# Improvement in the Flux Uniformity of the Solar Dish Concentrator System through a Concave Quartz Window

Authors: Du-zhong Nie; You-duo Peng; Jian Yan; Cheng-ji Mi; Yong-xiang Liu; Yong Tian
Journal: International Journal of Photoenergy
 (2020)

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## Abstract

A nonuniform and high-strength heat flux load would reduce the working efficiency, safety, and in-service life of a cavity receiver. Four types of concave quartz windows, including conical, spherical, sinusoidal, and hyperbolic tangent, were proposed to be used in the cylindrical cavity receiver of a solar dish concentrator system, which can improve the flux uniformity and reduce the peak concentration ratio of the receiver. For each concave quartz window, 36 structural schemes were offered. Based on the Monte Carlo ray-tracing method, the results showed that the nonuniformity coefficient of the receiver was 0.68 and the peak concentration ratio was 1320.21 by using a plane quartz window. At the same time, when the receiver is in the best optical performance, it is the receiver with sinusoidal, conical, spherical, and hyperbolic tangent quartz windows, respectively. The optical efficiency of the receiver with the above four types of quartz windows was basically the same as that of the receiver with the plane quartz window, but their nonuniformity coefficients were reduced to 0.31, 0.35, 0.36, and 0.39, respectively, and the peak concentration ratio was reduced to 806.82, 841.31, 853.23, and 875.89, respectively. Obviously, the concave quartz window was better than the plane quartz window in improving the flux uniformity. Finally, a further study on the sinusoidal quartz window scheme of all of the above optimal parameter schemes showed that when the installation position of the receiver relative to the dish concentrator was changed, the flux uniformity of the receiver could continue to improve. When the surface absorptivity of the receiver was reduced, the optical efficiency would be reduced. For the parabolic dish concentrator with different focal distance, the concave quartz window can also improve the uniformity of the flux distribution of the cylindrical cavity receiver.


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## Body

## 1. Introduction

Solar energy is a clean and environmentally friendly renewable energy. Due to the low density of solar radiation received by the earth’s surface, a solar concentrator is an indispensable core device in a solar energy high-grade utilization system. It can obtain high-density solar radiation on a small area of receivers to improve the energy utilization efficiency or reduce the construction cost [1, 2]. A solar dish concentrator/cavity receiver system (SDCR) is a typical high-grade concentrating and heat collecting device that is widely used in the field of solar thermal utilization [3]. However, the flux distribution of the cavity receiver is highly uneven [4], which results in a high temperature gradient on the inner wall of the receiver, a reduction in the working efficiency of the receiver, and, even more seriously, burning of key components such as the heat exchanger [5, 6].To improve the flux uniformity of the cavity receiver, the following work has been conducted. Chong et al. [7, 8] proposed a nonimaging planar concentrator and a design method to obtain a uniform flux distribution on a planar receiver. Yan et al. [9] proposed a mirror rearranging method for a parabolic dish concentrator and a novel discrete dish concentrator [10], which significantly improved the flux uniformity of the cavity receiver. Evangelos et al. [11] proposed cylindrical, rectangular, spherical, conical, and cylindrical-conical receivers, and the research shows that the cylindrical-conical receiver has the best flux uniformity. Shuai et al. [12] designed a pear-like cavity receiver, which has better flux uniformity than a hemispherical receiver. Jin et al. [13] proposed to add carbon particles in the boiling tube of the receiver, and Tao et al. [14] studied the use of phase-change materials in the cavity receiver, which can improve the uniformity of the temperature distribution on the heat exchange surface of the receiver. In addition, to enable a large amount of heat exchange and chemical reactions to be conducted in the cavity receiver [15, 16], the key condition for the receiver to form a closed high-pressure space [17] is installing a quartz window, which can make the solar radiation pass through at the front end of the receiver. Currently, the quartz windows that have been successfully applied are mainly planar [18, 19], hemispherical [20, 21], and semielliptical quartz windows [22] with equal thickness. Shuai et al. [23] designed a planoconvexo quartz window on a cavity receiver. The research showed that the redistribution effect of the convex quartz window on the solar radiation can also improve the uniformity of the flux distribution on the receiver surface, but their research did not involve the detailed influence of the concave quartz window on the flux distribution of the receiver.A previous literature review has shown that people have a strong interest in the flux uniformity of the solar cavity receiver; however, the research on how to improve the flux uniformity of the receiver by redistributing the solar radiation on the receiver through the concave quartz window is not very in-depth. Therefore, it is of great practical significance to study the concave quartz windows of various structures to improve the flux uniformity of the cavity receiver. In this paper, four typical concave quartz windows, including sinusoidal, conical, spherical, and hyperbolic tangent, are proposed to improve the flux uniformity of the cylindrical cavity receiver in the dish concentrator system. By analyzing the optical efficiency, the local concentration ratio, the nonuniformity coefficient and the peak concentration ratio of the cavity receiver with a concave quartz window, the flux distribution of the receiver with a concave quartz window, and a planar quartz window were compared with each other, and the receivers with different concave quartz windows were compared as well. Finally, the influence of the installation position of the receiver and the absorptivity of the cavity receiver surface on the flux distribution of the receiver is discussed.


