Uniform sunlight concentration reflectors for photovoltaic cells Rabi Ibrahim Rabady Electrical Engineering Department, Jordan University of Science and Technology, P.O. Box 3030, Irbid 22110, Jordan ([email protected]) Received 17 December 2013; revised 28 January 2014; accepted 12 February 2014; posted 13 February 2014 (Doc. ID 203096); published 19 March 2014
Sunlight concentration is essential to reach high temperatures of a working fluid in solar-thermal applications and to reduce the cost of photovoltaic (PV) electricity generation systems. Commonly, sunlight concentration is realized by parabolic or cylindrical reflectors, which do not provide uniform concentration on the receiver finite surface. Uniform concentration of sunlight is favored especially for the PV conversion applications since it not only enhances the conversion efficiency of sunlight but also reduces the thermal variations along the receiving PV cell, which can be a performance and life-span limiting factor. In this paper a reflector profile that uniformly infiltrates the concentrated sunlight into the receiving unit is attempted. The new design accounts for all factors that contribute to the nonuniform concentration, like the reflector curvature, which spatially reflects the sunlight nonuniformly, and the angular dependency of both the reflector reflectivity and the sunlight transmission through the PV cell. © 2014 Optical Society of America OCIS codes: (350.4600) Optical engineering; (080.4035) Mirror system design; (080.4228) Nonspherical mirror surfaces; (080.4295) Nonimaging optical systems; (080.4298) Nonimaging optics. http://dx.doi.org/10.1364/AO.53.001862
1. Introduction
The cost of the photovoltaic (PV) cells can be compromised by concentrating the sunlight using special reflectors before the PV conversion. Therefore, sunlight-reflecting concentrators can be useful components in order to reduce the cost of the PV solar systems. Nevertheless, some limitations and practical issues are encountered when applying sunlight concentration on a PV cell receiver. Usually a minimum concentration ratio needs to be accomplished in order to justify the additional cost that comes from using reflectors with suitable stand, sun-tracker system, and cooling system. Usually, the size and complexity, and therefore the cost, of the auxiliary systems increase with the concentration level. Moreover, using the common parabolic or cylindrical concentrators leads to a nonuniform illumination of the concentrated sunlight on the receiving finite PV 1559-128X/14/091862-07$15.00/0 © 2014 Optical Society of America 1862
APPLIED OPTICS / Vol. 53, No. 9 / 20 March 2014
cell, which can limit the conversion efficiency of the solar system [1–4]. For instance, a considerable amount of the solar energy is converted to heat energy instead of electricity in most available PV cells, which causes, if not handled, performance and life-span issues. The heating problem in the PV cell becomes even more prominent in the nonuniform sunlight concentration regime. The uneven heating of the PV cell leads to uneven temperature distribution inside the PV cell; this may cause internal thermal stress that could degrade the PV cell’s overall performance. Moreover, the uneven infiltration of the concentrated sunlight into the PV cell receiver causes uneven generation for the photocarriers throughout the PV absorbing medium of the cell, which may reduce the net photocurrent because of the greater internal recombination rate of photogenerated charges (i.e., the electrons and holes) because of the lateral diffusion of the unevenconcentration-free-charges. In this work, we attempt designing a reflecting concentrator profile that handles the uneven feeding
problem into a finite size PV cell. To achieve this goal we examine all factors that associate with the sunlight reflections and cause the uneven infiltration of the concentrated sunlight into the PV cell feeding surface. The previous works that treated this problem were not realistic because they accounted only for the uneven concentration, which is associated with the uneven spatial reflection that is caused by the curvature of the reflector [5–8], whereas other factors like the angular dependencies of both the sunlight reflectivity at the air–reflector interface and sunlight transmittance at the air–receiver interface were not accounted for. 2. Theory
In order to realize high-performance, uniformconcentration reflectors, there is more than one factor that needs be accounted for in concentrating the uniform flux of sunlight into a finite PV cell receiver. First is the angular uneven reflection of the uniform sunlight from the reflector curved surface because of the geometry effect on the one hand and the Fresnel reflection angular dependency on the other hand. Second is the concentrated sunlight transmittance angular dependency through the PV cell feeding surface since the sunlight will reach the receiver surface at various angles. Hence, the problem is to find the reflector profile yx such that the uniform flux of the incident sunlight is reflected, concentrated, and fed uniformly into the flat PV cell receiver that is oriented parallel to the incident sunlight as depicted in Fig. 1(a). The reflector curvature is responsible for the incidence of the uniform sunlight at various angles along the reflector surface, which not only leads to spatially different reflected angles of the sunlight rays toward the receiver but also causes different reflected power because of the Fresnel reflection angular dependency. Referring to Fig. 1(a), the spatially varied reflecting angles effect is accounted for by tracing a sunlight ray that reaches the vertical PV cell receiver at point 0; h upon reflecting off the reflector at point x; yx. Therefore, considering the triangle that comes between the three points x; y; x; h, and 0; h in Fig. 1(a), the tangent of the angle of this triangle at point x; y can be expressed in two ways: 2 tanθ 2y0 ; tan2θ 2 1 − tan θ 1 − y02
x ; h−y
(1)
Fig. 1. (a) Ray tracing of sunlight concentration and infiltration into a vertical PV cell receiver. (b) Ray tracing of sunlight incident at the starting point xo ; yo of the reflector.
