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Effect of multiple reflections on the performance of plane booster mirrors
Applied Energy It (1982) 307 318
EFFECT OF MULTIPLE REFLECTIONS PERFORMANCE O F P L A N E BOOSTER
ON THE
MIRRORS
A~IAy DANG, J. K. SnARMA and H. P. GARG
Centre of Energy Studies, Indian Institute of Technology, Haus Khas, New Delhi 110016 (India)
SUMMARY
This paper presents an analytical study of a stationao, V-trough concentrator. It consists of an array of east-west oriented trapezoidal channels with two side reflecting walls and a tubular absorber as a receiver at the base. The jormulae Jbr concentration factor and reflector surfaces hat'e been derired. It is explicitly shown that the concentration ratio and reflector surJace area depend upon the number ~[ reflections the solar rays undergo bejbre reaching the absorber, the cone apex angle, the coefficient of reflection and the acceptance angle. Results are presented graphically in such a way that one can choose the optimum configuration and the minimum material required to achiet'e a given concentration factor. The concentration ratio ranges from 1"2 to 3"6. The t'ariations of the collector's efficiem3' with temperature d(fference for different numbers of reflections, acceptance angles, cont,ective heat transfer coefficients and coe[flcients of re[tectit'ity have been predicted.
NOMENCLATURE
A B C R h
Collector aperture. Absorber base width. Concentration ratio. Reflector wall slant height. Effective overall convective heat transfer coefficient for the collector evaluated assuming the reference area is that of the base plate area. n Number of reflections. q.~ Direct solar flux. TB Base absorber temperature. Tj Ambient air temperature. 307 Applied Energy 0306-2619/82/0011-0307/$02.75 ~ Applied Science Publishers Ltd, England, t982 Printed in Great Britain
308 Ys 0ts r tr e~
AMAN DANG, J. K. SHARMA, H. P. GARG
Solar angle measured from the normal to the collector surface. Effective solar absorptivity of the absorber surface. Transmittance of the cover plate assembly. Stefan Boltzmann constant. Effective emittance of the absorber plate. Wall angle. Acceptance angle of the collector.
INTRODUCTION
The utilisation of solar energy for various applications such as solar cooling and air conditioning, power generation, the cooking of food, etc., requires heat at temperatures above = 90 °C. Conventional flat plate collectors cannot effectively supply heat at such temperatures. To provide heat at higher temperatures, concentration of the solar energy is needed. Various types of tracking and nontracking concentrator such as groove-like geometries (trough type), Fresnel lenses, Winston Collectors, paraboloids, parabolic troughs, segmented mirrors, centraltower heliostat arrays and spherical mirrors are used. Of course, the incorporation of curved surfaces into the design, and tracking of a concentrator, improves its performance to a certain degree but there are certain limitations of fabrication, handling and materials. Moderate concentration with fiat side booster mirrors is a technique to improve the collector's performance without tracking. To keep the incident solar radiations within the acceptance angle, the system can be seasonally tilted and adjusted. The V-trough concentrator has a number of advantages over other types of concentrator. It has the advantage of extreme ease of manufacture: it can, for example, be fabricated simply by folding a reflective aluminium sheet to the required opening angle and attaching a tubular absorber at the base. In the present paper we have studied the design parameters of a reflectorconcentrator with a tubular absorber at the base. The previous analyses of Tabor, Holland 2 and Mannan and Bannerot 3 are limited to single-beam reflection with flat absorbers. Bannerot and Howell 4 have calculated the collector efficiencies for plane absorbers taking single reflections into account. Burkhard et al. 5 have carried out an analysis for flat absorbers taking multiple reflections into account. A recent analysis by Garg e t al. 6 considers the tubular absorber as a receiver, but for single reflections only. In the present case we have found that the concentration ratio C is an optimum for a particular value of reflector surface area, R B . The concentration ratio C is higher for higher numbers of reflections. As expected, for a higher number of reflections to take place, the length of the reflection surface--and hence the input aperture--increases. However, the effect of the coefficient of reflection is to decrease the concentration. The solar radiation impinging on the reflected mirrors at the
309
EFFECT OF MULTIPLE REFLECTIONS
~1 Z
(aJ
(Yo,Zo)~!
