Acceptance angle and mirror deformations

Our second study considers the interactions between three different quantities: acceptance angle, vertical edge displacement Delta Z and collection efficiency maximum Emax. The results of these energetic and angular analyses are summarised in Fig. 7. The configuration examined in this study includes a linear parabolic mirror of focal length f=780mm. The absorber is a metal pipe of diameter D=50mm, enclosed into a glass tube of diameter G=70mm and thickness T=2mm.


The acceptance angle is estimated examining plots of collection efficiency versus angular misalignment (tilt angle), like the curve in Fig. 6. The solar trough collector is tilted for assessing the acceptance angle and, at the same time, the mirror is deformed, as described in Sections 2-3. The parameter chosen to indicate the deformation of mirror surface is again the vertical edge displacement Delta Z. Starting from the results of the first study, the range considered for Delta Z is (-2.5 mm; +2.5 mm).

The critical optical characteristic in this second study is the acceptance angle obtained for every solar trough with deformed mirror. It represents the maximum tilt angle for which the collection efficiency approximately maintains its maximum value Emax, presenting only very little energy losses (<1% of Emax). The acceptance angle basically depends on the respective positions of mirror surface and absorber, as well as, on other characteristics not examined in this paper.

Figure 7 reports acceptance angle and collection efficiency maximum Emax as a function of the vertical edge displacement Delta Z. The collection efficiency has been calculated for all examined cases and its maximum value is reported in the labels of Fig. 7; these Emax values are obtained without angular misalignment.

As it could be supposed, Figure 7 evidences that the acceptance angle reaches its maximum value for Delta Z = 0, corresponding to the parabolic profile (see Table 1). Then the acceptance angle decreases when the absolute value of Delta Z improves, but the curve is asymmetrical. In analogy to the different results obtained in the first study for the two cases of elliptic or hyperbolic deformations, the behaviour of the curve in Fig. 7 depends on the deformation type.

The previous study individuated two different limiting values for Delta Z: 2.5 mm for the elliptic case and -3.0 mm for the hyperbolic case. For edge deformations included in this range of Delta Z values (-3 mm; +2.5 mm), the collection efficiency maintains its maximum value, so these mirror deformations do not introduce energetic losses. The first study does not consider any angular misalignment of absorber and deformed mirror.

Whereas the second study simulates a more realistic situation, where misalignment errors interact with mirror deformations, and the consequences are assessed analysing the variations of acceptance angle and collection efficiency.

The maxima of collection efficiency present only minor variations, but their behaviour is not symmetrical with respect to Delta Z = 0. The optimum position corresponds to vertical edge displacement Delta Z = 1, for elliptic deformation of the mirror.

1. Conclusions

A solar trough collector has been analysed using ray tracing simulations. The main optical components of the system are linear parabolic mirror and absorber, composed of a metal pipe surrounded by a glass tube. The reference layout for the solar trough has focal length f=780mm and absorber dimensions D=50mm, G=70mm, T=2mm.

The extensive research has investigated several possibilities for the optical configuration, varying most of trough geometrical features. For the parabolic mirror, it has considered the dependence on mirror width, length and focal length; the effect of mirror deformations and errors in surface finishing. For the absorber, it has examined metallic pipe diameter and shape, glass tube diameter and thickness. Beside these geometrical features the research has studied angular misalignment (of solar trough) and absorber displacement with respect to parabolic mirror. All mentioned effects have been analysed taking into account the sun tracking. The main aspects considered in this latter analysis are solar trough positioning, with respect to Earth rotation axis, and errors in daily and monthly tracking.

This paper summarises the results concerning two studies developed in the framework of the extensive research on solar trough collectors. The first study introduces an original methodology to reproduce rigid deformations of the linear parabolic mirror. Then it examines the consequences of mirror deformations on the collection efficiency of solar trough. The second study analyses the interactions between mirror deformations, misalignment and tracking errors.

In both studies the light concentrated by the parabolic mirror and received by the absorber is expressed as collection efficiency, corresponding to the ratio between focused light and entering light. The other fundamental parameter introduced is the vertical displacement of deformed mirror extreme Delta Z, chosen to indicate the amount of mirror deformation.

The methodology to simulate the deformations is based on the introduction of conic constant K and conic equation to represent the mirror profiles. The deformation can be of two types: elliptic, for -1 < K < 0 or hyperbolic for K < -1. While for K = -1 the conic equation represents a parabolic profile. The vertical edge displacement Delta Z in our convention is positive in the elliptic case and negative in the hyperbolic case, while Delta Z = 0 for the parabola.

This procedure to replicate the deformations of a parabolic mirror is simple and efficient. But the most interesting result is that it seems to reproduce the imperfect rigidity and the flexibility of a real solar collector.

The result of the first study is the identification of two different limits of Delta Z for the elliptic case (2.5 mm) and for the hyperbolic case (-3.0 mm). For mirror deformations with Delta Zin the interval (-3 mm; +2.5 mm), the collection efficiency keeps its maximum value, indicating that the corresponding mirror deformations do not cause losses in the collected energy.

In the second study the deformation effects are combined with the angular misalignment, obtained by a rigid tilt of mirror and absorber. Here the fundamental optical characteristic is the acceptance angle, representing the maximum misalignment angle for which the collection efficiency maintains its maximum value.

The results of the second study is a plot summarising the interaction between collection efficiency, acceptance angle and mirror deformations. The collection efficiency is the most important quantity in the application to solar light exploitation, because it indicates the level of performance in energy collection of the solar system. While for the aspects of alignment and sun tracking the crucial parameter to be taken into account is the acceptance angle of solar collector.


The research has been developed in the framework of the S. A.L. T.O. project. S. A.L. T.O. (Solar assisted cooling Toscana) is a research integrated project POR Ob. 3 Toscana 2000/2006 Misura D4, partially financed by the REGIONE TOSCANA-settore Promozione e sostegno della Ricerca (Tuscany Region). Thanks are due to the industrial partners FAIT group and CEVIT for their support to our activities in developing Solar Cooling Systems.


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