Optical Performance Comparison of Different Shapes of Cavity Receiver in the Fixed Line-Focus Solar Concentrating System

: To optimize the ﬁxed-focus solar concentrating system (FLSCS) and linear cavity receiver of better optical performance, the effects of receiver parameters (geometric shape, receiver position f , receiver internal surface absorptivity α ab , and end reﬂection plane reﬂectivity ρ r ) on the relative optical efﬁciency loss η re-opt , loss , the maximum value of the local concentration ratio X max , and the non-uniformity factor σ non were studied in the present study. The results showed that the increases of sun declination angle δ in the range of 0–8 ◦ have a weak effect on the η re-opt , loss . The η re-opt , loss are 2.25%, 2.72%, 12.69% and 2.62%, 3.26%, 12.85%, respectively, when the solar hour angle ω is 0 ◦ , 30 ◦ , 60 ◦ as δ = 0 ◦ and 8 ◦ for linear rectangular cavity receiver. The X max mainly depends on the energy ﬂux distribution of ﬁrst intercepted sunlight on the cavity absorber inner wall. Increasing the distance between the cavity absorber inner wall and the focal line ∆ f can affect the X max . The smaller the ∆ f , the greater the X max , and vice versa. The changing trend of σ non is basically consistent with that of the X max . When the f is 600, 625, 650, 675, 700 mm and the ω = 0 ◦ , the σ non are 0.832, 0.828, 0.801, 0.747, and 0.671, respectively, for linear rectangular cavity receiver. This work could establish the foundation for further research on the optical to thermal energy conversion in the FLSCS.


Introduction
The technology of the concentrating solar collectors is promising for the solar heating system in residential housing [1,2]. The schematic of a concentrated solar heating system used to charge a storage tank of heat transfer fluid (HTF) is shown in Figure 1. Solar energy is harnessed in a fixed line-focus solar concentrating system (FLSCS) at a higher relative temperature compared to the ambient. It is utilized to heat the HTF in the cavity receiver. Note that the FLSCS can provide a relatively high optical efficiency for a whole year running based on the lens element's simple periodic slip adjustment [3,4]. The cavity receiver is fixedly installed in the system, which causes its structural parameters to have a significant impact on the system's optical performance during the sun-tracking process of the solar concentrator.
In the literature, diverse works on cavity receivers have been performed by many researchers to improve the performance of solar concentrating systems. Patil et al. [5] reported a 5 kW solar cavity receiver containing a reticulated porous ceramic structure. The results showed that the system achieved a peak efficiency of 0.69 at 1133 • C air outlet temperature. Liang et al. [6] presented a cavity receiver consisting of a centre tube and two inclined fins for the parabolic trough solar collector (PTC). It was found that the experimental collector 2 of 25 efficiency was in the range of 34.18-48.57%. Liang et al. [7] also studied the effects of the movable cover on reducing heat loss and overheating protection of a cavity receiver for PTC. They concluded that the heat loss reduction rate varied from 6.36% to 13.55% when turning off the movable cover was applied. Ebrahimpour et al. [8] investigated the influence of radiation and convection modes on the behavior of air for linear Fresnel reflector (LFR) with trapezoidal cavity receiver. Temperature difference augments about 180% when adiabatic wall angle enhances as temperature ratio is 0.8. Alipourtarzanagh et al. [9] analyzed the mechanism of heat losses from a cylindrical solar cavity receiver equipped with an air curtain. It was found that increasing the air curtain speed can reduce heat loss by up to 60% when the air curtain discharge angle is 30 • . Fang et al. [10] described the influence of surface optical and radiative properties on the thermal performance of a solar cavity receiver. The results indicated that the receiver efficiency is enhanced by about 12.6% as the solar absorptivity rises from 0.8 to 1.0. Li et al. [11] proposed a major arc-shaped linear cavity receiver with a lunate channel based on parabolic trough solar collectors' black cavity effect principle. The results show that it has a comparable or even better performance compared with evacuated tube collectors. Abbasi-Shavazi et al. [12] examined experimentally the heat loss from a model solar cavity receiver. The results show that the concept of stagnation and convection zone development in cavities are consistent with increasing cavity inclination angle.
Sustainability 2022, 13, x FOR PEER REVIEW 2 of 25 two inclined fins for the parabolic trough solar collector (PTC). It was found that the experimental collector efficiency was in the range of 34.18-48.57%. Liang et al. [7] also studied the effects of the movable cover on reducing heat loss and overheating protection of a cavity receiver for PTC. They concluded that the heat loss reduction rate varied from 6.36% to 13.55% when turning off the movable cover was applied. Ebrahimpour et al. [8] investigated the influence of radiation and convection modes on the behavior of air for linear Fresnel reflector (LFR) with trapezoidal cavity receiver. Temperature difference augments about 180% when adiabatic wall angle enhances as temperature ratio is 0.8. Alipourtarzanagh et al. [9] analyzed the mechanism of heat losses from a cylindrical solar cavity receiver equipped with an air curtain. It was found that increasing the air curtain speed can reduce heat loss by up to 60% when the air curtain discharge angle is 30°. Fang et al. [10] described the influence of surface optical and radiative properties on the thermal performance of a solar cavity receiver. The results indicated that the receiver efficiency is enhanced by about 12.6% as the solar absorptivity rises from 0.8 to 1.0. Li et al. [11] proposed a major arc-shaped linear cavity receiver with a lunate channel based on parabolic trough solar collectors' black cavity effect principle. The results show that it has a comparable or even better performance compared with evacuated tube collectors. Abbasi-Shavazi et al. [12] examined experimentally the heat loss from a model solar cavity receiver. The results show that the concept of stagnation and convection zone development in cavities are consistent with increasing cavity inclination angle. In addition, many researchers have analyzed the influence of cavity receiver parameters on the system's optical performance. Lin et al. [13] proposed and studied a linear Fresnel lens collector with four types of cavity receivers using east-west horizontal tracking mode. The analysis confirms that the Fresnel lens solar collector with triangular cavity receiver boasts best performance in terms of both optical and thermal characteristics. Sedighi et al. [14] proposed an indirectly irradiated cavity receiver. According to the result, a cylindrical cavity with an inverted conical base provides the highest optical efficiency, around 92%, compared to other cylindrical cavities with different base shapes. Sumit et al. [15] studied the performance of a tapered helical coil solar receiver with a nanostructured carbon florets coating. The efficiency increases from 48.4% to 68% for a closed cavity receiver at a flow rate of 20 L/min on the application of thermal coating. Abuseada et al. [16] surveyed a cavity receiver's energy-efficient variable aperture mechanism. The result indicated that the aperture blades captured 54% of intercepted surplus power. Slootweg et al. [17] presented a complex geometry solar tower molten salt cavity receiver and investigated its optical and thermal performance. It was found that optical efficiencies of the In addition, many researchers have analyzed the influence of cavity receiver parameters on the system's optical performance. Lin et al. [13] proposed and studied a linear Fresnel lens collector with four types of cavity receivers using east-west horizontal tracking mode. The analysis confirms that the Fresnel lens solar collector with triangular cavity receiver boasts best performance in terms of both optical and thermal characteristics. Sedighi et al. [14] proposed an indirectly irradiated cavity receiver. According to the result, a cylindrical cavity with an inverted conical base provides the highest optical efficiency, around 92%, compared to other cylindrical cavities with different base shapes. Sumit et al. [15] studied the performance of a tapered helical coil solar receiver with a nanostructured carbon florets coating. The efficiency increases from 48.4% to 68% for a closed cavity receiver at a flow rate of 20 L/min on the application of thermal coating. Abuseada et al. [16] surveyed a cavity receiver's energy-efficient variable aperture mechanism. The result indicated that the aperture blades captured 54% of intercepted surplus power. Slootweg et al. [17] presented a complex geometry solar tower molten salt cavity receiver and investigated its optical and thermal performance. It was found that optical efficiencies of the cavity receiver reached efficiencies over 93%. Soltani et al. [18] Sustainability 2022, 14, 1545 3 of 25 studied a helically baffled cylindrical cavity receiver in a parabolic dish collector. The results show that the performance can enhance up to 65% by selecting effective parameters. Aslfattahi et al. [19] evaluated the solar dish collector for three cavity receivers: cubical, hemispherical, and cylindrical shapes. According to the analysis, the hemispherical cavity receiver led to maximum thermal efficiency with the nanofluid used. Wang et al. [20,21] investigated the effects of receiver parameters on the optical performance of a Fresnel lens solar system. The analysis shows that average optical efficiencies using cavity receiver with the bottom reflective cone of spherical, cylindrical, and conical are 72.23%, 68.37%, and 76.40%, respectively. Lee et al. [22] assessed the effect of aspect ratio and head-on wind speed on the force and natural convective heat loss and area-averaged convective heat flux from a cylindrical solar cavity receiver. The numerical analysis indicates that the overall efficiency of the solar cavity receiver increases with the aspect ratio. Tu et al. [23] studied the effects of radiative surface properties inside a solar cavity receiver. They found that both the radiative and the convective heat loss of the present cavity receiver decrease with increasing thermal emissivity and improves the receiver's efficiency.
The literature shows that the study has been conducted extensively on cavity receivers for the concentrating solar collectors. However, there is no research on the comparative analysis of different shapes of linear cavity receivers used in fixed-focus solar concentrating systems in the published literature. Linear cavity receivers of various shapes provide different sunlight interception capabilities and form different energy flux distributions due to their design. Therefore, this work aims to investigate optical performance comparison of different shapes of cavity receivers in the FLSCS, which has not been reported earlier in the literature. The investigation of the optical performance comparison of different shapes of cavity receivers helps in understanding (a) the efficacy of the various shapes of cavity receiver design and (b) the optical behaviour of the receiver internal surface absorptivity and end reflection plane reflectivity on the receiver losses.
In this study, to optimize the FLSCS and linear cavity receiver of better optical performance, the present study concentrates on the effects of receiver parameters (geometric shape, receiver position f, receiver internal surface absorptivity α ab , and end reflection plane reflectivity ρ r ) on the optical efficiency of system and flux uniformity of linear cavity receiver. This work could provide a reference for the design and optimization of the optical matching between the fixed line-focus solar concentrator and linear cavity receiver and established the foundation for further research on the optical to thermal energy conversion in the FLSCS.

