Analysis of the Applicability of the Parabolic Trough Solar Thermal Power Plants in the Locations with a Temperate Climate

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Article Analysis of the Applicability of the Parabolic Trough Solar Thermal Power Plants in the Locations with a Temperate Climate

Tomasz Janusz Teleszewski 1,* , Mirosław Zukowski˙ 1,* , Dorota Anna Krawczyk 1,* and Antonio Rodero 2

1 Department of HVAC Engineering, Bialystok University of Technology, 15-351 Bialystok, Poland 2 Grupo de Física de Plasma: Modelos, Diagnosis y Aplicaciones, Campus of Rabanales, Universidad de Córdoba, E-14071 Córdoba, Spain; [email protected] * Correspondence: [email protected] (T.J.T.); [email protected] (M.Z.);˙ [email protected] (D.A.K.); Tel.: +48-797-995-926 (D.A.K.)

Abstract: Currently, intensive work is underway in Poland to increase the share of renewable energy sources in the overall energy balance. Therefore, this paper presents the possibilities of using concentrated solar power in zones with a temperate climate. A simplified model based on the energy balance in the solar collectors considering the main operating parameters of the typical solar power plant was developed. It should be noted here that the model does not take into account issues related to heat accumulation and electricity generation in a Solar Thermal Power Station. The simulation of forced convection inside the solar collector absorber was additionally included in the calculations to improve its accuracy. The model was verified using actual heat measurements at the outlet of the parabolic collector installation at a Solar Thermal Power Station located in the south of Spain. Citation: Teleszewski, T.J.; The heat generated by a similar solar collector system in a selected region with a temperate climate, ˙ Zukowski, M.; Krawczyk, D.A.; the city of Bialystok (north-eastern Poland, geographic coordinates: 53◦0800700 N 23◦0804400 E) was Rodero, A. Analysis of the determined by the developed simplified model for different months of the year. Based on the results Applicability of the Parabolic Trough of the analysis, it was found that the energy obtained from the same area of concentrated solar Solar Thermal Power Plants in the collectors located near Bialystok is eight times lower compared to the location in Cordoba depending Locations with a Temperate Climate. Energies 2021, 14, 3003. https:// on the variant of the power plant operation. doi.org/10.3390/en14113003 Keywords: concentrating solar power systems; solar gains; solar plants; parabolic trough collectors Academic Editors: Andrea Giostri and Marco Binotti

Received: 21 April 2021 1. Introduction Accepted: 20 May 2021 Solar energy is one of significantly developing branches of renewable energy sources. Published: 22 May 2021 Total annual solar radiation on the surface of the Earth is estimated to be about 7500 times higher than the annual global consumption of primary energy [1]. Trends in new invest- Publisher’s Note: MDPI stays neutral ments in solar systems sector show 23% increase, comparing to 2004, whereas a period with regard to jurisdictional claims in between 2013 and 2018 was marked by both: annual falls and growths [2]. According published maps and institutional affil- to forecasts [3] the share of fuels in overall energy balance will continue changes and iations. instead of two-thirds fossil fuels in 2018, two-thirds zero-carbon energy by 2050 will be reached, supplying almost 50% of world electricity from solar and wind energy, 50-by-50. On December 11, the Council of Europe approved a plan that will raise the CO2 reduction target from 40 percent to at least 55 percent by 2030 [4]. Thus far, concentrating technologies Copyright: © 2021 by the authors. using parabolic trough collectors (PTC) were used in countries with high radiation and Licensee MDPI, Basel, Switzerland. is responsible for more 96% of installed CSP power [5]. As reported by [6] PTC-based This article is an open access article solar thermal systems are mostly used in electricity generation systems, accounting for distributed under the terms and approximately 85% of total current installed capacity worldwide. conditions of the Creative Commons Currently the most parabolic trough solar thermal power plants (PTSTPP) are in Spain Attribution (CC BY) license (https:// and the USA [7]. In the literature, examples of models of such systems and analysis of creativecommons.org/licenses/by/ possible improvements were presented. Knysh [8] proposed a model of the flow dynamics 4.0/).

Energies 2021, 14, 3003. https://doi.org/10.3390/en14113003 https://www.mdpi.com/journal/energies Energies 2021, 14, 3003 2 of 19

and heat exchange in the tube receiver of a solar parabolic trough module (PTM) to be used during designing solar energy systems with PTC of different power. Tatebi et al. [9], using a numerical model and experimental data, investigated the effect of metal foams and nanofluids on the thermal performance of a direct absorption parabolic trough collector (DAPTC). Authors concluded that porous media with high absorption coefficient and scattering coefficient could absorb more incoming radiation and transfer to heat transfer fluid, thus the increase in the collector’s efficiency can be obtained. Possibilities to improve the absorber tube of a parabolic trough collector, resulting in a heat transfer enhancement was proposed by Aldulaimi [10]. Abbas et al. [11,12] analyzed different locations in Spain and compared efficiency of PTCs and Linear Fresnel Collectors (LFCs). The results of the optimization by means of a validated Monte Carlo Ray Trace code and analytic methods showed higher annual efficiency in the case of PTCs. Hongn et al. [13] studied end optical losses of collectors. Salazal et al. [14] developed and valuated an analytic modelling of the energy flows in parabolic trough solar thermal power plants that allows for evaluation of energy savings in a case of potential modifications in components, system design and location. Whereas Ma at al. [15] presented a thermal hydraulic model solved by a novel numerical approach based on graph theory and the Newton–Raphson method. Rogada et al. [16] focused on a heat transfer fluid (HTF) used to transfer the thermal energy of solar radiation through parabolic collectors to a water vapour Rankine cycle, and proposed a model to optimize the temperature of the fluid. Similar problems were analysed by Barcia at al. [17] and a dynamic model of the HTF heating process was proposed. It included main fluid properties such as density, thermal conductivity or specific heat, thus the model was not limited to commonly used synthetic oil. On the other hand Llamas et al. [18–20] developed of a mathematical model for the optimal operation of parabolic-trough power plants with different power in context of their integration into electricity markets with minimizing grid-connection costs. Recently, the solar market trend in developed countries seems to be stabilizing. In 2017, the solar investments in developing countries were much higher (USD 115.4 billion) than in developed economies (USD 45.4 billion) that comparing with 2016 is in line with 41% increase and 17% fall, respectively [2]. The vast majority of publications related to parabolic solar collectors concern a tropical or subtropical climate [21–24], which is understandable due to the significant DNI values in these regions. In the case of temperate climate, scientific publications focus mainly on flat plate collectors [25–28], which are most often used for domestic hot water heating. It should be emphasized here that parabolic solar collectors can generate high temperature of the medium in the absorber, which is why they are often used in thermal power plants. Flat solar collectors without a mirror system, which are used in temperate climates to heat domestic hot water, do not require such high temperature as parabolic solar collectors. No information has been found in the literature on the operation of parabolic solar collectors in the literature on the operation of parabolic solar collectors in temperate climates. In the case of electricity production in temperate climates, photovoltaic (PV) panels are commonly used [29]. The main advantage of photovoltaic panels is the use of diffusive radiation to generate electricity [30]. In areas with high values of solar radiation (tropical climate), CPS power plants based on parabolic solar collectors generate about 33% [31] more electricity than PV power plants. Power plants based on photovoltaic cells and parabolic solar collectors can be combined into hybrid systems [32]. The computational methods used in simulations of the operation of parabolic solar collectors can be divided according to the complexity of the model geometry. In the most extensive three-dimensional numerical methods, thermal parameters on the absorber walls are determined depending on the mirror system or the absorber shape, as a result of which it is possible to design new constructions of focusing solar collectors [33–35]. Three- dimensional calculations are most often performed with the use of complex computational programs Ansys-Fluent [34] or OpenFOAM [35]. The second group of thermal-flow calculations in solar collectors is related to the problem of the two-dimensional cross-section of the Energies 2021, 14, 3003 3 of 19