## 2. Model Description

In this paper, the solar dish concentrator/cavity receiver system (SDCR) was taken as the research object, as shown in Figure1(a). This system is composed of a parabolic dish concentrator and a cylindrical cavity receiver with a quartz window. The equation of the disc concentrator can be expressed as x2+y2=4fz, in which the focal length of the dish concentrator is f, the aperture radius of the dish concentrator is Rd, and the distance between the dish concentrator and the receiver is L. As shown in Figure 1, the quartz window is installed at the solar radiation entrance of the cylindrical cavity receiver. The central point of the quartz window is Oqw. When the focus point F of the dish concentrator coincides with Oqw, the distance between the dish concentrator and the receiver L is equal to the focal length of the dish concentrator, f. In the SDCR system, most of the solar radiation gathered by the dish concentrator enters the inner surface of the receiver through the quartz window, and some of the radiation is absorbed by the quartz window or reflected into the atmosphere through the quartz window, while some of the solar radiation that enters the receiver cavity will diffuse and then escape through the quartz window. To improve the flux uniformity of the cavity receiver, four typical concave quartz windows were proposed in this paper, which were conical, spherical, sinusoidal, and hyperbolic tangent.Figure 1
SDCR system. (a) Geometric parameters and working principle of the SDCR system. (b) Solar radiation transport of the cavity receiver.
(a)
(b)Figure2(a) describes that the absorption surface of the cylindrical cavity receiver is Dca in diameter and Hca in height. Figure 2(b) describes that the upper and lower surface of the quartz window are composed of symmetrical concave surfaces. In this paper, four types of quartz windows, including conical, spherical, sinusoidal and hyperbolic, are included. The center thickness of the quartz window is Hqw1, the edge thickness is Hqw, the outside diameter is Dqw1, and the diameter of the concave surface is Dqw. The contour curves of four types of concave sections in the first quadrant are represented by the following equations in the coordinate system of the diagram.
(1)Conicalcontourcurves:fx=kx0≤x≤Dqw2,(2)Sphericalcontourcurves:fx=t‐t2−x20≤x≤Dqw2,(3)Sinusoidalcontourcurves:fx=A+A⋅sinkx−0.5π0≤x≤Dqw2,(4)Hyperbolictangentcontourcurves:fx=Aekx−e−kxekx+e−kx0≤x≤Dqw2.Figure 2
(a) Cylindrical cavity receiver with quartz window. (b) Structure of quartz window.
(a)
(b)Here,k, t, and A are the parameters of the equation that must be given fixed values.Whenx is in the range of 0 to Dqw/2, according to the above contour curve equation, it can be concluded that the slope of the conical contour curve remains constant with any increases in the x value, while the slope of the spherical contour curve increases, and the slope of the sinusoidal contour curve increases first and then decreases. However, the slope of the hyperbolic tangent contour curve decreases, and the above four slope values are all positive. Therefore, the selection of these four concave quartz windows is very representative.In the specific case studied in this paper, the structure parameterDca of the receiver is 260 mm, and the height Hca is 330 mm. Among the four concave quartz windows in this study, the structural parameter Hqw1 is taken to be 6 mm, and the outside diameter Dqw1 is 240 mm. Hqw and Dqw had 6 sets of values, as shown in Table 1. The contour curve of the concave quartz window is determined by Equations (1)–(4), in such a way that 36 different structural schemes of the concave quartz window are formed.Table 1
The values ofDqw and Hqw of the concave quartz window.
Hqw, mm
Dqw, mm
18
140
160
180
200
220
240
20
140
160
180
200
220
240
22
140
160
180
200
220
240
24
140
160
180
200
220
240
26
140
160
180
200
220
240
28
140
160
180
200
220
240


## 3. Methodology and Model Validation

### 3.1. Methodology

In this paper, the optical software OptisWorks, which is based on the Monte Carlo ray-tracing method (MCRT), is used for analysis. This software has been widely used in optical analysis, and it can be reliable. For example, Daabo et al. [24] and Yan et al. [25] in our research group used OptisWorks based on MCRT to systematically study the optical performance of a cavity receiver.To evaluate the influence of a concave quartz window on the optical performance of a cavity receiver, it is usually necessary to evaluate the performances of the receiver, including the optical efficiency, local concentration ratio (LCR), and nonuniformity coefficient used to characterize the uniformity of the flux distribution.The optical efficiencyη of the receiver can be calculated according to Equation (5), where Eca is the total energy absorbed by the cylindrical cavity receiver, W0 is the value of the solar direct normal irradiance, and ρd is the reflectivity of the dish concentrator surface:
(5)η=EcaπRd2⋅W0⋅ρd⋅100%.The cavity receiver usually uses the nonuniform coefficientVσ to characterize the uniformity of its flux distribution, which can be calculated by Equation (6) [26]. In this paper, in the process of calculating the flux distribution of the receiver with the OptisWorks software, the solar radiation-receiving surface of the receiver will be meshed, where Ei is the flux distribution value received in the ith grid area and Eaverage is the average value of the flux distribution of the cavity receiver:
(6)Vσ=∑i=1nEi−Eaverage2/n‐1Eaverage.Then, the value of the local concentration ratioCi can be expressed as in Equation (7), where Ai is the area of the ith grid area:
(7)Ci=EiAi⋅W0.