Physical Model
The FLSCS is shown schematically in Figure 1. The parameters include the lens element length L, lens element width B of linear Fresnel lens and receiver position f, sun declination angle δ, solar hour angle ω, sunlight incidence angle θ and latitude angle ϕ. The linear Fresnel lens concentrates sunlight into the fixed linear cavity receiver through the polar tracking system. The polar axis of FLSCS is aligned parallel with that of the earth. The lens element slides on the lens frame according to the change of δ to ensure that the focal line does not exceed the end of the fixed linear cavity receiver. The incident sunlight is completely intercepted by the opening plane of the fixed linear cavity receiver under ideal conditions. The horizontal angle of the polar-axis depends on the ϕ. For daily tracking, linear Fresnel lens rotates around the polar-axis from east to west every day at the earth rotation angular velocity with the help of a driving motor. For seasonal tracking, the linear Fresnel lens manually adjusts at δ with the use of the lens frame. The working fluid flows in from the lower side of the cavity and flows out from the higher side, thereby taking away the heat. Heat transfer oil is used as the heat transfer working medium. The model and methodology can be extrapolated in other countries by modifying the design parameters and operating conditions. According to the previous work of our team [3,4], the θ in FLSCS shown in Figure 1, is determined by the following formula: The S is the horizontal angle of the polar axis, and the S = ϕ. In addition, the ω = 0 • under ideal tracking conditions. Applying them for Formula (1), we have: Moreover, the δ can be expressed as [24,25], The N is the day number of the year with N = 1 on 1 January.

Numerical Simulation Model
To estimate the optical performance of linear cavity receiver in the FLSCS, commercial software TracePro ® using the Monte Carlo ray tracing method was applied. Figure 2a displays the FLSCS system using one of the modelled receivers, linear triangular cavity receiver (LTCR), through the ray-tracing analysis. The incident sunlight is concentrated by the linear Fresnel lens, then intercepted and absorbed by the LTCR, and part of the sunlight escapes after being reflected. Figure 2b shows the energy flux distribution in the LTCR. The energy flux is mainly distributed at the bottom of the LTCR, and the energy flux density shows non-uniform characteristics. model and methodology can be extrapolated in other countries by modifying the design parameters and operating conditions. According to the previous work of our team [3,4], the θ in FLSCS shown in Figure 1, is determined by the following formula: The S is the horizontal angle of the polar axis, and the S = φ. In addition, the ω = 0° under ideal tracking conditions. Applying them for Formula (1), we have: Moreover, the δ can be expressed as [24,25], The N is the day number of the year with N = 1 on 1 January.

Numerical Simulation Model
To estimate the optical performance of linear cavity receiver in the FLSCS, commercial software TracePro ® using the Monte Carlo ray tracing method was applied. Figure 2a displays the FLSCS system using one of the modelled receivers, linear triangular cavity receiver (LTCR), through the ray-tracing analysis. The incident sunlight is concentrated by the linear Fresnel lens, then intercepted and absorbed by the LTCR, and part of the sunlight escapes after being reflected. Figure 2b shows the energy flux distribution in the LTCR. The energy flux is mainly distributed at the bottom of the LTCR, and the energy flux density shows non-uniform characteristics.  The receivers used in this study are LTCR, linear arc-shaped cavity receiver (LACR), and linear rectangular cavity receiver (LRCR). Moreover, three shapes of the linear cavity receiver are composed of coated copper tubes through which heat transfer fluid flows, as shown in Figure 3. All cavity receivers' opening size and internal surface area are the same, as shown in Table 1. The receivers used in this study are LTCR, linear arc-shaped cavity receiver (LACR), and linear rectangular cavity receiver (LRCR). Moreover, three shapes of the linear cavity receiver are composed of coated copper tubes through which heat transfer fluid flows, as shown in Figure 3. All cavity receivers' opening size and internal surface area are the same, as shown in Table 1.   The optical performance comparison of different shapes of linear cavity receivers in the FLSCS is carried out under ideal conditions. Therefore, engineering errors are not considered, including the slope, specular, and contour errors of a linear Fresnel lens, the installation error of the linear cavity receiver, the system's tracking error, etc. [26,27]. The half angular width of the sun is set to 0.27° during simulation to close to reality [28,29].

Optical Work Validation
The proposed method carried out in this research has been validated against the experimental work in our previous works [3,4], see Figure 4. After designing an optical model and setting up the TracePro ® software, the results of both works were compared, and a good agreement was achieved. According to the experimental and simulated results, the maximum flux density on the Lambert board is 18,137 W/m 2 and 19,052 W/m 2 , when the direct solar irradiance Id is 796 W/m 2 at 2:50 pm 29 July 2020. The main facula relative error between the filling area S0 under the experimental curves and the S0′ under the simulated curves is 3.24%. The relative error between the filling area S1 under the experimental curves and the S1′ under the simulated curves is 15.35%.  The optical performance comparison of different shapes of linear cavity receivers in the FLSCS is carried out under ideal conditions. Therefore, engineering errors are not considered, including the slope, specular, and contour errors of a linear Fresnel lens, the installation error of the linear cavity receiver, the system's tracking error, etc. [26,27]. The half angular width of the sun is set to 0.27 • during simulation to close to reality [28,29].

Optical Work Validation
The proposed method carried out in this research has been validated against the experimental work in our previous works [3,4], see Figure 4. After designing an optical model and setting up the TracePro ® software, the results of both works were compared, and a good agreement was achieved. According to the experimental and simulated results, the maximum flux density on the Lambert board is 18,137 W/m 2 and 19,052 W/m 2 , when the direct solar irradiance I d is 796 W/m 2 at 2:50 pm 29 July 2020. The main facula relative error between the filling area S 0 under the experimental curves and the S 0 under the simulated curves is 3.24%. The relative error between the filling area S 1 under the experimental curves and the S 1 under the simulated curves is 15.35%.

