Effects of Receiver Parameters on Solar Flux Distribution for Triangle Cavity Receiver in the Fixed Linear-Focus Fresnel Lens Solar Concentrator

: The objective of the study is to investigate and optimize the solar ﬂux uniformity of a ﬁxed linear-focus Fresnel lens solar concentrator using a triangle cavity receiver. The effects of receiver parameters including the vertical distance from the cavity opening plane to the Fresnel lens f , receiver internal surface absorptivity α ab , end reﬂection plane reﬂectivity ρ r , solar declination angle δ and solar angle ω on the uniformity factor ( UF ) of a triangle cavity receiver were carried out. The effects of receiver parameters are evaluated with a signiﬁcance test of critical factors. The results showed that the increase in f and δ would result in an increase in the UF . The average UF with f = 600, 625, 650, 675 and 700 mm, respectively, are 0.5030, 0.5858, 0.6337, 0.6576 and 0.6784 for ω in range of 0–60 ◦ . Moreover, the UF increases as α ab decreases when other receiver parameters are constant for the δ of 0–8 ◦ . The ρ r has a limited effect on the UF until δ becomes relatively larger and ω becomes relatively smaller. Furthermore, ω effects are most signiﬁcant on the UF , followed by δ , f and α ab . Setting a suitable f is the most economical and effective way to improve the UF .


Introduction
Based on the temperature of the heat transfer fluid, solar thermal collectors can be categorized into three categories: low temperature (50-80 • C), medium temperature (80-250 • C) and high temperature (400-1000 • C) solar thermal collectors [1]. High-temperature collectors require a high concentration of light (10-1000 X). Therefore, the cooling of these collectors is very important, because if the local temperature of the receiver is too high, it will cause the receiver to deform or even be burned [2,3]. The solar medium-temperature heat collection system has the characteristics of a simple structure, moderate energy density and flexible layout [4]. Combining it with buildings can effectively reduce the use of primary energy and improve land utilization [5,6]. In a solar energy medium-temperature heat collection system, the linear focusing devices, including parabolic trough collector, linear Fresnel reflector, linear Fresnel lens collector, etc., have been widely studied due to their relatively simple structure and easy manufacturing [7,8]. The efficiency and safe operation of the heat collection system are the two main research directions [9,10]. A main reason that threatens the safe operation of the system is the uneven distribution of the receiver flux.
To obtain a uniform solar flux distribution on the receiver surfaces, many design concepts of the linear concentrating solar collector have been studied by researchers. Qiu FLFLSC shown is assembled from five small Fresnel lenses of length, width and thickness of 320, 400 and 2 mm, respectively. The groove pitch is 1 mm and the focal length is 650 mm. The grooves are placed facing downwards to reduce dust accumulation. The lenses are mounted in a slidable structure with sufficient rail length. Two triangular reflecting surfaces inside the triangular cavity are arranged as reflecting surfaces at both ends to reduce energy loss. Two surfaces inside the triangular cavity are arranged as receiving surfaces, which are two elongated rectangular surfaces. The cavity receiver absorbs the concentrated solar energy through the receiving surfaces that are made up of the outer surfaces of many copper tubes seamlessly joined together and transfers the heat to the working fluid through the copper tube. The working fluid flows in from the lower side of the cavity and flows out from the higher side, thereby taking away the heat. The geometric parameters include the lens element width B, lens element length L, focal length f0 of FLFLSC, solar hour angle ω, sun declination angle δ and the horizontal angle of polar axis S0. In this study, the S0 is set to be 23°08′ based on the local latitude angle φ in Guangzhou, China. The linear Fresnel lens is aligned in a north-south direction and rotates about the polaraxis to track the daily sun motion at the rotation angular velocity of the earth. In most operational conditions, direct solar radiation is not incidentally perpendicular to the surface of the linear Fresnel lens. Therefore, the incidence angle θ of direct solar radiation on the single-axis tracking array with a polar north-south-orientated, fixed linear-focus Fresnel lens solar concentrator, as shown in Figure 1, is obtained by the following formula [20,21]: The key component, linear Fresnel lens is fabricated using polymethyl methacrylate (PMMA) through molding or extrusion with a clear texture. The digital photograph of FLFLSC shown is assembled from five small Fresnel lenses of length, width and thickness of 320, 400 and 2 mm, respectively. The groove pitch is 1 mm and the focal length is 650 mm. The grooves are placed facing downwards to reduce dust accumulation. The lenses are mounted in a slidable structure with sufficient rail length. Two triangular reflecting surfaces inside the triangular cavity are arranged as reflecting surfaces at both ends to reduce energy loss. Two surfaces inside the triangular cavity are arranged as receiving surfaces, which are two elongated rectangular surfaces. The cavity receiver absorbs the concentrated solar energy through the receiving surfaces that are made up of the outer surfaces of many copper tubes seamlessly joined together and transfers the heat to the working fluid through the copper tube. The working fluid flows in from the lower side of the cavity and flows out from the higher side, thereby taking away the heat. The geometric parameters include the lens element width B, lens element length L, focal length f 0 of FLFLSC, solar hour angle ω, sun declination angle δ and the horizontal angle of polar axis S 0 . In this study, the S 0 is set to be 23 • 08 based on the local latitude angle ϕ in Guangzhou, China.
The linear Fresnel lens is aligned in a north-south direction and rotates about the polaraxis to track the daily sun motion at the rotation angular velocity of the earth. In most operational conditions, direct solar radiation is not incidentally perpendicular to the surface of the linear Fresnel lens. Therefore, the incidence angle θ of direct solar radiation on the single-axis tracking array with a polar north-south-orientated, fixed linear-focus Fresnel lens solar concentrator, as shown in Figure 1, is obtained by the following formula [20,21]: cos θ = sin δ sin(ϕ − S 0 ) + cos δ cos(ϕ − S 0 ) cos ω (1) Note that S 0 is equal to the ϕ, i.e., S 0 = ϕ. Moreover, the ω of direct solar radiation on the surface of the linear Fresnel lens is tracked all the time under ideal tracking conditions, i.e., ω = 0 • . Applying the known conditions for Formula (1) we have: Furthermore, the δ for any day of the year (N) can be presented approximately as [22,23], For seasonal tracking, the element position of the linear Fresnel lens is manually adjusted with the aid of a screw rod mechanism. The lens element slips along the edge of the lens frame according to the δ until the offset linear focus does not exceed the end of the cavity receiver. For clarity, the principle sliding adjustment of Fresnel lens with change in incident angle of solar radiation is shown in Figure 2. When the sun rays shoot in with an oblique angle caused by the change of solar declination angle, the Fresnel lens is slid parallel until the concentrated sun rays reach one end of the triangle cavity receiver. Based on the linear Fresnel lens described in Table 1, offset distance changes of the linear focal spot on the focal plane with the δ as the values shown in Table 2. The above models and methods can be transferred and applied to other countries and regions by adjusting design parameters and changing operating conditions. Note that S0 is equal to the φ, i.e., S0 = φ. Moreover, the ω of direct solar radiation on the surface of the linear Fresnel lens is tracked all the time under ideal tracking conditions, i.e., ω = 0°. Applying the known conditions for Formula (1) For seasonal tracking, the element position of the linear Fresnel lens is manually adjusted with the aid of a screw rod mechanism. The lens element slips along the edge of the lens frame according to the δ until the offset linear focus does not exceed the end of the cavity receiver. For clarity, the principle sliding adjustment of Fresnel lens with change in incident angle of solar radiation is shown in Figure 2. When the sun rays shoot in with an oblique angle caused by the change of solar declination angle, the Fresnel lens is slid parallel until the concentrated sun rays reach one end of the triangle cavity receiver. Based on the linear Fresnel lens described in Table 1, offset distance changes of the linear focal spot on the focal plane with the δ as the values shown in Table 2. The above models and methods can be transferred and applied to other countries and regions by adjusting design parameters and changing operating conditions.