solar collector and it is most often a simplification of three-dimensional problems [36]. Both the two-dimensional and three-dimensional problems are determined using advanced computational fluid dynamics (CFD) programs with a heat exchange module. The third group are issues related to one-dimensional flow and they can be used to design solar collector installations with infrastructure. In the case of one-dimensional problems, specialized software, e.g., System Advisor Model (SAM) [22,24], TRNSYS [37] are also used. Some- times custom source codes are created [21,38]. They require an appropriate programming environment, e.g., Matlab or the ability to write own computer programs [39]. The most popular program System Advisor Model (SAM) [40] can be used for the initial analysis of the entire solar power plant, where three areas can be distinguished: solar collector installations, heat storage tanks and devices for generating thermal energy (solar field, thermal storage, power block). All these modules in a thermal power plant are interconnected into one system. In the case of the analysis of individual devices that are part of a thermal power plant, it is good practice to use classic equations resulting from thermal-flow issues. Due to the lack of data on the operation of parabolic collectors in temperate climates, a simplified model of estimating heat generated by parabolic collectors has been developed, which can be implemented in a spreadsheet, e.g., MS Excel or Apache OpenOffice. The developed simplified algorithm for the operation of parabolic solar collectors was determined on the basis of the problem of forced convection in the absorber [41]. The presented model does not take into account the dynamics of the system and devices included in the thermal power plant [17]. Dynamic modeling [17] of solar power plants introduces possibilities of optimizing the operation of solar parabolic collectors. In this paper, a simplified model to evaluate solar energy gains from PTSTPP in different locations was developed. Moreover, using this model it is possible to estimate an area of the parabolic through solar field (PTSF) necessary to deliver the energy to meet the assumed target. The validation of the developed model was performed by comparing its results with the heat generated by a set of parabolic solar collectors at a real thermal power plant in the south of Spain. The results of this type of simulations could be useful for preliminary analysis of the CSC systems applications. The presented model has been implemented in an MS Excel spreadsheet. The aim of the work is also to investigate the numerical work of parabolic solar collectors in a temperate climate based on the existing solar installation in Spain.

2. Simplified Model of Heat Production Forecasting by Concentrating Solar Collectors Figure1 shows the satellite view of a solar power plant. In this power plants (geographic coordinates: 37◦45018.0000 N 5◦3026.0000 W), five basic parts can be distinguished (Figure2): from parabolic solar collectors with pipelines, heat storage tanks, a set of heat exchangers in which steam is produced, steam turbines with an alternator and devices for cooling the medium from the steam turbine. The description of the elements of the collector field and the main parameters of this Solar Plant are presented in Tables1 and2 , respectively. The HTF parameters adopted in Table2 were determined for the average temperature at the outlet and inlet of a single loop (Tin + Tout)/2. The collector thermal efficiency ηCPS in Table2 is defined as the ratio of the useful energy delivered to the energy incident on the collector aperture. Parabolic solar collectors are the most effective and most widely used method of large-scale heat generation. Parabolic collectors (Figure3) are composed of mirrors that reflect and focus solar radiation on pipes (absorber) filled with HTF. The liquid is then transported to the steam generation system, then the steam is used in a steam turbine to generate electricity. The model focuses on the generation of heat by the collectors which is the part that depends strictly on the local climatic conditions. The location has a minimal impact on the operation and performance of the accumulation tanks and the power generators. Energies 2021, 14, 3003 4 of 19 Energies 2021, 14, 3003 4 of 19 Energies 2021, 14, 3003 4 of 19

FigureFigure 1. 1.View View of of solar solar cylinder cylinder parabolic parabolic collectors collectors at at the the Solar Solar Thermal Thermal Power Power Plant Plant “La “La Africana” Afri- cana”Figure (“satellite” 1. View of viewsolar fromcylinder googlemaps.com). parabolic collectors at the Solar Thermal Power Plant “La Afri- (“satellite” view from googlemaps.com). cana” (“satellite” view from googlemaps.com).

Figure 2. Simplified diagram of a thermal power plant with parabolic solar collectors: 1—parabolic Figure 2. Simplified diagram of a thermal power plant with parabolic solar collectors: 1—parabolic Figuresolar collectors, 2. Simpliﬁed 2—heat diagram storage of a tanks, thermal 3—steam power plantgeneration with parabolicsystem, 4 solar—steam collectors: turbine,1 —parabolic5—cooling solar collectors, 2—heat storage tanks, 3—steam generation system, 4—steam turbine, 5—cooling solartower, collectors, 6—water2 —heatstorage storagereservoirs. tanks, 3—steam generation system, 4—steam turbine, 5—cooling tower, 6—water storage reservoirs. tower, 6—water storage reservoirs. Table 1. Description of the elements of the collector field in the Solar Thermal Power Station of te Table 1. Description of the elements of the collector field in the Solar Thermal Power Station of te southThe of Spain. hourly heat supplied by the solar collector system is determined from the follow- south of Spain. ing energy balance: Description Value Unit q Description= q − q − q (kWh),Value Unit (1) Number of mirrorsnum in a CPSsingle solarR collector:P 28 - Number of mirrors in a single solar collector: 28 - Mirrors surface area in a single solar collector: 68.2 m2 where qCPS is theMirrors heat obtained surface area from in alla si solarngle solar collectors collector: in one hour, qR is the heat68.2 losses m2 at Number of solar collectors: 8064 - the receiver (absorber), andNumberqp is heatof solar losses collectors: at heating pipes. 8064 - Surface area of all mirrors: 549,965 m2 The heat obtained fromSurface the area entire of all solar mirrors: collector installation without 549,965 heat losses m2is Number of loops: 168 - given by: Number of loops: 168 - Number of solar collectors included in one loop: 48 - Number of solar collectorsqCPS = includednloop·qloop in (onekWh loop:), 48 - (2) Number of absorbers: 24,192 - Number of absorbers: 24,192 - with nloop, the number of loops, and qloop(t) is the hourly heat produced by a single loop. The length of a singlesingle loop:loop: 556.00 556.00 m m Absorber diameter 0.07 0.07 m m Diameters of pipes transporting HTFHTF 0.1143–0.61 0.1143–0.61 m m DiametersDiameters of the outer coats of thermal insulationtion ofof pipelinespipelines transportingtransporting HTF HTF 0.2–0.8 0.2–0.8 m m

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Table 1. Description of the elements of the collector ﬁeld in the Solar Thermal Power Station of te south of Spain.

