Commun. Theor. Phys. (Beijing, China) 45 (2006) pp. 165–166 c International Academic Publishers
Vol. 45, No. 1, January 15, 2006
Complete Solution of Sun Tracking for Heliostat CHEN Ying-Tian,∗ LIM Boon-Han, and LIM Chern-Sing Department of Astronomy and Applied Physics, University of Science and Technology of China, Hefei 230026, China
(Received April 14, 2005)
Abstract A general solution of sun tracking for an arbitrarily oriented heliostat towards an arbitrarily located target on the earth is published. With the most general form of solar tracking formulae, it is seen that the used azimuthelevation, spinning-elevation tracking formulae etc. are the special cases of it. The possibilities of utilizing the general solution and its significance in solar energy engineering are discussed. PACS numbers: 42.15.-i, 42.15.Fr, 42.79.Ek
Key words: solar tracking, azimuth-elevation, spinning-elevation, optical aberration, heliostat
The basic mathematics of sun tracking for heliostat has remained unchanged for many decades. It is mainly the azimuth-elevation tracking formulae[1] with two axes of rotations. However, following the earlier arguments made by Harald Ries[2] and Zaibel,[3] Chen et al.[4] made different move. They analyzed that if one of the rotational axes of heliostat can be made aligning with the target, the optical aberration can be reduced considerably with tiny segment corrections along the rows and columns of a mirror array. With this change of structure, a newly published spinning-elevation sun tracking formula has to be applied to replace the traditional one. New heliostat with dual function of tracking and focusing has brought many new applications[4−10] in the advancement of efficiency and cost. We further consider: Are there any other tracking methods which have been excluded by our study? Can we have a general solution of sun tracking for all kinds of heliostat to simplify and unite the designs of controller in solar energy collection? We believe the complete solution published here will serve this purpose. The solution, which should be the final one, can be used to build a universally applied controller for all forms of heliostat with an arbitrary orientation on the earth.
Sun tracking formula for a heliostat with a fixed target on the earth involves a sun vector (incident sun ray), a target vector (reflected ray) and a heliostat normal vector. However, unit position vector of sun is usually described in earth-center coordinate system (earth-center frame) while the unit position vector of fixed target is most conveniently expressed in earth-surface coordinate system (earth surface frame). For computing the individual rotation angle of each of the two axes attached to heliostat, it is necessary to perform the calculation in Heliostat-center coordinate system (Heliostat Frame). To study the vector rays from the sun and how they reflect into an arbitrarily located target on the earth with the orbital movements of the sun, the coordinate transformation method was utilized in our previous articles.[4−7] The same method is used in this article to obtain the complete solution of sun-tracking of an arbitrarily oriented heliostat towards an arbitrarily located target on the earth, which is βH = arcsin (A/2 cos θi ) ,
(1)
ρH = arcsin (B/2 cos θi cos βH ) ,
(2)
where
A = − sin α(cos Φ cos δ cos ω + sin Φ sin δ) − cos α sin ζ cos δ sin ω + cos α cos ζ(− sin Φ cos δ cos ω + cos Φ sin δ) + sin α sin λ + cos α sin ζ cos λ sin φ + cos α cos ζ cos λ cos φ , B = − cos ζ cos δ sin ω + sin ζ(sin Φ cos δ cos ω − cos Φ sin δ) cos ζ cos λ sin φ − sin ζ cos λ cos φ ; and the incident angle of sun ray, θi = 0.5 arccos(C) ,
(3)
where C = − sin λ(cos Φ cos δ cos ω + sin Φ sin δ) − cos λ sin φ cos δ sin ω + cos λ cos φ(cos Φ sin δ − sin Φ cos δ cos ω) . In the above, α is altitude angle of one of the axes of heliostat, ζ is azimuth angle of the same axis, λ is target angle of target, φ is facing angle of target, Φ is latitude, ω is solar hour angle, and δ is declination of the sun. Since the above formulae are the general solution, special cases can naturally branch into two categories: nonarbitrarily orientated heliostat with an arbitrarily located target (defined heliostat case) and an arbitrarily oriented ∗ Corresponding
author, E-mail:
[email protected]
166
CHEN Ying-Tian, LIM Boon-Han, and LIM Chern-Sing
Vol. 45
heliostat with non-arbitrarily located target (defined target case). The famous case of the former is azimuth-elevation tracking and the famous case of the latter is spinning-elevation tracking. Their differences are actually made from setting different conditions to the parameters: α, ζ and λ, φ in the above equations. Therefore, we can show that some of the established forms of solar tracking formula can be extracted from it with the conditions listed in Table 1. Derivations have been carried out to confirm that with different conditions, the derived formulae are indeed the same as those published ones.
Table 1 The conditions to obtain the established tracking formula using the complete solution published in this article. Category Defined Heliostat
Name Azimuth-Elevation Polar-Mounted
Defined Target
Conditions α=
−90◦
, ζ=
Major Advantages 0◦
α = Φ , ζ = φ = 180◦
Vertical mounted Nearly single axis tracking
General Spinning-Elevation
λα , φ = ζ
Minimum aberration effect
Latitude-Oriented
λ = α = Φ − 90◦ , φ = ζ = 180◦
Maximum solar power collection;
Polar-Oriented
λα = Φ , φ = ζ = 180◦
Single axis tracking minimum; aberration effect
minimum aberration effect
The publication of the above complete solution of solar tracking formula is not only in its mathematical interests; the application in solar engineering could be more significant. There are two possibilities of utilization which may alter our traditional approaches in the practice of solar energy: i) Specific method of sun tracking and the structure of heliostat are two related parts of the integral design of solar collector. A further study of the complete solution may give us more hints to change our present structure of heliostat to improve the feature of harnessing solar energy. Along this line, Chen et al. has moved forward in the direction of defined target case.[4−9] We hope there may be more studies in this respect. ii) The complete solution has provided a universal solution for the various control of different heliostat, which usually costs considerably for the project. If a universal micro-chip based on Eqs. (1) ∼ (3) can be made in a large quantity, they would become a cost-effective controller for all heliostats on the earth. What we need is only to input in seven parameters for individual heliostat, which are α, ζ, λ, φ, Φ, ω, and date to initial the tracking (represents the instantaneous declination angle of sun, δ. We hope this would be one of the steps to achieve cost-effectiveness in the application of solar energy for our nature.
References [1] C.E. Moeller, T.D. Brumleve, C. Grosskreutz, and L.O. Seamons, Solar Energy 25 (1980) 291. [2] H. Ries and M. Schubnell, Solar Energy Materials 21 (1990) 213. [3] R. Zaibel, E. Dagan, J. Karni, and H. Ries, Solar Energy Materials and Solar Cells 37 (1995) 191. [4] Y.T. Chen, et al., Solar Energy 71(3) (2001) 155. [5] Y.T. Chen, et al., Solar Energy 72(6) (2002) 531.
[6] Y.T. Chen, et al., J. of Solar Energy Engineering 126 (2004) 1. [7] Y.T. Chen, K.K. Chong, B.H. Lim, and C.S. Lim, Solar Energy Materials & Solar Cells 79 (2003) 1. [8] Y.T. Chen, et al., J. of Solar Energy Materials and Solar Cells, 80 (2003) 305. [9] Y.T. Chen, et al., Solar Energy 79 (2005) 280. [10] Y.T. Chen, B.H. Lim, and C.S. Lim, Solar Energy (2006), In Press.