Numerical Simulation Model
To estimate the solar flux distribution of triangle cavity receiver in the FLFLSC, commercial software TracePro ® using MCRT method was applied. It is an optomechanical software that provides professional service for design, optimization and evaluation for the optical and illumination systems [24]. Researchers use this tool extensively in solar optics applications [25]. The receiver employed in this study is a triangle cavity receiver constructed with a coated copper tube through which the heat transfer fluid flows, as shown in Figure 3a. The symbol B tr , L tr and α tr denote the opening width, the opening length and the apex angle of the triangle cavity receiver, respectively. Various 3D models of the FLFLSC with different receiver positions, solar declination angles and solar hour angles are firstly modeled and assembled in UG 10.0 [26] and then imported into the TracePro ® to simulate the ray tracing process. In the proposed model, the surface reflectivity, transmittance and absorptivity characteristics of the triangle cavity receiver are set for each of the specific simulation conditions. However, the above surface characteristics of the linear Fresnel lens are set as constant.

Numerical Simulation Model
To estimate the solar flux distribution of triangle cavity receiver in the FLFLSC, commercial software TracePro ® using MCRT method was applied. It is an optomechanical software that provides professional service for design, optimization and evaluation for the optical and illumination systems [24]. Researchers use this tool extensively in solar optics applications [25]. The receiver employed in this study is a triangle cavity receiver constructed with a coated copper tube through which the heat transfer fluid flows, as shown in Figure 3a. The symbol Btr, Ltr and αtr denote the opening width, the opening length and the apex angle of the triangle cavity receiver, respectively. Various 3D models of the FLFLSC with different receiver positions, solar declination angles and solar hour angles are firstly modeled and assembled in UG 10.0 [26] and then imported into the TracePro ® to simulate the ray tracing process. In the proposed model, the surface reflectivity, transmittance and absorptivity characteristics of the triangle cavity receiver are set for each of the specific simulation conditions. However, the above surface characteristics of the linear Fresnel lens are set as constant. It is necessary to establish some hypotheses before conducting the parametric study on the solar flux distribution of the solar concentrator since the optical performance research in this study is carried out in the perfect case. Engineering errors are not considered, including the slope, specular and contour errors of Fresnel lens; the alignment error of the triangular cavity receiver; and the tracking error of the system, etc. [27,28]. Once set the surface and geometry characteristics of the triangular cavity receiver will not change with the circumstances. To build a model close to reality, the half angular width of the sun is considered to be about 0.27° during simulation [29]. It is necessary to establish some hypotheses before conducting the parametric study on the solar flux distribution of the solar concentrator since the optical performance research in this study is carried out in the perfect case. Engineering errors are not considered, including the slope, specular and contour errors of Fresnel lens; the alignment error of the triangular cavity receiver; and the tracking error of the system, etc. [27,28]. Once set the surface and geometry characteristics of the triangular cavity receiver will not change with the circumstances. To build a model close to reality, the half angular width of the sun is considered to be about 0.27 • during simulation [29]. Figure 3b shows the ray-tracing analysis of a specific case. The ray-tracing process of a fixed linear-focus Fresnel lens coupled with a triangular cavity receiver can be seen clearly. The sunrays are refracted by the linear Fresnel lens and projected on the internal absorption surface of the triangle cavity receiver. Most of the incoming solar radiation energy can be absorbed after several reflections in the cavity. Finally, it is transferred to the working fluid through the copper tube after the absorption on the internal absorption surface. To minimize the optical loss at the end of the triangle cavity receiver, plane mirrors are arranged at both the ends of the triangle cavity receiver. The related parameters of FLFLSC using the triangle cavity receiver are listed in Table 1.

Optical Work Validation
To ensure the accuracy of the simulation, it is necessary to confirm the effectiveness of the simulation results based on the MCRT method by verifying the physical processes and methods used in this study. An indirect measurement system was designed for performance testing under real weather conditions to verify the simulation results.
In the experiment, the linear Fresnel lens is rotated to ensure that the solar radiation is incident vertically on the lens. The diffuse receiver is placed under the Fresnel lens parallel to the lens, or in other words, vertical to the direction of solar radiation. In order to verify the simulation results, an indirect measurement method was designed for performance testing under real weather conditions as shown in Figure 4. In the experiment, the linear Fresnel lens is rotated to ensure that the solar radiation is incident vertically with Fresnel lens grooves facing downwards. The target is a white Lambert board with dimensions of 300 mm by 320 mm square produced by Labsphere Company of U.S. The diffuse receiver is placed under the Fresnel lens parallel to the lens, or in other words, vertical to the direction of sunlight. The CCD camera is a detector with 2048 by 4096 pixels enabling high resolution images of focal line. There are two types of indirect measuring methods including on-axis and off-axis. The former means that the symmetry axis of the CCD camera overlaps with the vertical lines of the receiver plane. The other one is to make the vertical line of the receiving surface at a small angle to the central axis of the CCD camera. In order to avoid forming a CCD camera shadow on the diffuse receiver, the off-axis measurement is done. The deflection angle β is limited to 3 • by which way the error caused by Lambert attributes can be ignored [30]. absorption surface of the triangle cavity receiver. Most of the incoming solar radiation energy can be absorbed after several reflections in the cavity. Finally, it is transferred to the working fluid through the copper tube after the absorption on the internal absorption surface. To minimize the optical loss at the end of the triangle cavity receiver, plane mirrors are arranged at both the ends of the triangle cavity receiver. The related parameters of FLFLSC using the triangle cavity receiver are listed in Table 1.