hx; y y x
1 − y02 : 2y0
(4)
Next we consider the reflectivity of the unpolarized sunlight that strikes a medium with refractive index N 1 n1 jκ1 at an incident angle θ, which can be obtained from the following expression: 1 sinθ − ϕ 2 tanθ − ϕ 2 tanθ ϕ ; 2 sinθ ϕ
(5)
where ϕ is the angle of the transmitted sunlight and is related to θ by Snell’s law: (2)
where y0 tanθ:
(b)
Rθ
and tan2θ
(a)
(3)
Equating the right sides of Eqs. (1) and (2) and rearranging leads to
sinθ N 1 sinϕ:
(6)
Similarly, the sunlight transmission into the PV cell receiver, which has refractive index N 2 n2 jκ2 with incident angle α, is obtained by: 1 sinα − β 2 tanα − β 2 ; Tα 1 − 2 sinα β tanα β 20 March 2014 / Vol. 53, No. 9 / APPLIED OPTICS
(7)
1863
where β is the angle of the transmitted sunlight and is related to α by Snell’s law:
(a)
220 215 190
0.5
(8)
In order to obtain uniform sunlight feeding into the PV cell receiver, the concentration ratio should equal
140
y(x) (cm)
sinα N 2 sinβ:
Reflector profile when angular dependencies are accounted
115 90 65
1 dx : C RθTα dh
Profile difference when angular dependencies are not accounted (R=0.6130, T=0.6515)
40
(9)
dh 1 − y02 d 1 − y02 y0 x ; 2y0 2y0 dx dx
0
15 -10 10
(b)
20
30
40
50
60
70
80
90
100
110
x (cm)
45
Angular dependencies are not accounted (R=0.613, T=0.6515)
Sunlight infiltration (suns)
Therefore, the task is to find the reflector profile yx such that the right side of Eq. (9) is independent of x and equals the concentration ratio C. In order to do so, the three terms in the right side of Eq. (9) should be expressed in terms of yx and/ or y0 x. Differentiating Eq. (4) with respect to x yields the following:
profiles difference (cm)
165
40 Angular dependencies are accounted
35
(10) 30 0
which can be reduced to
(c)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
70
80
90
100
110
0.6
0.7
0.8
0.9
1
h (cm)
0.66 0.65
dh y02 1y0 − xy00 : dx 2y02
transmittance
0.64
(11)
0.63 0.62
" # tanϕ 2 1 − tanθ tanϕ 2 1 1 − tanθ Rθ : 1 1 tanθ tanϕ 2 1 tanϕ tanθ (12) Also, using Eqs. (3) and (6) we could express tanϕ as y0 tanϕ q : N 21 − 1y02 N 21
(13)
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APPLIED OPTICS / Vol. 53, No. 9 / 20 March 2014
0.6 Reflectivity
0.59 0.58 0.57 10
20
30
40
50
60
x (cm)
(d) 40 Sunlight divergence is accounted
39.5
Sunlight divergence is not accounted
39
38.5
38 0
0.1
0.2
0.3
0.4
0.5
h (cm)
Substituting Eqs. (3) and (13) in Eq. (12) yields 2 q N 2 − 1y02 N 2 − 1 2 1 1 1 Ry0 4 q 2 N 2 − 1y02 N 2 1 1 1 q 13 0 N 2 − 1y02 N 2 − y02 2 1 1 A5: × @1 q N 2 − 1y02 N 2 y02 1 1
0.61
Sunlight infiltration (suns)
Obviously, Eq. (11) presents a nonlinear function in terms of the first and second derivatives of yx. Next the terms Rθ and Tα should be expressed in terms of the function yx and/or its higher derivatives in order to homogenize the three components in the right side of Eq. (9); therefore, we could construct a differential equation for the target function yx. To accomplish this, Eq. (5) can be rewritten as follows:
(14)
Fig. 2. (a) 100 cm wide stainless steel reflector surface profile with uniform sunlight infiltration into silicon PV cell receiver (left scale) and the reflector profile difference when the angular dependences are ignored with R 0.613 and T 0.6515 (right scale) (C 40, xo 10, yo −10 cm). (b) Effect of reflection and transmittance angular dependencies on the sunlight infiltration into the PV silicon cell with R 0.613 and T 0.6515 for the case when angular dependencies are not accounted (C 40, xo 10, yo −10 cm). (c) Corresponding stainless steel reflectivity and transmittance through the silicon PV cell receiver (C 40, xo 10, yo −10 cm). (d) Effect of sunlight divergence (0.26°) on the sunlight infiltration into the PV silicon cell (C 40, xo 10, yo −10 cm).
(a)
250
Similarly, Eq. (7) can be rewritten as
200
2 2 3 tanβ 1 − tanα tanβ 2 1 4 1 − tanα 5: 1 Tα 1 − tanβ 2 1 tanα 1 tanα tanβ
y(x) (cm)
150 yo= -10 cm
100
50
(15)
yo= -40 cm
0 yo= -70 cm
-50
-100 10
20
30
40
50
60
70
80
90
100
110
x (cm)
Again, referring to the triangle that comes between the three points x; y; x; h, and 0; h in Fig. 1(a), we could write
(b) 0.62 yo= -70 cm
tanα
0.61 yo= -40 cm
1 1 − y02 : 2y0 tan2θ
(16)
0.6
Additionally, manipulating Eqs. (8) and (16) leads to R 0.59
1 − y02 tanβ q : N 22 1 y02 2 − 1 − y02 2
yo= -10 cm
0.58
0.57
10
20
30
40
50
60
70
80
90
100
110
(17)
Substituting Eqs. (16) and (17) into Eq. (15) yields
x (cm)
(c)
0.66
Ty0
yo= -10 cm
2 q N 2 1 y02 2 − 1 − y02 2 − 2y0 2 16 2 q 1− 6 4 2 N 2 1 y02 2 − 1 − y02 2 2y0 2
yo= -40 cm
0.64 yo= -70 cm
0.62
0.6
T
3 q 1 2y0 N 2 1 y02 2 − 1 − y02 2 − 1 − y02 2 2 7 2 A7 : × @1 q 5 0 2 02 2 02 2 02 2 2y N 2 1 y − 1 − y 1 − y
0.58
0
0.56
0.54 0.52 10
20
30
40
50
60
70
80
90
100
110
(18)
x (cm)
(d) 40
yo= -10 cm
Substituting Eq. (11) into Eq. (9) and rearranging for the sake of y00 yields
Sunlight infiltration (suns)
39 yo= -40 cm
38
yo= -70 cm
y00
37
36
(19)
This can be reduced to a first-order nonlinear differential equation by substituting
35
34 0
y0 y02 1Ry0 Ty0 C − 2y02 : xy02 1Ry0 Ty0 C
0.2
0.4
0.6
0.8
g y0 ;
1
(20)
h (cm)
Fig. 3. (a) Surface profiles of a 100 cm wide stainless steel reflector at three different relative positions of the receiver with respect to the reflector: yo −10 cm, yo −40, and yo −70 (C 40, xo 10). (b) Corresponding reflectivity along the stainless steel reflector with the three different relative positions of the receiver with respect to the reflector: yo −10 cm, yo −40, and yo −70 (C 40, xo 10). (c) Corresponding transmittance through the PV silicon receiver with the three different relative positions of the receiver with respect to the reflector: yo −10 cm, yo −40, and yo −70 (C 40, xo 10). (d) Corresponding sunlight infiltration into the PV silicon receiver with the three different relative positions of the receiver with respect to the reflector: yo −10 cm, yo −40, and yo −70 (C 40, xo 10).