Description Value Unit Number of mirrors in a single solar collector: 28 - Mirrors surface area in a single solar collector: 68.2 m2 Number of solar collectors: 8064 - Surface area of all mirrors: 549,965 m2 Number of loops: 168 - Number of solar collectors included in one loop: 48 - Number of absorbers: 24,192 - The length of a single loop: 556.00 m Absorber diameter 0.07 m Diameters of pipes transporting HTF 0.1143–0.61 m Diameters of the outer coats of thermal insulation of 0.2–0.8 m pipelines transporting HTF

Table 2. Operation parameters of the Solar Thermal Power Station of the south of Spain.

◦ Loop Inlet Temperature Tin 292 C ◦ Loop outlet temperature Tout 392 C Mass flow through the absorber m 7.06 kg/s The dynamic viscosity of the HTF in the solar installation µ 0.00017 kg/m/s Thermal conductivity of the HTF in the solar installation λm 0.0871 W/m/K HTF density in the solar installation ρ 757.85 kg/m3 ◦ Specific heat capacity of the HTF cp 2486.5 J/kg/ C External convective heat transfer coefficient from the outside 2 ha 25 W/m /K Energies 2021, 14, 3003 of the pipe 5 of 19 Thermal conductivity of solar thermal insulation λ 0.0871 W/m/K Collectors thermal efficiency ηCPS 0.47 -

FigureFigure 3. 3.Detail Detail of of a a cylindrical-parabolic cylindrical-parabolic solar solar collector collector of of a a Solar Solar Thermal Thermal Power Power Station Station (photo (photo by by T.J. Teleszewski). T.J. Teleszewski).

TableThe 2. O heatperation produced parameters by a loopof the of Solar collectors Thermal is: Power Station of the south of Spain.

Loop Inlet Temperature Tin 292 °C qloop = π · Da · Lloop · qw (kWh), (3) Loop outlet temperature Tout 392 °C Mass flow through the absorber m 7.06 kg/s The dynamic viscosity of the HTF in the solar installation μ 0.00017 kg/m/s Thermal conductivity of the HTF in the solar installation λm 0.0871 W/m/K HTF density in the solar installation ρ 757.85 kg/m3 Specific heat capacity of the HTF cp 2486.5 J/kg/°C External convective heat transfer coefficient from the outside of ha 25 W/m2/K the pipe Thermal conductivity of solar thermal insulation λ 0.0871 W/m/K Collectors thermal efficiency η CPS 0.47 -

The model focuses on the generation of heat by the collectors which is the part that depends strictly on the local climatic conditions. The location has a minimal impact on the operation and performance of the accumulation tanks and the power generators. The hourly heat supplied by the solar collector system is determined from the following energy balance: =−− qqqqnum CPS R P (kWh), (1)

where qCPS is the heat obtained from all solar collectors in one hour, qR is the heat losses at the receiver (absorber), and qp is heat losses at heating pipes. The heat obtained from the entire solar collector installation without heat losses is given by: ⋅ 𝑞 =𝑛 𝑞 (kWh), (2)

with nloop, the number of loops, and qloop(t) is the hourly heat produced by a single loop. The heat produced by a loop of collectors is: =⋅π ⋅ ⋅ qDLqloop a loop w (kWh), (3)

where Lloop is the length of the absorber in a single loop, Da is the diameter of the absorber, and qw is the heat flux per 1 m2 of the absorber, that can be calculated by:

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where Lloop is the length of the absorber in a single loop, Da is the diameter of the absorber, 2 and qw is the heat ﬂux per 1 m of the absorber, that can be calculated by:

DNI · Am · K · ηendloss · ηshadow · ηopt · Cl 2 qw = kWh/m , (4) π · Da · LSOL

where LSOL is the absorber length in a single solar collector, Am is the mirror surface area, DNI is the direct normal irradiation, ηendloss is the coefficient for the calculation of the relative end loss, ηshadow is the coefficient for solar shading, ηopt is the optical collector efficiency for perpendicular sun on collector which is assumed to be 0.75% based on research from the work [42], Cl is the mean cleanliness factor, while K is the Incident Angle Modifier. The Incident Angle Modifier was determined according to the following equation [43]:

2 K = cos(θi) − 0.000525 · θi − 0.0000286 · θi (5)

where θi is the angle of incidence on the collector, with continuous east–west tracking, which was determined by the equation [44–47]: q 2 2 2 θi = arccos cos (α) + cos (δs) · sin (ω) (6)

α = arcsin[sin(δs) sin(φ) + cos(φ) cos(δs) cos(ω)] (7) cos(90 − α) · sin φ − sin(δs) αs = sign(ω) arccos (8) sin(90 − α) cos φ

δs = 23.45 sin(280.11 + 0.984d) (9) ◦ ω = (tsol − 12) · 15 (10)

where α is solar altitude angle, αs is the solar azimuth angle, δs is the declination angle, ω is the hourly angle, tsol is the solar time angle, φ is the latitude, and d is the number of the day of the year (from 1 for 1 January to 365 for 31 December). Coefﬁcient for the end loss is determined according to the equation [48]:

L f tan(θi) ηendloss = 1 − (11) LPTC

where Lf is the focal length of the parabolic trough collector; LPTC is the length of the parabolic trough collector. The solar shading coefﬁcient of the shadow from row to row at low solar altitude is given by the equation [44]: L η = |cos(ρ )| s (12) shadow i w

where Ls is the distance from the center of the two PTCs, w is the aperture width, and ρ is the sun tracking angle described by the following relationship:

cos(α ) ρ = tan−1 s (13) tan(α)

The basic condition for the correct operation of a single loop is to obtain the minimum temperature of HTF, Tout, at the outlet of a single loop for a nominal mass ﬂow m:

qloop Tout = + Tin, (14) m · cp

where Tin is the inlet temperature of the HTF at the entrance to the loop. Energies 2021, 14, 3003 7 of 19 Energies 2021, 14, 3003 7 of 19

If the minimum temperature Tout is not achieved, in the presented simpliﬁed model it

was assumed that the qloop is equal to zero. Obtaining the set temperature at the outlet from the loopfrom is athe necessary loop is a condition necessary for condition the correct for operationthe correct of operation the solar powerof the solar plant. power In the plant. In the case of the analyzed solar power plant, the Tout temperature is in the range◦ of 360 case of the analyzed solar power plant, the Tout temperature is in the range of 360–392 C. An increase°C–392 in °C. DNI An causesincrease an in increase DNI causes in the an temperature increase in the of thetemperature medium inof the absorber.medium in the As theabsorber. temperature As the of thetemperature oil at the outlet of the of oil the at absorber the outlet increases, of the absorber the mass increases, flow through the mass a singleflow loop through increases a single (Figure loop4a,b). increases When (Figure the maximum 4a,b). When temperature the maximum at the temperature outlet from at the ◦ the loopoutletTout from= 392 theC loop is reached, Tout = 392 the °C mass is reached, flow reaches the mass the nominal flow reaches value ofthe 7.06 nominal kg/s. value In of the event7.06 ofkg/s. a decrease In the event in the ofT aout decreasetemperature in the inTout the temperature evening hours in the (which evening is associated hours (which is with aassociated decrease inwith DNI), a decrease the mass in flowDNI), decreases the mass to flow the decreases minimum to value the minimumm = 0.01 kg/s. value m = For practical0.01 kg/s. reasons, For practical it was assumed reasons, thatit wasqloop assumedis equal tothat zero qloop if is the equal required to zero temperature if the required ◦ Tout =temperature 360 C was not Toutachieved. = 360 °C was not achieved.