[b) ---- _----
102
~'~(O,~-Cot ~) \
Fig. l(a),(b).
L /
,,!/
Geometric design parameters showing a multiple reflected ray path for an acceptance angle, 6.
acceptance angle, 6, is considered to be received tangentially to the absorber surface, as shown in Fig. 1. All other solar radiations incident at the reflectors within the acceptance angle, 6, are received by the tubular absorber.
DERIVATION OF THE RAY TRACE EQUATION
(for p = 1)
As Shown in Fig. l(a), we consider that the solar radiations are incident at an angle, 6, with the symmetry axis of the trough. The apex of the trough is considered as the origin of the Cartesian co-ordinate system. The co-ordinates of the original incident ray are ( -Yo, Zo), where Yo = (A/2), A being the entrance aperture. The equation of the incident ray making an angle (90 ° - 6) with the horizontal and passing through ( -Yo, Zo) is written as: Z -
Z o -- m ( y
+Yo)
(1)
where: dZ m = ~ - = tan (qo - 6) = cot 6 oy Thus : Y = - Yo + (Z - Z o) tan 6
(2)
310
A M A N D A N G , J. K. S H A R M A , H. P. G A R G
The equation of the side of the trough is: y = + Z t a n ct
(3(a))
so that: Yo =
+
Zo tan
(3(b))
The equation of the ray after one reflection is: Z-
dZ
m =-- = - - t a n (qo - (6 + 2~)) dy
Z o = m ( y + Yo)
or: Y = -Yo + ( Z - Z o ) t a n ( - 0 )
(4)
Here 0 is the angle which the reflected ray makes with the y axis. The equation of the ray after two reflections is: Z-Z
l=m(y-y
1)
dZ m=~y=COt(6+4c0
or"
y =y~ + ( Z - Z 1 ) t a n ( 6 +4ct)
(5)
where: Z1 --'Yl c o t s
Here (y~, Z~) is the point of intersection o f y = -Yo - (Z - Zo) tan(fi + 2~) and: y = Z tan i.e. o f y = - Z t a n (fi + 2~) + (cot a. tan (6 + 2ct) - 1)yo and y = Z t a n ~. From this we obtain: y~ = - y ~ c o t a . t a n ( 6 +2~) + ( c o t ~ . t a n ( f i + 2 ~ ) - 1)yo or: cot e . t a n (6 + 2 e ) - 1 >'~-l+cot~.tan(f+2e) Y°
sin(6 + ~) sin(6+3~)Y°
(6)
Thus the equation of the ray after two reflections (i.e. eqn. (5)) takes the form: y = Z t a n ( f i +4c0 +yo[1 - C o t s . tan (6 +4~)] sin(6 + ~) sin (6 + 3~)
(7)
311
EFFECT OF MULTIPLE REFLECTIONS
Similarly, by finding the point of intersection (Yz, Zi), in terms ofyo, we write the equation of the ray after three reflections as: y=-Ztan(6+6ct)
y ° s i n ( 6 + c t ) [1 - cot ct. tan (6 + &t)] sin (6 + 5~)
(8)
The equation, after n reflections, is: sin (6 + ~) Ztan(2n~+5)+Yosin(6+(2n_l)~)
y-(-1)"
] [1 - cot ct. tan (6 + 2n~)] /
(9) Let n be odd or even according to the condition that the incident ray after n reflections passes tangentially to the absorber surface: i.e., B
c°t~'tan(6+2mt)+y°
sin (6 + ct) sin[6 + ( 2 n - 1]~] [1 - c o t , . t a n ( 6
[12 + tan 2 (6 + 2n~)] 1/2
+2n~)]
B -2 (10)
We obtain: A -
cos ~. sin (6 + 2n~) + sin sin(6 + ct)
(11)
For p 4: 1. Fig. l(b) depicts schematically the reflection process. Rays entering the aperture zone I undergo reflection before reaching the base. In zone II they undergo two reflections, etc. If D 1, D z, D, are the aperture widths required to undergo one reflection, two reflections and n reflections, respectively, we can write the concentration as: C-
