Abstract
This chapter presents a detailed mathematical analysis of PTCs. It is divided into three sections: tracking of the Sun, optical analysis and thermal analysis. The first section is an overview of equations and relationships used in solar geometry to determine the position of the Sun and, hence, the slope and the angle of incidence that in each instant must be assumed by a PTC to correctly follow the Sun. Since PTCs usually have one degree of freedom, correlations valid for east-west and north-south axis with continuous adjustment are discussed. The optical analysis starts by introducing the concentration ratio and continues presenting a thorough description of the geometry of a PTC. Optical errors and geometrical effects are also presented. Then, the optical analysis is concluded by taking into account the optical properties of the materials generally adopted in PTCs: the mirror, the cover and the absorber. The last section involves the thermal analysis of a PTC, i.e. the energy balance of the receiver. Each heat flux is described in detail, in order to determine the thermal efficiency of a PTC.
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Notes
- 1.
For a wide range of conditions encountered in solar collector applications, the equivalent angle for beam radiation, i.e. the angle which gives the same reflectance as for diffuse radiation, is essentially \({60}^{\circ }\) [1].
- 2.
This assumption is acceptable for all concentrators expect for those with low concentration ratio (i.e., for \(C=10\) or below). For systems with low concentration ratio, part of the diffuse radiation will be reflected to the receiver, with the amount depending on the acceptance angle of the concentrator [1].
- 3.
One should also consider the solar beam radiation which falls directly on the absorber tube, but this contribution can be ignored when the concentration ratio is high [9].
- 4.
Some authors define an effective transmittance-absorptance product which accounts for the reduced thermal losses due to absorption of solar radiation by the cover. However, this effect has been already considered in the energy balance of Eq. (2.48), thus it will be not reconsidered here.
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Coccia, G., Di Nicola, G., Hidalgo, A. (2016). Mathematical Modeling. In: Parabolic Trough Collector Prototypes for Low-Temperature Process Heat. SpringerBriefs in Applied Sciences and Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-27084-5_2
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DOI: https://doi.org/10.1007/978-3-319-27084-5_2
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