(a)

(b)

Figure 4. Changes in mass ﬂow m (a) and temperature Tin and Tout (b) in a single loop on 02/07/2019 Figure 4. Changes in mass flow m (a) and temperature Tin and Tout (b) in a single loop on for the Solar Thermal Power Station in Spain. 02/07/2019 for the Solar Thermal Power Station in Spain. The heat losses of pipelines are determined by the following equation: The heat losses of pipelines are determined by the following equation: N = · ( ⋅ ) qP ∑𝑞i==1 L∑i qi 𝐿kWh 𝑞 (kWh),, (15) (15)

i i wherewhereLi is the L lengthis the length of pipe ofi andpipeq i(t) andis theq (t) heat is the losses heat perlosses unit per length unit oflength this pipe, of this that pipe, is that givenis by: given by: ( − ) Tm T()aTTπ− π qi = = ma (kW/m), (16) 1 qi Di 1 (kW/m), ln 11+D 1000 2λ di lnhaiDi + 1000 (16) λ  2 dhDiai with Tm, the medium temperature, Ta, the average-hour outside temperature, di, the diameters of the pipes, D , the diameters of the thermal insulation around the pipelines, λ, with Tm, the mediumi temperature, Ta, the average-hour outside temperature, di, the diameters of the pipes, Di, the diameters of the thermal insulation around the pipelines, λ, the

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the thermal conductivity of the thermal insulation, and ha, the of the external convective heat transfer coefficient from the outside of the pipe. thermalThe conductivity heat losses in of the the absorber thermal wereinsulation, estimated and based ha, the on of the the SCHOTT external PTRconvective®70 Receiver heat transferdocumentation coefficient available from the on outside the website of the of pipe. the absorber manufacturer [49]. To determine heatThe losses heat in losses the absorber, in the absorber a heat loss were curve estimated was used based [49 ]on and the the SCHOTT loops were PTR® divided70 Re- ceiverinto 50 documentation elements. Calculations available ofon heat the losseswebsite in of the the absorber absorber were manufacturer made for the [49]. current To de- oil terminetemperature. heat losses Assuming in the a absorber, constant heata heat flux loss along curve the was absorber used walls,[49] and a linear the loops increase were in dividedthe average into temperature50 elements. ofCalculations the medium of flowing heat losses in the in absorber the absorber was obtained:were made for the current oil temperature. Assuming a constant heat flux along the absorber walls, a linear qw·π·Da ◦ increase in the average temperatureTm(x) of= theTin me+ dium flowingx ( C) in, the absorber was obtained: (17) m·cp ⋅ ⋅ 𝑇(𝑥) = 𝑇 + 𝑥 (°C), (17) ⋅ where Tm(x) is the temperature which varies with length x of the absorber. where Tm(x) is the temperature which varies with length x of the absorber. 3. Validation of the Presented Model through Heat Measurements in an Existing Solar Thermal Power Station in Spain 3. Validation of the Presented Model Through Heat Measurements in an Existing Solar TheThermal validation Power of Station the presented in Spain simplified model for estimating heat production by parabolic solar collectors was performed by comparing the heat determined from the The validation of the presented simplified model for estimating heat production by Equation (1) with the heat measured in the parabolic solar collectors at a thermoelectric parabolic solar collectors was performed by comparing the heat determined from the Equa- solar plant in the south of Spain (Solar Thermal Power Plant “La Africana”, Tables1 and2) tion (1) with the heat measured in the parabolic solar collectors at a thermoelectric solar plant three selected day from 2019 with different climate conditions: 07/02/2019, 03/22/2019, in the south of Spain (Solar Thermal Power Plant “La Africana”, Tables 1 and 2) three selected and 09/09/2019. day from 2019 with different climate conditions: 07/02/2019, 03/22/2019, and 09/09/2019. The actual values of direct solar radiation measured in Cordoba were used for the The actual values of direct solar radiation measured in Cordoba were used for the calculations. In the Solar Thermal Power Station of the south of Spain, the automation of calculations. In the Solar Thermal Power Station of the south of Spain, the automation of parabolic solar collectors is equipped with east–west tracking and is characterized by high parabolicpercentage solar tracking collectors performance. is equipped Percentage with east–west tracking tracking performance and is characterized [50] is a parameter by high percentagethat compares tracking the absorbed performance. energy Percentage for a given tracking type of tracking performance in relation [50] is to a the parameter absorbed thatenergy compares of full the tracking absorbed according energy to for the a equation:given type of tracking in relation to the absorbed energy of full tracking according to the equation: Absorbed energy for a given type of tracking Percentage𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 tracking performance 𝑡𝑟𝑎𝑐𝑘𝑖𝑛𝑔= 𝑝𝑒𝑟𝑓𝑜𝑟𝑚𝑎𝑛𝑐𝑒= Absorbed energy for full tracking 100100 (%) (18) (%) (18) FigureFigure 5 shows the percentage east–west tracking performance in southernsouthern SpainSpain comparedcompared to to full full tracking tracking mode mode for for the the four four se selectedlected days. days. East–west East–west tracking tracking is is relatively relatively similarsimilar compared compared to to full full tracking. tracking.

FigureFigure 5. 5. PercentagePercentage tracking tracking performance performance compared compared to to fu fullll tracking tracking mode mode on on selected selected days days at at the the SolarSolar Thermal Thermal Power Power Station Station in in Spain. Spain.