A nfl
1 n
Oo
Oo (12)
1
=-[p(C~-l)+p2(C 2-C l)+...+p"(C.-C. 1)+1] where
(13)
312
AMAN D A N G , J. K. S H A R M A , H. P. G A R G
For a single reflection and p = 1 the above expression reduces to our former expression. 6 Since R = [(A - B)/2 sin ~)], one obtains: R -
sin c~+ cos ~. sin (6 + 2m0 - sin (3 + c0 2sin ~. sin (3 + ~)
(14)
Equation (7) can also be written as: C-
1 =2 (R)sin~
(15)
C O L L E C T O R EFFICIENC1ES A N D E N E R G Y B A L A N C E
The efficiency and the equilibrium temperature are determined by solving an energy balance equation for the system as a whole. The energy input to the absorber is due to the incident solar flux. This energy input can be written as: G cos 7s~t's~d
(16)
Since, in equilibrium, this total energy input must be equal to the total energy output, then: , t 4 GCOSTsZCtsA = eBaT~A ~ + hAB(TB - To) + qtA
(17)
or:
qt
r/ . . . . G cos Ys
r~
dn'rT4An Aq~ cos 7s
hAB(Tn- To) AG cos 7s
(18)
So, for a given set of values of the parameters A, A B, G, T~, h, e~ and ~ , the efficiency can be determined from eqn. (18). Values for G, ?s and To~ are set by the local geographical and weather conditions. The results are given here in the form of graphs (Figs 4(a) to (d)). RESULTS A N D DISCUSSION
Figure 2(a) and (b) show R/B--the slant height, R, divided by the width, B, of the b a s e - - a s a function of ~ for troughs for which the acceptance angles are 3 = 1 and 9 degrees, respectively. These are the plots ofeqn. (14). When a concentration factor and associated values of R/B are chosen, these curves give associated values of ct for a given number n. The dotted lines are for R/B, using a plane absorber at the base, while the continuous lines are for a tubular absorber at the base. The value of R/B increases with an increase in the number of reflections which occur. This is true for 3 = 1 as well as for 6 = 9 degrees. Further, the value of R/B decreases with an increase in 3. This is also true for N = 1,2 and 3. It is easy to infer from Figs 2(a) and
313
EFFECT OF M U L T I P L E R E F L E C T I O N S
I
uo z
° °
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I
I 8//a
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~ ~
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o
~'~
~,~2
(a)
0
oL_
/
/
/
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d
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12
N:7
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/
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IB R/B
-- N=2
~N=3
N:!
N=2
24
N=3
30
: ! degrees
36
1,0
2.0
3,0
L 0 (b)
i/
/
[ 6
i///
//
N:2
N=I
I f/l-------
/
R}B
I 18
I 2~
N=I
S : I degrees 0..=.8
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I 36
1.0
2.o
3.0
> ~o C~
?
> ~o
> Z
>
>
L~J
6
I 12
1
k,~N-2
"\\\~ \\
R/B
--
~8
I
~N.3
24
I 30
1.0
o
2
3
(d)
6
I
\N:I
12
I
R/B
18
I
6= 9 degrees 0.=. 8
24
I
30
1.0
2.0
p = 0.8).