which leads to dg gg2 1RgTgC − 2g2 ; f x; g dx xg2 1RgTgC
(21)
where Rg and Tg are obtained from Eqs. (14) and (18), respectively, with the substitution y0 g, where gx is obtained from integrating Eq. (21) with an initial point (xo , gxo ), which is obtained, with reference to Fig. 1(b), from expressing tan2θ in two different ways as follows: 20 March 2014 / Vol. 53, No. 9 / APPLIED OPTICS
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xo 2gxo tan2θ : yo 1 − gxo 2
(22)
Accounting for the left and the right sides, Eq. (22) presents a quadratic equation with respect to gxo and has a useful solution with positive gxo as p y2o x2o yo : gxo go xo
(23)
Finally, yx could be found by integrating Eq. (20) with the initial point x xo ; y yo . 3. Simulations and Results
In this section some numerical examples are considered in order to verify the theory of producing a reflector profile that facilitates uniform infiltration of the concentrated sunlight into a finite PV cell receiver. The second-order Runge–Kutta, namely the midpoint method, was employed to solve the firstorder ordinary differential equation given by Eq. (21). Therefore, the differential Eq. (20) can be transferred using the midpoint method to the following set of difference equations: gi1 gi Δxf xi0.5 ; gi0.5 ;
xi0.5 xi
gi0.5 gi
f xi ; gi
Δx ; 2
Δx f xi ; gi ; 2
gi g2i 1Rgi Tgi C − 2g2i ; xi g2i 1Rgi Tgi C
(24a)
(24b)
(24c)
(24d)
where Δx is the step size. The initial go is found from substituting the point xo ; yo in Eq. (23), where the point xo ; yo identifies the bottom of the reflector relative to the bottom point of the vertical flat receiver as shown in Fig. 1. Given gx is determined numerically, which represents the slope of the reflector surface, the reflector profile surface yx can be determined by integrating gx with the initial point xo ; yo . In order to verify that the resulted reflector profile would yield the desired even sunlight infiltration profile into the receiver, the incident sunlight ray at point xi ; yxi on the reflector is traced every Δx to locate the incident point 0; yi at the receiver using Eq. (4). Additionally, the corresponding sunlight fraction that infiltrates the receiver, which is obtained by I i Ri T i, is also recorded. Therefore, the infiltrating sunlight concentration profile hi ; Ci can be obtained from yi ; I i as follows: 1866
APPLIED OPTICS / Vol. 53, No. 9 / 20 March 2014
hi
yi yi1 ; 2
Ci h
I i I i1 2
Δx
i; yi1 − yi
(25)
where Ci is intuitively estimated as the ratio between Δx and the trapezoidal area that comes between yi and yi1 under the curve that represents the data yi ; I i . Finally, it is worth mentioning that the sunlight divergence, which is about 0.26°, has minimal effect on the results; this is attributed to the fact we could apply the local linearity for small angles like 0.26° (about 4.6 mrad); therefore, the divergence around the incidence angle become symmetric and its effect cancels out; consequently, it leads to the same g y0 . However, in order to determine the actual sunlight infiltration profile because of the sunlight divergence, again, local linearity facilitates determining the resulting infiltration profile by averaging the three infiltration profiles that come from reflector slopes: g, g − 0.0023, and g 0.0023. The 0.0023 rad angle is the tangent of 0.13° which is half of the divergence angle since the reflector surface that is sloped at angle θ diverts the incident sunlight toward the receiver by 2θ. It is expected that such little divergence would even enhance the infiltration profile without altering the reflector surface profile since it acts as a narrow averaging window that yields a more flattened infiltration profile. In what follows, a numerical example that clarifies the above theory and discussions is considered. A. Examples
In this example we consider a stainless steel reflector and a silicon PV cell receiver with refractive indices as N 1 2.757 j3.792 and N 2 3.882 j0.019, respectively. Moreover, the reflector width is xf 100 cm, the concentration C equals 40, the initial point is (xo 10 cm, yo −10 cm), and the step size from the numerical method is Δx 0.01 cm. Substituting all these in Eqs. (23) and (24) yields the reflector profile that permits an even infiltration of the sunlight into the silicon PV cell receiver. Figure 2(a) shows the resulting reflector profile surface accounting for the angular dependencies of both the stainless steel reflectivity and the transmittance of the silicon PV cell in the right scale, whereas the curve that belongs to the left scale is the additive difference of the reflector profile when the stainless steel reflectivity and the silicon PV cell transmittance are not accounted for but are set to their normal incidence values, that is, R 0.613 and T 0.6515, respectively. Obviously, the difference between the two surfaces is relatively small; it is about 5 mm at its maximum value near the far edge of the reflector. Nevertheless, such a surface profile difference plays a major role in permitting even sunlight infiltration into the receiver, as is depicted in Fig. 2(b), which confirms that accounting for the angular dependencies provides a more even sunlight infiltration into the receiver. Obviously, the sunlight
(a) 250 200 C=1000
y(x) (cm)
150 C=200
100
C=40
50
0 10
20
30
40
50
60
70
80
90
100
110
x (cm)
(b) 900 800
Sunlight infiltration (suns)
700 C=1000
600 500 400 300
C=200
200 100 0 -0.2
C=40
0
0.2
0.4
0.6
0.8
1
h (cm)
Fig. 4. (a) Surface profiles of a 100 cm wide stainless steel reflector with three different concentration ratios: C 40, C 200, and C 1000 (xo 10, yo −10). (b) Corresponding sunlight infiltration into the PV silicon receiver with the three different concentration ratios: C 40, C 200, and C 1000. (xo 10, yo −10).