Optical Analysis
Since the photothermal conversion takes place on the internal wall of the linear cavity receiver, the optical efficiency of the FLSCS (ηopt) can be described by Formula (4) where Qabsorber (W) is the sunlight energy absorbed by the internal wall of the linear cavity receiver, Aabsorber (m 2 ) is the total area of the internal surface, Gmean (W/m 2 ) is the average flux density on the absorber, and Xmean is the mean value of the local energy flux concentration ratio. To facilitate comparison and avoid being affected by the cosine effect of the θ, the relative optical efficiency loss (ηre-opt,loss) is used to estimate the effects of receiver parameters on the optical efficiency of the FLSCS. ηre-opt,loss of the system can be calculated by: where Qabsorber,δ,ω (W) is the sunlight energy absorbed by the internal wall of the linear cavity receiver when the sun declination angle and solar hour angle is δ and ω, respectively. Qabsorber,ref (W) is the sunlight energy absorbed by the internal wall of the liner cavity receiver when the sun declination angle and solar hour angle are 0°, receiver internal surface absorptivity and end reflection plane reflectivity are 1.00 and receiver position is 650 mm. The mean value Xmean and maximum value Xmax of the local concentration ratio of the energy flux are simply obtained by applying Formula (5) and Formula (6), respectively [32,33]:

Optical Analysis
Since the photothermal conversion takes place on the internal wall of the linear cavity receiver, the optical efficiency of the FLSCS (η opt ) can be described by Formula (4) [30,31]: where Q absorber (W) is the sunlight energy absorbed by the internal wall of the linear cavity receiver, A absorber (m 2 ) is the total area of the internal surface, G mean (W/m 2 ) is the average flux density on the absorber, and X mean is the mean value of the local energy flux concentration ratio. To facilitate comparison and avoid being affected by the cosine effect of the θ, the relative optical efficiency loss (η re-opt,loss ) is used to estimate the effects of receiver parameters on the optical efficiency of the FLSCS. η re-opt,loss of the system can be calculated by: where Q absorber,δ,ω (W) is the sunlight energy absorbed by the internal wall of the linear cavity receiver when the sun declination angle and solar hour angle is δ and ω, respectively. Q absorber,ref (W) is the sunlight energy absorbed by the internal wall of the liner cavity receiver when the sun declination angle and solar hour angle are 0 • , receiver internal surface absorptivity and end reflection plane reflectivity are 1.00 and receiver position is 650 mm. The mean value X mean and maximum value X max of the local concentration ratio of the energy flux are simply obtained by applying Formulas (5) and (6), respectively [32,33]: where G max (W/m 2 ) is the maximum energy flux density, respectively. For comparison, the flux uniformity of the receiver internal surface is defined by a non-uniformity factor (σ non ) as indicated below [34,35],

Results and Discussion
To investigate the optical performance of different shapes of linear cavity receivers in the FLSCS, we studied the optical efficiency of the solar system and the energy flux distribution in linear cavity receivers under different receiver parameters. Three various shapes of linear cavity receiver-LTCR, LACR, and LRCR-were studied (see Figure 3 and Table 1). The results were compared, which were obtained through simulation realized by the use of TracePro ® on different the f, θ, α ab , and ρ r . Cases differ in the η re-opt,loss , X max , and σ non , allowing us to elucidate sensitivity analysis of the effective parameters in a linear cavity receiver. Essentially, the linear cavity receivers have the same area of internal wall and same size of the opening, but different shapes.