### 3.2. Model Validation

To verify the accuracy of the above methods in this study, the verification of this paper is divided into two steps. The first step is to verify the flux distribution on the focal plane of Jeter’s model [25] by using the OptisWorks simulation software based on MCRT. Through the comparison between the results of the OptisWorks simulation and Jeter’s calculation [27], it is found that the two methods are highly consistent, and the verification process is shown in the previous work of the same research group [23]; the present paper will not elaborate.Because the receiver in the SDCR system analyzed in this paper is a cavity receiver with a quartz window, the model with a quartz window must be further verified by the method mentioned above. Therefore, this paper uses the improved Jeter’s model for further verification. As shown in Figure3, an 8 mm thick quartz window is added above the focal plane of Jeter’s verification model, and a test flux receiving plane is set at a distance of 10 mm from the quartz window. When the solar radiation is directly on the parabolic dish concentrator, the gathered solar radiation will be refracted, reflected, and absorbed when passing through the quartz window. At the same time, because of the symmetry of the model, when the solar radiation reaches the test plane, a circular spot will be formed. At this time, the diameter of the spot and the total energy at the spot calculated by this method are compared with the theoretical results. If the data is highly consistent, then the method is reliable. The parameters in the validation model are shown in Table 2, and the other parameters refer to the Jeter model [27] data.Figure 3
(a) Improved Jeter’s model. (b) Transmission route of solar radiation through a quartz window.
(a)
(b)Table 2
Summary of the validation model.
Parameter
Symbol
Value
Thickness of the quartz window
Hf
6 mm
Distance between the receiving plane for the test and quartz window
Ht
1.8 mm
Wavelength of the solar radiation
λ
587 nm
Absorption coefficient of the quartz window
kλ
1.4 m-1
Refractive index of the quartz window
n
1.5
Reflectivity of the quartz window
ρqw
0.04First, the theoretical calculation of the spot diameter reaching the test plane is conducted, and the parameters of the dish concentrator can be referred to [27].As shown in Figure3(a), the spot diameter of Jeter’s model is calculated as follows, where ϕrim=arctan4f⋅Rd/4f2−Rd2, and δ is the solar half angle:
(8)Df=2Rdcosϕrim+δ×sinδsinϕrim.As shown in Figure3(b), when the incident light enters and exits the quartz window, refraction will occur on the interface surface, and the direction of the refraction follows Snell’s law. Because the incident angle of the solar radiation at the limit position of the incident quartz window is i1=ϕrim+δ, the value of the spot diameter of the test plane can be expressed as follows:
(9)Dt=Df+2Hftanarcsinsinϕrim+δn+2Httanϕrim+δ.The above is calculated atDt=69.8 mm.Secondly, the total energy that reaches the test plane spot is calculated theoretically, which can be expressed as follows:(10)Efocus=π⋅Rd2⋅W0⋅ρd.When the solar radiation passes through the quartz window, it will cause a loss of solar radiation due to the reflection and absorption of the quartz window. Then, the total energyEtest transmitted through the quartz window to the test plane can be expressed as follows:
(11)Etest=Efocus⋅T,where the transmissivity of the quartz window is T=1‐ρqw2⋅e−kλ⋅l [28]. Here, l is the distance of the solar radiation passing through the quartz window. In this paper, l is the average distance of the light radiation passing through the quartz window, which is calculated as 9.8 mm.Through Equations (10) and (11), it can be calculated that the total energy in the test plane is 3433 W.The results of the flux distribution of the focal spot in the test plane are shown in Figure4 through the simulation calculation with the OptisWorks software. The diameter of the spot is approximately 69.8 mm, which is highly consistent with the theoretical calculation result of Equation (9). The simulation results show that the total energy of the test plane is 3436 W, and the error is 0.09% compared with the theoretical results of Equation (11), and the two results are very close to each other. To summarize, in this paper, using OptisWorks software for the simulation calculations is reliable.Figure 4
Flux distribution of test plane.