Fig. 3. (a) C o n c e n t r a t i o n factor versus R/B for different n u m b e r s o f reflections (6 = 1 degree, p = 1). T h e right-hand side scale c o r r e s p o n d s to a tubular absorber. (b) C o n c e n t r a t i o n factor versus R/B for different n u m b e r s of reflections (5 = 1 degree, p = 0.8). (c) C o n c e n t r a t i o n factor versus R/B for different n u m b e r s o f reflections (6 = 9 degrees, p = I). (d) C o n c e n t r a t i o n factor versus RIB for different n u m b e r s o f reflections (0 = 9 aegrees,
(c)
/
2.0
z
¢3 ,.-]
t--
t--"
r--' .-4
~e
©
-tq
316
AMAN DANG, J. K. SHARMA, H. P. GARG
2(b) that R/B for N = 3 is higher than for N = 1 or 2 corresponding to similar values of acceptance angle, tS. Figure 3(a) shows the variations of the concentration with RIB for different values of RIB for f i = l d e g r e e , p = l and N = 1 , 2 , 3 . The concentrations are higher for higher values of N. This is due to the fact that, for a higher N (i.e. number of reflections), the input aperture is greater. Further, there is an optimum value of R/B at which the concentration is a maximum. The continuous line and right-hand scale are for a tubular absorber. The dashed line and left-hand side are for a plane absorber. For a fixed value of R/B the concentration for a plane absorber at the base is higher. This is the expected result as the plane surface area is less than that of the tubular one. Hence, the concentration is higher for a plane absorber. Figure 3(b) shows the effect of the reflective coefficient on the concentration. The nature of the curves is similar to that of Fig. 3(a), but the concentration is less because of the reflectivity (p = 0.8). Figure 3(c) shows the variation of the concentration with R/B for N = 1,2, 3 (p = 1, 6 = 9 degrees). This is the counterpart of Fig. 3(a) with the acceptance angle, 6, now equal to 9 degrees. The concentrations are lower in comparison with Fig. 3(a), but choosing 6 = 9degrees minimises the tilting routine. Again, for a fixed value of R/B the concentrations are higher for a plane absorber. Further, there is an optimum value of R/B for each N = 1,2, 3 at which the concentrations are a maximum. Figure 3(d) is similar to Fig. 3(c) except that there is a difference in the coefficient of reflectivity (p =0.8). The nature of the curves is the same as that for Fig. 3(c). The concentrations are less in comparison with Fig. 3(c) because of the reflectivity. In Fig. 4(a) the variation of the collector's efficiency (6 = l degree, p = 1) with ( T ~ - T~) is shown taking into account the convective and radiative losses. The efficiency is higher for a higher value ofn. This is because of the fact that the value of the concentration is higher for higher n--i.e, corresponding to less absorber surface and small losses. Figure 4(b) is similar to Fig. 4(a) except that 6 = 9 degrees: the efficiencies are smaller than for Fig. 4(a) but increase with an increase in the number of reflections, as in Fig. 4(a). In Fig. 4(c) we have shown the effect of h on the efficiency of the collector. The efficiencies decrease with an increase in the value of the convective heat transfer coefficient. The increase in convective heat transfer causes an increase in the thermal losses, thus less energy is used and hence the efficiency (r/) is lower. In Fig. 4(d) the effect of the coefficient of reflectivity is shown. The efficiency is lower for a lower coefficient of reflectivity, as is expected.
ACKNOWLEDGEMENTS
The present work is part of the programme covering solar concentrators being carried out at the Centre of Energy Studies, IIT, New Delhi, India. The authors are thankful to Professor M. S. Sodha and Professor S. S. Mathur for encouragement and guidance.
o
0
1oo
(Q)
(To (b)
- T ~)*C
200
EFFECT O{ N)
,
I 200 ( TB - T ~ )*C
I 100
N=I
I
1 200
300
o
.9
o
0
X~
100
I
1oo "C
(d)
( T8 -Too ) *c
OF R)
200
N=])
I
200
(EFFECT
(¢)
( TB - T ~ }
h=I0 ~ h = B
h= O'tW m - 2 " C - I
300
I
I
300
Fig. 4. (a) Variation o f the collector efficiency with ( T B - T , ) for different n u m b e r s of reflections (6 = 1 degree, p = 1). (b) V a r i a t i o n s of the c o l l e c t o r efficiency with ( T B - T , ) for different n u m b e r s of reflections (6 = 9 degrees, p = 1). (c) V a r i a t i o n s of the collector efficiency with ( T a - To~) for different values of heat transfer coefficient (6 = 9 degrees, p = 1 a n d N = 3). (d) V a r i a t i o n s of the collector efficiency with ( T B - T~) for different values o f coefficient of reflectivity (6 = 9 degrees, N = 3).