Optical Work Validation
To ensure the accuracy of the simulation, it is necessary to confirm the effectiveness of the simulation results based on the MCRT method by verifying the physical processes and methods used in this study. An indirect measurement system was designed for performance testing under real weather conditions to verify the simulation results.
In the experiment, the linear Fresnel lens is rotated to ensure that the solar radiation is incident vertically on the lens. The diffuse receiver is placed under the Fresnel lens parallel to the lens, or in other words, vertical to the direction of solar radiation. In order to verify the simulation results, an indirect measurement method was designed for performance testing under real weather conditions as shown in Figure 4. In the experiment, the linear Fresnel lens is rotated to ensure that the solar radiation is incident vertically with Fresnel lens grooves facing downwards. The target is a white Lambert board with dimensions of 300 mm by 320 mm square produced by Labsphere Company of U.S. The diffuse receiver is placed under the Fresnel lens parallel to the lens, or in other words, vertical to the direction of sunlight. The CCD camera is a detector with 2048 by 4096 pixels enabling high resolution images of focal line. There are two types of indirect measuring methods including on-axis and off-axis. The former means that the symmetry axis of the CCD camera overlaps with the vertical lines of the receiver plane. The other one is to make the vertical line of the receiving surface at a small angle to the central axis of the CCD camera. In order to avoid forming a CCD camera shadow on the diffuse receiver, the off-axis measurement is done. The deflection angle β is limited to 3° by which way the error caused by Lambert attributes can be ignored [30].  We calibrated the angle and position of the CCD camera before taking the shot, and then captured the focused linear facula. The flux distribution of the focus facula can be obtained by the data acquisition system, which can correct the distortion of the image, change the coordinate unit and convert the grayscale of the image into the flux intensity We calibrated the angle and position of the CCD camera before taking the shot, and then captured the focused linear facula. The flux distribution of the focus facula can be obtained by the data acquisition system, which can correct the distortion of the image, change the coordinate unit and convert the grayscale of the image into the flux intensity [29]. The flux distribution of the focus facula measured under actual conditions is shown in Figure 5a. Referring to Figure 5b, it will be understood that the simulated flux distribution has a similar result, the width of the main facula and the facula, respectively, are about 26 mm and 84 mm. When the direct radiation value is 796 W/m 2 , the maximum flux density on the planar receiver according to the experimental and simulated results are 18,137 and 19,052 W/m 2 , respectively.
In order to compare the experimental and simulation results, the flux distribution at five horizontal lines of the focus facula was selected for analysis, and the curves of the average flux density on five horizontal lines are shown in Figure 6 [31]. In the area within the main facula, the experimental and simulation results are basically consistent with each other. The areas enclosed by the experimental and simulated curves, respectively, are S 0 Sustainability 2021, 13, 6139 7 of 21 and S 0 '. The relative error between them is 3.24%. In the area within the width of the facula, the experimental results of side facula are larger than that of simulation, but the error is not significant. The areas enclosed by the experimental and simulated curves, respectively, are S 1 and S 1 ', and the relative error between them is 15.35%. It may be due to a manufacturing error, surface scattering and the teeth rounded edges of the linear Fresnel lens and the disparity between the response of the solar spectrum and the CCD camera, etc. The present numerical procedure matched closely with the experimental data.
Sustainability 2021, 13, 6139 7 of 22 [29]. The flux distribution of the focus facula measured under actual conditions is shown in Figure 5a. Referring to Figure 5b, it will be understood that the simulated flux distribution has a similar result, the width of the main facula and the facula, respectively, are about 26 mm and 84 mm. When the direct radiation value is 796 W/m 2 , the maximum flux density on the planar receiver according to the experimental and simulated results are 18,137 and 19,052 W/m 2, respectively. In order to compare the experimental and simulation results, the flux distribution at five horizontal lines of the focus facula was selected for analysis, and the curves of the average flux density on five horizontal lines are shown in Figure 6 [31]. In the area within the main facula, the experimental and simulation results are basically consistent with each other. The areas enclosed by the experimental and simulated curves, respectively, are S0 and S0'. The relative error between them is 3.24%. In the area within the width of the facula, the experimental results of side facula are larger than that of simulation, but the error is not significant. The areas enclosed by the experimental and simulated curves, respectively, are S1 and S1', and the relative error between them is 15.35%. It may be due to a manufacturing error, surface scattering and the teeth rounded edges of the linear Fresnel lens and the disparity between the response of the solar spectrum and the CCD camera, etc. The present numerical procedure matched closely with the experimental data.
TracePro ® is incorporated into the simulation of the flux distribution on the triangular cavity receiver. As the resulting precision is decided by the total number of trace rays emitted by the light source, it is necessary to find out the most appropriate number of rays. Initial ray numbers of 7300, 32,100, 72,100, 128,900 and 203,900 were set during the simulation on the optical efficiency of the FLFLSC using a triangular cavity receiver with θ = 0°. For each of rays, the total energy is the same. As the total energy of sun rays for five different ray numbers is the same, the energy allocated to each ray is different. The less the number of sun rays, the more energy each ray carries, and the more likely it is to cause a high concentration of energy. Furthermore, the fewer the number of sun rays, the greater the possibility of error. Conversely, if the number of sun rays is larger, the less likely it is to cause errors, and the more the simulation result closes to the real situation. Figure 7 shows the flux distribution along the width of the receiver internal surface. Compared to other numbers of rays, the value of heat flux for 7300 rays is more than twice due to the more carried energy of each ray causing a high concentration of energy.  TracePro ® is incorporated into the simulation of the flux distribution on the triangular cavity receiver. As the resulting precision is decided by the total number of trace rays emitted by the light source, it is necessary to find out the most appropriate number of rays. Initial ray numbers of 7300, 32,100, 72,100, 128,900 and 203,900 were set during the simulation on the optical efficiency of the FLFLSC using a triangular cavity receiver with θ = 0 • . For each of rays, the total energy is the same. As the total energy of sun rays for five different ray numbers is the same, the energy allocated to each ray is different. The less the number of sun rays, the more energy each ray carries, and the more likely it is to cause a high concentration of energy. Furthermore, the fewer the number of sun rays, the greater the possibility of error. Conversely, if the number of sun rays is larger, the less likely it is to cause errors, and the more the simulation result closes to the real situation. Figure 7 shows the flux distribution along the width of the receiver internal surface. Compared to other numbers of rays, the value of heat flux for 7300 rays is more than twice due to the more carried energy of each ray causing a high concentration of energy.   Figure 8 is about 0.126, when 128,900 rays are set. Moreover, the calculated optical efficiency of the FLFLSC using the triangular cavity receiver in the case of θ = 0° in the 128,900 rays and the 203,900 rays is 77.93%. Therefore, 128,900 rays are used to simulate the optical performance of FLFLSC using a triangular cavity receiver.   Figure 8 is about 0.126, when 128,900 rays are set. Moreover, the calculated optical efficiency of the FLFLSC using the triangular cavity receiver in the case of θ = 0 • in the 128,900 rays and the 203,900 rays is 77.93%. Therefore, 128,900 rays are used to simulate the optical performance of FLFLSC using a triangular cavity receiver.