Effect of the Sunlight Incident Angle θ
To study the induced propagation and collection process of sunlight changes by the effect of the θ, we compare the optical performance index of the system when the δ is 0 • , 8 • , 16 • , and 23.45 • respectively, simulated under the f = 650 mm, α ab = 0.85 and ρ r = 0.85. In sun tracking, the change of the ω will also affect the propagation and collection of sunlight inside the cavity absorber. To monitor the optical performance index changes of the system during the sun tracking, we simulated the case of different ω and different δ. Figure 5a shows the effect of the θ on the optical performance of the FLSCS using LTCR. The η re-opt,loss is basically unchanged for ω of 0-30 • and linearly increases for ω of 30-60 • , when the δ = 0 • . The η re-opt,loss are 2.25%, 2.72%, and 12.69%, respectively, when the ω is 0 • , 30 • , 60 • . A similar change in the η re-opt,loss can be seen in Figure 5a, when the δ = 8 • . Noted that the η re-opt,loss are 2.62%, 3.26%, and 12.85%, respectively, when the ω is 0 • , 30 • , and 60 • . It means that the increases of δ in the range of 0-8 • have a weak effect on the η re-opt,loss . However, as shown in Figure 5a, when the δ is 16 • and 23.45 • , the η re-opt,loss increases as ω raises in the range of 0-60 • , and the rate of increase becomes ever larger. When the δ is 16 • and 23.45 • and the ω is 0 • , 30 • , and 60 • , the η re-opt,loss are 2.94%, 4.69%, 23.1% and 4.68%, 14.43%, 41.73%, respectively. The maximum η re-opt,loss of the latter is basically twice that of the former. It is mainly due to the upward movement of the focus line as the ω increases. Figure 6 shows the upward movement distance of the focal line with the θ. It can be seen that the focal line moves slowly upward for δ of 0-8 • and rises sharply for δ of 8-23.45 • .
To clearly observe the influence of the δ and ω on the propagation of sunlight, we analyze the propagation of sunlight within twice-intercepted by the cavity absorber through the simple diagram in Figure 7. The upward movement distance for δ = 8 • is 10.7 mm, and the analysis is carried out with reference to Figure 7d-i. It reveals that the sunlight did not escape during the two interception processes of the cavity absorber for ω of 0-30 • . Still, part of the first reflected sunlight escapes from the cavity absorber after ω = 30 • , when δ is in the range of 0-8 • . The upward movement distance for δ = 16 • is 46.7 mm, and the analysis is carried out with reference to Figure 7a-c. It is inferred that part of the first reflected sunlight escapes from the cavity absorber for ω of 30-60 • and even part of the incident sunlight fails to be intercepted by the cavity absorber and escapes directly as ω = 60 • . The number of escaped first reflected sunlight increases with the increasing of ω and δ for δ of 16-23.45 • . Therefore, in addition to the cosine loss caused by the δ, the existence of the above-mentioned escape sunlight exacerbates the increase in the η re-opt,loss . To clearly observe the influence of the δ and ω on the propagation of sunlight, we analyze the propagation of sunlight within twice-intercepted by the cavity absorber through the simple diagram in Figure 7. The upward movement distance for δ = 8° is 10.7 mm, and the analysis is carried out with reference to Figure 7d-i. It reveals that the sunlight did not escape during the two interception processes of the cavity absorber for ω of 0-30°. Still, part of the first reflected sunlight escapes from the cavity absorber after ω = 30°, when δ is in the range of 0-8°. The upward movement distance for δ = 16° is 46.7 mm, and the analysis is carried out with reference to Figure 7a-c. It is inferred that part of the first reflected sunlight escapes from the cavity absorber for ω of 30-60° and even part of the incident sunlight fails to be intercepted by the cavity absorber and escapes directly as ω = 60°. The number of escaped first reflected sunlight increases with the increasing of ω and δ for δ of 16-23.45°. Therefore, in addition to the cosine loss caused by the δ, the existence of the above-mentioned escape sunlight exacerbates the increase in the ηre-opt,loss. Moreover, as shown in Figure 5a, the Xmax is increasing for ω of 0-60°, but the growth rate of Xmax decreases for δ of 0-23.45°. The Xmax mainly depends on the energy flux distribution of first intercepted sunlight on the cavity absorber inner wall. The symmetry plane of the linear Fresnel lens and the cavity receiver inner wall form an intersection line, and in the ηre-opt,loss can be seen in Figure 5b, when the δ = 8°. Noted that the ηre-opt,loss is 15.12%, 14.33%, and 3.89%, respectively, when the ω is 0°, 15°, and 60°. This means that the increase of δ in the range of 0-8° has a weak effect on the ηre-opt,loss.  Moreover, as shown in Figure 5a, the X max is increasing for ω of 0-60 • , but the growth rate of X max decreases for δ of 0-23.45 • . The X max mainly depends on the energy flux distribution of first intercepted sunlight on the cavity absorber inner wall. The symmetry plane of the linear Fresnel lens and the cavity receiver inner wall form an intersection line, and ∆f represents the distance between the intersection line and the focal line. The energy flux distribution of the focus facula shows that the energy flux density in the main facula is much higher than that in the side facula [3,4]. The energy flux density of the focus facula is negatively related to the distance between the focal plane and the focal line. Therefore, ∆f is used to describe the change process of the X max intuitively. The smaller the ∆f, the greater the energy flux density value on the intersection line, the greater the X max , and vice versa. It can be seen in Figure 7a-i that when the δ is a fixed value, the ∆f decreases with the increase of ω, increasing the X max . With the increasing of δ, the focal line moves upward, and the ∆f increases, finally leading to a decrease in the growth trend of ∆f for ω of 0-60 • , leading to a decline in the growth trend of the X max . In addition, the X max decreases with the increase of δ, when the ω is a constant value. For example, when the δ is 0 • , 8 • , 16 • and 23.45 • and the ω = 60 • , the X max are 6.461, 5.950, 4.428, and 2.773, respectively. This means that the δ has a greater influence than the ω on the X max . In other words, the X max can be reduced by moving the focal line upward. Furthermore, as shown in Figure 5a, the σ non is increasing for ω of 0-60 • . However, it can be seen in Figure 7a-i that the distribution area of the twice-intercepted sunlight does not monotonously increase or decrease with the change of ω. This means that the X max is far greater than the X mean , thus the σ non mainly depends on the X max . Note that the growth rate of σ non increases for δ of 0-23.45 • , while the X max is the opposite. In addition, the σ non decreases with the increase of δ, when the ω is a constant value. For example, when the δ is 0 • , 8 • , 16 • , and 23.45 • and the ω = 60 • , the σ non are 0.737, 0.715, 0.662, and 0.591, respectively. It indicates that the rate of escaped sunlight increases for δ of 0-23.45 • , resulting in a sharp drop in the X mean . It is consistent with the change of η re-opt,loss . Analysis showed that the influence of δ on optical performance is extremely greater than that of ω.
As in the case of the LTCR, Figure 5b shows the effect of θ on the optical performance of the FLSCS using LACR. As shown in Figure 5b, the η re-opt,loss decrease slowly for ω of 0-15 • and decreases significantly for ω of 15-60 • , when the δ = 0 • . The η re-opt,loss are 14.99%, 14.72%, and 2.61%, respectively, when the ω is 0 • , 15 • , and 60 • . Referring to Figure 8g-i, the first intercepted sunlight escapes completely after being reflected as ω = 0 • . After that, part of the first reflected sunlight is intercepted for the second time by the cavity absorber and the number of second intercepted sunlight also increases with the ω. A similar change in the η re-opt,loss can be seen in Figure 5b, when the δ = 8 • . Noted that the η re-opt,loss is 15.12%, 14.33%, and 3.89%, respectively, when the ω is 0 • , 15 • , and 60 • . This means that the increase of δ in the range of 0-8 • has a weak effect on the η re-opt,loss .
However, the η re-opt,loss decreases for ω of 0-45 • and increases for ω of 45-60 • when the δ = 16 • , as shown in Figure 5b. When the ω is 0 • , 45 • , and 60 • , the η re-opt,loss is 12.20%, 9.28%, and 18.03%, respectively. Referring to Figure 8a-c, part of the first reflected sunlight escapes after being reflected as ω = 0 • . After that increases with the ω, the number of escaped first reflected sunlight decreases, but part of the incident sunlight is blocked by the cavity absorber, increasing optical loss. It is worth noting that the η re-opt,loss increases for ω of 0-60 • and the growth rate of η re-opt,loss increases with ω, when the δ = 23.45 • , as shown in Figure 5b. The η re-opt,loss is 10.19%, 17.48%, and 40.53%, respectively, when the ω = 0 • , 30 • , 60 • . Referring to Figure 8a-i, it can be inferred that the upward movement distance of the focal line is 97.2 mm as δ = 23.45 • , causing the incident sunlight not to be completely intercepted by the cavity absorber and escape. With the increasing of ω, the number of escaped incident sunlight increases. This means that the η re-opt,loss mainly depends on the first intercepted sunlight.
Moreover, as shown in Figure 5b, the X max increases for ω of 0-45 • but decreases for ω of 45-60 • . Referring to Figure 8a-i, it can be seen that the ∆f decreases during the tracking of ω, increasing the X max of the energy flux distribution of the first intercepted sunlight. The secondary intercepted sunlight is reflected again to the distribution area of the first intercepted sunlight, resulting in the X max being further increased for ω of 0-45 • . However, the distribution area of the third intercepted sunlight gradually deviates from that of the first intercepted sunlight for ω of 45-60 • , which can cause the X max to decrease. Even though the focal line continues to approach the cavity absorber inner wall during the tracking of ω and the X max increases, it is not enough to compensate for the decrease in the X max caused by the deviation of the secondary reflected sunlight, which ultimately leads to a decrease in the X max . The X max mainly depends on the energy flux distribution of first intercepted sunlight, but the energy flux distribution of the subsequently intercepted sunlight can also affect the X max .  Furthermore, as shown in Figure 5b, the σ non increases for ω of 0-45 • and decreases for ω of 45-60 • when δ = 0 • and 8 • , which is similar to the change in the X max . However, it can be seen in Figure 8d-i that the distribution area of the twice-intercepted sunlight increased for ω of 0-60 • . This means that the X max is far greater than the X mean , thus the σ non mainly depends on the X max . Noted that the σ non increases for ω of 0-60 • when δ = 16 • and 23.45 • , as shown in Figure 5b, which is different from the previous two cases. The growth rate of σ non increases for δ of 16-23.45 • . It can be seen in Figure 8c that part of the incident sunlight failed to be intercepted by the cavity absorber for the first time as ω = 60 • , resulting in a decrease in the X mean . The X max and σ non decrease and increase respectively for ω of 45-60 • , which means that the incident sunlight failed to be intercepted by the cavity absorber has a greater impact on the X mean than the X max . As the δ increases, the number of incident sunlight that fails to be intercepted by the cavity absorber increases, making the effects mentioned above more apparent. Figure 5c shows the effect of the θ on the optical performance of the LFR using LRCR. As shown in Figure 5c, when the δ = 0 • , the η re-opt,loss decreases slowly for ω of 0-15 • , then decreases sharply for ω of 15-45 • , and finally decreases slowly for ω of 45-60 • . The η re-opt,loss are 15.19%, 2.80%, and 2.47%, respectively, when the ω is 0 • , 45 • , and 60 • . A similar change in the η re-opt,loss can be seen in Figure 5c, when the δ = 8 • . Note that the η re-opt,loss is 15.42%, 3.66%, and 2.94%, respectively, when the ω is 0 • , 45 • , and 60 • . This means that the increases of δ in the range of 0-8 • have a weak effect on the η re-opt,loss . However, as shown in Figure 5c, when the δ = 16 • , the η re-opt,loss decreases approximately linearly for ω of 0-45 • and then increases for ω of 45-60 • . The η re-opt,loss are 14.09%, 5.23%, and 11.10%, respectively, when the ω is 0 • , 45 • , and 60 • . The upward movement distance for δ = 16 • is 46.7 mm. The analysis is carried out with reference to Figure 9a-c. It is inferred that the number of escaped first reflected sunlight decreases for ω of 0-45 • . Part of the incident sunlight fails to be intercepted and escapes for ω of 45-60 • , and the η re-opt,loss rises sharply. In other words, the increased energy of second intercepted sunlight is much smaller than that of escaped incident sunlight.
As shown in Figure 5c, when the δ = 23.45 • , the η re-opt,loss increases for ω of 0-60 • , and the growth rate of η re-opt,loss becomes ever larger. When the ω is 0 • , 30 • , and 60 • , the η re-opt,loss are 12.86%, 16.72%, and 37.75%, respectively. This is mainly due to the upward movement of the focus line as the δ increases. The upward movement distance for δ = 23.45 • is 97.2 mm. It is more than twice that in Figure 5c as the δ = 16 • . Thus, the η re-opt,loss mainly depends on the first intercepted sunlight. Moreover, as shown in Figure 5c, the X max is decreasing for ω of 0-45 • and increasing for ω of 45-60 • . The X max mainly depends on the energy flux distribution of first intercepted sunlight. It can be seen in Figure 9a-i that when the δ is a fixed value, with increasing ω, ∆f have an increase, followed by a decrease, resulting in a similar change in the X max . With the increasing of δ, the focal line moves upward and the ∆f increases, leading to a decrease in X max with the rise of δ when the ω is a constant value. For example, when the δ is 0 • , 8 • 16 • , and 23.45 • and the ω = 60 • , the X max are 7.007, 6.144, 6.056, and 3.986 respectively. Furthermore, as shown in Figure 5c, the σ non is decreasing for ω of 0-45 • and increasing for ω of 45-60 • . It is consistent with the changing trend of the X max . It can be seen in Figure 9a-i that the distribution area of the twice-intercepted sunlight does not monotonously increase or decrease with the change of ω. This means that the X max is far greater than the X mean , thus the σ non mainly depends on the X max . In addition, the σ non decreases with the increase of δ for ω of 0-45 • , when the ω is a constant value. However, the σ non variation of ω = 60 • is not monotonous for δ of 0-23.45 • , because part of the incident sunlight fails to be intercepted and escapes.

