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

of the Applicability of the Parabolic Trough Solar Plants in the Locations Climate. 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. The heat generated by a similar solar collector system in a selected region with a temperate climate, the city of Bialystok (north-eastern Poland, geographic coordinates: 53 ◦ 08 (cid:48) 07 (cid:48)(cid:48) N 23 ◦ 08 (cid:48) 44 (cid:48)(cid:48) E) was determined by the developed simpliﬁed model for different months of the year. Based on the results of the analysis, it was found that the energy obtained from the same area of concentrated solar collectors located near Bialystok is eight times lower compared to the location in Cordoba depending on the variant of the power plant operation. heat transfer coefﬁcient as a function of mass ﬂow in the absorber.


Introduction
Solar energy is one of significantly developing branches of renewable energy sources. 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 investments in solar systems sector show 23% increase, comparing to 2004, whereas a period between 2013 and 2018 was marked by both: annual falls and growths [2]. According to forecasts [3] the share of fuels in overall energy balance will continue changes and 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 CO 2 reduction target from 40 percent to at least 55 percent by 2030 [4]. Thus far, concentrating technologies using parabolic trough collectors (PTC) were used in countries with high radiation and is responsible for more 96% of installed CSP power [5]. As reported by [6] PTC-based solar thermal systems are mostly used in electricity generation systems, accounting for approximately 85% of total current installed capacity worldwide.
Currently the most parabolic trough solar thermal power plants (PTSTPP) are in Spain and the USA [7]. In the literature, examples of models of such systems and analysis of possible improvements were presented. Knysh [8] proposed a model of the flow dynamics 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][19][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][22][23][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][26][27][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][34][35]. Threedimensional 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 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. Sometimes 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. Figure 1 shows the satellite view of a solar power plant. In this power plants (geographic coordinates: 37 • 45 18.00 N 5 • 3 26.00 W), five basic parts can be distinguished ( Figure 2): 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 Tables 1 and 2, respectively. The HTF parameters adopted in Table 2 were determined for the average temperature at the outlet and inlet of a single loop (T in + T out )/2. The collector thermal efficiency η CPS in Table 2 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 (Figure 3) 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.