Uniformity Factor (UF)
For comparison, the flux uniformity of the receiver internal surface is defined by a uniformity factor (UF) as indicated below [32]:

Uniformity Factor (UF)
For comparison, the flux uniformity of the receiver internal surface is defined by a uniformity factor (UF) as indicated below [32]: The maximum flux is defined as the peak value collected during the experiment or simulation, while the average flux is calculated by dividing the incident and absorbed energy by the area of receiver internal surface.

Simulation Method
The simulation process is completed by the software TracePro ® . During the process, the parameters such as solar declination angle δ, solar hour angle ω, the vertical distance between the opening plane of the cavity and the Fresnel lens f, the internal surface absorptivity α ab and the end plane reflectivity ρ r are changed individually. The value of δ varies from −23.45 • to +23.45 • throughout the year. Due to symmetry factors, only δ values in the range of 0 to +23.45 • are studied. The observation points are set at an interval of 8 • . Thus, the δ is set as 0 • , +8 • , +16 • and +23.45 • . The performance of the FLFLSC is examined for planning to explore the performance of the small FLFLSC during the four hours before and after noon, which means a total of eight hours of sunlight during the day. Similarly, due to symmetry, the solar hour angle range of 0 • -60 • is investigated. The observation points are set at an interval of 15 • . Thus, the solar hour angle is set as 0 • , 15 • , 30 • , 45 • and 60 • . Since the focal length of the Fresnel lens is 650 mm, the vertical distance between the opening plane of the cavity and the Fresnel lens f is set as 600 mm, 625 mm, 650 mm, 675 mm and 700 mm. In order to simulate the real situation as much as possible, the internal surface absorptivity α ab is set as 0.75, 0.85 and 1.00. The end plane reflectivity ρ r is set as 0.75, 0.85 and 1.00.
For each case, the controlling variable method is adopted. For example, when the date varies to a specific day, the value of δ is +8 • and the solar hour angle is gradually varied, with other parameters kept constant. The software is run, and the light distribution and the cavity receiver energy distribution are obtained through the built-in program. The energy distribution on the area of two elongated rectangular surfaces inside the triangle cavity is counted. The performance of the small FLFLSC on a specific day can be obtained in this way. The influence of every parameter can be acquired by changing the parameter value and rerunning the software. Note that when the δ changes, the Fresnel lens is slipped parallel for an appropriate distance to make sure that the focal line slide does not exceed the cavity receiver.

Results and Discussion
The primary purpose of this study is to investigate the solar flux uniformity of the triangle cavity receiver in FLFLSC under various sunlight incidence angles and to find the optimal parameters of the receiver aiming to improve the UF of the triangle cavity receiver. The parameters are receiver position (f ), receiver internal surface absorptivity (α ab ) and end reflection plane reflectivity (ρ r ). The ω and the δ have been taken into consideration in the following cases for showing the effect of receiver parameters on the UF of the triangle cavity receiver objectively.