## 3.1. Methodology

In this paper, the optical software OptisWorks, which is based on the Monte Carlo ray-tracing method (MCRT), is used for analysis. This software has been widely used in optical analysis, and it can be reliable. For example, Daabo et al. [24] and Yan et al. [25] in our research group used OptisWorks based on MCRT to systematically study the optical performance of a cavity receiver.To evaluate the influence of a concave quartz window on the optical performance of a cavity receiver, it is usually necessary to evaluate the performances of the receiver, including the optical efficiency, local concentration ratio (LCR), and nonuniformity coefficient used to characterize the uniformity of the flux distribution.The optical efficiencyη of the receiver can be calculated according to Equation (5), where Eca is the total energy absorbed by the cylindrical cavity receiver, W0 is the value of the solar direct normal irradiance, and ρd is the reflectivity of the dish concentrator surface:
(5)η=EcaπRd2⋅W0⋅ρd⋅100%.The cavity receiver usually uses the nonuniform coefficientVσ to characterize the uniformity of its flux distribution, which can be calculated by Equation (6) [26]. In this paper, in the process of calculating the flux distribution of the receiver with the OptisWorks software, the solar radiation-receiving surface of the receiver will be meshed, where Ei is the flux distribution value received in the ith grid area and Eaverage is the average value of the flux distribution of the cavity receiver:
(6)Vσ=∑i=1nEi−Eaverage2/n‐1Eaverage.Then, the value of the local concentration ratioCi can be expressed as in Equation (7), where Ai is the area of the ith grid area:
(7)Ci=EiAi⋅W0.


## 3.2. Model Validation

To verify the accuracy of the above methods in this study, the verification of this paper is divided into two steps. The first step is to verify the flux distribution on the focal plane of Jeter’s model [25] by using the OptisWorks simulation software based on MCRT. Through the comparison between the results of the OptisWorks simulation and Jeter’s calculation [27], it is found that the two methods are highly consistent, and the verification process is shown in the previous work of the same research group [23]; the present paper will not elaborate.Because the receiver in the SDCR system analyzed in this paper is a cavity receiver with a quartz window, the model with a quartz window must be further verified by the method mentioned above. Therefore, this paper uses the improved Jeter’s model for further verification. As shown in Figure3, an 8 mm thick quartz window is added above the focal plane of Jeter’s verification model, and a test flux receiving plane is set at a distance of 10 mm from the quartz window. When the solar radiation is directly on the parabolic dish concentrator, the gathered solar radiation will be refracted, reflected, and absorbed when passing through the quartz window. At the same time, because of the symmetry of the model, when the solar radiation reaches the test plane, a circular spot will be formed. At this time, the diameter of the spot and the total energy at the spot calculated by this method are compared with the theoretical results. If the data is highly consistent, then the method is reliable. The parameters in the validation model are shown in Table 2, and the other parameters refer to the Jeter model [27] data.Figure 3
(a) Improved Jeter’s model. (b) Transmission route of solar radiation through a quartz window.
(a)
(b)Table 2
Summary of the validation model.
Parameter
Symbol
Value
Thickness of the quartz window
Hf
6 mm
Distance between the receiving plane for the test and quartz window
Ht
1.8 mm
Wavelength of the solar radiation
λ
587 nm
Absorption coefficient of the quartz window
kλ
1.4 m-1
Refractive index of the quartz window
n
1.5
Reflectivity of the quartz window
ρqw
0.04First, the theoretical calculation of the spot diameter reaching the test plane is conducted, and the parameters of the dish concentrator can be referred to [27].As shown in Figure3(a), the spot diameter of Jeter’s model is calculated as follows, where ϕrim=arctan4f⋅Rd/4f2−Rd2, and δ is the solar half angle:
(8)Df=2Rdcosϕrim+δ×sinδsinϕrim.As shown in Figure3(b), when the incident light enters and exits the quartz window, refraction will occur on the interface surface, and the direction of the refraction follows Snell’s law. Because the incident angle of the solar radiation at the limit position of the incident quartz window is i1=ϕrim+δ, the value of the spot diameter of the test plane can be expressed as follows:
(9)Dt=Df+2Hftanarcsinsinϕrim+δn+2Httanϕrim+δ.The above is calculated atDt=69.8 mm.Secondly, the total energy that reaches the test plane spot is calculated theoretically, which can be expressed as follows:(10)Efocus=π⋅Rd2⋅W0⋅ρd.When the solar radiation passes through the quartz window, it will cause a loss of solar radiation due to the reflection and absorption of the quartz window. Then, the total energyEtest transmitted through the quartz window to the test plane can be expressed as follows:
(11)Etest=Efocus⋅T,where the transmissivity of the quartz window is T=1‐ρqw2⋅e−kλ⋅l [28]. Here, l is the distance of the solar radiation passing through the quartz window. In this paper, l is the average distance of the light radiation passing through the quartz window, which is calculated as 9.8 mm.Through Equations (10) and (11), it can be calculated that the total energy in the test plane is 3433 W.The results of the flux distribution of the focal spot in the test plane are shown in Figure4 through the simulation calculation with the OptisWorks software. The diameter of the spot is approximately 69.8 mm, which is highly consistent with the theoretical calculation result of Equation (9). The simulation results show that the total energy of the test plane is 3436 W, and the error is 0.09% compared with the theoretical results of Equation (11), and the two results are very close to each other. To summarize, in this paper, using OptisWorks software for the simulation calculations is reliable.Figure 4
Flux distribution of test plane.