infiltration profile at the receiver is higher for the case R 0.613 and T 0.6515, which is attributed to the fact that the stainless steel surface reflectivity decreases notably below its maximum value 0.613 as sunlight reflects near the far edge of the reflector, as is depicted in Fig. 2(c), which shows the angular dependencies of stainless steel reflectivity and the transmittance of the silicon PV cell as sunlight encounters various incidence angles along the reflector and the receiver. The extremes that are shown in both curves of Fig. 3(c) are attributed to the sunlight, which is partially magnetically polarized, passing through the Brewster angle as the incidence angle varies across the reflector and the receiver surfaces. Figure 3(d) shows how the resulting sunlight infiltration profile is affected by the sunlight divergence, which is about 0.26°; obviously, the very small divergence of sunlight has very little impact on the result; it even helps flatten the profile by acting like a smoothing averaging window. Next, for the sake of comparison and better understanding how the relative positioning of the PV cell receiver with respect to the reflecting surface could affect the sunlight concentration and infiltration into the receiver, three different reflectors are considered with the same previous givens (i.e., C 40, stainless steel reflector, silicon PV cell that is oriented vertically and starts at the origin) but with different
starting points xo ; yo for the reflector. Figure 3(a) shows the resulting three reflecting surfaces: the upper surface profile belongs to a reflector that starts at xo 10 cm; yo −10 cm, and the middle surface profile belongs to a reflector that starts at (xo 10 cm, yo −40 cm), whereas the last surface profile belongs to a reflector that starts at xo 10 cm; yo −70 cm. Expectedly, the farther the receiver is from the reflector, the flatter is the resulting reflecting surface. Figure 3(b) shows the corresponding sunlight reflectivity of the stainless steel reflectors with the surface profiles shown in Fig. 3(a), and Fig. 3(c) shows the corresponding sunlight transmittance through the silicon PV cell using stainless steel reflectors with the surface profiles shown in Fig. 3(a). A tradeoff scenario is observed between the sunlight reflectivity and the sunlight transmittance as the incidence angle of sunlight at the receiver starting points varies. Figure 3(d) shows that the infiltration profile becomes more uniform when such angle becomes small, that is, for the reflector case when the starting point is xo 10 cm; yo −10 cm. This leads us to conclude that an optimal positioning of the receiver that permits maximum uniform infiltration of the concentrated sunlight into the receiver is a useful aspect that is worth further investigation, which can be a future research. Finally, as with most concentrators, our proposed reflector has limitations on the concentration ratio; this comes from the fact that the infiltration profile is not perfectly uniform, especially at the start, because of the low sunlight transmittance through the PV cell at sharp incidence angles, as depicted from Figs. 2(b) and 3(d). Therefore, the higher the concentration ratio, the more the corresponding reflector concentrates the sunlight at small values of h where the uneven infiltration is more prominent; consequently, it strays away from achieving even sunlight infiltration into the receiver as illustrated by Fig. 4. Three reflecting surface profiles with xo 10 cm; yo −10 cm and various concentration ratios (40, 200, and 1000) are shown in Fig. 4(a) with the corresponding sunlight infiltration profiles in Fig. 4(b). Obviously, examining Fig. 4(b) leads us to conclude that there is a trade-off between sunlight infiltration uniformity and the concentration ratio, which needs be accounted for before employing such a reflector design in practice. 4. Summary
A uniform concentration and infiltration of sunlight into the PV cells is useful to achieve better performance and longer life service. In this paper a reflector profile that permits uniform infiltration of the concentrated sunlight into a PV cell receiver was attempted and realized theoretically. The nonimaging optical derivation of the reflector profile accounted for not only the uneven spatial reflection of the sunlight but also the angular dependency of both the reflectivity at the reflector surface and the transmittance through the PV cell receiver. In the 20 March 2014 / Vol. 53, No. 9 / APPLIED OPTICS
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proposed reflector design it was found that the relative position of the receiver with respect to the reflector affects, to a certain degree, the resulting sunlight infiltration profile, especially when the sunlight incidence angle at the receiver is close to 90° where the transmittance is significantly small, which also imposes a limitation on the concentration ratio. However, by careful consideration of the relative positioning between the reflector and the receiver and the concentration ratio limitations, our proposed reflectors are expected to enhance the conversion efficiency and the overall performance of the concentrating PV systems better than the known reflectors that were previously attempted for realizing an even sunlight infiltration into the PV cells. References 1. H. Pfeiffer and M. Bihler, “The effects of non-uniform illumination of solar cells with concentrated light,” Sol. Cells 5, 293–299 (1982).
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2. H. Baig, K. C. Heasman, and T. K. Mallick, “Non-uniform illumination in concentrating solar cells,” Renew. Sust. Energ. Rev. 16, 5890–5909 (2012). 3. H. Baig, N. Sarmah, K. C. Heasman, and T. K. Mallick, “Numerical modelling and experimental validation of a low concentrating photovoltaic system,” Sol. Energy Mater. Sol. Cells 113, 201–219 (2013). 4. N. Sellami and T. K. Mallick, “Optical characterisation and optimisation of a static window integrated concentrating photovoltaic system,” Sol. Energy 91, 273–282 (2013). 5. J. J. O’Gallagher and R. Winston, “Nonimaging solar concentrator with near-uniform irradiance for photovoltaic arrays,” Proc. SPIE 4446, 60–64 (2001). 6. D. G. Jenkings, “High-uniformity solar concentrators for photovoltaic systems,” Proc. SPIE 4446, 52–59 (2001). 7. A. Akbarzadeh and T. Wadowski, “Heat pipe-based cooling systems for photovoltaic cells under concentrated solar radiation,” Appl. Therm. Eng. 16, 81–87 (1996). 8. H. Ries, J. M. Gordon, and M. Laxen, “High-flux photovoltaic solar concentrators with kaleidoscope based optical designs,” Sol. Energy 60, 11–16 (1997).