Effect of Receiver Position f
The effect of shifting the receiver position is intensely investigated. Figures 9-12 show the results of five studied positions for the triangle cavity receiver when the α ab and ρ r are both 0.85. As shown in Figure 9, when the δ is 0 • , the effect of the receiver position is clearly seen. The UF has a sensitivity to various values of f as the UF between them is significantly different for ω of 0-60 • . Moreover, the larger the value of f, the larger the UF is. As can be seen from the size of the interval between the curves that after f is greater than 650 mm, the UF rises faster. It may be explained that when f is in the range of 600-650 mm, the linear focus can be basically enclosed in the cavity, but when f exceeds 650 mm, the width of focused linear facula raises, resulting in a higher UF value. 650 mm, the linear focus can be basically enclosed in the cavity, but when f exceeds 650 mm, the width of focused linear facula raises, resulting in a higher UF value.
It is worth noting that the interval between the f = 600 mm curve and other f mainly increase first for the ω of 0-15° and then decreases as ω further increases in the range of 15-60°, even if there is overlap between the curves f = 600 and 625 mm for ω of 35-60°. It is because that the focused light is initially distributed on the two absorption surfaces of the cavity, and then gradually concentrated on only one of the surfaces. The apex angle of triangle cavity receiver plays an essential role during the transferring since the sun rays concentrate here causing the low ebb on the curves of f = 600 and 625 mm. The low ebb can be observed only in the above two curves, which suggests that the concentrating effect should be considered while using a small f.  As mentioned previously, UF is positively correlated with f for the ω of 0-60°. The reason is that the larger f is, the farther the focal line is from the bottom of the triangular cavity. On the contrary, the smaller the f, the closer the focal line is to the apex angle of triangle cavity, which causes the received solar energy to accumulate since the bottom area of the triangular cavity is small. In other words, increasing f can make the energy flow density uniform. The UF curves with f = 600 and 625 mm decrease first for ω about 0-20°, then increase for ω about 20-30°, then finally decrease with the increasing of ω between 30° and 60°. It can be inferred that the amount of sun rays received by the two sides of the receiver's internal surface is basically unchanged for ω about 20-30°, but the total area receiving the concentrated sun rays becomes larger, which leads to the decrease of the maximum flux. The average UF for the triangle cavity receiver with f = 600, 625, 650, 675 and 700 mm, respectively, are 0.3054, 0.3166, 0.3539, 0.4096 and 0.4813 for the ω of 0-60°. The triangle cavity receiver with f = 700 mm has the highest average UF in one day as the δ = 0°, compared to other receiver positions.
Similar behavior can also be observed in Figure 10, when the value of δ is 8°. Note that UF increases with f and the interval of the UF between two neighboring curves increases with f in the range of 600-700 mm for the ω of 0-50°. In addition, except for the UF curve of f = 600 mm, the other curves basically decrease as ω raises in the range of 0-60°. Additionally, the rate of decline is gradually slowed down. The UF curve with f = 700 mm drops approximately linearly. The overall change in the trend of each curve with a δ value of 8° is similar to the curve with a δ value of 0°, but there are slight differences. It is observed that the value of δ also influences UF in Figures 9 and 10. Additionally, the average UF for the triangle cavity receiver with f = 600, 625, 650, 675 and 700 mm, respectively, are 0.3081, 0.3316, 0.3860, 0.4460 and 0.5173 under the ω between 0° and 60° when the value of δ is 8°. They are higher than the corresponding average UF values when the δ is 0°. This is because when the sun declination angle increases, the focal line has an upward displacement making the focal line farther from apex at the bottom, but this may result in a decrease in optical efficiency. In the same way, Figure 11 shows the effect of changing the triangle cavity receiver position under different ω when the δ is 16°. As can be observed, the UF is sensitive to the different preset f as the difference between UF curves are significant for ω about 0-60°. The interval between the curves is different before and after a ω value of 30°. The δ is It is worth noting that the interval between the f = 600 mm curve and other f mainly increase first for the ω of 0-15 • and then decreases as ω further increases in the range of 15-60 • , even if there is overlap between the curves f = 600 and 625 mm for ω of 35-60 • . It is because that the focused light is initially distributed on the two absorption surfaces of the cavity, and then gradually concentrated on only one of the surfaces. The apex angle of triangle cavity receiver plays an essential role during the transferring since the sun rays concentrate here causing the low ebb on the curves of f = 600 and 625 mm. The low ebb can be observed only in the above two curves, which suggests that the concentrating effect should be considered while using a small f.
As mentioned previously, UF is positively correlated with f for the ω of 0-60 • . The reason is that the larger f is, the farther the focal line is from the bottom of the triangular cavity. On the contrary, the smaller the f, the closer the focal line is to the apex angle of triangle cavity, which causes the received solar energy to accumulate since the bottom area of the triangular cavity is small. In other words, increasing f can make the energy flow density uniform. The UF curves with f = 600 and 625 mm decrease first for ω about 0-20 • , then increase for ω about 20-30 • , then finally decrease with the increasing of ω between 30 • and 60 • . It can be inferred that the amount of sun rays received by the two sides of the receiver's internal surface is basically unchanged for ω about 20-30 • , but the total area receiving the concentrated sun rays becomes larger, which leads to the decrease of the maximum flux. The average UF for the triangle cavity receiver with f = 600, 625, 650, 675 and 700 mm, respectively, are 0.3054, 0.3166, 0.3539, 0.4096 and 0.4813 for the ω of 0-60 • . The triangle cavity receiver with f = 700 mm has the highest average UF in one day as the δ = 0 • , compared to other receiver positions.
cially the curves with f values of 650, 675 and 700 mm. This is because the focal line moves up further with the increase of δ. This upward movement combined with the reflection of the cavity reduces the influence of f. The average UF values for the triangle cavity receiver with f = 600, 625, 650, 675 and 700 mm, respectively, are 0.3534, 0.4143, 0.4818, 0.5341 and 0.5865 for the ω of 0-60° with a δ value of 16°. Figure 12 illustrates the effect of changing the triangle cavity receiver position under different ω when the δ is 23.45°. It can be found that the UF is sensitive to the different preset f for ω in the range of 0-60°. Additionally, the intervals in the UF between two neighboring curves decrease with the increasing of the f in the range of 600-700 mm. It is because the linear focus is outside the triangle cavity receiver for the f of 600-700 mm, the area receiving the concentrated sun rays on internal absorption surface does not increase significantly with the increase of the f in the range of 600-700 mm. It is worth noting that the differences of the UF between f = 600 mm and other f mainly increase firstly for the ω of 0-30° and then decrease in the range of 30-60°, but the size of the interval hardly fluctuates. In addition, the UF curves basically decrease linearly with the increase in ω for the range of 0-60°. Moreover, the average UF for the triangle cavity receiver with f = 600, 625, 650, 675 and 700 mm, respectively, are 0.5030, 0.5858, 0.6337, 0.6576 and 0.6784 under the ω between 0° and 60° with a δ value of 23.45°. From the changes in the UF at different δ in Figures 9-12, it can be found that the solar flux uniformity of triangle cavity receiver improves as the δ increases, and deteriorates as the ω increases.   Figures 13 and 14 show the results of the receiver internal surface absorptivity influences for the UF of triangle cavity receiver based on ρr = 0.85 and f = 650 mm. Figure 13 illustrates the effect of αab on the UF of triangle cavity receiver under various ω, when the Similar behavior can also be observed in Figure 10, when the value of δ is 8 • . Note that UF increases with f and the interval of the UF between two neighboring curves increases with f in the range of 600-700 mm for the ω of 0-50 • . In addition, except for the UF curve of f = 600 mm, the other curves basically decrease as ω raises in the range of 0-60 • . Additionally, the rate of decline is gradually slowed down. The UF curve with f = 700 mm drops approximately linearly. The overall change in the trend of each curve with a δ value of 8 • is similar to the curve with a δ value of 0 • , but there are slight differences. It is observed that the value of δ also influences UF in Figures 9 and 10. Additionally, the average UF for the triangle cavity receiver with f = 600, 625, 650, 675 and 700 mm, respectively, are 0.3081, 0.3316, 0.3860, 0.4460 and 0.5173 under the ω between 0 • and 60 • when the value of δ is 8 • . They are higher than the corresponding average UF values when the δ is 0 • . This is because when the sun declination angle increases, the focal line has an upward displacement making the focal line farther from apex at the bottom, but this may result in a decrease in optical efficiency.