qq
N=3
EFFECT OF N )
t'rl
---.I
Z
t-
t" m
t" -4
~S
.-4 ©
318
AMAN DANG, J. K. SHARMA, H. P. GARG REFERENCES
1. H. TABOR, Stationary mirror systems for solar collectors, Solar Energy, 2 (1958), p. 27. 2. K. G. T. HOLLAND, A concentrator for thin film solar cells, Solar Energy, 13 (1971), p. 149. 3. K.D. MANNANand R. B. BANNEROT,Optimal geometries for one and two faceted symmetric sidewall booster mirrors, Solar Energy, 21 0978), p. 385. 4. R.B. BANNEROTand J. R. HOWELL, Moderately concentrating flat plate solar energy collectors. Paper presented at the AIChe-ASME Heat Transfer Conference, San Francisco, California, August, 1975. 5. DONALDG. BURKHARD,GEORGE L. STROBELand DARYL L. BURKHARD,Flat sided rectilinear trough as a solar concentrator, Applied Optics, 17 (1978), p. 1870. 6. H. P. GARG, J. K. SHARMAand A. DANG, Optimal geometries of plane mirror solar collectors, International Journal of Energy Research (1982). (In press.)
EFFECT OF MULTIPLE REFLECTIONS PERFORMANCE O F P L A N E BOOSTER
ON THE
MIRRORS
A~IAy DANG, J. K. SnARMA and H. P. GARG
Centre of Energy Studies, Indian Institute of Technology, Haus Khas, New Delhi 110016 (India)
SUMMARY
This paper presents an analytical study of a stationao, V-trough concentrator. It consists of an array of east-west oriented trapezoidal channels with two side reflecting walls and a tubular absorber as a receiver at the base. The jormulae Jbr concentration factor and reflector surfaces hat'e been derired. It is explicitly shown that the concentration ratio and reflector surJace area depend upon the number ~[ reflections the solar rays undergo bejbre reaching the absorber, the cone apex angle, the coefficient of reflection and the acceptance angle. Results are presented graphically in such a way that one can choose the optimum configuration and the minimum material required to achiet'e a given concentration factor. The concentration ratio ranges from 1"2 to 3"6. The t'ariations of the collector's efficiem3' with temperature d(fference for different numbers of reflections, acceptance angles, cont,ective heat transfer coefficients and coe[flcients of re[tectit'ity have been predicted.
NOMENCLATURE
A B C R h
Collector aperture. Absorber base width. Concentration ratio. Reflector wall slant height. Effective overall convective heat transfer coefficient for the collector evaluated assuming the reference area is that of the base plate area. n Number of reflections. q.~ Direct solar flux. TB Base absorber temperature. Tj Ambient air temperature. 307 Applied Energy 0306-2619/82/0011-0307/$02.75 ~ Applied Science Publishers Ltd, England, t982 Printed in Great Britain
308 Ys 0ts r tr e~
AMAN DANG, J. K. SHARMA, H. P. GARG
Solar angle measured from the normal to the collector surface. Effective solar absorptivity of the absorber surface. Transmittance of the cover plate assembly. Stefan Boltzmann constant. Effective emittance of the absorber plate. Wall angle. Acceptance angle of the collector.