Simplified Model of Heat Production Forecasting by Concentrating Solar Collectors
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: where q CPS is the heat obtained from all solar collectors in one hour, q R is the heat losses at the receiver (absorber), and q p is heat losses at heating pipes. The heat obtained from the entire solar collector installation without heat losses is given by: with n loop , the number of loops, and q loop (t) is the hourly heat produced by a single loop.     The heat produced by a loop of collectors is: where L loop is the length of the absorber in a single loop, D a is the diameter of the absorber, and q w is the heat flux per 1 m 2 of the absorber, that can be calculated by: where L SOL is the absorber length in a single solar collector, A m is the mirror surface area, DNI is the direct normal irradiation, η endloss is the coefficient for the calculation of the relative end loss, η shadow is the coefficient for solar shading, η opt is the optical collector efficiency for perpendicular sun on collector which is assumed to be 0.75% based on research from the work [42], C l is the mean cleanliness factor, while K is the Incident Angle Modifier. The Incident Angle Modifier was determined according to the following equation [43]: where θ i is the angle of incidence on the collector, with continuous east-west tracking, which was determined by the equation [44][45][46][47]: where α is solar altitude angle, α s is the solar azimuth angle, δ s is the declination angle, ω is the hourly angle, t sol 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]: where L f is the focal length of the parabolic trough collector; L PTC 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]: where L s 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: The basic condition for the correct operation of a single loop is to obtain the minimum temperature of HTF, T out , at the outlet of a single loop for a nominal mass flow m: where T in is the inlet temperature of the HTF at the entrance to the loop. If the minimum temperature T out is not achieved, in the presented simplified model it was assumed that the q loop is equal to zero. Obtaining the set temperature at the outlet from the loop is a necessary condition for the correct operation of the solar power plant. In the case of the analyzed solar power plant, the T out temperature is in the range of 360-392 • C. An increase in DNI causes an increase in the temperature of the medium in the absorber. As the temperature of the oil at the outlet of the absorber increases, the mass flow through a single loop increases (Figure 4a,b). When the maximum temperature at the outlet from the loop T out = 392 • C is reached, the mass flow reaches the nominal value of 7.06 kg/s. In the event of a decrease in the T out temperature in the evening hours (which is associated with a decrease in DNI), the mass flow decreases to the minimum value m = 0.01 kg/s. For practical reasons, it was assumed that q loop is equal to zero if the required temperature T out = 360 • C was not achieved. from the loop is a necessary condition for the correct operation of the solar power plant.
In the case of the analyzed solar power plant, the Tout temperature is in the range of 360 °C-392 °C. An increase in DNI causes an increase in the temperature of the medium in the absorber. As the temperature of the oil at the outlet of the absorber increases, the mass flow through a single loop increases (Figure 4a,b). When the maximum temperature at the outlet from the loop Tout = 392 °C is reached, the mass flow reaches the nominal value of 7.06 kg/s. In the event of a decrease in the Tout temperature in the evening hours (which is associated with a decrease in DNI), the mass flow decreases to the minimum value m = 0.01 kg/s. For practical reasons, it was assumed that qloop is equal to zero if the required temperature Tout = 360 °C was not achieved.  The heat losses of pipelines are determined by the following equation: where Li is the length of pipe i and qi(t) is the heat losses per unit length of this pipe, that is given by: with Tm, the medium 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 The heat losses of pipelines are determined by the following equation: where L i is the length of pipe i and q i (t) is the heat losses per unit length of this pipe, that is given by: with T m , the medium temperature, T a , the average-hour outside temperature, d i , the diameters of the pipes, D i , the diameters of the thermal insulation around the pipelines, λ, the thermal conductivity of the thermal insulation, and h a , the of the external convective heat transfer coefficient from the outside of the pipe. The heat losses in the absorber were estimated based on the SCHOTT PTR ® 70 Receiver documentation available on the website of the absorber manufacturer [49]. To determine heat losses in the absorber, a heat loss curve was used [49] and the loops were divided into 50 elements. Calculations of heat losses in the absorber were made for the current oil temperature. Assuming a constant heat flux along the absorber walls, a linear increase in the average temperature of the medium flowing in the absorber was obtained: where T m (x) is the temperature which varies with length x of the absorber.

Validation of the Presented Model through Heat Measurements in an Existing Solar Thermal Power Station in Spain
The validation of the presented simplified model for estimating heat production by parabolic solar collectors was performed by comparing the heat determined from the Equation (1) Figure 5 shows the percentage east-west tracking performance in southern Spain compared to full tracking mode for the four selected days. East-west tracking is relatively similar compared to full tracking. thermal conductivity of the thermal insulation, and ha, the of the external convective heat transfer coefficient from the outside of the pipe. The heat losses in the absorber were estimated based on the SCHOTT PTR ® 70 Receiver documentation available on the website of the absorber manufacturer [49]. To determine heat losses in the absorber, a heat loss curve was used [49] and the loops were divided into 50 elements. Calculations of heat losses in the absorber were made for the current oil temperature. Assuming a constant heat flux along the absorber walls, a linear increase in the average temperature of the medium flowing in the absorber was obtained: where Tm(x) is the temperature which varies with length x of the absorber.