Sunlight concentration is essential to reach high temperatures of a working fluid in solar-thermal applications and to reduce the cost of photovoltaic (PV) electricity generation systems. Commonly, sunlight concentration is realized by parabolic or cylindrical reflectors, which do not provide uniform concentration on the receiver finite surface. Uniform concentration of sunlight is favored especially for the PV conversion applications since it not only enhances the conversion efficiency of sunlight but also reduces the thermal variations along the receiving PV cell, which can be a performance and life-span limiting factor. In this paper a reflector profile that uniformly infiltrates the concentrated sunlight into the receiving unit is attempted. The new design accounts for all factors that contribute to the nonuniform concentration, like the reflector curvature, which spatially reflects the sunlight nonuniformly, and the angular dependency of both the reflector reflectivity and the sunlight transmission through the PV cell. © 2014 Optical Society of America OCIS codes: (350.4600) Optical engineering; (080.4035) Mirror system design; (080.4228) Nonspherical mirror surfaces; (080.4295) Nonimaging optical systems; (080.4298) Nonimaging optics. http://dx.doi.org/10.1364/AO.53.001862
1. Introduction
The cost of the photovoltaic (PV) cells can be compromised by concentrating the sunlight using special reflectors before the PV conversion. Therefore, sunlight-reflecting concentrators can be useful components in order to reduce the cost of the PV solar systems. Nevertheless, some limitations and practical issues are encountered when applying sunlight concentration on a PV cell receiver. Usually a minimum concentration ratio needs to be accomplished in order to justify the additional cost that comes from using reflectors with suitable stand, sun-tracker system, and cooling system. Usually, the size and complexity, and therefore the cost, of the auxiliary systems increase with the concentration level. Moreover, using the common parabolic or cylindrical concentrators leads to a nonuniform illumination of the concentrated sunlight on the receiving finite PV 1559-128X/14/091862-07$15.00/0 © 2014 Optical Society of America 1862
APPLIED OPTICS / Vol. 53, No. 9 / 20 March 2014
cell, which can limit the conversion efficiency of the solar system [1–4]. For instance, a considerable amount of the solar energy is converted to heat energy instead of electricity in most available PV cells, which causes, if not handled, performance and life-span issues. The heating problem in the PV cell becomes even more prominent in the nonuniform sunlight concentration regime. The uneven heating of the PV cell leads to uneven temperature distribution inside the PV cell; this may cause internal thermal stress that could degrade the PV cell’s overall performance. Moreover, the uneven infiltration of the concentrated sunlight into the PV cell receiver causes uneven generation for the photocarriers throughout the PV absorbing medium of the cell, which may reduce the net photocurrent because of the greater internal recombination rate of photogenerated charges (i.e., the electrons and holes) because of the lateral diffusion of the unevenconcentration-free-charges. In this work, we attempt designing a reflecting concentrator profile that handles the uneven feeding
problem into a finite size PV cell. To achieve this goal we examine all factors that associate with the sunlight reflections and cause the uneven infiltration of the concentrated sunlight into the PV cell feeding surface. The previous works that treated this problem were not realistic because they accounted only for the uneven concentration, which is associated with the uneven spatial reflection that is caused by the curvature of the reflector [5–8], whereas other factors like the angular dependencies of both the sunlight reflectivity at the air–reflector interface and sunlight transmittance at the air–receiver interface were not accounted for. 2. Theory
In order to realize high-performance, uniformconcentration reflectors, there is more than one factor that needs be accounted for in concentrating the uniform flux of sunlight into a finite PV cell receiver. First is the angular uneven reflection of the uniform sunlight from the reflector curved surface because of the geometry effect on the one hand and the Fresnel reflection angular dependency on the other hand. Second is the concentrated sunlight transmittance angular dependency through the PV cell feeding surface since the sunlight will reach the receiver surface at various angles. Hence, the problem is to find the reflector profile yx such that the uniform flux of the incident sunlight is reflected, concentrated, and fed uniformly into the flat PV cell receiver that is oriented parallel to the incident sunlight as depicted in Fig. 1(a). The reflector curvature is responsible for the incidence of the uniform sunlight at various angles along the reflector surface, which not only leads to spatially different reflected angles of the sunlight rays toward the receiver but also causes different reflected power because of the Fresnel reflection angular dependency. Referring to Fig. 1(a), the spatially varied reflecting angles effect is accounted for by tracing a sunlight ray that reaches the vertical PV cell receiver at point 0; h upon reflecting off the reflector at point x; yx. Therefore, considering the triangle that comes between the three points x; y; x; h, and 0; h in Fig. 1(a), the tangent of the angle of this triangle at point x; y can be expressed in two ways: 2 tanθ 2y0 ; tan2θ 2 1 − tan θ 1 − y02
x ; h−y
(1)
Fig. 1. (a) Ray tracing of sunlight concentration and infiltration into a vertical PV cell receiver. (b) Ray tracing of sunlight incident at the starting point xo ; yo of the reflector.