## 4. Results and Discussion

In this study, the 17.7 m SDCR [29] is taken as the research object. To conduct the research effectively, the following boundary conditions are assumed:
(i)
The influence of the cloud cover on the system is not considered
(ii)
The dish concentrator is in an ideal working condition, and there is no error in the installation, mirror slope, or tracking. The surface of the concentrator is a mirror reflection
(iii)
The surface of the quartz window is specular reflectionThe parameters of the SDCR system in this study are shown in Table3. The optical properties of the quartz window are shown in Table 4 [30]. In the table, n is the refractive index of the quartz window, ρqw is the reflectivity of the quartz window, and kλ is the absorption coefficient of the quartz window. In addition, the structural parameters of the cavity receiver and the quartz window are shown in Section 2.Table 3
Summary of the SDCR system.
Parameter
Symbol
Value
Aperture radius of the dish concentrator
Rd
8.85 m
Focal length of the dish concentrator
f
9.49 m
Reflectivity of the dish concentrator surface
ρd
0.92
Absorptivity of the dish concentrator surface
σd
0.08
Absorptivity of the cavity receiver surface
σca
0.95
Reflectivity of the cavity receiver surface
ρca
0.05
The value of the solar direct normal irradiance
W0
1000 W/m2
Rim angle of the dish concentrator
δ
4.65 mradTable 4
Optical properties of quartz glass.
λ, um
n
ρqw
kλ, m-1
0.2-2.7
1.51
0.04
1.4
2.7-∞
1.48
0.04
1000


### 4.1. Flux Distribution of the Receiver with a Quartz Window

At present, planar quartz windows are widely used in cavity receivers. In this paper, the optical performance of a cavity receiver with a planar quartz window is studied. The diameter of the planar quartz window is 240 mm, and the thickness is 8 mm. The following conclusions are obtained by simulation with the OptisWorks software. The optical efficiency of the cavity receiver is 90.80%. The LCR distribution on the sidewall of the receiver is shown in Figure5(b). The nonuniformity coefficient of the flux distribution on the sidewall of the receiver is 0.68, and the peak concentration ratio (PCR) on the sidewall of the receiver is 1320.21. The results show that the flux distribution of the receiver is highly nonuniform and the PCR is relatively large.Figure 5
Optical performance of the cylindrical cavity receiver with a conical quartz window. (a) Optical efficiency. (b) LCR. (c) Nonuniformity coefficient. (d) PCR.
(a)
(b)
(c)
(d)


#### 4.1.1. Conical Quartz Window

When a conical quartz window is used in a cavity receiver, there are 36 different structural schemes of the quartz window according to the dimensionsHqw and Dqw. The values of dimension Hqw and Dqw of the quartz window refer to Table 1 (the same as below). The results of the optical performance of the receiver obtained by the simulation calculation are shown in Figure 5 and Table 5. As shown in Figure 5(a), the optical efficiency of the receiver using the various quartz window schemes mentioned above is between 87.08% and 90.08%. The optical efficiency of some of the receivers is greatly reduced due to the total reflection of the solar radiation when it passes through the quartz window. Considering that the optical efficiency is an important factor to measure the optical performance of the receiver, this paper discusses only the optical performance of a quartz window scheme whose optical efficiency is equal to or higher than 89% (the same as below). Therefore, the optical efficiency of the other quartz window receivers is 89.01-90.10%, which is close to the optical efficiency of the 8 mm planar quartz window receiver. Figure 5(b) shows the LCR distribution of the receiver’s sidewall when a planar quartz window and a conical concave quartz window are used. Considering the large amount of data, only the LCR distribution curve of four extreme working conditions is thickened (the same as in the following sections). It can be seen from the curve in Figure 5(b) that the PCR of the receiver with a conical concave quartz window is much smaller than that with a planar quartz window, and Figure 5(c) shows the nonuniformity coefficient distribution of the receiver sidewall when the conical quartz window is used. It can be seen from Figure 5(c) that the nonuniformity coefficient Vσ is between 0.35 and 0.53; while keeping Hqw constant, the nonuniformity coefficient Vσ monotonically increases, and it is in the range of 22.06-48.53% and lower than that of the receiver with the 8 mm planar quartz window receiver. Figure 5(d) shows the PCR on the sidewall of the receiver when the conical quartz window is used. It can be seen from the figure that the value range of the PCR on the receiver is 838.90-997.68. While keeping Hqw constant, the PCR increased monotonically, and it decreased by 24.43-36.46% compared with the PCR with the 8 mm planar quartz window receiver. Therefore, in terms of the overall optical performances of the receiver, the conical quartz window is the best when Hqw is 26 mm and Dqw is 160 mm.Table 5
Optical performance of the cylindrical cavity receiver with a conical quartz window.
Hqw, mm
Parameters
Dqw, mm
140
160
180
200
220
240
18
η (%)
89.60
89.77
89.90
89.96
90.01
90.10
PCR
887.64
894.03
909.89
941.54
971.94
997.68
Vσ
0.46
0.47
0.49
0.51
0.52
0.53
20
η (%)
89.41
89.59
89.72
89.82
89.93
89.96
PCR
872.76
885.35
892.65
896.25
923.06
948.15
Vσ
0.43
0.44
0.47
0.48
0.50
0.51
22
η (%)
89.21
89.44
89.59
89.71
89.78
89.89
PCR
856.22
870.52
879.89
890.62
894.74
914.51
Vσ
0.40
0.42
0.44
0.46
0.48
0.49
24
η (%)
89.01
89.26
89.43
89.56
89.67
89.76
PCR
838.80
854.42
869.97
880.01
886.83
892.79
Vσ
0.37
0.39
0.41
0.44
0.46
0.47
26
η (%)
88.54
89.09
89.28
89.43
89.54
89.66
PCR
821.33
841.31
859.92
867.25
877.88
887.27
Vσ
0.33
0.35
0.39
0.41
0.43
0.45
28
η (%)
87.08
88.88
89.12
89.28
89.42
89.53
PCR
806.83
825.80
845.64
858.37
869.27
878.43
Vσ
0.32
0.31
0.36
0.39
0.41
0.43