Effect of the Receiver Position f
To investigate the effect of the f on the optical performance more detail, four cases of f = 600, 625, 675, and 700 mm are selected for comparative analysis during the downward and upward shift of the linear Fresnel lens. To avoid the influence of the upward movement of the focal line caused by the change of δ, the situation of δ = 0 • is selected for analysis and simulated under the α ab = 0.85 and ρ r = 0.85, as shown in Figure 10. Figure 10a shows the effect of the f on the optical performance of the FLSCS using LTCR. As shown in Figure 10a, the η re-opt,loss is basically unchanged for ω of 0-30 • and increases for ω of 30-60 • when the f = 600 mm and 625 mm. However, the difference from the change in Figure 5a of δ = 0 • is that the growth rate of η re-opt,loss increases for ω of 30-60 • . When the f is 600 mm, 625 mm and the ω is 30 • , 45 • , and 60 • , the η re-opt,loss are 2.25%, 5.81%, 24.16%, and 2.25%, 6.65%, 14.92%, respectively. It means that the η re-opt,loss decreases as f decreases for ω of 0-45 • , but the η re-opt,loss increases as f decreases when ω = 60 • . The downward shift of the linear Fresnel lens can be referenced in Figure 7j-o. It reveals that the sunlight is completely intercepted by the cavity absorber twice for ω of 0-45 • , and more of the secondary reflected sunlight is intercepted by the cavity absorber again, resulting in a decrease in the η re-opt,loss . However, part of the incident sunlight may be blocked by the cavity absorber before entering when the ω = 60 • , and the optical blocking loss becomes more serious as f decreases. As shown in Figure 10a, the η re-opt,loss increases for ω of 0-60 • when the f = 675 mm and 700 mm. However, the difference from the change in Figure 5a of f = 650 mm is that the η re-opt,loss increases for ω of 0-30 • . When the f is 675 mm, 700 mm and the ω is 0 • , 30 • , the η re-opt,loss are 2.25%, 2.91% and 2.25%, 3.83% respectively. It means that the η re-opt,loss remains substantially unchanged with the f of 600-700 mm as δ = 0 • and ω = 0 • , but the η re-opt,loss increases for f of 650-700 mm as δ = 0 • and ω = 30 • . The upward shift of the linear Fresnel lens can be referenced to Figure 7a-f. It reveals that when the ω = 0 • , the sunlight is completely intercepted by the cavity absorber twice, and when the ω = 30 • , part of the first reflected sunlight escapes from the cavity absorber, becoming more obvious as f increases. Figure 10c shows the effect of the f on the optical performance of the LFR using LRCR. As shown in Figure 10c, when the f = 600 mm and 625 mm, the η re-opt,loss decreases slowly for ω of 0-15 • , then drops sharply for ω of 15-45 • , and finally increases for ω of 45-60 • . However, the difference from the change in Figure 5c of f = 650 mm as δ = 0 • is that the η re-opt,loss increases for ω of 45-60 • . When the f is 600 mm, 625 mm and the ω is 0 • , 45 • , and 60 • , the η re-opt,loss are 15%, 2.8%, 16.72% and 15%, 2.26%, 5.48%, respectively. It means that the change of f has a weak influence on the η re-opt,loss for ω of 0-45 • . The downward shift of the linear Fresnel lens can be referenced in Figure 9j-o. It reveals that in tracking the ω, the distribution area of first intercepted sunlight is concentrated on the bottom, then on the bottom and one side, and finally on one side of it. The distribution area of the first intercepted sunlight is concentrated on the bottom, and one side increases the number of the reflection of sunlight and reduces the number of escaped secondary reflected sunlight, which is similar to the role of an LTCR. The sharp increase in the η re-opt,loss of f = 600 mm as the ω = 60 • is due to part of the incident sunlight being blocked by the cavity absorber before entering, and the optical blocking loss becomes more serious as f decreases. As shown in Figure 10c, when the f = 675 mm and 700 mm, except for the η re-opt,loss of f = 700 mm as the ω = 60 • , the η re-opt,loss decreases with the increasing of ω. It is similar to the change of the η re-opt,loss in Figure 5c of f = 650 mm as δ = 0 • . When the f is 675 mm, 700 mm and the ω is 0 • , 30 • , 60 • , the η re-opt,loss are 15%, 6.64%, 3.17% and 14.43%, 8.15%, 9.74, respectively. The upward shift of the linear Fresnel lens can be referenced in Figure 9a-f. It reveals that the number of second intercepted sunlight increases with the increase of f for ω of 0-15 • , but the number of incident sunlight which fails to be intercepted and escapes increases with the increase of f for ω of 15-60 • . Moreover, the change of the X max in Figure 10c is similar to that in Figure 5c of f = 650 mm as δ = 0 • , the X max decreases for ω of 0-45 • and then decreases for ω of 45-60 • . It can be seen in Figure 9a-i that when the ω is a fixed value, ∆f increases with the increasing of f, resulting in a similar change in the X max . For example, when the f = 600, 625, 675 and 700 mm and the ω is 60 • , the X max are 8.341, 8.686, 5.514 and 4.395, respectively. Furthermore, as shown in Figure 10c As shown in Figure 5c, when the δ = 23.45°, the ηre-opt,loss increases for ω of 0-60°, and the growth rate of ηre-opt,loss becomes ever larger. When the ω is 0°, 30°, and 60°, the ηre-opt,loss are 12.86%, 16.72%, and 37.75%, respectively. This is mainly due to the upward movement of the focus line as the δ increases. The upward movement distance for δ = 23.45° is 97.2 mm. It is more than twice that in Figure 5c as the δ = 16°. Thus, the ηre-opt,loss mainly depends Xmean is greater than the Xmax as f = 600 mm. In addition, except for the σnon of the ω = 45° and 60° as f = 600 mm, the σnon decreases with the increase of f when the ω is a constant value. For example, when the f is 600 mm, 625 mm, 675 mm, and 700 mm as the ω = 0°, the σnon is 0.527, 0.501, 0.413, and 0.346, respectively. Referring to Figure 7a-o, it can be seen that the distribution area of the incident sunlight intercepted by the cavity absorber for the first time increases for f of 600-700 mm when the ω is a constant value, resulting in a drop in the σnon.  Moreover, different from the change of the X max in Figure 5a of f = 650 mm, the X max of f = 600 mm increases first and then decreases with the increase of ω, and the situation change occurs as the ω = 30 • . Referring to Figure 7m-o, it can be seen that the focal line first approaches the cavity absorber inner wall and then moves away from it during the tracking of ω. Combined with the changing trend of X max in Figure 10a as the f = 600 mm, it can be seen that the X max reaches its maximum value when the focal line falls on the cavity absorber inner wall, and the corresponding ω is in the range of 15 • to 30 • . The changing trend of X max as f = 625, 675, and 700 mm is similar to that of f = 650 mm, the X max increase for ω of 0-60 • . Referring to Figures 7a-f,j-l, it can be seen that the ∆f decreases during the tracking of ω. With the increasing of ω, part of the first reflected sunlight escapes from the cavity absorber. The number of second reflected sunlight reflected to the distribution area of the first intercepted sunlight decreases. This would originally cause the X max to decrease. Still, the actual X max increases with the increase of ω, which indicates that the X max depends on the energy flux distribution of first intercepted sunlight. Furthermore, as shown in Figure 10a, the σ non is increasing for ω of 0-60 • . Note that the variation of η re-opt,loss , X max , and σ non with the ω and f in Figure 10a of f = 625 mm, 675 mm and 700 mm are similar to those in Figure 5a of δ = 16 • and 23.45 • , and thus detailed analysis is omitted herein. However, the changing trend of σ non in Figure 10a as the f = 600 mm is inconsistent with that of the X max for ω of 30-60 • . It shows that the decrease rate of X mean is much greater than that of X max ; that is, the influence of optical blocking loss on the X mean is greater than the X max as f = 600 mm. In addition, except for the σ non of the ω = 45 • and 60 • as f = 600 mm, the σ non decreases with the increase of f when the ω is a constant value. For example, when the f is 600 mm, 625 mm, 675 mm, and 700 mm as the ω = 0 • , the σ non is 0.527, 0.501, 0.413, and 0.346, respectively. Referring to Figure 7a-o, it can be seen that the distribution area of the incident sunlight intercepted by the cavity absorber for the first time increases for f of 600-700 mm when the ω is a constant value, resulting in a drop in the σ non .
As in the case of the LTCR, Figure 10b shows the effect of f on the optical performance of the FLSCS using LACR. As shown in Figure 10b, when the f = 600 mm and 625 mm, the η re-opt,loss is basically unchanged for ω of 0-15 • , then decreases for ω of 15-45 • and finally increased for ω of 45-60 • . However, the difference from the change in Figure 5b of δ = 0 • is that the of η re-opt,loss increases for ω of 45-60 • . When the f = 600 mm and 625 mm and the ω = 45 • and 60 • , the η re-opt,loss are 6.04%, 13.18% and 3.37%, 3.55% respectively. It means that the η re-opt,loss increases as f decreases for ω of 45-60 • . The downward shift of the linear Fresnel lens can be referenced in Figure 8l,o. It reveals that with the decreasing of f, part of the incident sunlight may be blocked by the cavity absorber before entering for ω of 45-60 • , and the optical blocking loss becomes more severe as f decreases. Figure 10b of the f = 675 mm shows that the η re-opt,loss decreases for ω of 0-60 • . It is similar to the change of η re-opt,loss in Figure 5b of δ = 0 • . Nevertheless, the η re-opt,loss decreased for ω of 0-45 • and increased for ω of 45-60 • in Figure 10b as f = 700 mm. When the f = 675 mm and 700 mm and the ω = 45 • and 60 • , the η re-opt,loss are 7.26%, 6.29% and 8.33%, 16.65%, respectively. Referring to Figure 8a-f, it can be seen that with the increase of ω, the number of escaped first intercepted sunlight gradually decreases, thus the η re-opt,loss decreases. However, with the increase of f, the spatial distribution area of the light after passing the focal line increases. Part of the incident sunlight failed to be intercepted by the cavity absorber for the first time as ω = 60 • , increasing the η re-opt,loss . Note that the η re-opt,loss decreases as f increases for ω of 0-30 • . The distribution area of first intercepted sunlight increases as f increases, increasing the number of second intercepted sunlight, and thus the η re-opt,loss decreases.
Moreover, the changing trend of X max as f = 600, 625, 675, and 700 mm in Figure 10b is similar to that of f = 650 mm in Figure 5b as δ = 0 • , the X max increase for ω of 0-45 • and decreases for ω of 45-60 • . Thus, detailed analysis is omitted herein. The X max should generally increase as f decreases when the ω is fixed. However, when the f = 600, 625, 675, and 700 mm and the ω = 45 • , 60 • , the X max are 7.477, 7.626, 6.740, 5.912 and 5.832, 6.949, 6.509, 5.699, respectively. Referring to Figure 8m-o, it can be seen that when f = 600 mm, the trajectory of the focal line intersects with the LACR, causing the focal line to approach and then move away from the cavity absorber inner wall during the tracking of ω, thus the X max decrease as the ω = 45 • , 60 • for the f = 600 mm and 625 mm. Furthermore, as shown in Figure 10b, when the f = 600 mm, the σ non is unchanged for ω of 0-15 • , then decreases for ω of 15-60 • . It is inconsistent with the change of the X max . Referring to Figure 8m-o, it can be seen that the distribution area of the twice-intercepted sunlight increased for ω of 0-60 • . In addition, the η re-opt,loss decreases for ω of 0-45 • and increase for ω of 45-60 • . Thus, the X mean has a greater impact on the σ non than that of the X max for ω of 0-45 • , and the situation is reversed for ω of 45-60 • . As shown in Figure 10b, when the f = 625 mm, the σ non increase for ω of 0-30 • , then decrease for ω of 30-60 • . It is consistent with the change of the X max for ω of 0-30 • . This is because the η re-opt,loss is almost unchanged for ω of 0-30 • , thus the X mean basically unchanged. However, the η re-opt,loss drops sharply for ω of 30-45 • , while the X max rises sharply. It means that the X mean has a greater impact on the σ non than the X max for ω of 30-45 • . Noted that the X mean almost unchanged for ω of 45-60 • due to the stable η re-opt,loss . Therefore, the change of σ non depends on that of X max . As shown in Figure 10b, when the f = 675 mm, the σ non increase for ω of 0-45 • , then decrease for ω of 45-60 • . It is consistent with the change of the X max for ω of 0-60 • , and the η re-opt,loss decrease for ω of 0-60 • . It means that the X max has a greater impact on the σ non than the X mean for ω of 0-60 • . A similar situation can be seen in Figure 10b as the f = 700 mm for ω of 0-45 • . However, the σ non and η re-opt,loss increase for ω of 45-60 • during the X max decrease. It means that the X mean has a greater impact on the σ non than the X max for ω of 45-60 • .