hx; y y x
1 − y02 : 2y0
(4)
Next we consider the reflectivity of the unpolarized sunlight that strikes a medium with refractive index N 1 n1 jκ1 at an incident angle θ, which can be obtained from the following expression: 1 sinθ − ϕ 2 tanθ − ϕ 2 tanθ ϕ ; 2 sinθ ϕ
(5)
where ϕ is the angle of the transmitted sunlight and is related to θ by Snell’s law: (2)
where y0 tanθ:
(b)
Rθ
and tan2θ
(a)
(3)
Equating the right sides of Eqs. (1) and (2) and rearranging leads to
sinθ N 1 sinϕ:
(6)
Similarly, the sunlight transmission into the PV cell receiver, which has refractive index N 2 n2 jκ2 with incident angle α, is obtained by: 1 sinα − β 2 tanα − β 2 ; Tα 1 − 2 sinα β tanα β 20 March 2014 / Vol. 53, No. 9 / APPLIED OPTICS
(7)
1863
where β is the angle of the transmitted sunlight and is related to α by Snell’s law:
(a)
220 215 190
0.5
(8)
In order to obtain uniform sunlight feeding into the PV cell receiver, the concentration ratio should equal
140
y(x) (cm)
sinα N 2 sinβ:
Reflector profile when angular dependencies are accounted
115 90 65
1 dx : C RθTα dh
Profile difference when angular dependencies are not accounted (R=0.6130, T=0.6515)
40
(9)
dh 1 − y02 d 1 − y02 y0 x ; 2y0 2y0 dx dx
0
15 -10 10
(b)
20
30
40
50
60
70
80
90
100
110
x (cm)
45
Angular dependencies are not accounted (R=0.613, T=0.6515)
Sunlight infiltration (suns)
Therefore, the task is to find the reflector profile yx such that the right side of Eq. (9) is independent of x and equals the concentration ratio C. In order to do so, the three terms in the right side of Eq. (9) should be expressed in terms of yx and/ or y0 x. Differentiating Eq. (4) with respect to x yields the following:
profiles difference (cm)
165
40 Angular dependencies are accounted
35
(10) 30 0
which can be reduced to
(c)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
70
80
90
100
110
0.6
0.7
0.8
0.9
1
h (cm)
0.66 0.65
dh y02 1y0 − xy00 : dx 2y02
transmittance
0.64
(11)
0.63 0.62
" # tanϕ 2 1 − tanθ tanϕ 2 1 1 − tanθ Rθ : 1 1 tanθ tanϕ 2 1 tanϕ tanθ (12) Also, using Eqs. (3) and (6) we could express tanϕ as y0 tanϕ q : N 21 − 1y02 N 21
(13)
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0.6 Reflectivity
0.59 0.58 0.57 10
20
30
40
50
60
x (cm)
(d) 40 Sunlight divergence is accounted
39.5
Sunlight divergence is not accounted
39
38.5
38 0
0.1
0.2
0.3
0.4
0.5
h (cm)
Substituting Eqs. (3) and (13) in Eq. (12) yields 2 q N 2 − 1y02 N 2 − 1 2 1 1 1 Ry0 4 q 2 N 2 − 1y02 N 2 1 1 1 q 13 0 N 2 − 1y02 N 2 − y02 2 1 1 A5: × @1 q N 2 − 1y02 N 2 y02 1 1
0.61
Sunlight infiltration (suns)
Obviously, Eq. (11) presents a nonlinear function in terms of the first and second derivatives of yx. Next the terms Rθ and Tα should be expressed in terms of the function yx and/or its higher derivatives in order to homogenize the three components in the right side of Eq. (9); therefore, we could construct a differential equation for the target function yx. To accomplish this, Eq. (5) can be rewritten as follows:
(14)
Fig. 2. (a) 100 cm wide stainless steel reflector surface profile with uniform sunlight infiltration into silicon PV cell receiver (left scale) and the reflector profile difference when the angular dependences are ignored with R 0.613 and T 0.6515 (right scale) (C 40, xo 10, yo −10 cm). (b) Effect of reflection and transmittance angular dependencies on the sunlight infiltration into the PV silicon cell with R 0.613 and T 0.6515 for the case when angular dependencies are not accounted (C 40, xo 10, yo −10 cm). (c) Corresponding stainless steel reflectivity and transmittance through the silicon PV cell receiver (C 40, xo 10, yo −10 cm). (d) Effect of sunlight divergence (0.26°) on the sunlight infiltration into the PV silicon cell (C 40, xo 10, yo −10 cm).
(a)
250
Similarly, Eq. (7) can be rewritten as
200
2 2 3 tanβ 1 − tanα tanβ 2 1 4 1 − tanα 5: 1 Tα 1 − tanβ 2 1 tanα 1 tanα tanβ
y(x) (cm)
150 yo= -10 cm
100
50
(15)
yo= -40 cm
0 yo= -70 cm
-50
-100 10
20
30
40
50
60
70
80
90
100
110
x (cm)
Again, referring to the triangle that comes between the three points x; y; x; h, and 0; h in Fig. 1(a), we could write
(b) 0.62 yo= -70 cm
tanα
0.61 yo= -40 cm
1 1 − y02 : 2y0 tan2θ
(16)
0.6
Additionally, manipulating Eqs. (8) and (16) leads to R 0.59
1 − y02 tanβ q : N 22 1 y02 2 − 1 − y02 2
yo= -10 cm
0.58
0.57
10
20
30
40
50
60
70
80
90
100
110
(17)
Substituting Eqs. (16) and (17) into Eq. (15) yields
x (cm)
(c)
0.66
Ty0
yo= -10 cm
2 q N 2 1 y02 2 − 1 − y02 2 − 2y0 2 16 2 q 1− 6 4 2 N 2 1 y02 2 − 1 − y02 2 2y0 2
yo= -40 cm
0.64 yo= -70 cm
0.62
0.6
T
3 q 1 2y0 N 2 1 y02 2 − 1 − y02 2 − 1 − y02 2 2 7 2 A7 : × @1 q 5 0 2 02 2 02 2 02 2 2y N 2 1 y − 1 − y 1 − y
0.58
0
0.56
0.54 0.52 10
20
30
40
50
60
70
80
90
100
110
(18)
x (cm)
(d) 40
yo= -10 cm
Substituting Eq. (11) into Eq. (9) and rearranging for the sake of y00 yields
Sunlight infiltration (suns)
39 yo= -40 cm
38
yo= -70 cm
y00
37
36
(19)
This can be reduced to a first-order nonlinear differential equation by substituting
35
34 0
y0 y02 1Ry0 Ty0 C − 2y02 : xy02 1Ry0 Ty0 C
0.2
0.4
0.6
0.8
g y0 ;
1
(20)
h (cm)
Fig. 3. (a) Surface profiles of a 100 cm wide stainless steel reflector at three different relative positions of the receiver with respect to the reflector: yo −10 cm, yo −40, and yo −70 (C 40, xo 10). (b) Corresponding reflectivity along the stainless steel reflector with the three different relative positions of the receiver with respect to the reflector: yo −10 cm, yo −40, and yo −70 (C 40, xo 10). (c) Corresponding transmittance through the PV silicon receiver with the three different relative positions of the receiver with respect to the reflector: yo −10 cm, yo −40, and yo −70 (C 40, xo 10). (d) Corresponding sunlight infiltration into the PV silicon receiver with the three different relative positions of the receiver with respect to the reflector: yo −10 cm, yo −40, and yo −70 (C 40, xo 10).