#### 4.1.2. Spherical Quartz Window

When the spherical quartz window is used in the cavity receiver, the results of the optical performance of the receiver obtained by the simulation are shown in Figure6 and Table 6. As shown in Figure 6(a), the scheme with the optical efficiency lower than 89% is first removed. Figure 6(b) shows the LCR distribution of the receiver’s sidewall when the planar quartz window and the spherical concave quartz window are used. According to the curve in Figure 6(b), it can be found that the PCR of the receiver with the spherical concave quartz window is much smaller than that of the receiver with the planar quartz window. Figure 6(c) shows the nonuniformity coefficient distribution on the sidewall of the receiver when the spherical quartz window is used. It can be seen from the figure that the nonuniformity coefficient Vσ is between 0.36 and 0.55. While keeping Hqw constant, the nonuniformity coefficient Vσ increases monotonically with Dqw, and it decreases by 19.12-47.06% compared with the nonuniformity coefficient of the receiver with the 8 mm planar quartz window receiver. Figure 6(d) shows the PCR on the sidewall of the receiver when the spherical quartz window is used. It can be seen from the figure that the value range of the receiver PCR is 856.42-1070. While keeping Hqw constant, the PCR increases monotonically with Dqw, and it decreases by 18.95-35.13% compared with the PCR using the 8 mm planar quartz window receiver. Therefore, in terms of the overall optical performance of the receiver, the spherical quartz window is the best when Hqw is 22 mm and Dqw1 is 160 mm.Figure 6
Optical performance of the cylindrical cavity receiver with the spherical quartz window. (a) Optical efficiency. (b) LCR. (c) Nonuniformity coefficient. (d) PCR.
(a)
(b)
(c)
(d)Table 6
Optical performance of the cylindrical cavity receiver with a spherical quartz window.
Hqw, mm
Parameters
Dqw, mm
140
160
180
200
220
240
18
η (%)
89.62
90.15
90.28
90.36
90.44
90.48
PCR
857.19
910.50
957.90
1000.00
1040.00
1070.00
Vσ
0.37
0.42
0.47
0.51
0.54
0.55
20
η (%)
88.66
89.86
90.20
90.32
90.36
90.41
PCR
825.84
880.92
926.45
971.53
1010.00
1040.00
Vσ
0.35
0.38
0.44
0.48
0.51
0.54
22
η (%)
87.45
89.25
90.10
90.25
90.32
90.37
PCR
801.34
853.23
900.65
942.97
981.63
1010.00
Vσ
0.35
0.36
0.40
0.45
0.49
0.52
24
η (%)
86.13
88.43
89.81
90.16
90.28
90.34
PCR
774.57
831.48
878.94
919.50
958.04
995.78
Vσ
0.34
0.35
0.38
0.43
0.47
0.50
26
η (%)
84.74
87.52
89.30
90.08
90.23
90.29
PCR
752.89
807.73
856.42
898.27
937.46
976.56
Vσ
0.34
0.35
0.36
0.40
0.45
0.48
28
η (%)
83.38
86.52
88.72
89.85
90.17
90.28
PCR
733.21
787.74
838.66
881.20
919.11
953.81
Vσ
0.34
0.35
0.36
0.38
0.42
0.46


