Commun. Theor. Phys. (Beijing, China) 51 (2009) pp. 315–320 c Chinese Physical Society and IOP Publishing Ltd
Vol. 51, No. 2, February 15, 2009
Derivative Form of Off-axis Aberration Correction Surface and Its Application in Solar Energy Concentration LI Li,1,2,∗ CHEN Ying-Tian,3,† and HU Sen4,‡ 1
Joint Advanced Research Center in Suzhou, University of Science and Technology of China and City University of Hong Kong, Suzhou 215123, China 2 Suzhou Institute for Advanced Study, University of Science and Technology of China, Suzhou 215123, China 3
Department of Astronomy and Applied Physics, University of Science and Technology of China, Heifei 230026, China
4
Department of Mathematics, University of Science and Technology of China, Hefei 230026, China
(Received March 25, 2008)
Abstract By using the derivative method, we obtained the same result with that of the previous work of Chen et al. in 2006. Different from the integral form, the derivative form of the surface expression published in this paper is derived from differential equation and based on the theory of non-imaging focusing heliostat proposed by Chen et al. in 2001. The comparison of the derivative form of fixed aberration correction surface has been made with that of integral form surface as well as that of spherical surface in concentrating the solar ray. PACS numbers: 42.15.Eq, 42.15.Fr
Key words: off-axis aberration correction, canting, continuous surface, asymmetric curvature
1 Introduction Although in most of the ray concentration, the traditional geometries of the reflective mirror are used, some authors, particular Ries,[1] Zaibel,[2] and Chen[3] pointed out that the shape of the reflective mirror surface is important to reduce the size of the images. If the incident ray is on axis, a parabolic surface is the ideal reflective surface to focus parallel rays on one point. However, in the case of off-axis, not any shape of mirror, among the traditional geometries, would be able to focus the parallel beams on one point, or we say, aberration-free. Off-axis aberration correction has been a problem to be solved for a long time in optics. Until recently, Chen[3,4] achieves first order aberration correction for the concentration of solar rays, which gives a residual image spread much smaller than that of the solar-disc effect. In Chen’s focusing heliostat, the mirror array is combined by n rows and m columns of movable facets driven by row and column actuators. Therefore, the movement of facets can be driven by line movements, and the number of actuators can be reduced from 2mn to m + n. In Chen’s work before 2005,[3−7] he mainly discussed the results related to the active control of the facets, a kind of single parameter adaptive optical surface. Then, he further extended his theory into fixed geometry surface in aiming to reduce the manufacture cost in the application of solar energy concentration. In 2006, a new fixed aberration correction surface was derived by Chen et al.[8] with the integral method. The mirror surface has the form of sin 2θ 3 cos θ + 11 cos 3θ 4 cos θ 2 x − x − x z(x, y) = 2(2L) 3(2L)2 32(2L)3 ∗ E-mail
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2 sin 2θ + 17 sin 4θ 5 x 40(2L)4 −10 cos θ + 59 cos 3θ + 447 cos 5θ 6 + x 768(2L)5 1 3 + y2 − y4 4L cos θ 8(2L cos θ)3 31 + y6 . 48(2L cos θ)5 +
(1)
Chen has proved with Spin-Elevation sun tracking formula, that this new surface can significantly improve the efficiency by introducing the canting angle so that the size of the solar images is considerably reduced and the shape of images become uniform. In this paper, we publish a new expression of the mirror surface (which we refer to as the derivative form in later context) with the form (3). And equation (1) is referred as the integral form, correspondingly. It is found that the two forms of reflective surface have the similar behavior.
2 Derivation of the New Surface After the adjustment of the main plane of the heliostat, the mirror can be coordinated by origin O, and x, y, z axes (Fig. 1), where O is the center of the mirror, i.e. the nearest point to the supporting plane. z-axis is the normal direction at the origin. x and y axes are two directions naturally generated in the adjustment of the main plane, where x-axis is the asymmetric axis, with the sun, origin O, and the target in the x-z plane, y is the symmetric axis perpendicular to the x-axis along the plane. In the continuous mirror case, it is supposed that our surface can reflect the parallel solar rays with the incident angle θ focusing on one point in the position of the target. Denote
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the distance between the target T and origin O by L, then the coordinate of the target is (2L sin θ, 0, L cos θ).
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second-order ordinary differential equation and does not have analytic solution for general θ. However, we can approximate the solution numerically. There are many methods to do the approximation. For the convenience of the industrial application, we choose polynomial functions. Since the ratio of L/H is small (H is the diameter of the circular reflector), this approximation is meaningful in the sense that the higher-order terms become much and much smaller. Let z˜(x) = a2 x2 + a3 x3 + a4 x4 + a5 x5 + a6 x6 . Use it to replace z˜(x) in Eq. (2). Then compare the coefficients on both sides of the equation. We obtain cos θ 2 cos θ sin θ 3 5 cos θ sin2 θ 4 x − x + x 4L 8L2 64L3 7 cos θ sin3 θ 5 21 cos θ sin4 θ 6 x + x . − 128L4 512L5 This is the equation of the truncated curve on the x-z cross section plane. Now, we will consider the other direction along y-axis. Take a cross section parallel to the y-z plane at position x, then in the ideal situation, that mirror surface should concentrate the parallel solar rays on the target. Denote the truncated curve equation by zx (y), here the sub index x shows the dependence on the x position, where the cross section was taken. Then similar arguments as in the x-z plane give us the equation of zx (y): z˜0 (x) =
Fig. 1 Choosing of the coordinate and two cross section planes.
zx (y) − L cos θ 1 − zx0 (y)2 = − cot(2 arctan zx0 (y)) = y 2zx0 (y) with conditions zx (0) = z˜(x),
Fig. 2
zx0 (0) = 0 .
