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 (αab) and end reflection plane reflectivity (ρr). 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.
3.1. Effect of Receiver Position f
The effect of shifting the receiver position is intensely investigated.
Figure 9,
Figure 10,
Figure 11 and
Figure 12 show the results of five studied positions for the triangle cavity receiver when the
αab and
ρr 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.
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
Figure 9 and
Figure 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 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
Figure 9,
Figure 10,
Figure 11 and
Figure 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.
3.2. Effect of Receiver Internal Surface Absorptivity αab
Figure 13 and
Figure 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
δ is 0°. The simulation results were obtained with
αab values of 1.00, 0.85 and 0.75. It can be detected that the different preset
αab 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
αab values of 1.00 and 0.85 is larger than that between
αab values of 0.85 and 0.75 for the
ω in the range of 0–60°. Moreover, the differences of the
UF between
αab value of 1.00 and other
αab 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
αab, 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
αab 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°.
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 them are significant. Note that the differences of the
UF between
αab 1.00 and other
αab 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
αab 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
αab, 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 αab = 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 αab = 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
αab as the
UF between them are significant for
ω about 0–60°. However, the differences of the
UF between
αab = 1.00 and other
αab 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
αab 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 23.45°.
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
Figure 13,
Figure 14,
Figure 15 and
Figure 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.
3.3. Effect of End Reflection Plane Reflectivity ρr
Figure 17 and
Figure 18 present the effect of end reflection plane reflectivity
ρr on the
UF of the triangle cavity receiver under various values of
ω at four different solar declination angle
δ = 0°, 8°, 16° and 23.45°, respectively. In
Figure 17, the effect of changing
ρr under 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°.
3.4. 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 4 × 5
2 × 3
2 times (i.e., 900 times).
Table 4 lists the ANOVA of the quadratic model on
UF. Adequate precision measures the signal to noise ratio, and
R2 indicates the coefficient of multiple determination [
33,
34].
R2 = 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 (
ρr), 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 (
αab). 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
αab 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
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.
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.