#### 4.1.3. Sinusoidal Quartz Window

When the sinusoidal quartz window is used in the cavity receiver, the results of the optical performance of the receiver obtained by the simulation are shown in Figure7 and Table 7. As shown in Figure 7(a), the scheme with an optical efficiency lower than 89% is first removed. Figure 7(b) shows the LCR distribution of the receiver’s sidewall when the planar quartz window and the sinusoidal concave quartz window are used. According to the curve shown in Figure 7(b), it can be found that the PCR of the receiver with the sinusoidal concave quartz window is much smaller than that of the receiver with the planar quartz window. Figure 7(c) shows the nonuniformity coefficient distribution on the sidewall of the receiver when the sinusoidal quartz window is used. It can be seen from the figure that the nonuniformity coefficient Vσ is between 0.31 and 0.47, and it decreases by 30.88-54.41% compared with the nonuniformity coefficient of the receiver with the 8 mm planar quartz window receiver. Figure 7(d) shows the PCR on the sidewall of the receiver when the spherical quartz window is used. It can be seen from the figure that the value range of the receiver PCR is 798.09-923.42. While keeping Hqwconstant, the PCR decreases first and then increases with Dqw, and it decreases by 30.06-39.55% compared with the PCR using the 8 mm planar quartz window receiver. Therefore, in terms of the overall optical performance of the receiver, the sinusoidal quartz window is the best when Hqw is 28 mm and Dqw1 is 240 mm.Figure 7
Optical performance of the cylindrical cavity receiver with the sinusoidal quartz window. (a) Optical efficiency. (b) LCR. (c) Nonuniformity coefficient. (d) PCR.
(a)
(b)
(c)
(d)Table 7
Optical performance of the cylindrical cavity receiver with a sinusoidal quartz window.
Hqw, mm
Parameters
Dqw, mm
140
160
180
200
220
240
18
η (%)
89.62
89.74
89.89
90.04
90.10
90.19
PCR
1020.00
842.89
848.35
872.79
895.15
923.42
Vσ
0.45
0.43
0.43
0.45
0.46
0.47
20
η (%)
89.07
89.56
89.82
89.90
90.02
90.09
PCR
1050.00
864.82
824.05
843.68
866.86
890.32
Vσ
0.41
0.39
0.39
0.40
0.42
0.44
22
η (%)
88.00
89.03
89.59
89.77
89.91
90.02
PCR
1070.00
898.06
800.60
818.04
840.02
868.37
Vσ
0.39
0.36
0.35
0.37
0.39
0.41
24
η (%)
86.49
87.69
88.64
89.60
89.77
89.91
PCR
1090.00
903.88
782.94
798.09
823.02
846.22
Vσ
0.38
0.3
0.32
0.32
0.35
0.38
26
η (%)
84.54
86.13
87.69
88.87
89.62
89.81
PCR
1060.00
876.82
759.26
780.13
801.78
823.77
Vσ
0.38
0.32
0.30
0.30
0.31
0.35
28
η (%)
82.53
84.25
86.47
87.58
88.84
89.64
PCR
968.15
851.38
753.77
762.26
784.89
806.82
Vσ
0.35
0.31
0.29
0.28
0.29
0.31


#### 4.1.4. Hyperbolic Tangent Quartz Window

When the hyperbolic tangent quartz window is used in the cavity receiver, the results of the optical performance of the receiver obtained by the simulation are shown in Figure8 and Table 8 As shown in Figure 8(a), the scheme with an optical efficiency of lower than 89% is first removed. Figure 8(b) shows the LCR distribution of the receiver’s sidewall when the planar quartz window and the hyperbolic tangent concave quartz window are used. According to the curve in Figure 8(b), it can be found that the PCR of the receiver with the hyperbolic tangent concave quartz window is much smaller than that of the receiver with the planar quartz window. Figure 8(c) shows the nonuniformity coefficient distribution on the sidewall of the receiver when the hyperbolic tangent quartz window is used. It can be seen from the figure that the nonuniformity coefficient Vσ is between 0.39 and 0.53, and it decreases by 22.06-42.65% compared with the nonuniformity coefficient of the receiver with an 8 mm planar quartz window receiver. Figure 8(d) shows the PCR on the sidewall of the receiver when the spherical quartz window is used. It can be seen from the figure that the value range of the receiver PCR is 875.89-983.03, and it decreases by 25.54-33.66% compared with the PCR using the 8 mm planar quartz window receiver. Therefore, in terms of the overall optical performance of the receiver, the hyperbolic tangent quartz window is the best when Hqw is 28 mm and Dqw1 is 240 mm.Figure 8
Optical performance of the cylindrical cavity receiver with the hyperbolic tangent quartz window. (a) Optical efficiency. (b) LCR. (c) Nonuniformity coefficient. (d) PCR.
(a)
(b)
(c)
(d)Table 8
Optical performance of the cylindrical cavity receiver with a hyperbolic tangent quartz window.
Hqw, mm
Parameters
Dqw, mm
140
160
180
200
220
240
18
η (%)
89.55
89.68
89.80
89.87
89.94
90.01
PCR
935.45
929.29
923.92
944.31
964.25
983.03
Vσ
0.48
0.49
0.50
0.51
0.52
0.53
20
η (%)
89.34
89.51
89.62
89.70
89.80
89.87
PCR
902.86
899.55
900.57
906.09
917.23
935.43
Vσ
0.46
0.46
0.47
0.48
0.49
0.50
22
η (%)
89.12
89.32
89.47
89.57
89.66
89.73
PCR
897.65
890.99
894.18
894.15
900.04
902.07
Vσ
0.43
0.44
0.45
0.46
0.47
0.48
24
η (%)
88.76
89.11
89.27
89.39
89.48
89.61
PCR
915.02
887.99
889.89
888.34
891.25
898.07
Vσ
0.41
0.42
0.43
0.44
0.45
0.46
26
η (%)
88.02
88.73
89.07
89.24
89.32
89.44
PCR
936.02
880.12
881.76
884.29
885.88
885.70
Vσ
0.39
0.39
0.40
0.41
0.43
0.44
28
η (%)
86.97
88.05
88.72
89.04
89.18
89.30
PCR
941.77
871.41
870.93
875.89
879.66
884.90
Vσ
0.37
0.37
0.38
0.39
0.41
0.42