Effect of Receiver Internal Surface Absorptivity α ab
After the specific case study, the parametric study was conducted to quantify the effects of α ab on the optical performance of the system. The existence of δ causes the sunlight to obliquely enter the linear Fresnel lens and then move the focal line upwards, which leads to a decrease in the number of rays intercepted by the cavity absorber. Therefore, the α ab has a great influence on the η re-opt,loss . In addition, the reflection number of sunlight inside the cavity absorber is also greatly affected by the α ab , which ultimately affects the σ non , especially when the δ = 23.45 • . In this section, for the receiver internal surfaces, three kinds of α ab (0.75, 0.85, and 1.00) were considered, and the ω varying from 0 • to 60 • at the particular α ab was numerically analyzed under the f = 650 mm (except for LACR) and ρ r = 0.85. Figure 11a shows the effect of the α ab on the optical performance of the FLSCS using LTCR.As shown in Figure 11a, the η re-opt,loss increases with the decrease of α ab , but its increasing trend gradually decreases with the increase of ω. For example, when the ω = 0 • and 60 • , the α ab is 1.00, 0.85, and 0.75, the η re-opt,loss is 2.47%, 4.68%, 6.32% and 35.56%, 41.73%, 46.53%, respectively. Referring to Figure 7a-c, as the ω increases, the distribution area of first intercepted sunlight gradually moves from two sides to one side, resulting in a gradual decrease in the number of secondary intercepted sunlight. Thus, the absorbed sunlight energy is increasingly dependent on the first interception, and the ratio of η re-opt,loss is close to the ratio of α ab when the ω = 60 • . In other words, the cavity structure can be optimized to increase the number of incident sunlight reflections on the inner wall, thereby reducing the requirement for high α ab . Moreover, as shown in Figure 11a, the X max increases with the decrease of α ab as the ω = 0 • , while the X max decreases with the decrease of α ab for ω of 15-60 • . In addition, the decreasing trend of X max with the α ab becomes more obvious for ω of 15-60 • . This is because the incident sunlight is symmetrically distributed on both bottom sides of the cavity absorber when the ω = 0 • . With the decreasing of α ab , the number of the reflection of the sunlight on both bottom sides increases to form energy accumulation, increasing X max . Referring to Figure 7a-c, as the ω increases, the distribution area of first intercepted sunlight gradually shifts to one side, part of the second reflected sunlight escapes, and the energy-concentration effect decreases. The X max mainly depends on the energy flux distribution of the first intercepted sunlight. The greater the α ab , the greater the X max . The ∆f decreases with the increase of ω, thus the energy flux density of first intercepted sunlight increases, resulting in a more noticeable difference in X max with different α ab . As shown in Figure 11a, the variation of σ non is similar to that of the X max . The σ non increases with the decrease of α ab as the ω = 0 • , and the situation is the opposite for ω of 15-60 • . It further shows that the X max is far greater than the X mean , and the σ non mainly depends on the X max . However, unlike the change of X max , the decreasing trend of σ non with the α ab becomes gentle as ω increases in the range of 15-60 • . This is because as the ω increases, the energy absorbed of first intercepted sunlight accounts for an increasing proportion of the total absorbed energy. In other words, the energy flux distribution mainly depends on the distribution of first intercepted sunlight. Therefore, the influence of α ab on the σ non is weakened. of 15-60°. It further shows that the Xmax is far greater than the Xmean, and the σnon mainly depends on the Xmax. However, unlike the change of Xmax, the decreasing trend of σnon with the αab becomes gentle as ω increases in the range of 15-60°. This is because as the ω increases, the energy absorbed of first intercepted sunlight accounts for an increasing proportion of the total absorbed energy. In other words, the energy flux distribution mainly depends on the distribution of first intercepted sunlight. Therefore, the influence of αab on the σnon is weakened. As in the case of the LTCR, Figure 11b shows the effect of αab on the optical performance of the FLSCS using LACR. To avoid being affected by the occluded and escaped As in the case of the LTCR, Figure 11b shows the effect of α ab on the optical performance of the FLSCS using LACR. To avoid being affected by the occluded and escaped incident sunlight, the case of f = 675 mm is selected. As shown in Figure 11b, the η re-opt,loss increases with the decrease of α ab , but its increasing trend gradually decreases with the increase of ω. For example, when the ω = 0 • and 60 • , the α ab is 1.00, 0.85, and 0.75, the η re-opt,loss is 12.75%, 19.11%, 23.71% and 48.27%, 52.44%, 55.61%, respectively. Since the upward movement distance of focal line is 97.2 mm as δ = 23.45 • and the f = 675 mm, referring to Figure 8a-c, as the ω increases, the distribution area of first intercepted sunlight gradually moves from the bottom to one side, resulting in a gradual increase in the number of secondary intercepted sunlight. Therefore, as the number of the reflection of sunlight in the cavity absorber increases, the effect of α ab on the absorption of sunlight energy decreases gradually. Moreover, as shown in Figure 11b, the X max decreases with the decrease of α ab for ω of 0-60 • . In addition, the decreasing trend of X max with the α ab is basically stable before ω = 45 • but becomes obvious as ω increases in the range of 45-60 • . Referring to Figure 8a-c, it can be inferred that the energy flux distribution mainly depends on the first intercepted sunlight, and with the increase of ω, there is an increase in the amount of escaped sunlight. Therefore, the influence of α ab on the X max is intensified, and the amount of escaped sunlight increases sharply for ω of 45-60 • . When the ω = 60 • , the α ab is 1.00, 0.85 and 0.75, the X max is 4.782, 4.034, 3.567, respectively, which is close to the ratio between the different α ab . As shown in Figure 11b, the variation of σ non is similar to that of the X max . The σ non decreases with the decrease of α ab . Before ω = 45 • , the variation of the σ non with the α ab can basically be ignored, but it becomes obvious as ω increases in the range of 45-60 • . This is because the energy flux distribution mainly depends on the distribution of first intercepted sunlight, especially for ω of 0-45 • . Therefore, the influence of α ab on the σ non is weakened. However, the amount of escaped sunlight increases sharply for ω of 45-60 • , the distribution of second or more intercepted sunlight has an increased influence on the energy flux distribution. When the ω = 0 • and 60 • , the α ab is 1.00, 0.85, and 0.75, the σ non is 0.4065, 0.404, 0.4016 and 0.6205, 0.5864, 0.5636, respectively.
As in the case of the LTCR, Figure 11c shows the effect of α ab on the optical performance of the FLSCS using LRCR as f = 650 mm. As shown in Figure 11c, the η re-opt,loss increases with the decrease of α ab , but its increasing trend gradually decreases with the increase of ω. For example, when the ω = 0 • and 60 • , the α ab is 1.00, 0.85, and 0.75, the η re-opt,loss is 2.47%, 12.86%, 20.99% and 36.08%, 37.75%, 39.06%, respectively. Referring to Figure 9a-c, as the ω increases, the distribution area of first intercepted sunlight gradually moves from the bottom to one side, resulting in a gradual increase in the number of secondary intercepted sunlight. Therefore, as the number of the reflection of sunlight in the cavity absorber increases, the effect of α ab on the absorption of sunlight energy decreases gradually. Moreover, as shown in Figure 11c, the X max decreases with the decrease of α ab for ω of 0-60 • . In addition, the decreasing trend of X max with the α ab decrease for ω of 0-60 • . Referring to Figure 9a-c, it can be inferred that the energy flux distribution mainly depends on the first intercepted sunlight as ω = 0 • , and with the increase of ω, the amount of secondary intercepted sunlight increases. Especially as ω = 60 • , part of the incident sunlight fails to be intercepted and escapes. The influence of the secondary intercepted sunlight on the energy flux distribution gradually increases. Therefore, the influence of α ab on the X max decreases as the ω increases. When the ω = 60 • , the α ab is 1.00, 0.85 and 0.75, the X max is 4.221, 3.986, 3.751, respectively. As shown in Figure 11c, the variation of σ non is similar to that of the X max . The σ non decreases with the decrease of α ab . It further shows that the X max is far greater than the X mean , and the σ non mainly depends on the X max . Noted that the decrease rate of σ non presents continuous fluctuations for ω of 0-60 • . However, the increase rate of the η re-opt,loss and the decrease rate of the X max is decreased for ω of 0-60 • . In other words, the α ab can affect the change rate of σ non , but not the changing trend. When the ω = 0 • and 60 • , the α ab is 1.00, 0.85, and 0.75, the σ non is 0.682, 0.6663, 0.6558 and 0.7039, 0.6945, 0.6822, respectively.

