# A Simplified Method to Avoid Shadows at Parabolic-Trough Solar Collectors Facilities

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}+ 0.0121 − Lat + 10.9 with R

^{2}of 99.8%. Finally, the model has been simplified to obtain in the standard case the shadows in the running time of a PTC facility.

## 1. Introduction

## 2. Classical Methods for the Sizing of PTC: A Brief Overview

_{ZS}= (π/2) − γs = |δ| + Φ

_{ZS}as zenith angle.

## 3. Standard Methods for Determining the Spacing between Collectors in PTC Facilities

#### 3.1. Standard Method 1

#### 3.2. Standard Method 2

## 4. Proposed Method

#### 4.1. Solar Angle Calculation

- (90° − δ) ⇒ hour angle (H)
- (90° − Φ) ⇒ the azimuth supplement (A*)
- (90° − h) ⇒ solar height (h)

_{0}) and the hours that are equally divided backwards and forwards from this central hour. The central peak is assumed to be H

_{0}= 0° for the Greenwich meridian, with each hour corresponds to 15 degrees, with the adding on the right of 15 degrees per hour of solar gain and the subtracting on the left of 15° per hour (with 0 assumed to be 360° to prevent values of angles as negative). e.g., in Spain, for a setting of four hours of sun, at the time of 10:00 h, there would be an H of 330°, and at 12:00 h, it is reached an H of 30° (see Figure 9 as guidance).

#### 4.2. The Extent of the Shade

_{i}). To plot these shading distances on the floor, polar coordinates are applied, whereby the angle H

_{i}is the hourly angle of the point i. The critical event is the shortest day of one year, where h represents the shortest day of the solar field design. The shading envelope of a PTC is obtained, and from this envelope, the shadow projected for each whole row of the PTC assembly can be drawn so that no shadow can be cast between the rows.

_{1}− H

_{3})), meaning the difference in the hour angles of the other two vertexes, and the sides from point P to first point and to third point are d

_{1}and d

_{3}, respectively. Before for h

_{1}and h

_{3}are calculated the values of d

_{1}and d

_{3}. Then, it is possible to determine an effective distance of shade considering the projection angle of the sun, as shown in Equation (14) (Figure 11),

_{ORTHO}° and H

_{SUNSET}°.

## 5. Results and Discussion

#### 5.1. Case Study: Results

_{ORTHO}and H

_{SUNSET}are calculated using Equation (17), obtaining H

_{ORTHO}= 289.09° and H

_{SUNSET}= 70.91°.

_{1}= 330°; H

_{3}= 30°, angles (h

_{1}, h

_{3}), and distances d (d

_{1}y d

_{3}) are computed, considering the dimensions of a standard PTC of the commonly used Eurotrough model [29] (the aperture plane size W = 5.760 m and focal distance f = 1.710 m) for the calculation of the distance of the vertex of the collector perpendicular to the aperture plane according to Equation (18), resulting in that z = 1.212 m.

#### 5.2. Extension of the Case Study to Worldwide

^{2}= 0.9769

^{2}+ 0.0121 Lat + 10.9

^{2}= 0.9984

^{2}greater than 99.8%.

## 6. Conclusions

^{2}of 97.69% and a polynomial model with R

^{2}of 99.8%. Both run well within the range of 20 to 45 degrees latitude, but outside this zone, the polynomial model works best. In short, it is proposed to use the polynomial model obtained. Furthermore, this work opens new perspectives for the calculation of shadows in CSPs plants since the methodology developed in this work can be used to establish simple shadow calculation models when the dimensions of the PTC are different from the one used in this work (or even if other type of CSP collectors are studied) or when the operating times of the installation are different.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Scheme of working of a PTC facility. (1) Reflector; (2) Absorber tube; (3) Structure; (4) Solar Field Piping.

**Figure 6.**Proposed method: isometric view of PTC. (

**A**) Shadowing at 10 a.m.; (

**B**) Shadowing at 12 a.m.

**Table 1.**Calculations for every shadow of the three points at south of Spain on 22 December 2019 (latitude 37.091° N).

Solar Hour | Time Angle (° (H) | Elevation or Solar Height (°) (h) | Flat Tilt Opening (°) (α) | Distance between Pylons (m) d‴ + d′ = W (sinα + cosα (sin H/tg h)) |
---|---|---|---|---|

10:00:00 | 330.000 | 0.416 | 40.666 | 8.847 |

12:00:00 | 300.000 | 0.500 | 89.999 | 5.760 |

14:00:00 | 270.000 | 0.583 | 40.665 | 8.847 |

**Table 2.**Calculations according to the proposed modelling the main PTC facilities in the northern hemisphere.

Country | Emplacement | Latitude (° N) | Solar Hour (2 h after Sunrise) | Calculated Shadow Distance d‴ + d′ = W (sinα + cosα (sin H/tg h)) (m) |
---|---|---|---|---|

Thailand | Kanchanaburi | 14.022 | 8:25:07 | 11.294 |

USA | Kailua-Kona (Hawai) | 19.639 | 8:35:43 | 11.546 |

UEA | Medinat Zayed (Abu Dabi) | 23.660 | 8:43:00 | 11.749 |

USA | Indiantown (Florida) | 27.027 | 8:51:07 | 11.957 |

Algeria | HassiR’mel | 32.928 | 9:05:07 | 12.407 |

Morocco | Ain Beni Mathar (Oujda) | 34.088 | 9:08:00 | 12.508 |

USA | Mojave Desert (California) | 35.031 | 9:10:00 | 12.593 |

Spain | Almeria | 37.051 | 9:16:25 | 12.787 |

Italy | Massa Martana | 42.776 | 10:08:59 | 13.374 |

Canada | Kingsey Falls (Québec) | 45.860 | 9:46:00 | 13.699 |

Germany | Jülich | 50.922 | 10:08:59 | 14.147 |

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**MDPI and ACS Style**

Novas, N.; Fernández-García, A.; Manzano-Agugliaro, F.
A Simplified Method to Avoid Shadows at Parabolic-Trough Solar Collectors Facilities. *Symmetry* **2020**, *12*, 278.
https://doi.org/10.3390/sym12020278

**AMA Style**

Novas N, Fernández-García A, Manzano-Agugliaro F.
A Simplified Method to Avoid Shadows at Parabolic-Trough Solar Collectors Facilities. *Symmetry*. 2020; 12(2):278.
https://doi.org/10.3390/sym12020278

**Chicago/Turabian Style**

Novas, Nuria, Aránzazu Fernández-García, and Francisco Manzano-Agugliaro.
2020. "A Simplified Method to Avoid Shadows at Parabolic-Trough Solar Collectors Facilities" *Symmetry* 12, no. 2: 278.
https://doi.org/10.3390/sym12020278