Effect of Receiver Internal Surface Absorptivity αab
In the same way, Figure 11 shows the effect of changing the triangle cavity receiver position under different ω when the δ is 16 • . As can be observed, the UF is sensitive to the different preset f as the difference between UF curves are significant for ω about 0-60 • . The interval between the curves is different before and after a ω value of 30 • . The δ is inferred to play a crucial role on the UF. The intervals of the UF between f = 600 mm curves and other f values increase firstly for the ω of 0-15 • and then decrease in the range of 15-60 • . The interesting thing is that can be found that the larger the f is, the larger the decrease in UF for ω in 0-60 • range. It can be seen that the curves are dispersed at the beginning and converge gradually. When ω is 60 • , the curves are relatively close to each other, especially the curves with f values of 650, 675 and 700 mm. This is because the focal line moves up further with the increase of δ. This upward movement combined with the reflection of the cavity reduces the influence of f. The average UF values for the triangle cavity receiver with f = 600, 625, 650, 675 and 700 mm, respectively, are 0.3534, 0.4143, 0.4818, 0.5341 and 0.5865 for the ω of 0-60 • with a δ value of 16 • . Figure 12 illustrates the effect of changing the triangle cavity receiver position under different ω when the δ is 23.45 • . It can be found that the UF is sensitive to the different preset f for ω in the range of 0-60 • . Additionally, the intervals in the UF between two neighboring curves decrease with the increasing of the f in the range of 600-700 mm. It is because the linear focus is outside the triangle cavity receiver for the f of 600-700 mm, the area receiving the concentrated sun rays on internal absorption surface does not increase significantly with the increase of the f in the range of 600-700 mm. It is worth noting that the differences of the UF between f = 600 mm and other f mainly increase firstly for the ω of 0-30 • and then decrease in the range of 30-60 • , but the size of the interval hardly fluctuates. In addition, the UF curves basically decrease linearly with the increase in ω for the range of 0-60 • . Moreover, the average UF for the triangle cavity receiver with f = 600, 625, 650, 675 and 700 mm, respectively, are 0.5030, 0.5858, 0.6337, 0.6576 and 0.6784 under the ω between 0 • and 60 • with a δ value of 23.45 • . From the changes in the UF at different δ in Figures 9-12, it can be found that the solar flux uniformity of triangle cavity receiver improves as the δ increases, and deteriorates as the ω increases. Figures 13 and 14 show the results of the receiver internal surface absorptivity influences for the UF of triangle cavity receiver based on ρ r = 0.85 and f = 650 mm. Figure 13 illustrates the effect of α ab on the UF of triangle cavity receiver under various ω, when the δ is 0 • . The simulation results were obtained with α ab values of 1.00, 0.85 and 0.75. It can be detected that the different preset α ab of the triangle cavity receiver has an influence on the UF as the values of UF between them are significantly different. Note that the interval of the UF between α ab values of 1.00 and 0.85 is larger than that between α ab values of 0.85 and 0.75 for the ω in the range of 0-60 • . Moreover, the differences of the UF between α ab value of 1.00 and other α ab mainly increase first for the ω of 0-25 • and then decrease in the range of 25-60 • . It is because that the concentrated sun rays can be basically received after many times of absorption and reflection between the two sides of the receiver internal surface in the triangle cavity receiver when the ω is between 0 • and 25 • . Since the triangular cavity has two receiving surfaces, the light concentrates on one of the receiving surfaces when the value of ω changes. By lowering the value of α ab , more energy can be reflected to the other receiving surface, improving the uniformity of energy receiving and thus increasing the UF. The improvement was more pronounced when the light was more concentrated on one surface. In addition, the concentrated sun rays are mainly concentrated on one side of the receiver internal surface for the ω in the range of 25-60 • , sun rays that have not been absorbed for one time begin to escape from the triangle cavity receiver and the number of escaped sun rays increases with the increasing of ω. Thus, the influence of α ab on the UF decreases gradually with the increase of ω. It may be revealed that the ω also influences UF. It also can be found that the UF decreases with the increase of αab for the ω in the range of 0-60°. This was because the concentrated sun rays are mainly received, absorbed and reflected between the two sides of the receiver internal surface in the triangle cavity receiver, thus, the lower the αab, the more the number of concentrated sun rays reflected and absorbed, and the greater the distribution of the concentrated sun rays absorbed on the receiver internal surface. The UF of different value of αab firstly falls sharply and decreases slowly with the increase of ω before and after a ω value of 30°. It may be inferred that the influence of ω on the UF before the ω reaches 30° is stronger than when it exceeds 30°. The average UF for the triangle cavity receiver of αab = 1.00, 0.85 and 0.75, respectively, are 0.3302, 0.3539 and 0.3719 under the ω between 0° and 60°.