TheThe calculations calculations assumed assumed the the mass mass flow flow in inthe the range range from from 5 kg/s 5 kg/s to 7.06 to 7.06 kg/s kg/s with with the minimumthe minimum temperature temperature condition condition (Equation (Equation (14)) at (14)) the atoutflow the outflow from the from loop the equal loop to equal 360 ◦ °C.to 360In factC. the In factflow the increases flow increases gradually gradually with the withincrease the increasein direct insolar direct radiation, solar radiation, reaches its maximum value equal to 7.06 kg/s and then decreases in the evening with the decrease in

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reaches its maximum value equal to 7.06 kg/s and then decreases in the evening with solather decreaseradiation. in An solar example radiation. of the course An example of changes of thein the course mass offlow changes measured in the on 02/07/2018 mass flow inmeasured a single loop on02/07/2018 is shown in Figure in a single 4a,b. loopIt should is shown be noted in Figure here that4a,b. in Itpractice, should solar be noted collectors here arethat used in practice, to produce solar heat collectors for a solar are usedpower to plant produce only heat when for the a solar required power temperature plant only at when the outletthe required from the temperature solar system at is achieved the outlet and from at nominal the solar mass system flow. is achieved and at nominal mass flow. WithWith these these assumptions, assumptions, the the heat heat flux flux per per square square meter meter of of the the absorber absorber can can be be calcu- calcu- latedlated by by Equation Equation (4). (4). Figure Figure 6 shows calculationscalculations of this heat fluxflux density together with DNIDNI for for selected selected days days in in the the Spanish Spanish Solar Solar Plant. Plant. The The maximum maximum unit unit heat heat flux flux is about is about 25 kW25 kWper persquare square meter meter of the of theabsorber. absorber.

FigureFigure 6. 6. CalculatedCalculated heat heat flux ﬂux density density per one square metermeter ofof anan absorberabsorber and and DNI DNI for for selected selected days daysfor the for Solar the Solar Thermal Thermal Power Power Station Station in Spain. in Spain.

TheThe next next stage stage of of model model validation validation consisted consisted in in comparing comparing the the numerical numerical values values of of annualannual heat heat qqsuppliedsupplied by by the the solar solar collector collector system, system, calculated calculated by by Equation Equation (1), (1), and and actual actual valuesvalues measured measured in in Solar Solar Plant Plant for for selected selected days days of of the the year: year: 22/03/2019, 22 March 2019, 02/07/2019 2 July 2019and 29/09/2019.and 29 September The results 2019. of Thethis resultscomparison of this are comparison shown in Figure are shown 7a–c. in A Figure good 7agreementa–c. A good is foundagreement between is found the measured between theand measured calculated and values calculated of annual values heat ofsupplied annual heatby the supplied collec- torby field the collector for the different field for thedates. different This agr dates.eement This shows agreement the goodness shows the of goodnessthe model. of The the observedmodel. The differences observed between differences these between values is these caused values by the is control caused of by the the flow control in the of ab- the sorberflow in by the the absorber automatic by control the automatic system. control The simulations system. The in this simulations work were in thisperformed work were for theperformed flow in the for range the flow from in the5 kg/s range to 7.06 from kg/s. 5 kg/s to 7.06 kg/s. According this comparison, the relative error of the developed model was determined according to the following relationship:

qreal − qnum δqnum = 100%, (19) qreal

where qnum is the heat determined from Equation (1) and qreal is the heat measured in the thermoelectric solar plant in Spain, Cordoba province. Table3 presents the calculation results(a) of the model and the actual values of heat produced by parabolic solar collectors in March, July and September. The calculations also considered the shutdown time of parabolic solar collectors intended for conservation works and for March, July and Septem- ber, respectively, 11, 3 and 4 days. The relative error of the presented model for the selected months did not exceed 7.1%. Energies 2021, 14, 3003 9 of 19

solar radiation. An example of the course of changes in the mass flow measured on 02/07/2018 in a single loop is shown in Figure 4a,b. It should be noted here that in practice, solar collectors are used to produce heat for a solar power plant only when the required temperature at the outlet from the solar system is achieved and at nominal mass flow.

With these assumptions, the heat flux per square meter of the absorber can be calculated by Equation (4). Figure 6 shows calculations of this heat flux density together with DNI for selected days in the Spanish Solar Plant. The maximum unit heat flux is about 25 kW per square meter of the absorber.

Figure 6. Calculated heat flux density per one square meter of an absorber and DNI for selected days for the Solar Thermal Power Station in Spain.

The next stage of model validation consisted in comparing the numerical values of annual heat q supplied by the solar collector system, calculated by Equation (1), and actual values measured in Solar Plant for selected days of the year: 22/03/2019, 02/07/2019 and 29/09/2019. The results of this comparison are shown in Figure 7a–c. A good agreement is found between the measured and calculated values of annual heat supplied by the collector field for the different dates. This agreement shows the goodness of the model. The Energies 2021, 14, 3003 observed differences between these values is caused by the control of the flow in the10 of ab- 19 sorber by the automatic control system. The simulations in this work were performed for the flow in the range from 5 kg/s to 7.06 kg/s.

(a)

Energies 2021, 14, 3003 10 of 19

(b)

(c)

Figure 7.7. Comparison ofof heatheat producedproduced by by solar solar collectors collectors with with heat heat determined determined from from the the developed devel- modeloped model the Solar the Solar Thermal Thermal Power Power Station Station in Spain in Spain for selectedfor selected days: days: (a) ( 22.03.2019,a) 22.03.2019, (b )(b 02.07.2019,) (02.07.2019,c) 29.09.2019. (c) 29.09.2019.

FigureAccording8 shows this the comparison, temperature the dependence relative error on theof outputthe developed of a single model loop consistingwas deter- ofmined 48 solar according collectors to the as following a function relationship: of the density of DNI and mass flow in the absorber. With the decrease in mass flow in the absorber and the increase in solar radiation intensity, qq− the temperature at the outlet fromδ a single= real loop num increases. The minimum temperature at qnum 100% , (19) ◦ q the outlet of a single loop is 360 C and is achievedreal for DNI of 450 Wh/m2, 670 Wh/m2 and 905 Wh/m2 for flow rates of 5 kg/s, 7 kg/s and 10 kg/s, respectively. In the case of a where qnum is the heat determined from Equation (1) and qreal is the heat measured in the working temperature of 392 ◦C, it is obtained for 680 Wh/m2, 920 Wh/m2 for 5 kg/s and thermoelectric solar plant in Spain, Cordoba province. Table 3 presents the calculation 7 kg/s, respectively. With a mass flow of 10 kg/s, the operating temperature is not reached results of the model and the actual values of heat produced by parabolic solar collectors in March, July and September. The calculations also considered the shutdown time of parabolic solar collectors intended for conservation works and for March, July and Septem- ber, respectively, 11, 3 and 4 days. The relative error of the presented model for the selected months did not exceed 7.1%.

Table 3. Relative Errors of the model in the selected days.