This equation has analytic solution 1 zx (y) = z˜(x) + y2 . 4(L cos θ − z˜(x))
Reflection of solar rays in x-z plane.
Firstly, focus on the cross section plane y = 0 (Fig. 2). Denote the truncated curve of the mirror surface by z˜(x), which is a function of the coordinate x. The normal direction is perpendicular to the tangent at every point x. Compared with the supporting plane of the heliostat, the tangent at position x changed an angle arctan z˜0 (x), therefore the normal at position x changed arctan z˜0 (x) with respect to the z-axis. The incident angle of the sun with respect to the z-axis is θ. Therefore, the incident angle of the sun with respect to the normal at position x is θ + arctan z˜0 (x), so does the reflective angle. Thus, the angle between the reflective ray and the z axis is −(θ + 2 arctan z˜0 (x)). The reflective line should satisfies
Replace z˜(x) into it. We obtain the expression of the mirror surface: cos θ 2 cos θ sin θ 3 5 cos θ sin2 θ 4 z(x, y) = zx (y) = x − x + x 4L 8L2 64L3 7 cos θ sin3 θ 5 21 cos θ sin4 θ 6 x + x − 128L4 512L5 h 1 x2 sin θx3 + + − 4 cos θL 16 cos θL3 32 cos θL4 2 4i (4 + 5 sin θ)x + y2 . (3) 256 cos θL5
z˜(x) − L cos θ = − cot(θ + 2 arctan z˜0 (x)) x + L sin θ cos θ(1 − z˜0 (x)2 ) − 2 sin θ˜ z 0 (x) =− sin θ(1 − z˜0 (x)2 ) + 2 cos θ˜ z 0 (x) with conditions
3 Comparison of the Simulated Results
z˜(0) = 0,
(2)
z˜0 (0) = 0 .
The formulas of the triangle functions are used to derive the last line of Eq. (2). Note that it is a non-linear
The performances of the derivative form in this paper, of the integral form obtained by Chen,[8] and of the spherical surface are simulated and compared in the charts. The distance L between target and origin O is set to be 4.8 m, and H = 2.4 m. When the incident angle is in the range of 0◦ to 33◦ , the canting angle is 23.9◦ for derivative form and 24.6◦ for integral form while when the incident angle is in the range 33◦ to 57◦ , the canting angle is 47.75◦ for derivative form and 48.55◦ for integral form.
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Derivative Form of Off-axis Aberration Correction Surface and Its Application in Solar Energy Concentration
Fig. 3 Reflected images of the new derivative form surface of this paper. The incident angle is 0◦ (a), 10◦ (b), 24◦ (c), and 33◦ (d), respectively.
Fig. 4 Reflected images of the integral form surface in 2006. The incident angle is 0◦ (a), 10◦ (b), 24◦ (c), and 33◦ (d), respectively.
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Fig. 5
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Reflected images of the spherical surface. The incident angle is 0◦ (a) , 10◦ (b), 24◦ (c), and 33◦ (d), respectively.
Fig. 6 Reflected images of the new derivative form surface of this paper. The incident angle is 33◦ (a), 40◦ (b), 48◦ (c), and 57◦ (d), respectively.
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Derivative Form of Off-axis Aberration Correction Surface and Its Application in Solar Energy Concentration
Fig. 7 Reflected images of the integral form surface in 2006. The incident angle is 33◦ (a), 40◦ (b), 48◦ (c), and 57◦ (d), respectively.
Fig. 8
Reflected images of the spherical surface. The incident angle is 33◦ (a), 40◦ (b), 48◦ (c), and 57◦ (d), respectively.
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time (with small incident angle) to early morning and late afternoon (with big incident angle). From Figs. 9 and 10, we can see that although we use a different method to derive the expression of mirror surface, the area of images goes in a similar way in the derivative form and in the integral form.
Fig. 9 Diagram of the area of images with respect to the incident angle for three different kinds of surfaces: new derivative form surface of this paper, integral form surface in 2006, and spherical surface, where the incident angle is in the range 0◦ to 33◦ . The canting angle is chosen to be 23.9◦ for derivative form and 24.6◦ for integral form.
It was shown in Figs. 3 ∼ 8 that the spherical surface gives extremely huge images when the incident angle is big. Our surface, as well as the integral form surface, improves the efficiency not only in the way to reduce the area of solar image, but also to change the hot spot from noon
References [1] H. Ries and M. Schubnell, Solar Energy Materials 21 (1990) 213. [2] R. Zaibel, E. Dagan, J. Karni, and H. Ries, Solar Energy Materials and Solar Cells 37 (1995) 191. [3] Y.T. Chen, K.K. Chong, T.P. Bligh, et al., Solar Energy 71 (2001) 155. [4] Y.T. Chen, K.K. Chong, B.H. Lim, and C.S. Lim, Solar Energy Materials and Solar Cells 79 (2003) 1.
Fig. 10 Diagram of the area of images with respect to the incident angle for three different kinds of surfaces: new derivative form surface of this paper, integral form surface in 2006, and spherical surface, where the incident angle is in the range 33◦ to 57◦ . The canting angle is chosen to be 47.75◦ for derivative form and 48.55◦ for integral form.
[5] Y.T. Chen, K.K. Chong, C.S. Lim, et al., Solar Energy 72 (2002) 531. [6] Y.T. Chen, A. Kribus, B.H. Lim, et al., J. of Solar Energy Engineering 126 (2004) 638. [7] Y.T. Chen, K.K. Chong, C.S. Lim, B.H. Lim, B.K. Tan, and Y.F. Lu, Solar Energy 79 (2005) 280. [8] Y.T. Chen, B.H. Lim, and C.S. Lim, Solar Energy 80 (2006) 268.