INTRODUCTION
The utilisation of solar energy for various applications such as solar cooling and air conditioning, power generation, the cooking of food, etc., requires heat at temperatures above = 90 °C. Conventional flat plate collectors cannot effectively supply heat at such temperatures. To provide heat at higher temperatures, concentration of the solar energy is needed. Various types of tracking and nontracking concentrator such as groove-like geometries (trough type), Fresnel lenses, Winston Collectors, paraboloids, parabolic troughs, segmented mirrors, centraltower heliostat arrays and spherical mirrors are used. Of course, the incorporation of curved surfaces into the design, and tracking of a concentrator, improves its performance to a certain degree but there are certain limitations of fabrication, handling and materials. Moderate concentration with fiat side booster mirrors is a technique to improve the collector's performance without tracking. To keep the incident solar radiations within the acceptance angle, the system can be seasonally tilted and adjusted. The V-trough concentrator has a number of advantages over other types of concentrator. It has the advantage of extreme ease of manufacture: it can, for example, be fabricated simply by folding a reflective aluminium sheet to the required opening angle and attaching a tubular absorber at the base. In the present paper we have studied the design parameters of a reflectorconcentrator with a tubular absorber at the base. The previous analyses of Tabor, Holland 2 and Mannan and Bannerot 3 are limited to single-beam reflection with flat absorbers. Bannerot and Howell 4 have calculated the collector efficiencies for plane absorbers taking single reflections into account. Burkhard et al. 5 have carried out an analysis for flat absorbers taking multiple reflections into account. A recent analysis by Garg e t al. 6 considers the tubular absorber as a receiver, but for single reflections only. In the present case we have found that the concentration ratio C is an optimum for a particular value of reflector surface area, R B . The concentration ratio C is higher for higher numbers of reflections. As expected, for a higher number of reflections to take place, the length of the reflection surface--and hence the input aperture--increases. However, the effect of the coefficient of reflection is to decrease the concentration. The solar radiation impinging on the reflected mirrors at the
309
EFFECT OF MULTIPLE REFLECTIONS
~1 Z
(aJ
(Yo,Zo)~!
[b) ---- _----
102
~'~(O,~-Cot ~) \
Fig. l(a),(b).
L /
,,!/
Geometric design parameters showing a multiple reflected ray path for an acceptance angle, 6.
acceptance angle, 6, is considered to be received tangentially to the absorber surface, as shown in Fig. 1. All other solar radiations incident at the reflectors within the acceptance angle, 6, are received by the tubular absorber.
DERIVATION OF THE RAY TRACE EQUATION
(for p = 1)
As Shown in Fig. l(a), we consider that the solar radiations are incident at an angle, 6, with the symmetry axis of the trough. The apex of the trough is considered as the origin of the Cartesian co-ordinate system. The co-ordinates of the original incident ray are ( -Yo, Zo), where Yo = (A/2), A being the entrance aperture. The equation of the incident ray making an angle (90 ° - 6) with the horizontal and passing through ( -Yo, Zo) is written as: Z -
Z o -- m ( y
+Yo)
(1)
where: dZ m = ~ - = tan (qo - 6) = cot 6 oy Thus : Y = - Yo + (Z - Z o) tan 6
(2)
310
A M A N D A N G , J. K. S H A R M A , H. P. G A R G
The equation of the side of the trough is: y = + Z t a n ct
(3(a))
so that: Yo =
+
Zo tan
(3(b))
The equation of the ray after one reflection is: Z-
dZ
m =-- = - - t a n (qo - (6 + 2~)) dy
Z o = m ( y + Yo)
or: Y = -Yo + ( Z - Z o ) t a n ( - 0 )
(4)
Here 0 is the angle which the reflected ray makes with the y axis. The equation of the ray after two reflections is: Z-Z