Effect of End Reflection Plane Reflectivity ρ r
The increase of the δ causes the light to enter the cavity absorber obliquely. The end loss can be effectively reduced by sliding the mirror element, and the sunlight reflected by the cavity absorber inner wall tends to propagate to one end. By setting the end reflection plane, the sunlight incident on the end can be reflected again so that it has a chance to be intercepted again by the cavity absorber to reduce the optical loss further. The end reflection plane itself has a role in the amount of sunlight reflected and lost. Thus, the effect of the end reflection plane in reducing the optical loss and its influence on the energy flux distribution is explained by studying end reflection planes with different ρ r . In this section, for the end reflection planes, three kinds of ρ r (0.75, 0.85 and 1.00) were considered, and the ω varying from 0 • to 60 • at the particular ρ r was numerically analyzed. The f, α ab and δ are 650 mm, 0.85, and 23.45 • , respectively. Figure 12a shows the effect of the ρ r on the optical performance of the FLSCS using LTCR. It can be seen from Figure 12a that the η re-opt,loss increases slightly as the ρ r decreases when the ω is fixed. The average η re-opt,loss for ρ r = 1.00, 0.85, and 0.75 are 18.49%, 18.56%, and 18.60%, respectively, for ω of 0-60 • . However, the average η re-opt,loss is reduced by 0.46%, 0.39%, and 0.35%, respectively, compared to 18.95% if the end reflection plane is not installed. It can be inferred that the number of incident sunlight on the end reflection plane can be ignored compared to intercepted sunlight. The results prove that sliding the lens element can effectively solve the problem of end loss. In addition, as shown in Figure 12a, the X max decreases with the decrease of ρ r for ω of 0-60 • . It is because part of the incident sunlight on the end reflection plane is reflected on the cavity absorber inner wall again, which causes the X max to increase. However, the sunlight mentioned above energy decreases as the ρ r decreases, decreasing the X max . The average X max for ρ r = 1.00, 0.85, and 0.75 are 2.629, 2.586, and 2.565, respectively, for ω of 0-60 • . The average X max is increased by 3.91%, 2.21%, and 1.38%, respectively, compared to 2.531 if the end reflection plane is not installed. As shown in Figure 12a, the variation of σ non with ρ r is similar to that of the X max . The σ non decreases with the decrease of the ρ r for ω of 0-60 • . It is because the η re-opt,loss of different ρ r is almost constant for ω of 0-60 • , resulting in a similar X mean ; thus, the σ non depends on the X max . The average σ non for ρ r = 1.00, 0.85, and 0.75 are 0.385, 0.372, and 0.366, respectively, for ω of 0-60 • . The average σ non is increased by 7.84%, 4.20%, and 2.52%, respectively, compared to 0.357 if the end reflection plane is not installed. This indicates that the optical loss can be slightly reduced by setting the end reflection plane. Still, compared with the cost of the end reflection plane and the increased non-uniformity of the energy flux distribution, the cost performance of setting the end reflection plane is low. the ω is fixed. The average ηre-opt,loss at ρr of 1.00, 0.85, and 0.75 are 20.78%, 20.94%, and 21.04%, respectively for ω of 0-60°. It can be inferred that the number of sunlight incidents on the end reflection plane is negligible compared to the amount of intercepted sunlight by the cavity absorber. In addition, as shown in Figure 12c, the Xmax decreases with the decrease of ρr. The average Xmax at ρr of 1.00, 0.85, and 0.75 are 4.798, 4.459, and 4.244, respectively, for ω of 0-60°. The average Xmax difference of different ρr is obvious, but the difference of average ηre-opt,loss of that is small. As shown in Figure 12c, the variation of σnon with ρr is similar to that of the Xmax. The σnon decreases with the decrease of the ρr for ω of 0-60°. It is because the ηre-opt,loss of different ρr is almost constant, resulting in a similar Xmean, and the σnon depends on the Xmax. The average σnon at ρr of 1.00, 0.85, and 0.75 are 0.676, 0.652, and 0.635, respectively, for ω of 0-60°. Therefore, introducing the end reflection plane will aggravate the non-uniformity of the energy flux distribution without significantly increasing the optical efficiency.