which leads to dg gg2 1RgTgC − 2g2 ; f x; g dx xg2 1RgTgC
(21)
where Rg and Tg are obtained from Eqs. (14) and (18), respectively, with the substitution y0 g, where gx is obtained from integrating Eq. (21) with an initial point (xo , gxo ), which is obtained, with reference to Fig. 1(b), from expressing tan2θ in two different ways as follows: 20 March 2014 / Vol. 53, No. 9 / APPLIED OPTICS
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xo 2gxo tan2θ : yo 1 − gxo 2
(22)
Accounting for the left and the right sides, Eq. (22) presents a quadratic equation with respect to gxo and has a useful solution with positive gxo as p y2o x2o yo : gxo go xo
(23)
Finally, yx could be found by integrating Eq. (20) with the initial point x xo ; y yo . 3. Simulations and Results
In this section some numerical examples are considered in order to verify the theory of producing a reflector profile that facilitates uniform infiltration of the concentrated sunlight into a finite PV cell receiver. The second-order Runge–Kutta, namely the midpoint method, was employed to solve the firstorder ordinary differential equation given by Eq. (21). Therefore, the differential Eq. (20) can be transferred using the midpoint method to the following set of difference equations: gi1 gi Δxf xi0.5 ; gi0.5 ;
xi0.5 xi
gi0.5 gi
f xi ; gi
Δx ; 2
Δx f xi ; gi ; 2
gi g2i 1Rgi Tgi C − 2g2i ; xi g2i 1Rgi Tgi C
(24a)
(24b)
(24c)
(24d)
where Δx is the step size. The initial go is found from substituting the point xo ; yo in Eq. (23), where the point xo ; yo identifies the bottom of the reflector relative to the bottom point of the vertical flat receiver as shown in Fig. 1. Given gx is determined numerically, which represents the slope of the reflector surface, the reflector profile surface yx can be determined by integrating gx with the initial point xo ; yo . In order to verify that the resulted reflector profile would yield the desired even sunlight infiltration profile into the receiver, the incident sunlight ray at point xi ; yxi on the reflector is traced every Δx to locate the incident point 0; yi at the receiver using Eq. (4). Additionally, the corresponding sunlight fraction that infiltrates the receiver, which is obtained by I i Ri T i, is also recorded. Therefore, the infiltrating sunlight concentration profile hi ; Ci can be obtained from yi ; I i as follows: 1866
APPLIED OPTICS / Vol. 53, No. 9 / 20 March 2014
hi
yi yi1 ; 2
Ci h
I i I i1 2
Δx
i; yi1 − yi
(25)
where Ci is intuitively estimated as the ratio between Δx and the trapezoidal area that comes between yi and yi1 under the curve that represents the data yi ; I i . Finally, it is worth mentioning that the sunlight divergence, which is about 0.26°, has minimal effect on the results; this is attributed to the fact we could apply the local linearity for small angles like 0.26° (about 4.6 mrad); therefore, the divergence around the incidence angle become symmetric and its effect cancels out; consequently, it leads to the same g y0 . However, in order to determine the actual sunlight infiltration profile because of the sunlight divergence, again, local linearity facilitates determining the resulting infiltration profile by averaging the three infiltration profiles that come from reflector slopes: g, g − 0.0023, and g 0.0023. The 0.0023 rad angle is the tangent of 0.13° which is half of the divergence angle since the reflector surface that is sloped at angle θ diverts the incident sunlight toward the receiver by 2θ. It is expected that such little divergence would even enhance the infiltration profile without altering the reflector surface profile since it acts as a narrow averaging window that yields a more flattened infiltration profile. In what follows, a numerical example that clarifies the above theory and discussions is considered. A. Examples
In this example we consider a stainless steel reflector and a silicon PV cell receiver with refractive indices as N 1 2.757 j3.792 and N 2 3.882 j0.019, respectively. Moreover, the reflector width is xf 100 cm, the concentration C equals 40, the initial point is (xo 10 cm, yo −10 cm), and the step size from the numerical method is Δx 0.01 cm. Substituting all these in Eqs. (23) and (24) yields the reflector profile that permits an even infiltration of the sunlight into the silicon PV cell receiver. Figure 2(a) shows the resulting reflector profile surface accounting for the angular dependencies of both the stainless steel reflectivity and the transmittance of the silicon PV cell in the right scale, whereas the curve that belongs to the left scale is the additive difference of the reflector profile when the stainless steel reflectivity and the silicon PV cell transmittance are not accounted for but are set to their normal incidence values, that is, R 0.613 and T 0.6515, respectively. Obviously, the difference between the two surfaces is relatively small; it is about 5 mm at its maximum value near the far edge of the reflector. Nevertheless, such a surface profile difference plays a major role in permitting even sunlight infiltration into the receiver, as is depicted in Fig. 2(b), which confirms that accounting for the angular dependencies provides a more even sunlight infiltration into the receiver. Obviously, the sunlight
(a) 250 200 C=1000
y(x) (cm)
150 C=200
100
C=40
50
0 10
20
30
40
50
60
70
80
90
100
110
x (cm)
(b) 900 800
Sunlight infiltration (suns)
700 C=1000
600 500 400 300
C=200
200 100 0 -0.2
C=40
0
0.2
0.4
0.6
0.8
1
h (cm)
Fig. 4. (a) Surface profiles of a 100 cm wide stainless steel reflector with three different concentration ratios: C 40, C 200, and C 1000 (xo 10, yo −10). (b) Corresponding sunlight infiltration into the PV silicon receiver with the three different concentration ratios: C 40, C 200, and C 1000. (xo 10, yo −10).