Effect of Receiver Internal Surface Absorptivity α ab
Similar behavior also can be noticed in Figure 14, when the δ is 8°. The introduced triangle cavity receivers of different αab have an influence on the UF as the differences of UF between them are significant. The differences in the UF between αab = 1.00 and 0.85 are larger than that between αab = 0.85 and 0.75 for the ω of 0-60°. Moreover, the differences of the UF between αab = 1.00 and other αab mainly increase first for the ω of 0-25° and then decrease with the increasing of ω in the range of 25-60°. It also can be seen that the UF decreases as the αab increases for the ω of 0-60°. Additionally, the UF of different αab first falls sharply and decreases slowly with the increase of ω before and after a ω value of 30°. The average UF for the triangle cavity receiver of αab = 1.00, 0.85 and 0.75, respectively, are 0.3552, 0.3818 and 0.4005 under the ω between 0° and 60°. In short, the curves of the UF with different αab when the δ is 8° are basically the same as that when the δ is 0°. It means that the increase in the δ cannot significantly improve the UF based on ρr = 0.85 and f = 650 mm when the δ is between 0° and 8°. In other words, the δ has little influence on UF when δ is 0-8°. Likewise, Figure 15 shows the effect of changing receiver internal surface absorptivity of the triangle cavity receiver under different ω when the δ is 16°. The different preset αab of triangle cavity receivers has an influence on the UF as the differences of UF between It also can be found that the UF decreases with the increase of α ab for the ω in the range of 0-60 • . This was because the concentrated sun rays are mainly received, absorbed and reflected between the two sides of the receiver internal surface in the triangle cavity receiver, thus, the lower the α ab , the more the number of concentrated sun rays reflected and absorbed, and the greater the distribution of the concentrated sun rays absorbed on the receiver internal surface. The UF of different value of α ab firstly falls sharply and decreases slowly with the increase of ω before and after a ω value of 30 • . It may be inferred that the influence of ω on the UF before the ω reaches 30 • is stronger than when it exceeds 30 • . The average UF for the triangle cavity receiver of α ab = 1.00, 0.85 and 0.75, respectively, are 0.3302, 0.3539 and 0.3719 under the ω between 0 • and 60 • .
Similar behavior also can be noticed in Figure 14, when the δ is 8 • . The introduced triangle cavity receivers of different α ab have an influence on the UF as the differences of UF between them are significant. The differences in the UF between α ab = 1.00 and 0.85 are larger than that between α ab = 0.85 and 0.75 for the ω of 0-60 • . Moreover, the differences of the UF between α ab = 1.00 and other α ab mainly increase first for the ω of 0-25 • and then decrease with the increasing of ω in the range of 25-60 • . It also can be seen that the UF decreases as the α ab increases for the ω of 0-60 • . Additionally, the UF of different α ab first falls sharply and decreases slowly with the increase of ω before and after a ω value of 30 • . The average UF for the triangle cavity receiver of α ab = 1.00, 0.85 and 0.75, respectively, are 0.3552, 0.3818 and 0.4005 under the ω between 0 • and 60 • . In short, the curves of the UF with different α ab when the δ is 8 • are basically the same as that when the δ is 0 • . It means that the increase in the δ cannot significantly improve the UF based on ρ r = 0.85 and f = 650 mm when the δ is between 0 • and 8 • . In other words, the δ has little influence on UF when δ is 0-8 • .
Likewise, Figure 15 shows the effect of changing receiver internal surface absorptivity of the triangle cavity receiver under different ω when the δ is 16 • . The different preset α ab of triangle cavity receivers has an influence on the UF as the differences of UF between them are significant. Note that the differences of the UF between α ab 1.00 and other α ab mainly decrease first for the ω of 0-3 • , then increase for the ω of 3-25 • , and finally decrease with the increasing of ω in the range of 25-60 • . It also can be seen that the UF increases and decreases, respectively, as the α ab increases before and after the ω value of 3 • . It may be because that when the δ is 16 • , the concentrated sun rays after being refracted by the linear Fresnel lens obliquely enter the triangle cavity receiver, and the inclination angles of the concentrated sun rays are relatively large, causing the sun rays to be concentrated toward the end of receiver internal surface after being reflected. When the ω is fixed, the lower the α ab , the more obvious this situation is.
However, as the ω increases, the concentrated sun rays gradually gather from two sides to one side of receiver internal surface, thus the above situation is weakened. It may be inferred that the ω has a more significant influence on the UF, compared to the δ. Moreover, the UF of α ab = 1.00 and 0.85 firstly falls sharply and decreases slowly with the increasing of ω before and after a ω value of 30 • . The average UF for the triangle cavity receiver of α ab = 1.00, 0.85 and 0.75, respectively, are 0.4517, 0.4818 and 0.5009 under the ω between 0 • and 60 • . Figure 16 shows the effect of changing receiver internal surface absorptivity of the triangle cavity receiver at different ω when the δ is 23.45 • . It can be found that the UF has a sensitivity to the introduced triangle cavity receiver of various α ab as the UF between them are significant for ω about 0-60 • . However, the differences of the UF between α ab = 1.00 and other α ab mainly decrease first for the ω of 0-10 • , then increase for the ω of 10-30 • , and finally decrease as the ω increases in the range of 30-60 • . It also can be seen that the UF increase and decrease, respectively, with the increasing of α ab before and after the ω value of 10 • . It means that more sun rays have been concentrated toward the end of receiver internal surface after being reflected when the δ is 23. Besides, the UF curves decrease with the increasing of ω in the range of 0-60°, but the downward trend of the UF curves with αab = 1.00, 0.85 and 0.75, respectively, decreases when unchanged and increases with the increase of ω in the range of 0-60°. It means that the δ also influences UF. Moreover, the average UF for the triangle cavity receiver of αab = 1.00, 0.85 and 0.75, respectively, are 0.6024, 0.6277 and 0.6397 under the ω between 0° and 60°. From the changes in the UF at different δ in Figures 13-16, it can be concluded that the solar flux uniformity is improved by the lower αab as the concentrated sun rays are reflected mainly on both sides of the receiver internal surface.  Besides, the UF curves decrease with the increasing of ω in the range of 0-60°, but the downward trend of the UF curves with αab = 1.00, 0.85 and 0.75, respectively, decreases when unchanged and increases with the increase of ω in the range of 0-60°. It means that the δ also influences UF. Moreover, the average UF for the triangle cavity receiver of αab = 1.00, 0.85 and 0.75, respectively, are 0.6024, 0.6277 and 0.6397 under the ω between 0° and 60°. From the changes in the UF at different δ in Figures 13-16, it can be concluded that the solar flux uniformity is improved by the lower αab as the concentrated sun rays are reflected mainly on both sides of the receiver internal surface.  Besides, the UF curves decrease with the increasing of ω in the range of 0-60 • , but the downward trend of the UF curves with α ab = 1.00, 0.85 and 0.75, respectively, decreases when unchanged and increases with the increase of ω in the range of 0-60 • . It means that the δ also influences UF. Moreover, the average UF for the triangle cavity receiver of α ab = 1.00, 0.85 and 0.75, respectively, are 0.6024, 0.6277 and 0.6397 under the ω between 0 • and 60 • . From the changes in the UF at different δ in Figures 13-16, it can be concluded that the solar flux uniformity is improved by the lower α ab as the concentrated sun rays are reflected mainly on both sides of the receiver internal surface.  Figure 17, the effect of changing ρ r under different ω can be observed as the δ is set to 0 • and 8 • . The different preset ρ r of triangle cavity receiver does not influence the UF because the UF curves of them basically coincide. Moreover, the UF of different ρ r first falls sharply and decreases slowly with the increasing of ω before and after a ω value of 30 • , respectively. However, the differences of the UF between the triangle cavity receivers with different ρ r are significant for ω about 0-30 • and 0-40 • when the δ, respectively, are 16 • and 23.45 • , as shown in Figure 18. It may be inferred that the concentrated sun rays are mainly reflected and absorbed between the two sides of receiver internal surface when the δ is between 0 • and 8 • . different ω can be observed as the δ is set to 0° and 8°. The different preset ρr of triangle cavity receiver does not influence the UF because the UF curves of them basically coincide. Moreover, the UF of different ρr first falls sharply and decreases slowly with the increasing of ω before and after a ω value of 30°, respectively. However, the differences of the UF between the triangle cavity receivers with different ρr are significant for ω about 0-30° and 0-40° when the δ, respectively, are 16° and 23.45°, as shown in Figure 18. It may be inferred that the concentrated sun rays are mainly reflected and absorbed between the two sides of receiver internal surface when the δ is between 0° and 8°. However, the concentrated sun rays gradually concentrate toward the end of receiver internal surface after being reflected as the δ increases. Some sun rays are absorbed by the end of receiver internal surface after being reflected by the end reflection plane. When the δ and ω are fixed, the lower the ρr, the more pronounced this situation is. There is a difference in the effect of ρr on UF at a δ value of 16° and 23.45° at a ω value of 0°. It means that as the value of δ increases, the number of sun rays reflected from the end of receiver internal surface to the end reflection plane increases, resulting in an increase in the number of sun rays reflecting back to the end of receiver internal surface through the end reflection plane, increasing the local energy flow density, and finally resulting in a decrease in UF. Nevertheless, as the ω increases, the concentrated sun rays gradually gather from both sides to one side of receiver internal surface, thus the above situation is weakened. It may be inferred that the ω has a more significant influence on the UF, compared to the δ. In addition, the UF of different αab basically declines linearly with the increasing of ω between 0° and 60°. Nevertheless, for comparison's sake, the average UF under different ρr were calculated when the δ were 0°, 8°, 16° and 23.45°. The average UF under different ρr for the triangle cavity receiver of δ = 0°, 8°, 16° and 23.45°, respectively, are 0.4074, 0.4403, 0.5550 and 0.7349 under the ω between 0° and 30°. Moreover, the average UF under different ρr for the triangle cavity receiver of δ = 0°, 8°, 16° and 23.45°, respectively, are 0.3539, 0.3810, 0.4809 and 0.7349 under the ω between 0° and 60°.

