Analytic solutions for the geometric and optical properties of stationary compound parabolic concentrators with fully illuminated inverted V receiver

摘要

Stationary low concentrator collectors (C < 2), of the CPC type, are of great interest to supply thermal energy for industrial processes, at temperatures below or around 90℃. In particular, concentrators with fully illuminated inverted V absorbers have attractive properties for thermal energy production. Two classes of CPC's with inverted V absorber are identified, according to the relationship between the vertex angle of the absorber (γ) and the acceptance angle of the cavity (θ_a), (γ ≥ θ_a) or (γ < θ_a). The first class of CPC's (γ ≥ θ_a) converge to the fully illuminated CPC with horizontal fiat receiver when (γ = 90°). The second class of CPC's (γ < θ_a) converge to the fully illuminated CPC with vertical flat receiver when (γ = 0°). Both limiting cases have been published in the technical literature. This paper analyzes the class of concentrators satisfying (γ ≥ θ_a). The ideal concentrator corresponding to a fully illuminated wedge absorber and (γ ≥ θ_a) is a circular involute plus three parabolic segments. Closed-form analytic solutions are derived for its geometric and optical properties: reflector geometry, aperture, height, reflector length, angular acceptance function and average number of reflections for any degree of truncation. The equations obtained can be used as important design tools, for simulation techniques and optimization purposes. The collectible energy for North-South and East-West oriented collectors, for various receiver vertex angle and acceptance angle, was calculated. A cost-benefit figure, given by the relationship between collectible energy and reflector surface, is also estimated. Numerical results for any degree of cavity truncation are presented. As the degree of truncation varies, a clear minimum of the length over aperture ratio (L/A) occurs. The geometric and optical characteristics of different low concentration CPC's (C, between 1 and 2, range of interest of stationary concentrators) show that the cavities with the minimal relationship between the length or height of the reflector surface and the aperture, (L/A) and (H/A), and the lower average number of reflections < n > correspond to the lowest angular acceptance concentrator (highest nominal concentration). If a concentration of 1.2 is desired, then the smallest ratios of (L/A), (H/A) and < n >, within the set of concentrators with maximal concentration (C) between 1 and 2, occur for (C = 2) (nominal acceptance half angle θ_a = 30°). Collectible energy results together with a cost-benefit relationship enable to conclude that a good choice for a well-designed collector for the city of Recife-PE, Brazil is: (a) East-West orientation; (b) receiver vertex angles (7) of the order of (65°); (c) acceptance angle of the CPC (θ_a = 30°) and (d) concentration of the truncated cavity (C_t) in the interval (1.0-1.2).