l=m(y-y
1)
dZ m=~y=COt(6+4c0
or"
y =y~ + ( Z - Z 1 ) t a n ( 6 +4ct)
(5)
where: Z1 --'Yl c o t s
Here (y~, Z~) is the point of intersection o f y = -Yo - (Z - Zo) tan(fi + 2~) and: y = Z tan i.e. o f y = - Z t a n (fi + 2~) + (cot a. tan (6 + 2ct) - 1)yo and y = Z t a n ~. From this we obtain: y~ = - y ~ c o t a . t a n ( 6 +2~) + ( c o t ~ . t a n ( f i + 2 ~ ) - 1)yo or: cot e . t a n (6 + 2 e ) - 1 >'~-l+cot~.tan(f+2e) Y°
sin(6 + ~) sin(6+3~)Y°
(6)
Thus the equation of the ray after two reflections (i.e. eqn. (5)) takes the form: y = Z t a n ( f i +4c0 +yo[1 - C o t s . tan (6 +4~)] sin(6 + ~) sin (6 + 3~)
(7)
311
EFFECT OF MULTIPLE REFLECTIONS
Similarly, by finding the point of intersection (Yz, Zi), in terms ofyo, we write the equation of the ray after three reflections as: y=-Ztan(6+6ct)
y ° s i n ( 6 + c t ) [1 - cot ct. tan (6 + &t)] sin (6 + 5~)
(8)
The equation, after n reflections, is: sin (6 + ~) Ztan(2n~+5)+Yosin(6+(2n_l)~)
y-(-1)"
] [1 - cot ct. tan (6 + 2n~)] /
(9) Let n be odd or even according to the condition that the incident ray after n reflections passes tangentially to the absorber surface: i.e., B
c°t~'tan(6+2mt)+y°
sin (6 + ct) sin[6 + ( 2 n - 1]~] [1 - c o t , . t a n ( 6
[12 + tan 2 (6 + 2n~)] 1/2
+2n~)]
B -2 (10)
We obtain: A -
cos ~. sin (6 + 2n~) + sin sin(6 + ct)
(11)
For p 4: 1. Fig. l(b) depicts schematically the reflection process. Rays entering the aperture zone I undergo reflection before reaching the base. In zone II they undergo two reflections, etc. If D 1, D z, D, are the aperture widths required to undergo one reflection, two reflections and n reflections, respectively, we can write the concentration as: C-
A nfl
1 n
Oo
Oo (12)
1
=-[p(C~-l)+p2(C 2-C l)+...+p"(C.-C. 1)+1] where
(13)
312
AMAN D A N G , J. K. S H A R M A , H. P. G A R G
For a single reflection and p = 1 the above expression reduces to our former expression. 6 Since R = [(A - B)/2 sin ~)], one obtains: R -
sin c~+ cos ~. sin (6 + 2m0 - sin (3 + c0 2sin ~. sin (3 + ~)
(14)
Equation (7) can also be written as: C-
1 =2 (R)sin~
(15)
C O L L E C T O R EFFICIENC1ES A N D E N E R G Y B A L A N C E
The efficiency and the equilibrium temperature are determined by solving an energy balance equation for the system as a whole. The energy input to the absorber is due to the incident solar flux. This energy input can be written as: G cos 7s~t's~d
(16)
Since, in equilibrium, this total energy input must be equal to the total energy output, then: , t 4 GCOSTsZCtsA = eBaT~A ~ + hAB(TB - To) + qtA
(17)
or:
qt
r/ . . . . G cos Ys
r~
dn'rT4An Aq~ cos 7s
hAB(Tn- To) AG cos 7s
(18)
So, for a given set of values of the parameters A, A B, G, T~, h, e~ and ~ , the efficiency can be determined from eqn. (18). Values for G, ?s and To~ are set by the local geographical and weather conditions. The results are given here in the form of graphs (Figs 4(a) to (d)). RESULTS A N D DISCUSSION
Figure 2(a) and (b) show R/B--the slant height, R, divided by the width, B, of the b a s e - - a s a function of ~ for troughs for which the acceptance angles are 3 = 1 and 9 degrees, respectively. These are the plots ofeqn. (14). When a concentration factor and associated values of R/B are chosen, these curves give associated values of ct for a given number n. The dotted lines are for R/B, using a plane absorber at the base, while the continuous lines are for a tubular absorber at the base. The value of R/B increases with an increase in the number of reflections which occur. This is true for 3 = 1 as well as for 6 = 9 degrees. Further, the value of R/B decreases with an increase in 3. This is also true for N = 1,2 and 3. It is easy to infer from Figs 2(a) and
313