Validation of the Presented Model Through Heat Measurements in an Existing Solar Thermal Power Station in Spain
The validation of the presented simplified model for estimating heat production by parabolic solar collectors was performed by comparing the heat determined from the Equation (1) Figure 5 shows the percentage east-west tracking performance in southern Spain compared to full tracking mode for the four selected days. East-west tracking is relatively similar compared to full tracking. The calculations assumed the mass flow in the range from 5 kg/s to 7.06 kg/s with the minimum temperature condition (Equation (14)) at the outflow from the loop equal to 360 °C. In fact the flow increases gradually with the increase in direct solar radiation, reaches its maximum value equal to 7.06 kg/s and then decreases in the evening with the decrease in The calculations assumed the mass flow in the range from 5 kg/s to 7.06 kg/s with the minimum temperature condition (Equation (14)) at the outflow from the loop equal to 360 • C. In fact the flow increases gradually with the increase in direct solar radiation, reaches its maximum value equal to 7.06 kg/s and then decreases in the evening with the decrease in 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 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 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 observed differences between these values is caused by the control of the flow in the absorber 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. According this comparison, the relative error of the developed model was determined according to the following relationship: where q num is the heat determined from Equation (1) and q real is the heat measured in the thermoelectric solar plant in Spain, Cordoba province. Table 3 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 observed differences between these values is caused by the control of the flow in the absorber 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. According this comparison, the relative error of the developed model was determined according to the following relationship: where qnum is the heat determined from Equation (1) and qreal is the heat measured in the thermoelectric solar plant in Spain, Cordoba province. Table 3 presents the calculation 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 September, respectively, 11, 3 and 4 days. The relative error of the presented model for the selected months did not exceed 7.1%.  With the decrease in mass flow in the absorber and the increase in solar radiation intensity, the temperature at the outlet from a single loop increases. The minimum temperature at the outlet of a single loop is 360 • C and is achieved for DNI of 450 Wh/m 2 , 670 Wh/m 2 and 905 Wh/m 2 for flow rates of 5 kg/s, 7 kg/s and 10 kg/s, respectively. In the case of a working temperature of 392 • C, it is obtained for 680 Wh/m 2 , 920 Wh/m 2 for 5 kg/s and 7 kg/s, respectively. With a mass flow of 10 kg/s, the operating temperature is not reached for the maximum DNI. The maximum operating temperature of 400 • C is achieved with a flow of 5 kg/s.  Figure 8 shows the temperature dependence on the output of a single loop consisting of 48 solar collectors as a function 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, the temperature at the outlet from a single loop increases. The minimum temperature at the outlet of a single loop is 360 °C and is achieved for DNI of 450 Wh/m 2 , 670 Wh/m 2 and 905 Wh/m 2 for flow rates of 5 kg/s, 7 kg/s and 10 kg/s, respectively. In the case of a working temperature of 392 °C, it is obtained for 680 Wh/m 2 , 920 Wh/m 2 for 5 kg/s and 7 kg/s, respectively. With a mass flow of 10 kg/s, the operating temperature is not reached for the maximum DNI. The maximum operating temperature of 400 °C is achieved with a flow of 5 kg/s.      Figure 8 shows the temperature dependence on the output of a single loop consisting of 48 solar collectors as a function 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, the temperature at the outlet from a single loop increases. The minimum temperature at the outlet of a single loop is 360 °C and is achieved for DNI of 450 Wh/m 2 , 670 Wh/m 2 and 905 Wh/m 2 for flow rates of 5 kg/s, 7 kg/s and 10 kg/s, respectively. In the case of a working temperature of 392 °C, it is obtained for 680 Wh/m 2 , 920 Wh/m 2 for 5 kg/s and 7 kg/s, respectively. With a mass flow of 10 kg/s, the operating temperature is not reached for the maximum DNI. The maximum operating temperature of 400 °C is achieved with a flow of 5 kg/s.   The results of calculations of the convective heat transfer coefficient inside the absorber are presented below. For this purpose, a simplified model of forced convection for The results of calculations of the convective heat transfer coefficient inside the absorber are presented below. For this purpose, a simplified model of forced convection for turbulent flow was used. The local convective heat transfer coefficient is described by the following equation: where D is the absorber diameter, λ is the conductivity coefficient of the medium, while the Nusselt number is described by the Dittus-Boelter correlation: Nu D = 0.023Re 4/5 Pr 0.4 , Re ≥ 10, 000, 0.6 ≤ Pr ≤ 1600, Pr = c p µ where Re is the Reynolds number, while Pr is the Prandtl number. Figure 10 shows the relationship of convective heat transfer coefficient. The convection coefficient increases with increasing flow through the absorber. With a nominal flow value of m = 7.06 kg/s, the convective heat transfer coefficient is 2714 W/(m 2 K). The determined value of the convective heat transfer coefficient is relatively high compared to the values determined in the literature [33,38], which is the result of much larger real Reynolds numbers in relation to the Reynolds numbers from works [33,38]. Increasing the mass flow of the medium in the absorber to 10 kg/s increases the convective heat transfer coefficient by about 24%. turbulent flow was used. The local convective heat transfer coefficient is described by the following equation: where D is the absorber diameter, λ is the conductivity coefficient of the medium, while the Nusselt number is described by the Dittus-Boelter correlation: where Re is the Reynolds number, while Pr is the Prandtl number. Figure 10 shows the relationship of convective heat transfer coefficient. The convection coefficient increases with increasing flow through the absorber. With a nominal flow value of m = 7.06 kg/s, the convective heat transfer coefficient is 2714 W/(m 2 K). The determined value of the convective heat transfer coefficient is relatively high compared to the values determined in the literature [33,38], which is the result of much larger real Reynolds numbers in relation to the Reynolds numbers from works [33,38]. Increasing the mass flow of the medium in the absorber to 10 kg/s increases the convective heat transfer coefficient by about 24%.