Heat Determined Relative Er- Number of Days in the Month Heat Measured Selected from the Model ror When Solar Collectors Were qreal Month qnum δqnum Turned off MWh MWh % March 2019 11 22,043 20,508 6.96 July 2019 3 61,076 64,210 −5.13 September 2019 4 43,312 40,249 7.07

EnergiesEnergies 20212021, 14, 14, 3003, 3003 1111 of of 19 19 Energies 2021, 14, 3003 11 of 19

for the maximum DNI. The maximum operating temperature of 400 ◦C is achieved with a ﬂowFigure of 5 kg/s. 8 shows the temperature dependence on the output of a single loop consisting of 48 solarFigure collectors 8 shows asthe a temperature function of thedependence density of on DNI the outputand mass of aflow single in loopthe absorber. consisting Withof 48 Tablethe solar decrease 3. collectorsRelative in Errorsmass as a offlowfunction the in model the of absorberinthe the density selected and of days.the DNI increase and mass in solar flow radiation in the absorber. intensity,With the the temperature decrease in at mass the outlet flow fromin the a absorbersingle loop and increases. the increase The minimumin solar radiation temperature inten- Heat Determined sity, the temperature at the outlet from a single loop increases. The minimum2 temperature2 Numberat the outlet of Days of in a thesingle loopHeat is 360 Measured °C and is achieved for DNI of 450 Wh/mRelative, 670 ErrorWh/m from the Model 2 2 at the outlet of2 a single loop is 360qreal °C and is achieved for DNI of 450 Wh/m , 670 Wh/m Selected Month andMonth 905 When Wh/m Solar for flow rates of 5 kg/s, 7 kg/s and 10 kg/s,qnum respectively. In theδq numcase of a Collectorsworkingand 905 Were temperature Wh/m Turned2 for off flow of 392 rates °C, ofit is5 kg/s,obtained 7 kg/s for and 680 10Wh/m kg/s,2, 920respectively. Wh/m2 for In 5 the kg/s case and of 7 a MWh MWh % kg/s,working respectively. temperature With of a mass392 °C, flow it is of obtained 10 kg/s, forthe 680operating Wh/m 2temperature, 920 Wh/m2 foris not 5 kg/s reached and 7 March 2019forkg/s, the respectively.maximum 11 DNI. With The a mass maximum 22,043flow of operating 10 kg/s, thetemperature operating 20,508 of temperature 400 °C is achieved is 6.96not reached with July 2019 3 61,076 64,210 −5.13 a forflow the of maximum 5 kg/s. DNI. The maximum operating temperature of 400 °C is achieved with September 2019a flow of 4 5 kg/s. 43,312 40,249 7.07

Figure 8. Medium temperature in the absorber at the outlet of a single loop as a function of mass Figure 8. Medium temperature in the absorber at the outlet of a single loop as a function of mass flowFigure and 8. solar Medium radiation temperature (calculations in the performed absorber atfor the the outl conditionset of a single in Cordoba). loop as a function of mass flowflow and solar radiation (calculations performed for the conditions in Cordoba). FigureFigure 99 shows shows an an example example of of the the temperat temperatureure distribution distribution of of the the medium medium flowing flowing Figure 9 shows an example of the temperature distribution of the medium flowing throughthrough the the absorber absorber as as a afunction function of of the the absorber absorber length length for for a asingle single loop loop for for a aconstant constant through the absorber as a function of the absorber2 length2 for a single2 loop for a constant2 massmass flow flow of of 6 6kg/s kg/s and and selected selected DNI: DNI: 1000 1000 W/m W/m, 8002, 800 W/m W/m, 6002, W/m 600 W/m and2 400and W/m 400 W/m. The 2. mass flow of 6 kg/s and selected DNI: 1000 W/m2, 800 W/m2, 600 W/m2 and 400 W/m2. The calculationsThe calculations were made were according made according to the Equation to the Equation (8). The (8).resulting The resultinglinear rise linear in temper- rise in calculations were made according to the Equation (8). The resulting linear rise in temperaturetemperature of the medium of the medium in the absorber in the absorber is typica isl typicalof the constant of the constant condition condition of the ofheat the flux heat ature of the medium in the absorber is typical of the constant condition of the heat flux alongflux alongthe walls the [41] walls of [ 41the] ofabsorber. the absorber. along the walls [41] of the absorber.

FigureFigure 9. 9.TemperatureTemperature distribution distribution of of the the medium medium flow flowinging through through the the absorber absorber as as a a function function of of the theFigureabsorber absorber 9. lengthTemperature length for for different different distribution DNI. DNI. of the medium flowing through the absorber as a function of the absorber length for different DNI. TheThe results results of of calculations calculations of of the the convective convective heat heat transfer transfer coefficient coefficient inside inside the the ab- ab- sorbersorberThe are are presentedresults presented of below.calculations below. For For this of this thepurpose, purpose, convective a asimplified simplified heat transfer model model ofcoefficient of forced forced convection convectioninside the for ab- for sorber are presented below. For this purpose, a simplified model of forced convection for

Energies 2021, 14, 3003 12 of 19

turbulent flow was used. The local convective heat transfer coefficient is described by the turbulentfollowing flow equation: was used. The local convective heat transfer coefficient is described by the following equation: Nu λ h Nu= DλD h =f ,, (20)(20) f DD wherewhereD Dis is the the absorber absorber diameter, diameter,λ isλ theis the conductivity conductivity coefficient coefficient of the of the medium, medium, while while the Nusseltthe Nusselt number number is described is described by the by Dittus–Boelterthe Dittus–Boelter correlation: correlation: c μ =≥≤≤=4/5 0.44/5 0.4 cpµ 4m4m Nu = 0.023ReNuD 0.023RePr , Re Pr≥ , 10, Re 000, 10000, 0.6 0.6 ≤ Pr ≤ Pr1600, 1600, Pr Pr= = , Re Re= ?, , (21)(21) D λπμλ πDDµ

wherewhere ReRe isis thethe ReynoldsReynolds number,number, while while Pr Pr is is the the Prandtl Prandtl number. number. FigureFigure 10 10 shows shows the the relationship relationship of of convective convective heat heat transfer transfer coefficient. coefficient. The The convection convec- coefficienttion coefficient increases increases with increasingwith increasing flow throughflow through the absorber. the absorber. With a With nominal a nominal flow value flow ofvaluem = of 7.06 m kg/s,= 7.06 thekg/s, convective the convective heat transfer heat transfer coefficient coefficient is 2714 is W/(m 2714 2W/(mK). The2K). determined The deter- valuemined of value the convective of the convective heat transfer heat transfer coefficient coefficient is relatively is relatively high compared high compared to the values to the determinedvalues determined in the literaturein the literatur [33,38e ],[33,38], which which is the is result the result of much of much larger larger real real Reynolds Reyn- numbersolds numbers in relation in relation to the to Reynolds the Reynolds numbers numb fromers worksfrom works [33,38 [33,38].]. Increasing Increasing the mass the mass flow offlow the of medium the medium in the in absorber the absorber to 10 kg/sto 10 increaseskg/s increases the convective the convective heat transfer heat transfer coefficient coef- byficient about by 24%. about 24%.

FigureFigure 10.10. ConvectiveConvective heatheat transfertransfer coefﬁcientcoefficient asas aa functionfunction ofof massmass ﬂowflowin inthe theabsorber. absorber.

