*5.1. Case Study: Results*

The case study will be the most unfavourable day, i.e., on December 22, the winter solstice. The shadow projected by each PTC was estimated, allowing to determine the minimum distance of the next row of PTC. The first case of study was situated in the southern of Spain, CIEMAT-PSA research centre (latitude 37.091◦ N; longitude 2.355◦ W). The data used were declination δ = −23◦ 27; latitude Φ = 37.093◦ N, for Equations (11) and (12). It is estimated that in this location the PTCs do not reach adequate temperature to start working until two hours after sunrise. Therefore, in this case, the shadows will be calculated for the period of time in which the installation is in service. That is, from 10 to 14 h.

HORTHO and HSUNSET are calculated using Equation (17), obtaining HORTHO = 289.09◦ and HSUNSET = 70.91◦.

Table 1 shows the results obtained for each shadow of the three points (first and third) as shown in Figures 9 and 10. These outputs are considered to be valid for every point in time at which the shading distance was calculated. The known angles H1 = 330◦; H3 = 30◦, angles (h1, h3), and distances d (d1 y d3) 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.

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


As can be observed in the results presented in Table 1, the distance calculated between PTC pylons shows a perfect symmetry throughout the day with respect to the moon, as was expected. This fact proves that a first and essential requirement to check the validity of the proposed model is accomplished.

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

The proposed model has been calculated in several key locations for PTC facilities in the northern hemisphere, from a latitude of 14 degrees to almost 51 degrees. Table 2 summarizes the results obtained. Clearly, the distance increases with increasing latitude.

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


If, from the data obtained in Table 2, a simple model is established to calculate the shadow of a standard concentrator (the opening plane size *W* = 5.760 m and focal distance *f* = 1.710 m), where S is the calculated shadow and Lat is the northern latitude, the following models are obtained:

The Linear Estimation:

$$\text{S} = 0.0796 \,\text{L} \,\text{at} + 9.9243 \,\text{} \tag{19}$$

with R<sup>2</sup> = 0.9769

> The Polynomial Estimation:

$$\text{S} = 0.001 \text{ L} \text{at}^2 + 0.0121 \text{ L} \text{at} + 10.9 \tag{20}$$

with R<sup>2</sup> = 0.9984

If the solutions that would be obtained with each model are represented. Figure 12 is obtained, where, the area marked in green, which is to say between 20 and 45 degrees north latitude, there is scarce di fference between both models. Outside this area, the linear model underestimates the magnitude of the shadow and therefore its use would not be advisable. For example, at 14 degrees north latitude, the linear model underestimates the shadow by 25 cm, while the polynomial model underestimates it by less than 3 cm. In short, the results sugges<sup>t</sup> the use of the polynomial model obtained for the calculation of the shadows since it o ffers very good results as it has an R<sup>2</sup> greater than 99.8%.

**Figure 12.** Di fferent models obtained for the calculation of shadows in depending on the latitude.

If the maximum value obtained for the distance between the PTC pylons is noted, a separation of around 11 to 14 m must be selected for the layout of the solar field during the design phase. This would involve a significantly lower land occupation (around 50% lower) compared to the total area to be covered if the thumb rule (of four times the aperture area) is considered. Consequently, the model presented is this work is a very useful tool for CSP plant designer because it is easy to apply, and the investment costs are significantly reduced thanks to the reduction in the land occupation for the solar field.