Analysis of Variance
Though the UF of FLFLSC using triangle cavity receiver is the-larger-the-better, it is also affected by various factors. To evaluate the level of each impact factor on the UF of triangle cavity receiver and provide a more intuitive judgment basis for the significance of various factors, the analysis of variance (ANOVA) was used. Table 3 shows the design of various controlling factors and corresponding levels based on the full-factorial orthogonal array. Five factors that include one 4-level factor (δ), two 5-level factors (f and ω) and two 3-level factors (αab and ρr) can be seen in it. Therefore, the experiment was conducted However, the concentrated sun rays gradually concentrate toward the end of receiver internal surface after being reflected as the δ increases. Some sun rays are absorbed by the end of receiver internal surface after being reflected by the end reflection plane. When the δ and ω are fixed, the lower the ρ r , the more pronounced this situation is. There is a difference in the effect of ρ r on UF at a δ value of 16 • and 23.45 • at a ω value of 0 • . It means that as the value of δ increases, the number of sun rays reflected from the end of receiver internal surface to the end reflection plane increases, resulting in an increase in the number of sun rays reflecting back to the end of receiver internal surface through the end reflection plane, increasing the local energy flow density, and finally resulting in a decrease in UF. Nevertheless, as the ω increases, the concentrated sun rays gradually gather from both sides to one side of receiver internal surface, thus the above situation is weakened. It may be inferred that the ω has a more significant influence on the UF, compared to the δ. In addition, the UF of different α ab basically declines linearly with the increasing of ω between 0 • and 60 • . Nevertheless, for comparison's sake, the average UF under different ρ r were calculated when the δ were 0

Analysis of Variance
Though the UF of FLFLSC using triangle cavity receiver is the-larger-the-better, it is also affected by various factors. To evaluate the level of each impact factor on the UF of triangle cavity receiver and provide a more intuitive judgment basis for the significance of various factors, the analysis of variance (ANOVA) was used. Table 3 shows the design of various controlling factors and corresponding levels based on the full-factorial orthogonal array. Five factors that include one 4-level factor (δ), two 5-level factors (f and ω) and two 3-level factors (α ab and ρ r ) can be seen in it. Therefore, the experiment was conducted 4 × 5 2 × 3 2 times (i.e., 900 times). Table 3. Controlling factors of the analysis of variance and their levels.   Table 4 lists the ANOVA of the quadratic model on UF. Adequate precision measures the signal to noise ratio, and R 2 indicates the coefficient of multiple determination [33,34]. R 2 = 0.685 indicates that the quadratic model can explain 68.5% of the variance in the response. The large F-value of the model (127.954) implies the great significance of the regression model. The associated p-value is less than 0.05 and greater than 0.1, respectively, indicating that the model item is statistically significant and the effects of the model terms are not significant. Therefore, in addition to the factor C (ρ r ), other factors have a very significant impact on UF.
According to the SS in Table 4, factor E (ω) has the most significant influence on the UF, followed by factor D (δ), factor A (f ) and factor B (α ab ). Though δ and ω affect the most in the UF, by the time the solar concentrator is completed, the δ and ω have been fixed, and the high α ab of the coating material will lead to high costs of solar concentrator production. Thus, combined with the annual and temporal variability of solar radiation in the location of the solar energy system, placing the triangle cavity receiver in an appropriate position is a means that considers both economy and effectiveness to improve the UF. Consequently, the FLFLSC using triangle cavity receiver with a α ab value of 0.85 and a ρ r value of 0.75 is selected, and the average UF for the ω of 0-60 • at different f and different δ is calculated, as shown in Figure 19. location of the solar energy system, placing the triangle cavity receiver in an appropriate position is a means that considers both economy and effectiveness to improve the UF. Consequently, the FLFLSC using triangle cavity receiver with a αab value of 0.85 and a ρr value of 0.75 is selected, and the average UF for the ω of 0-60° at different f and different δ is calculated, as shown in Figure 19.  Figure 19. The average UF of FLFLSC using triangle cavity receiver at different f and different δ.
It can be found that when the f is fixed, the average UF increases with the increasing of the δ, but the average UF increases with the increasing of the f when the δ is fixed, except for the δ value of 23.45°. Note that when the δ is 23.45°, the average UF firstly increases then decreases and finally increases with the increasing of the f. It is mainly because when the f is between 600 and 650 mm, some areas at the concentrated side of triangle cavity receiver have no incident concentrated sun rays, but it decreases as f increases, resulting in an increase in average UF. When the f is between 650 and 675 mm, the area of no incident concentrated sun rays at the non-concentrated side of triangle cavity receiver increases with the increasing of the f, resulting in a decrease in average UF. When the f is between 600 and 650 mm, although the area of no incident concentrated sun rays at the non-concentrated side of triangle cavity receiver is further increased. However, the density of incident concentrated sun rays decreases with the increase of the f, resulting in a decrease in the maximum flux density formed at the concentrated side of triangle cavity receiver, which in turn leads to an increase in average UF. It can be found that when the f is fixed, the average UF increases with the increasing of the δ, but the average UF increases with the increasing of the f when the δ is fixed, except for the δ value of 23.45 • . Note that when the δ is 23.45 • , the average UF firstly increases then decreases and finally increases with the increasing of the f. It is mainly because when the f is between 600 and 650 mm, some areas at the concentrated side of triangle cavity receiver have no incident concentrated sun rays, but it decreases as f increases, resulting in an increase in average UF. When the f is between 650 and 675 mm, the area of no incident concentrated sun rays at the non-concentrated side of triangle cavity receiver increases with the increasing of the f, resulting in a decrease in average UF. When the f is between 600 and 650 mm, although the area of no incident concentrated sun rays at the non-concentrated side of triangle cavity receiver is further increased. However, the density of incident concentrated sun rays decreases with the increase of the f, resulting in a decrease in the maximum flux density formed at the concentrated side of triangle cavity receiver, which in turn leads to an increase in average UF.

Conclusions
In the present research, based on the application of linear Fresnel lens in the linear concentrating solar collector, a fixed linear-focus Fresnel lens solar concentrator (FLFLSC) is proposed. In order to investigate and optimize the solar flux uniformity of FLFLSC using triangle cavity receiver under various sun incidence angles, the effects of receiver parameters were studied. The main results are summarized as follows: