the mirror elements.
elements, the designing is carried out in such way that a ray coming from the centre of the sun and impinging normally on the mid point of each mirror element is reflected to a common point F in the focal zone of the concentrator (Fig. 2). Obviously, the size of the spreading produced by the solar radiation reflected from the mirror elements of equal width will be different for different mirror elements. Hence, an appropriate value of the diameter of a tubular absorber which would intercept all the solar radiation reflected from all the constituent mirror elements is calculated for the desired aperture diameter of the concentrator. In each concentrator design, an appropriate space is introduced between two constituent mirror elements so that the radiation reflected from a constituent mirror element is not blocked by the adjacent mirror element. Detailed design procedures for two different concentrator designs are given below.
3. LFR!K design derived from prespecified diameter of the absorber In this concentrator design, the angle of inclination of each mirror element with respect to the base of the concentrator (say nth mirror element) is determined so that the extreme outer ray P,Q, of the cone impinging on the lower edge of the mirror element, after reflection just touches the tubular absorber of prespecified diameter 2R, tangentially at point A,. The extreme inner ray of the cone after reflection meets the absorber at point B,.Using simple geometrical optics the
G.D. Sootha, RS. Negi~Solar Materials and Solar Cells 32 (1994) 169-186
174
generalised expression for the angle of inclination of the nth mirror element can be given by:
Xa 2
tan- 1
,
[L~f-R~(x~,~-RZ)+(Y.,~-f)
O"=3[
I
- ~o ,
z
(1)
where tan(90 + 2 0 n + G0)R Xa'n
=
~/1 + tan2(90 + 2On + G0)
R and Ya,n----f--
~/1 + tan2(90 + 2 0 n + G0)
are the ( x , y ) coordinates of the point A n on the circular cross-section of the absorber, R is the radius of the tubular absorber, f is the distance between the centre of the absorber and the concentrator base (i.e. point O), L n is the location of the nth mirror element on the concentrator plane with respect to the centre of the concentrator aperture, and G0 is the half of the angular subtense of the sun on the earth ( = 16'). L n is calculated from: L n •Ln_ 1 + W~_ 1 cos On_ 1 + S n,
(2)
with L 0 ffi W0 = S~ = 0 and O 0 = 0 as initial values. However, since the concentrator is perfectly tracked, the tubular absorber will east a shadow on the concentrator plane and, hence, the location of the first mirror element, L 1, is determined so that it is just beyond the edge of the shadow. If 2G0 ( = 32') represents the angular subtense of the sun at any point on the earth, then the location of the first mirror element on either half of the concentrator is defined as: L 1 = R + f tan G0.
(3)
The width of the mirror element under consideration is determined by drawing a line parallel to BnQ n (i.e., the reflecting ray associated with the extreme inner ray of the cone incident on the extreme lower edge of the first mirror element) such that it is tangential to the tubular absorber at point Cn and intersects the line making an angle On with the aperture plane of the concentrator, at point Tn. Hence, QnTn should be the required width of the mth mirror element and the line CnTn corresponds to the reflected ray associated with the extreme inner ray SnTn of the cone incident on the extreme upper edge of the first mirror element. This condition ensures that all the solar radiation reflected from the mirror element is intercepted by the tubular absorber (i.e., from point A n to point Cn). The width of the nth mirror element may be calculated from:
IV,, = ~(Xc,n --Xb.n) 2 + (Yc,n --Yb,n) 2 COS(2On -- Go - an),
(4)
where (Xo,n,yb, n) and (Xc,n,yc, n) are the (x,y) coordinates of the points B n and Cn on the circular cross section of the absorber, respectively, and a n is the angle of inclination of the line joining the points B n and C n. a n is calculated from:
1(,c an
-- tan
-
k
Xc,n _ X b , n
.
G.D. Sootha, B.S. Negi/ Solar Materials and Solar Cells 32 (1994) 169-186
175
The shift associated with the nth mirror element is expressed as:
=
(Ln_ 1 + Wn_ 1 cos O~_ 1 +x~.a)W~_ 1 sinO~_ 1 y.,, - W~_ 1 sin O,_x
(5)
The aperture diameter, D, of the concentrator for a desired number of mirror elements is defined by: D=2
1
Ll + E (W~ cos On + S~) ,
[
'
n=l
(6)
where n varies from 1 to k, and k is the total number of mirror elements employed on each half of the concentrator. The surface area factor of the concentrator defined as the ratio of reflective surface area to aperture area is calculated from: k
2Ew. SA
D
(7)
The surface area factor will indicate the cost of the reflective material as a multiple of the absolute minimum possible. From the geometry of the concentrator it is clear that for a certain given aperture diameter of the concentrator the reflective surface area will vary according to the distance the focal region is from the base of the concentrator.
4. LFRSC design derived from mirror elements of equal width In this design approach, the angle of inclination of mirror elements of equal width, W, with respect to the base of the concentrator, is chosen such that a ray incident normally on the concentrator aperture and striking the mid point of the mirror element, is reflected to a common point F in the focal zone of the concentrator (Fig. 2). In the present analysis, since the incident solar rays are being considered in form of cones, the above ray corresponds to the central ray of the cone emanating from the sun. A tubular absorber of an appropriate diameter is placed in the focal zone along the length of the concentrator in such a way that the centre of the absorber coincides with the point F, and hence captures all the rays reflected from the mirror elements. Since, in this case, the absorber size (i.e., the radius of the tubular absorber) is not given, the absorber may be determined from the size of the spreadings produced by the rays reflected from the mirror elements, and measured with respect to the point F in the focal zone of the concentrator of a desired aperture diameter, so that all rays reflected from the mirror elements are intercepted by the tubular absorber. Thus, the radius of the tubular absorber may be determined by drawing perpendiculars from the point F on the extreme outer reflecting rays associated with the extreme rays PnQn and SnT~ of the cones incident on the extreme upper and lower edges of the mirror elements. The largest
G.D. Sootha, B.S. Negi/ SolarMaterialsand Solar Cells32 (1994) 169-186
176
perpendicular distance from the point F is taken to be diameter of the tubular absorber. However, as a first approximation the designing may be initiated considering the radius of the absorber as equal to the half of the width of the mirror element. Similar to the case of the first design of LFRSC, an appropriate space is introduced between two consecutive mirror elements to avoid blocking of solar radiations reflected from the mirror elements. This effectively means that the extreme outer ray of the cone impinging on the lower edge of the nth mirror element, after reflection, just touches the extreme upper age of the ( n - 1)th mirror element and is finally intercepted by the absorber. Based on the simple geometrical considerations as for the first design, the following generalised expressions for the shift S~, location L~ and the angle of inclination with respect to the base of the concentrator parameters associated with the nth mirror element can be derived: S~ = W sin 0._ 1 tan(20~ + ~o),
(8)
L~ =L~_ 1 + W sin 0~_1 + S . ,
(9)
and
O.
I
=
~- tan-1
( L,, + W/2 cos O,, ) (10)
f - W/2 sin On
'
with O 0 = 0, L 0 -- So -- 0.0 and L i = W / 2 + f tan O 0 as initial values. Eqs. (9-11) are solved iteratively, n varies from 1,2. . . . . k, and k is the total number of mirror elements on each half of the concentrator. The radius of the tubular absorber, the maximum length of the perpendicular dropped from the point F on the extreme outer rays of the cone reflected from the extreme edges of the mirror element, which would intercept all the solar rays reflected from all the constituent mirror elements, is calculated from: [ L . + W cos On _ f ] sin(20, R = W sin O~ + [ ~ a n ( - 2 ~ - ~-~ , ¢0)
(11)
Since the above design equations have been derived presuming the diameter of the absorber equal to the width of the mirror element, the actual size of the shading caused by the absorber on the concentrator plane and hence the actual location of the first mirror element will be larger than assumed. Thus, the actual value of L has to be calculated so that no part of the mirror element is shaded by the absorber. This effectively means that the shading caused by the absorber of radius given by Eq. (11) would extend beyond the assumed value of L 1 used for initiating the designing of the concentrator. This necessitates the use of an iterative procedure in which the value of R obtained from Eq. (11) is used to calculate another value of L 1 a s : L 1 = r + f tan s%.
(12)
Using the value of L 1 given by Eq. (12) the design procedure is repeated (i.e., Eqs. (8-10)) to get another set of values for shift, angle of inclination and location
G.D. Sootha, B.S. Negi / Solar Materials and Solar Cells 32 (1994) 169-186
177
parameters for each constituent mirror element. This results in a new value of R. The new value of R is again used to calculate another value of L 1. The iterations are continued until the difference between the two consecutive values of L 1 is within a prespecifled convergence limit. The aperture diameter D of the concentrator is defined by: D=2
L~+
+S~) .
(13)
The surface area fact or, S A, in this case is calculated from k
2~w SA ~
n=l
D
(14)
5. Distribution of local concentration ratio (LCR) The distribution of the local concentration ratio (LCR) on the surface of the tubular absorber has been determined using the ray trace technique. In the ray trace technique, equations for the incident rays and the corresponding reflected rays are derived and then the intersection points of the reflected rays with the absorber placed in the focal cone of the concentrator are calculated. The aperture diameter, D, of the concentrator is divided into a large number of devises (ray N ) of equal width. On each division a cone containing 33 rays with an angular interval of 1' has been considered. The coordinates of the point of intersection of the incident rays of each cone with the mirror elements are calculated. This is followed by deriving equations for the reflected rays associated with the incident rays for which the intersection points are calculated, and finally calculating the points of intersection of the reflected rays with the circular cross-section of the absorber. Obviously, the incident cones falling within the space introduced between mirror elements are assumed to be lost. Similarly, incident rays directly striking the tubular absorber have not been considered. The circular cross-section of the absorber is also divided into equal angular divisions. Thus, the ratio of the number of rays striking a particular division of the absorber of arc length equal to the width of the division on the concentrator aperture to the number of rays incident on a division of equal width on the concentrator plane gives the LCR for that particular division of the absorber. This provides a smooth LCR distribution curve with a peak at the point (f-R) on the circular cross section of the absorber. Calculations of the ray trace technique were done on a computer.
6. Consideration of limb darkening effect As explained earlier the limb darkening describes a variation in the brightness on the solar disc; the brightness is maximum at the centre and decreases consider-
178
G.D. Sootha, B.S. Negi / Solar Materials and Solar Cells 32 (1994) 169-186
ably with the distance from the Centre of the sun disc. This effectively indicates that ff I 0 is the intensity at the centre of the solar disc, and if the solar disc were of uniform intensity, it would radiate the same energy as a non-uniform disc with its intensity at the Centre, Io, NuO, equal to 1.256 Io,~o as Evans [15] has suggested the relationship between two as: 0.785 lo,uo = 0.625 IO,NuO.
(15)
Obviously, in this case each ray of the cone will carry a different intensity. The procedure provided in Ref. [16] for studying LCR distribution on a flat absorber employed with a cylindrical parabolic trough and a polygal trough, respectively, has been used to assign the appropriate value of weight factor to each of the rays emanating from the sun, and thus determine the LCR on the surface of the absorber. The ray trace equations derived for determining the LCR on the surface of the tubular absorber are described below. The distance of an incident point of a cone on any division on either half of the concentrator from the centre of the plane of the concentrator (xx) is given by O x, = 2
( m - 1) N '
(16)
where N is the total number of divisions of equal size on the aperture plane of the concentrator and m varies from 1 to N + 1. In the present work 1000 divisions were made on the concentrator aperture for an aperture diameter D equal to 1.0 m. As discussed above, the projection of a cone is simulated by 33 rays, each separated by an angular interval of 1'. The incident cone would strike a typical mirror element, say the nth, if x lies between L n and L n + Wn cos On, i.e., between the lower and the upper edge of the mirror element. Any typical ray of the incident cone under consideration may be represented by an equation as: Y =Yi,n + (X --X,,n) tan(90 + ~),
(17)
where (xi, n, Yi,n) are the ( x , y ) coordinates of the point of intersection of the incident cone with the nth mirror element, and is the angular deviation of a ray of the incident cone from the ray emanating from the centre of the sun (g varies from 0 to + 16'). The positive sign with ~ is taken for those rays of the cone which make an angle greater than 90 ° with the concentrator plane and the negative sign for those rays which make angle less than 90°. Y~,n is calculated from: Y,,n = ( xi,n - Ln) tan On.
(18)
Once the intersection points (xi,n,y~, n) of the incident cone with the mirror element are calculated, the equation of the reflected ray of a cone associated with the ray of the incident cone under consideration can be derived. The equation of the reflected ray is: Y =Y,,n +~v,n( x - x i , n ) , where /3v,n = tan[90 + (2On + g)]
(19)
179
G.D. Sootha, B.S. Negi / Solar Materials and Solar Cells 32 (1994) 169-186
is the slope of the reflected ray represented by Eq. (19). The positive sign before (20 n -t-~) corresponds to the rays falling on the mirror elements placed on the right hand-half of the concentrator and the negative sign corresponds to the reflected rays associated with the incident rays on the mirror elements on the left half of the concentrator. All the 33 rays of each cone reflected from all mirror elements have to be intercepted by an absorber placed in the focal zone and for which the concentrator has been designed. Thus the procedures is followed by the calculation of intersection points of the reflected rays with the tubular absorber cross-section. The circular cross-section of the tubular absorber can be represented by an equation as: x2 + ( Y _ f ) 2 = R 2.
(20)
The x-y coordinates of the point of intersection of a ray reflected from the nth mirror element represented by Eq. (19), with the circular cross-section of the absorber described by Eq. (21), are: Zn + ~ z2 -- Sec2(2On + ~){[ Xi,n + Yi,n tan(20~ ± g)]2 + f 2 _ R 2) Yip,n =
Sec2(20n ± ~)
, (21)
and Xia,n = xi,n - tan(20~ + g )( y,p,~ - Yi,~),
(22)
with Zn = f + x , , n
tan(20, +~:) +y,,, tan2(2On + ~ ) .
The above procedure is repeated for all the 33 rays of the cones incident on the different divisions made on the aperture diameter of the concentrator to obtain the LCR curve on the tubular absorber cross-section.
7. Results and discussion
In order to carry out a comparative study of optical design and performance characteristics of two LFRSC designs derived using two different approaches for tubular absorber configuration, some numerical calculations have been done, and the results are presented in this section. For the purpose of the analysis, being presented herein, the aperture diameter of each design of LFRSC has been taken as D -- 1.0 m, and 2R (for the first design)= W (for the second design). As the concentrator can use an integral number of mirror elements, the actual value of D may, however, not be the same as chosen. The actual value of D also varies as the height of the absorber varies with respect to the base of the concentrator. The variation in the actual value of D in the case of a LFRSC design generated from the prespecified field diameter of the absorber is found to be very small (i.e., about
180
G.D. Sootha, B.S. Negi / Solar Materials and Solar Cells 32 (1994) 169-186
Table 1 Variation in D, S n and SA with f for three different values of R, for the LFRSC design derived from the prespeeified diameter of the tubular absorber f (m)
0.3 0.5 0.7 0.9 1.1
2R = 0.03 m
2R = 0.04 m
2R = 0.05 m
D (m)
2S n (m)
SA
D (m)
2S n (m)
SA
D (m)
2S n (m)
SA
0.994 0.985 1.027 1.017 1.003
0.22 0.11 0.07 0.046 0.03
0.787 0.88 0.91 0.93 0.94
0.99 0.99 0.964 1.03 1.023
0.22 0.108 0.059 0.047 0.032
0.78 0.87 0.906 0.92 0.93
1.028 0.96 0.976 1.007 1.038
0.235 0.098 0.044 0.042 0.036
0.77 0.864 0.897 0.91 0.92
0.018 m). Whereas in the case of LFRSC using mirror elements of equal width the variation in the value of D is found to be more than for the former. The variation in the actual value of D and the total shift in the concentrator and the surface area factor SA for two designs of LFRSC, with the height of the absorber, f , is shown in Tables 1 and 2. As expected, in the case of the LFRSC design generated from a prespecified diameter of the absorber, the width of the mirror elements varied. However, in the case of the concentrator using a tubular absorber the variation in the width is very small, so small that the variation can be ignored, and an equal width of the mirror elements can be employed with almost negligible loss in flux concentration. For example, for 2R = 0.03 m, the width of the mirror elements vary from 0.0373 m (i.e., W1) to 0.0399 m (Wk = 14) for f - - 0 . 3 m, from 0.0254 m (W t) to 0.0255 m ( W k - - 1 7 ) for f = 0.5 m, from 0.0235 m (W 1) to 0.0231 (Wk = 20) for f - - 0 . 7 m from 0.0216 m (W 1) to 0.0211 m ( Wk = 22) for f = 0.9 m and from 0.0198 m (W 1) to 0.0192 m ( W k = 24) for f = 1.1 m. The angle of inclination of mirror elements with respect to the base of the concentrator, for each case, increases as one moves from the centre towards the rim of the concentrator aperture. The angle of inclination as well as the shift associated with each mirror element, however, decreases with an increase in the value of f. As a result, the number of mirror elements that can be employed in a concentrator of a given aperture, diameter, increases with an increase in f. The total shift 2ES n, however, first decreases drastically with increase in f and finally as can be seen in Table 1, the decrease is very small.
Table 2 Variation in D, 2S n, R and SA with f for three different values of W for the LFRSC design derived using mirror elements of equal width f (m) D (m)
W = 0.03 m 2S n (m) R (m) SA
D (m) 2S n (m) R (m) SA
D (m) 2S n (m) R (m) SA
0.3 0.5 0.7 0.9 1.1
0.227 0.101 0.064 0.039 0.032
0.9126 1.018 0.974 0.96 1.032
0.8914 0.9258 1.007 0.99 0.98
1.0016 0.96 0.99 0.97 1.02
0.016 0.017 0.018 0.019 0.02
W = 0.04 m
0.78 0.875 0.91 0.93 0.94
0.18 0.117 0.06 0.037 0.032
0.021 0.022 0.023 0.024 0.025
W = 0.05 m
0.79 0.865 0.904 0.92 0.93
0.17 0.089 0.064 0.0386 0.026
0.026 0.027 0.028 0.029 0.031
0.785 0.864 0.894 0.909 0.92
G.D. Sootha, B.S. Negi / Solar Materials and Solar Cells 32 (1994) 169-186
181
D:l.Om d : O.O3m
t~
50'
90°
~
UNIFORM INTENSITY SOLAR DISC
....
NON UNIFORM I N T E N S I T Y S O L A R
90°
DISC
0
#It ' ,,"l~1~ ~
~ " ~
F =O.'m
,=o.,°
#///'/20
jY
~
"
~
,.o.,.
10
////
/
160
120
80
40
ANGULARPOSITIONON
0
40
THE ABSORBER
80
120
i
160
(DEGREES)
Fig. 3. Distribution of local concentration ratio (LCR) on the surface of the tubular absorber for LFRSC design derived from the prespecified diameter (2R = 0.03 m) of the absorber for f = 0.3, 0.5 and 0.7 m.
Thus, the increase in the number of mirror elements slows down for f > 1.1 m. Consequently, the surface area factor S A increases with an increase in f . The increase in the value of S A is very small for values of f > 0.7 m. This is contrary to the case of the cylindrical parabolic trough in which for a fixed aperture area of the concentrator as the distance between the focal region and the concentrator base decreases the deepness and surface area increases, and hence SA increases. The reflector tends to a flat mirror when the focal region is moved far away from the base. In the case of a L F R S C design employing mirror elements of equal width, the actual aperture diameter of the concentrator varies considerably when f increases (Table 2). Nevertheless, the values of SA for each value of f and W are more or less the same as for the corresponding values of 2R and f in the case of LFRSC employing mirror elements of varying width. Hence, the cost aspects of both the L F R S C designs appear to be same even for smaller aperture diameters in the case of the concentrator design using equal width of the mirror elements. However, for later the diameter of the tubular absorber increases with increase in f .
182
G.D. Sootha, B.S. Negi I Solar Materiab and Solar Cells 32 (1994) 169-186 D : 1.Om d = O.O4m
40....
UNIFORM INTENSITY SOLAR DISC NON UNIFORM INTENSITY SOLAR DISC
30. F : O.7m
/,//..., , ~
10"
,'.f/.t"
E
/'/~
i
160
120
/
"~
I
80
F:O.5m --
40
0
40
80
120
160
ANGULAR POSITION ON THE ABSORBER (DEGREES) Fig. 4. Distribution of local concentration ratio (LCR) on the surface o f the tubular absorber for LFRSC design derived from the prespecfficd diameter (2R = 0.04 m) of the absorber for f = 0.3, 0.5 and 0.7 m. F=l.0m d : O.OSm
40-
----
30-
//
/'20"
s'*--
-',,xx ~
~
UNIFORM INTENSITY SOLAR DIsC NON UNIFORM INTENSITY SOLAR DISC
F= 0.7m F:O.5m
0.3m
!
160
__,,,m4r" ~"~.,,,dv
!
!
120 80 40 0 40 80 120 ANGULAR POSITION ON THE ABSORBER (DEGREES)
I
160
Fig. 5. Distribution of local concentration ratio (LCR) on the surface of the tubular absorber for LFRSC design derived from the prespecified diameter (2R ffi 0.04 m) of the absorber for f = 0.3, 0.5 and 0.7 m.
G.D. Sootha, B.S. Negi / Solar Materials and Solar Cells 32 (1994) 169-186
t 40
W : O.03m O:l.Om UNIFORMINTENSITYSOLAR DISC . . . . NONUNIFORMINTENSITYSOLAR DISC
N
-
160
120
80
40
183
0
40
80
F = O.3m
120
160
ANGULAR POSITION ON THE ABSORBER (DEGREES) Fig. 6. Distribution of local concentration ratio (LCR) on the surface of the tubular absorber for LFRSC design derived from the prespecified equal width (W= 0.03 m) of the mirror elements for f = 0.3, 0.5 and 0.7 m.
The second aspect of the present study is the comparison of the flux concentration on the absorber surface for the two designs of concentrator under consideration. The distribution of the local concentration ratio (LCR) on the circular cross-section of the absorber has been determined using the ray trace technique. Both uniform intensity and non-uniform intensity on the solar disc have been considered while determining the LCR distribution. Figs. (3-5) illustrates for the L F R S C design using mirror elements of varying width for three values of d, and Figs. (6-8) for the L F R S C design using mirror elements of equal width for three values of W, the distribution of LCR on the surface of the tubular absorber, for three values of f , e.g., 0.3, 0.5 and 0.7 m. It may be noted that in each case the LCR is maximum at the centre of the circular cross-section, i.e., at the point (f-R) on the absorber and the LCR decreases with an increase in the arc length on both sides with respect to the distance f-R on the absorber. Further, the consideration of the solar limb darkening, as expected, resulted in a different LCR, showing a considerable increase in the peak value of LCR near the centre of the circular cross-section (i.e., at the point (f-R) on the absorber) in comparison to the LCR obtained considering the uniform intensity on the solar disc. The peak value of LCR and the total concentrated flux density on the absorber surface, in each ease, increases with an increase in f. The increase in the peak value of LCR as well as
184
G.D. Sootha, B.S. Negi /Solar Materials and Solar Cells 32 (1994) 169-186 40-
D = 1.Ore W : O.O4m
30-
-
UNIFORM INTENSITY SOLAR DISC NON UNIFORM INTENSITY SOLAR DISC
....
- ~ ' ' 2 ~ . . . - ~ " F : 0.7m ~
F:O.Sm
0.3m
I
I
160
120
80
40
ANGULAR
0
I
I
20
80
-" - " ~
I
120
160
P O S I T I O N ON THE A B S O R B E R ( D E G R E E S )
Fig. 7. Distribution of local concentration ratio (LCR) on the surface of the tubular absorber for LFRSC design derived from the prespecLfied equal width (W = 0.04 m) of the mirror elements for f = 0.3, 0.5 and 0.7 m.
40.
D = 1.Ore W = O.05m
30"
UNIFORM INTENSITY SOLAR DISC NON UNIFORM INTENSITY SOLAR DISC
~
....
20: p/
~ F : O . , m
..
@ ~
~
~ . ~ - F : O.5m
~
O
.
~
3
- -
I
160
,11I~ ~ ' r
120
I
I
80
40
m _
_
_
i
0
40
80
120
160
ANGULAR POSITION ON THE ABSORBER (DEGREES) Fig. 8. Distribution of local concentration ratio (LCR) on the surface of the tubular absorber for LFRSC design derived from the prespecified equal width (W= 0.05 m) of the mirror elements for f = 0.3, 0.5 and 0.7 m.
G.D. Sootha, B.S. Negi / Solar Materials and Solar Cells 32 (1994) 169-186
185
total concentrated flux density decreases when f increases from 0.5 to 0.7 m and for values of f > 0.7 m the increase in concentration is very small and hence the same has not been plotted. This is due to the fact that when f increases from 0.3 to 0.7 m, the total shift decreases considerably, as explained above. As a result the number of mirror elements that can be employed in a given aperture diameter increases, hence the flux concentration on the absorber increases. For f > 0.7 m, since the decrease in the total shift becomes slow, and hence results in a small increase in the number of mirror elements. As a result of which the increase in flux concentration on the circular cross-section of the absorber is also not much. The concentration decreases when the diameter of the absorber (in the case of the first design) or the width of the mirror elements (in the case of the second design) increases. As can be seen in Figs. 3-8, due to plane geometrical configuration of the LFRSC, a considerable portion of the circular cross-section is not illuminated from the solar rays reflected from the constituent mirror elements. This portion of the absorber which does not contribute to the absorption of solar energy reflected from the mirror elements has to be insulated to prevent heat losses. A comparison of the results of the LCR distribution indicates that the peak value as well as the total flux density of the concentrated flux on the surface of tubular absorber for the first design of LFRSC is more than that for the second LFRSC design. The surface area factor aspect, however, does not show much variation for the two designs of concentrator. This effectively means the cost of the reflector and, hence, the total cost of the concentrator may be the same for two designs. The fabrication of mirror elements of varying width, in the case of the first design, may be cumbersome and hence may add upto the cost of the system. However, as indicated earlier that variation in the width of the mirror elements in the first LFRSC design is so small that the variation can be ignored and a common width of the mirror elements can be taken with a negligible loss in concentration. The LFRSC employing mirror elements of equal widths, on the other hand, yield an increase in the diameter of the absorber with an increase in the value of f, for a given value of IV. Thus, the increased diameter would result in an increase in the heat losses as well as the cost of the system compared to the first design of LFRSC. It may be concluded that the LFRSC generated from the prespecified diameter of the absorber is a suitable design for solar thermal applications.
8. References [1] IC Harry and J.R. Charles, Solar photovoltaic energy systems, Handbook of Energy Technology and Economics, Ch. 6 (Wiley, NY, 1983). [2] J.A. Duffle and W.A. Beckman, Solar Engineering of Thermal Process (Wiley, NY, 1980). [3] J.C. Duran and R.O. Nicolas, Comparative optical analysis of cylindrical parabolic solar concentrators, Appl. Optics, 26(3) (1987) 578. [4] D.L. Evans and L.W. Florschuetz, Terrestrial concentrating photovoltaic power system studies, Solar Energy, 20 (1978) 37. [5] Marshall E. Alper, Photovoltaics and solar thermal conversion to electricity status and prospects, J. Energy, 3(5) (1979) 263.
186
G.D. Sootha, RS. Negi / Solar Materials and Solar Cells 32 (1994) 169-186
[6] J. Sangrador and G. Sala, Static concentrator for two sided photovoltaic solar cells, Solar Energy, 23 (1979) 53. [7] W.B. Ittner, An array of directable mirrors as a photovoltalc solar concentrator, Solar Energy, 24 (1980) 221. [8] U.Z. KurzWeg, Characteristics of axic concentrators for use in photovoltaic energy conversion, Solar Energy, 24 (1980) 411. [9] F.A. Akhmedov, Sh.Z. Mirtursunov and R.A. Mumnov, An investigation of the possibility of using inexpensive concentrating systems in photovoltaic power modules, Appl. Solar Energy, 17 (1981) 10. [10] O.I. Chesta and V.A. Grilikhes, Analysis of characteristics of flat film concentrators for solar photoelectric units, Appl. Solar Energy, 23(6) (1987) 20. [11] R.Z. Zakhidov, Sh.I. Klycber, I.A. Ogneva and M.N. Shulman, Optical and energy characteristics of fresnel mirrors, Appl. Solar Energy, 26(4) (1990) 47. [12] I.I. Tkachenko, N.N. Kllmova, Yu. I. Meshchanov and P.P. Lavrov, Experimental solar unit for power supply to autonomous agricultural consumers, Appl. Solar Energy, 26(6) (1990) 17. [13] E.V. Tveryanovich, Optical concentrating systems for solar electric stations, Appl. Solar Energy, 25(3) (1989) 18. [14] A.V. Vartanyan and L.A. Gagiyan, Linear concentrator of solar radiation with uniform energy distribution in plane of generatrix's optical axix, Appl. Solar Energy, 26(4) (1990) 50. [15] D.L. Evans, On the performance of cylindrical parabolic solar concentrators with flat absorber, Solar Energy, 19 (1977) 379. [16] B.S. Negi, N.C. Bhowmik, S.S. Mathur and T.C. Kandpal, Ray trace evaluation of solar concentrators including limb darkening effects, Solar Energy, 36(3) (1986) 293. [17] R.M. Cosby, Concentration characteristics of fresnel solar strip reflection concentrator, NASA Report NASA-CR-120336, 1974. [18] S.S. Mathur, B.S. Negi and T.C. Kandpal, Geometrical designs and performance analysis of a linear Fresnel reflector solar concentrator with flat horizontal absorber, Int. J. Energy Res., 14(1) (1990) 107. [19] B.S. Negi, T.C. Kandpal and S.S. Mathur, Designs and performance characteristics of a linear Fresnel reflector solar concentrator with a flat vertical absorber, Solar Wind Techn., 7(4) (1990) 379. [20] R.P. Goswami, B.S. Negi, H.IC Sehgal and G.D. Sootha, Optical design and concentration characteristics of a linear Fresnel reflector solar concentrator with a triangular absorber, Sol. Energy Mater., 21(2,3) (1990) 237. [21] B.S. Negi and H.K. Sehgal, Performance characteristics of spray pyrolysed selective cobalt oxide coated tubular absorber operated with a linear solar concentrator, Int. J. Energy Res., 15(9) (1991) 715.