Application of the Model for the Performance Prediction of a Solar Thermal Power Station Located in a Temperate Climate Location
This chapter presents the results of calculations of heat produced by a system of parabolic solar collectors for a selected region of temperate climate: in the city of Bialystok located in the north-eastern part of Poland. The climatic conditions of this localization are presented in Figure 11 and compared with Spanish conditions. The main differences with the south of Spain are much lower values of direct solar radiation and temperature, which is mainly due to the different latitude. The annual value of DNI in Cordoba was 1938 kWh/m 2 /year, while in the case of Bialystok, the annual DNI is 467 kWh/m 2 /year.
The correct operation of a thermoelectric power plant depends on the minimum temperature at the outlet of the solar installation, giving by Equation (14), which is 360 °C for a thermoelectric power plant in Spain in the Cordoba province, therefore two variants were adopted for the calculations:

Application of the Model for the Performance Prediction of a Solar Thermal Power Station Located in a Temperate Climate Location
This chapter presents the results of calculations of heat produced by a system of parabolic solar collectors for a selected region of temperate climate: in the city of Bialystok located in the north-eastern part of Poland. The climatic conditions of this localization are presented in Figure 11 and compared with Spanish conditions. The main differences with the south of Spain are much lower values of direct solar radiation and temperature, which is mainly due to the different latitude. The annual value of DNI in Cordoba was 1938 kWh/m 2 /year, while in the case of Bialystok, the annual DNI is 467 kWh/m 2 /year.
The correct operation of a thermoelectric power plant depends on the minimum temperature at the outlet of the solar installation, giving by Equation (14), which is 360 • C for a thermoelectric power plant in Spain in the Cordoba province, therefore two variants were adopted for the calculations:

B.
Many publications [18][19][20][51][52][53][54] describe studies of the use of parabolic colle for thermoelectric power plants in countries such as Greece, Spain, China and Nig which are located in the subtropical and tropical climatic zone, i.e., in the area where d solar radiation is significant. The territory of Poland is located in a temperate climate z which is not recommended by producers of parabolic solar collectors used to pro heat for the operation of typical steam turbines. According to the SCHOTT PTR ® 70 Receiver brochure [49], the worldwide use divided into four categories: "Excellent", "Good", "Satisfactory", "Unsuitable". The of the use of solar collectors [49] is consistent with the map of annual solar radiation i world [55]. Cordoba is in the area marked "Good", while Bialystok is labelled "Unsuita The above calculation results using the simplified model confirm this fact. Comparing the in SCHOTT PTR ® 70 Receiver [49] with the map of the distribution of climatic zones [53], be noticed that the best location of parabolic solar collectors is the climate of the tropica subtropical zones, excluding parts of South America and Africa with tropical rainforest. 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 Bialystok (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.

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  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 (Figure 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][19][20][51][52][53][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.

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. 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. Prandtl number (-) q CPS heat obtained from all solar collectors during one hour (kWh) q loop hourly heat produced by a single loop (kWh) q num hourly heat supplied by the solar collector system (kWh) q P heat losses from heating pipes (kWh) q R heat loss from the absorber (kWh) Re

Nomenclature
Reynolds number (-) t sol solar time angle (h) T a average-hour outside temperature ( • C) T in single loop inlet fluid temperature ( • C) T m medium temperature ( • C) T out single loop outlet fluid temperature ( • C) Greek symbols α solar altitude angle (degree) α s solar azimuth angle (degree) δ S declination angle (degree) η endloss coefficient for the calculation of the relative end loss (-) η opt optical collector efficiency for perpendicular sun position (-) η shadow coefficient for solar shading (-) θ i angle of incidence on the collector surface (degree) λ thermal conductivity of the thermal insulation (W/m/K) λ m 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 (