infiltration profile at the receiver is higher for the case R 0.613 and T 0.6515, which is attributed to the fact that the stainless steel surface reflectivity decreases notably below its maximum value 0.613 as sunlight reflects near the far edge of the reflector, as is depicted in Fig. 2(c), which shows the angular dependencies of stainless steel reflectivity and the transmittance of the silicon PV cell as sunlight encounters various incidence angles along the reflector and the receiver. The extremes that are shown in both curves of Fig. 3(c) are attributed to the sunlight, which is partially magnetically polarized, passing through the Brewster angle as the incidence angle varies across the reflector and the receiver surfaces. Figure 3(d) shows how the resulting sunlight infiltration profile is affected by the sunlight divergence, which is about 0.26°; obviously, the very small divergence of sunlight has very little impact on the result; it even helps flatten the profile by acting like a smoothing averaging window. Next, for the sake of comparison and better understanding how the relative positioning of the PV cell receiver with respect to the reflecting surface could affect the sunlight concentration and infiltration into the receiver, three different reflectors are considered with the same previous givens (i.e., C 40, stainless steel reflector, silicon PV cell that is oriented vertically and starts at the origin) but with different
starting points xo ; yo for the reflector. Figure 3(a) shows the resulting three reflecting surfaces: the upper surface profile belongs to a reflector that starts at xo 10 cm; yo −10 cm, and the middle surface profile belongs to a reflector that starts at (xo 10 cm, yo −40 cm), whereas the last surface profile belongs to a reflector that starts at xo 10 cm; yo −70 cm. Expectedly, the farther the receiver is from the reflector, the flatter is the resulting reflecting surface. Figure 3(b) shows the corresponding sunlight reflectivity of the stainless steel reflectors with the surface profiles shown in Fig. 3(a), and Fig. 3(c) shows the corresponding sunlight transmittance through the silicon PV cell using stainless steel reflectors with the surface profiles shown in Fig. 3(a). A tradeoff scenario is observed between the sunlight reflectivity and the sunlight transmittance as the incidence angle of sunlight at the receiver starting points varies. Figure 3(d) shows that the infiltration profile becomes more uniform when such angle becomes small, that is, for the reflector case when the starting point is xo 10 cm; yo −10 cm. This leads us to conclude that an optimal positioning of the receiver that permits maximum uniform infiltration of the concentrated sunlight into the receiver is a useful aspect that is worth further investigation, which can be a future research. Finally, as with most concentrators, our proposed reflector has limitations on the concentration ratio; this comes from the fact that the infiltration profile is not perfectly uniform, especially at the start, because of the low sunlight transmittance through the PV cell at sharp incidence angles, as depicted from Figs. 2(b) and 3(d). Therefore, the higher the concentration ratio, the more the corresponding reflector concentrates the sunlight at small values of h where the uneven infiltration is more prominent; consequently, it strays away from achieving even sunlight infiltration into the receiver as illustrated by Fig. 4. Three reflecting surface profiles with xo 10 cm; yo −10 cm and various concentration ratios (40, 200, and 1000) are shown in Fig. 4(a) with the corresponding sunlight infiltration profiles in Fig. 4(b). Obviously, examining Fig. 4(b) leads us to conclude that there is a trade-off between sunlight infiltration uniformity and the concentration ratio, which needs be accounted for before employing such a reflector design in practice. 4. Summary
A uniform concentration and infiltration of sunlight into the PV cells is useful to achieve better performance and longer life service. In this paper a reflector profile that permits uniform infiltration of the concentrated sunlight into a PV cell receiver was attempted and realized theoretically. The nonimaging optical derivation of the reflector profile accounted for not only the uneven spatial reflection of the sunlight but also the angular dependency of both the reflectivity at the reflector surface and the transmittance through the PV cell receiver. In the 20 March 2014 / Vol. 53, No. 9 / APPLIED OPTICS
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proposed reflector design it was found that the relative position of the receiver with respect to the reflector affects, to a certain degree, the resulting sunlight infiltration profile, especially when the sunlight incidence angle at the receiver is close to 90° where the transmittance is significantly small, which also imposes a limitation on the concentration ratio. However, by careful consideration of the relative positioning between the reflector and the receiver and the concentration ratio limitations, our proposed reflectors are expected to enhance the conversion efficiency and the overall performance of the concentrating PV systems better than the known reflectors that were previously attempted for realizing an even sunlight infiltration into the PV cells. References 1. H. Pfeiffer and M. Bihler, “The effects of non-uniform illumination of solar cells with concentrated light,” Sol. Cells 5, 293–299 (1982).
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2. H. Baig, K. C. Heasman, and T. K. Mallick, “Non-uniform illumination in concentrating solar cells,” Renew. Sust. Energ. Rev. 16, 5890–5909 (2012). 3. H. Baig, N. Sarmah, K. C. Heasman, and T. K. Mallick, “Numerical modelling and experimental validation of a low concentrating photovoltaic system,” Sol. Energy Mater. Sol. Cells 113, 201–219 (2013). 4. N. Sellami and T. K. Mallick, “Optical characterisation and optimisation of a static window integrated concentrating photovoltaic system,” Sol. Energy 91, 273–282 (2013). 5. J. J. O’Gallagher and R. Winston, “Nonimaging solar concentrator with near-uniform irradiance for photovoltaic arrays,” Proc. SPIE 4446, 60–64 (2001). 6. D. G. Jenkings, “High-uniformity solar concentrators for photovoltaic systems,” Proc. SPIE 4446, 52–59 (2001). 7. A. Akbarzadeh and T. Wadowski, “Heat pipe-based cooling systems for photovoltaic cells under concentrated solar radiation,” Appl. Therm. Eng. 16, 81–87 (1996). 8. H. Ries, J. M. Gordon, and M. Laxen, “High-flux photovoltaic solar concentrators with kaleidoscope based optical designs,” Sol. Energy 60, 11–16 (1997).