4.4. ApplicationApplication ofof thethe ModelModel forfor thethe PerformancePerformance PredictionPrediction ofof aa SolarSolar ThermalThermal PowerPower Station Located in a Temperate Climate Location Station Located in a Temperate Climate Location This chapter presents the results of calculations of heat produced by a system of This chapter presents the results of calculations of heat produced by a system of par- parabolic solar collectors for a selected region of temperate climate: in the city of Bialystok locatedabolic solar in the collectors north-eastern for a selected part of Poland.region of The temperate climatic climate: conditions in the of thiscity localizationof Bialystok arelocated presented in the innorth-eastern Figure 11 and part compared of Poland. with The Spanish climatic conditions. conditions of The this main localization differences are withpresented the south in Figure of Spain 11 and are compared much lower with values Spanish of direct conditions. solar radiation The main and differences temperature, with whichthe south is mainly of Spain due are to much thedifferent lower values latitude. of direct The solar annual radiation value ofand DNI temperature, in Cordoba which was 1938is mainly kWh/m due2/year, to the while different in the latitude. case of Bialystok,The annual the value annual of DNIDNI isin 467 Cordoba kWh/m was2/year. 1938 2 2 kWh/mThe/year, correct while operation in the case of a of thermoelectric Bialystok, the powerannualplant DNI is depends 467 kWh/m on the/year. minimum temperatureThe correct at the operation outlet of of the a thermoelectric solar installation, power giving plant by depends Equation on (14), the which minimum is 360 tem-◦C forperature a thermoelectric at the outlet power of the plant solar in installation, Spain in the giving Cordoba by Equation province, (14), therefore which two is 360 variants °C for werea thermoelectric adopted for power the calculations: plant in Spain in the Cordoba province, therefore two variants were adopted for the calculations: • Variant A, the result of the calculations is the heat produced by parabolic solar collec- • Varianttors without A, the considering result of the the calculations minimum temperatureis the heat produced at the outlet by parabolic from a single solar loop collec- of tors48 solar without collectors, considering the minimum temperature at the outlet from a single loop of • 48Variant solar B,collectors, the result of the calculations is the heat produced by parabolic solar collec- • Varianttors, considering B, the result the of minimum the calculations temperature is the heat at the produced outlet of by a single parabolic loop solar of 48 collec- solar tors,collectors considering equal to the 360 minimum◦C. temperature at the outlet of a single loop of 48 solar collectors equal to 360 °C.

Energies 2021, 14, 3003 13 of 19

In the case of variant A (Figure 12a), the heat generated in the south of Spain by the presented parabolic collectors is approximately four times greater than in the case of Bi- alystok (Table 4) Parabolic collectors require direct radiation, which is also four times lower in Bialystok compared to South of Spain (Figure 11a). Diffuse solar radiation (Figure 11b) in Spain and Bialystok is similar but, unfortunately, it does not play a significant role in the production of heat in the case of concentrating solar collectors.

Table 4. Annual heat produced by solar collectors in Spain (Cordoba) and Poland (Bialystok).

Case A Case B Region Heat (MWh) South of Spain 528,388 406,383 Poland (Bialystok) 132,536 52,269

In the case of variant B (Figure 12b), the estimated amount of heat generated by solar collectors in Bialystok is eight times lower than the amount of heat produced by solar collectors in the south of Spain (Table 4). The greatest differences between the heat produced in Spain and Bialystok are in the fall and winter months, and the smallest in the summer. As in variant A, these differences are mainly caused by a small share of direct solar radiation and low outdoor temperature in Poland compared to southern Spain (Fig- ure 11c). Higher temperature of the medium in the absorber and lower temperature outside causes greater heat losses in the absorber and the solar pipes. In December, the expected heat production by parabolic solar collectors in Bialystok is close to zero for variant B. Many publications [18–20,51–54] describe studies of the use of parabolic collectors for thermoelectric power plants in countries such as Greece, Spain, China and Nigeria, which are located in the subtropical and tropical climatic zone, i.e., in the area where direct solar radiation is significant. The territory of Poland is located in a temperate climate zone, Energies 2021, 14, 3003 13 of 19 which is not recommended by producers of parabolic solar collectors used to produce heat for the operation of typical steam turbines.

(a)

Energies 2021, 14, 3003 14 of 19

(b)

(c)

Figure 11. Hourly annual distribution of direct solar radiation (a), diffuse solar radiation (b) and Figure 11. Hourly annual distribution of direct solar radiation (a), diffuse solar radiation (b) and outdoor temperature (c). outdoor temperature (c).

® In the case ofAccording variant A (Figureto the SCHOTT 12a), the heatPTR generated70 Receiver in brochure the south [49], of Spain the worldwide by the uses are presented parabolicdivided collectors into four iscategories: approximately “Excellent”, four times “Good”, greater “Satisfactory”, than in the case “Unsuitable”. of Bia- The map lystok (Table4of) Parabolicthe use of collectors solar collectors require [49] direct is consistent radiation, with which the is map also of four annual times solar lower radiation in the world [55]. Cordoba is in the area marked “Good”, while Bialystok is labelled “Unsuitable”. The above calculation results using the simplified model confirm this fact. Comparing the map in SCHOTT PTR®70 Receiver [49] with the map of the distribution of climatic zones [53], it can be noticed that the best location of parabolic solar collectors is the climate of the tropical and subtropical zones, excluding parts of South America and Africa with tropical rainforest. One solution to increase the heat production of parabolic solar collectors in temperate climates is to use more collectors in a single loop, which allows for higher heat production with a lower DNI value. The disadvantages of this solution are additional investment costs, higher hydraulic losses in the loops and significant fluctuations in the medium flow in the absorber. It should be noted that the simplified algorithm based on the forced convection presented in this paper may be useful for estimating the heat production by solar collectors in a solar thermal power station under various climatic conditions.

Energies 2021, 14, 3003 14 of 19

Energies 2021, 14, 3003 15 of 19 in Bialystok compared to South of Spain (Figure 11a). Diffuse solar radiation (Figure 11b) in Spain and Bialystok is similar but, unfortunately, it does not play a signiﬁcant role in the production of heat in the case of concentrating solar collectors.

(a)

(b)

Figure 12. Annual heat produced by parabolic solar collectors for variants A (a) and B (b). Figure 12. Annual heat produced by parabolic solar collectors for variants A (a) and B (b). Table 4. Annual heat produced by solar collectors in Spain (Cordoba) and Poland (Bialystok). 5. Conclusions A simplified model for evaluation ofCase solar A energy gains from PTSTPP Case was B developed Region and verified on based on data measurements data. TheHeat relative (MWh) error of the presented al- gorithmSouth based of on Spain the forced convection for 528,388 the selected months did not 406,383 exceed 7.1%, thus it shouldPoland be noted (Bialystok) that the model can be used 132,536 to estimate the heat production 52,269 by parabolic solar collectors, considering metrological conditions and temperature at the outlet from the solar installation for different climatic zones. The simplified method of estimating heat In the case of variant B (Figure 12b), the estimated amount of heat generated by production by parabolic solar collectors presented in this paper may be an alternative to the solar collectors in Bialystok is eight times lower than the amount of heat produced by most commonly used computer programs, such as SAM, in which simulations are performed solar collectors in the south of Spain (Table4). The greatest differences between the heat for ready-made diagrams of thermal power plants. The presented method makes it possible produced in Spain and Bialystok are in the fall and winter months, and the smallest in the summer.to estimate As the in variant heat generated A, these differencesby parabolic are sola mainlyr collectors, caused regardless by a small of share the typical of direct schemes solar radiationof a thermal and power low outdoor plant with temperature a part of inheat Poland accumulation compared in tostorage southern tanks Spain and (Figurea part related 11c). to the production of electricity. The main advantage of the presented method is the possibility of quick implementation of the presented equations in spreadsheets. Thermoelectric plants equipped with parabolic solar collectors are dynamically developed as alternative energy sources in the area of tropical and subtropical climate. The