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2(b) that R/B for N = 3 is higher than for N = 1 or 2 corresponding to similar values of acceptance angle, tS. Figure 3(a) shows the variations of the concentration with RIB for different values of RIB for f i = l d e g r e e , p = l and N = 1 , 2 , 3 . The concentrations are higher for higher values of N. This is due to the fact that, for a higher N (i.e. number of reflections), the input aperture is greater. Further, there is an optimum value of R/B at which the concentration is a maximum. The continuous line and right-hand scale are for a tubular absorber. The dashed line and left-hand side are for a plane absorber. For a fixed value of R/B the concentration for a plane absorber at the base is higher. This is the expected result as the plane surface area is less than that of the tubular one. Hence, the concentration is higher for a plane absorber. Figure 3(b) shows the effect of the reflective coefficient on the concentration. The nature of the curves is similar to that of Fig. 3(a), but the concentration is less because of the reflectivity (p = 0.8). Figure 3(c) shows the variation of the concentration with R/B for N = 1,2, 3 (p = 1, 6 = 9 degrees). This is the counterpart of Fig. 3(a) with the acceptance angle, 6, now equal to 9 degrees. The concentrations are lower in comparison with Fig. 3(a), but choosing 6 = 9degrees minimises the tilting routine. Again, for a fixed value of R/B the concentrations are higher for a plane absorber. Further, there is an optimum value of R/B for each N = 1,2, 3 at which the concentrations are a maximum. Figure 3(d) is similar to Fig. 3(c) except that there is a difference in the coefficient of reflectivity (p =0.8). The nature of the curves is the same as that for Fig. 3(c). The concentrations are less in comparison with Fig. 3(c) because of the reflectivity. In Fig. 4(a) the variation of the collector's efficiency (6 = l degree, p = 1) with ( T ~ - T~) is shown taking into account the convective and radiative losses. The efficiency is higher for a higher value ofn. This is because of the fact that the value of the concentration is higher for higher n--i.e, corresponding to less absorber surface and small losses. Figure 4(b) is similar to Fig. 4(a) except that 6 = 9 degrees: the efficiencies are smaller than for Fig. 4(a) but increase with an increase in the number of reflections, as in Fig. 4(a). In Fig. 4(c) we have shown the effect of h on the efficiency of the collector. The efficiencies decrease with an increase in the value of the convective heat transfer coefficient. The increase in convective heat transfer causes an increase in the thermal losses, thus less energy is used and hence the efficiency (r/) is lower. In Fig. 4(d) the effect of the coefficient of reflectivity is shown. The efficiency is lower for a lower coefficient of reflectivity, as is expected.
ACKNOWLEDGEMENTS
The present work is part of the programme covering solar concentrators being carried out at the Centre of Energy Studies, IIT, New Delhi, India. The authors are thankful to Professor M. S. Sodha and Professor S. S. Mathur for encouragement and guidance.
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AMAN DANG, J. K. SHARMA, H. P. GARG REFERENCES
1. H. TABOR, Stationary mirror systems for solar collectors, Solar Energy, 2 (1958), p. 27. 2. K. G. T. HOLLAND, A concentrator for thin film solar cells, Solar Energy, 13 (1971), p. 149. 3. K.D. MANNANand R. B. BANNEROT,Optimal geometries for one and two faceted symmetric sidewall booster mirrors, Solar Energy, 21 0978), p. 385. 4. R.B. BANNEROTand J. R. HOWELL, Moderately concentrating flat plate solar energy collectors. Paper presented at the AIChe-ASME Heat Transfer Conference, San Francisco, California, August, 1975. 5. DONALDG. BURKHARD,GEORGE L. STROBELand DARYL L. BURKHARD,Flat sided rectilinear trough as a solar concentrator, Applied Optics, 17 (1978), p. 1870. 6. H. P. GARG, J. K. SHARMAand A. DANG, Optimal geometries of plane mirror solar collectors, International Journal of Energy Research (1982). (In press.)