Conclusions
Optical performance comparison of different cavity receiver shapes in the FLSCS has been investigated. The analysis was conducted by studying the effects of sunlight incident angle θ, receiver position f, receiver internal surface absorptivity αab, and end reflection  Figure 12b shows the effect of ρ r on the optical performance of the FLSCS using LACR. It can be seen from Figure 12b that the η re-opt,loss increases slightly as the ρ r decreases when the ω is fixed. The average η re-opt,loss at ρ r of 1.00, 0.85, and 0.75 are 31.43%, 31.55%, and 31.64%, respectively for ω of 0-60 • . It can be inferred that the number of sunlight incidents on the end reflection plane is negligible compared to the amount of intercepted sunlight by the cavity absorber. In other words, the end reflection plane can be replaced by insulating cotton, which reduces the cost of the system and reduces heat loss. In addition, as shown in Figure 12b, the X max decreases with the decrease of ρ r . The changing trend of X max as f = 675 mm using LACR is similar to that of f = 650 mm using LTCR, and thus detailed analysis is omitted herein. The average X max at ρ r of 1.00, 0.85, and 0.75 are 4.755, 4.456 and 4.262, respectively for ω of 0-60 • . As shown in Figure 12b, the variation of σ non with ρ r is similar to that of the X max . The σ non decreases with the decrease of the ρ r for ω of 0-60 • . It is because the η re-opt,loss of different ρ r is almost constant, resulting in a similar X mean , and the σ non depends on the X max . The average σ non at ρ r of 1.00, 0.85, and 0.75 are 0.498, 0.465, and 0.441, respectively, for ω of 0-60 • . This indicates that the effect of setting the end reflection plane on the optical efficiency of the system is negligible, and it will increase the hot spot effect on the cavity receiver inner wall. Figure 12c shows the effect of ρ r on the optical performance of the FLSCS using LRCR. It can be seen from Figure 12c that the η re-opt,loss increases slightly as the ρ r decreases when the ω is fixed. The average η re-opt,loss at ρ r of 1.00, 0.85, and 0.75 are 20.78%, 20.94%, and 21.04%, respectively for ω of 0-60 • . It can be inferred that the number of sunlight incidents on the end reflection plane is negligible compared to the amount of intercepted sunlight by the cavity absorber. In addition, as shown in Figure 12c, the X max decreases with the decrease of ρ r . The average X max at ρ r of 1.00, 0.85, and 0.75 are 4.798, 4.459, and 4.244, respectively, for ω of 0-60 • . The average X max difference of different ρ r is obvious, but the difference of average η re-opt,loss of that is small. As shown in Figure 12c, the variation of σ non with ρ r is similar to that of the X max . The σ non decreases with the decrease of the ρ r for ω of 0-60 • . It is because the η re-opt,loss of different ρ r is almost constant, resulting in a similar X mean , and the σ non depends on the X max . The average σ non at ρ r of 1.00, 0.85, and 0.75 are 0.676, 0.652, and 0.635, respectively, for ω of 0-60 • . Therefore, introducing the end reflection plane will aggravate the non-uniformity of the energy flux distribution without significantly increasing the optical efficiency.

Conclusions
Optical performance comparison of different cavity receiver shapes in the FLSCS has been investigated. The analysis was conducted by studying the effects of sunlight incident angle θ, receiver position f, receiver internal surface absorptivity α ab , and end reflection plane reflectivity ρ r on the optical efficiency of system and flux uniformity of linear cavity receiver. The main results are summarized as follows: (1) The increases of δ in the range of 0-8 • have a weak effect on the η re-opt,loss . The η re-opt,loss are 2.25%, 2.72%, 12.69% and 2.62%, 3.26%, 12.85%, respectively when the ω is 0 • , 30 • , 60 • for δ = 0 • and 8 • for LTCR. The increase of ω can affect the number of secondary intercepted sunlight. The η re-opt,loss are 14.99%, 14.72%, and 2.61%, respectively when the ω is 0 • , 15 • , 60 • when the δ = 0 • for LACR. (2) The increase of f can affect the number of first intercepted sunlight. When the f is 600, 625, 650, 675, 700 mm and the ω = 60 • , the η re-opt,loss are 16.72%, 5.48%, 2.47%, 3.17%, 14.43%, respectively, for LRCR. The increase of α ab can affect the reflection number of sunlight. When the ω = 0 • and 60 • , the α ab is 1.00, 0.85, and 0.75, the η re-opt,loss is 2.47%, 12.86%, 20.99% and 36.08%, 37.75%, 39.06%, respectively for LRCR. The increase of ρ r has little effect on the η re-opt,loss. When the ω = 0 • and 60 • , the α ab is 1.00, 0.85, and 0.75, the η re-opt,loss is 12.75%, 19.11%, 23.71% and 48.27%, 52.44%, 55.61%, respectively, for LACR. The increase of ω can affect the X max due to the ∆f change with the increase of ω. The X max are 5.896, 7.140, and 6.969, respectively, when the ω is 0 • , 45 • , and 60 • when the δ = 0 • for LACR. (4) The increase of f can affect the X max significantly. When the f is 600, 625, 650, 675, and 700 mm and the ω = 60 • , the X max are 5.832, 6.949, 6.969, 6.509, and 5.699, respectively, for LRCR. The increase of α ab can affect the X max significantly as the reflection number of sunlight is small. When the ω = 0 • , the α ab is 1.00, 0.85, and 0.75, the X max is 5.999, 5.108, 4.511, respectively for LRCR. The increase of ρ r has little effect on the X max . For LACR, the average X max at ρ r of 1.00, 0.85, and 0.75 are 4.755, 4.456, and 4.262, respectively, for ω of 0-60 • . Finally, through comparison with cavity receiver such as LFR and PTC, it is found that LTCR is suitable for FLSCS because of its better optical performance. Since the FLSCS is installed obliquely north-south, the I d increases significantly for the Tropic of Cancer and its north when the δ is +23.45 • (the summer solstice of the Chinese calendar). When the δ is between 8 • and 23.45 • , the focal line moves up obviously with the increase of δ, which leads to the decrease of the system's optical efficiency and becomes more obvious with the increasing of the δ. To obtain the maximum annual total solar radiation for the solar system, these factors need to be considered in the future to match the optical performance of the FLSCS with the annual variation of I d . Opening length of linear arc-shaped cavity receiver (mm) L tr

Nomenclatures
Opening length of linear triangle cavity receiver (mm) L re Opening length of linear rectangular cavity receiver (mm) N The day number of the year p Prism size of linear Fresnel lens (mm) Q absorber Sunlight energy absorbed by the internal wall of the linear cavity receiver (W) Q absorber,δ,ω Sunlight energy absorbed by the internal wall of the linear cavity receiver when the sun declination angle and solar hour angle is δ and ω, respectively (W) Q absorber,ref Sunlight energy absorbed by the internal wall of the liner cavity receiver when the sun declination angle and solar hour angle are 0 • , receiver internal surface absorptivity and end reflection plane reflectivity are 1.00 and receiver position is 650 mm (W)