Energies 2021, 14, 3003 15 of 19

Higher temperature of the medium in the absorber and lower temperature outside causes greater heat losses in the absorber and the solar pipes. In December, the expected heat production by parabolic solar collectors in Bialystok is close to zero for variant B. Many publications [18–20,51–54] describe studies of the use of parabolic collectors for thermoelectric power plants in countries such as Greece, Spain, China and Nigeria, which are located in the subtropical and tropical climatic zone, i.e., in the area where direct solar radiation is significant. The territory of Poland is located in a temperate climate zone, which is not recommended by producers of parabolic solar collectors used to produce heat for the operation of typical steam turbines. According to the SCHOTT PTR®70 Receiver brochure [49], the worldwide uses are divided into four categories: “Excellent”, “Good”, “Satisfactory”, “Unsuitable”. The map of the use of solar collectors [49] is consistent with the map of annual solar radiation in the world [55]. Cordoba is in the area marked “Good”, while Bialystok is labelled “Unsuitable”. The above calculation results using the simplified model confirm this fact. Comparing the map in SCHOTT PTR®70 Receiver [49] with the map of the distribution of climatic zones [53], it can be noticed that the best location of parabolic solar collectors is the climate of the tropical and subtropical zones, excluding parts of South America and Africa with tropical rainforest. One solution to increase the heat production of parabolic solar collectors in temperate climates is to use more collectors in a single loop, which allows for higher heat production with a lower DNI value. The disadvantages of this solution are additional investment costs, higher hydraulic losses in the loops and significant fluctuations in the medium flow in the absorber. It should be noted that the simplified algorithm based on the forced convection presented in this paper may be useful for estimating the heat production by solar collectors in a solar thermal power station under various climatic conditions.

5. Conclusions A simplified model for evaluation of solar energy gains from PTSTPP was developed and verified on based on data measurements data. The relative error of the presented algorithm based on the forced convection for the selected months did not exceed 7.1%, thus it should be noted that the model can be used to estimate the heat production by parabolic solar collectors, considering metrological conditions and temperature at the outlet from the solar installation for different climatic zones. The simplified method of estimating heat production by parabolic solar collectors presented in this paper may be an alternative to the most commonly used computer programs, such as SAM, in which simulations are performed for ready-made diagrams of thermal power plants. The presented method makes it possible to estimate the heat generated by parabolic solar collectors, regardless of the typical schemes of a thermal power plant with a part of heat accumulation in storage tanks and a part related to the production of electricity. The main advantage of the presented method is the possibility of quick implementation of the presented equations in spreadsheets. Thermoelectric plants equipped with parabolic solar collectors are dynamically developed as alternative energy sources in the area of tropical and subtropical climate. The results of calculations using a simplified calculation model, which were verified on the existing thermoelectric plant, indicate that the thermal efficiency of parabolic solar collectors working with standard steam turbines is eight times lower than in the case of the existing thermoelectric plant in the south of Spain. Therefore, it can be concluded that in the climatic conditions of Poland, the use of solar plant based on concentrated solar collectors is not justified. In the opinion of the authors the solution to the problem is the use of low-temperature technologies, which will be the subject of further research on the viability of using Parabolic Trough Solar Thermal Power Plants in moderate climate conditions. Energies 2021, 14, 3003 16 of 19

Author Contributions: T.J.T. and M.Z.˙ are co-author of the concept, T.J.T. developed the model, M.Z.,˙ D.A.K. and A.R. proposed corrections in algorithm, all co-authors conducted analysis of meteorological data, T.J.T. wrote a draft of the paper and prepared figures, while M.Z.,˙ D.A.K. and A.R. corrected the text and conclusions. All authors have read and agreed to the published version of the manuscript. Funding: This scientific project was financed from funds of BUT InterAcademic Partnerships (PPI/APM/2018/1/00033/DEC/1) NAWA project and WZ/WBiIS/9/2019 scientific research at Bialystok University of Technology. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. Acknowledgments: Authors would like to thank the Polish NAWA Agency for financial support. Also, the authors also are grateful to Juan Manuel Vizcaino García and Rosa Rumi Palomo, Plant Director and Head of Operation of the Solar Thermal Power Plant “La Africana”, respectively, for their collaboration in this research. Conflicts of Interest: The authors declare no conflict of interest.

Nomenclature

2 Am mirror surface area(m ) Cl mean cleanliness factor ◦ cp specific heat capacity of the HTF (J/kg/ C) d number of the day of the year (from 1 for January 1 to 365 for December 31) di diameters of the pipes (m) Da absorber diameter (m) Di diameters of the thermal insulation of the pipelines (m) DNI direct normal irradiance for hourly values (Wh/m2) 2 ha external convective heat transfer coefficient from the outside of the pipe (W/m /K) 2 hf local convective heat transfer coefficient (W/m /K) K Incident Angle Modifier (-) Lf focal length of the parabolic trough collector (m) Lloop absorber length in a single loop (m) LPTC length of the parabolic trough collector (m) LS distance from the center of the two PTCs (m) m mass flow rate through the absorber (kg/s) nloop number of loops (-) NuD Nusselt number (-) Pr Prandtl number (-) qCPS heat obtained from all solar collectors during one hour (kWh) qloop hourly heat produced by a single loop (kWh) qnum hourly heat supplied by the solar collector system (kWh) qP heat losses from heating pipes (kWh) qR heat loss from the absorber (kWh) Re Reynolds number (-) tsol solar time angle (h) ◦ Ta average-hour outside temperature ( C) ◦ Tin single loop inlet fluid temperature ( C) ◦ Tm medium temperature ( C) ◦ Tout single loop outlet fluid temperature ( C) Greek symbols α solar altitude angle (degree) αs solar azimuth angle (degree) δS declination angle (degree) Energies 2021, 14, 3003 17 of 19

ηendloss coefficient for the calculation of the relative end loss (-) ηopt optical collector efficiency for perpendicular sun position (-) ηshadow coefficient for solar shading (-) θi angle of incidence on the collector surface (degree) λ thermal conductivity of the thermal insulation (W/m/K) λm thermal conductivity of the medium (W/m/K) µ dynamic viscosity of the HTF in the solar installation (kg/m/s) ρ sun tracking angle (degree) φ Latitude (degree) ω hourly angle (degree) Abbreviations CSC Concentrated Solar Collector HTF Heat Transfer Fluid PTC Parabolic Trough Collector PTR Parabolic trough reflector PTSTPP Parabolic trough solar thermal power plant PTSF Parabolic trough solar field

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