### 4.2. Comprehensive Analysis

#### 4.2.1. Comparison of the Different Designs

According to the calculation results in Section4.1, when the cavity receiver uses four types of quartz windows with different structures, i.e., conic, spherical, sinusoidal, and hyperbolic tangent, its optical performance is evaluated. The optimal structure schemes of the four types of quartz windows are shown in Table 9, Figure 9 shows the LCR distribution of the receiver sidewall under the four quartz window structure schemes, and Figure 10 shows the LCR distribution of the receiver sidewall under the four quartz window structure schemes. According to Table 9 and Figure 9, further analysis of the quartz windows of the four structural schemes shows that the optical efficiency of all of the structural schemes is basically the same. From the comparison of the nonuniformity coefficient and the peak value of the concentration ratio, it can be found that the optimal optical properties of the four types of quartz windows are sinusoidal, conical, spherical, and hyperbolic tangent quartz windows. The nonuniformity coefficient of the receiver with the sinusoidal quartz window is 0.31 and the PCR is 806.82.Table 9
Optical performance parameters of a cavity receiver with a quartz window with four optimal structure schemes.
Name
Hqw (mm)
Dqw (mm)
η (%)
Vσ
PCR
Receiver with sinusoidal quartz window
28
240
89.64
0.31
806.82
Receiver with conical window
26
160
89.09
0.35
841.31
Receiver with spherical quartz window
22
160
89.25
0.36
853.23
Receiver with hyperbolic tangent quartz window
28
200
89.04
0.39
875.89Figure 9
LCR distribution of a receiver with a concave quartz window with four optimal structure schemes.Figure 10
LCR contours of a receiver with a concave quartz window with four optimal structure schemes: (a) sinusoidal, (b) conical, (c) spherical, and (d) hyperbolic tangent.
(a)
(b)
(c)
(d)


#### 4.2.2. Influence of the Installation Position of the Cavity Receiver

According to the analysis results in Section4.2.1, the sinusoidal quartz window receiver with the optimal structure scheme is further discussed, and the quartz window’s dimensions are Hqw=28mm and Dqw=240mm when the distance L between the apex of the dish concentrator and the point Oqw on the receiver is changed (shown in Figure 1), in other words, when L=f+0.06m, L=f+0.012m, L=f−0.006m, and L=f−0.012m, respectively. The optical performance of the receiver in these four cases is simulated and calculated, and the results are shown in Table 10 and Figure 11. The results show that the optical efficiency basically remains unchanged when L increases, the nonuniformity coefficient increases, and the PCR increases. When L is equal to f−0.006m, the optical efficiency of the receiver basically remains unchanged, but the nonuniformity coefficient drops to 0.31, and the PCR drops to 766.93. The optical performance of the receiver has been further improved.Table 10
Optical performance parameters of the receiver when changing the installation position of the sinusoidal quartz window (Hqw=28 mm, Dqw=240 mm).
Parameters
L=f
L=f+0.06
L=f+0.012
L=f−0.006
L=f−0.012
η (%)
89.64
89.60
89.59
89.62
89.64
PCR
806.82
854.75
909.41
766.93
737.93
Vσ
0.31
0.38
0.34
0.31
0.33Figure 11
LCR distribution of the receiver when changing the installation position of the sinusoidal quartz window (Hqw=28mm, Dqw=240mm).


#### 4.2.3. Influence of the Absorptivity of the Cavity Receiver Surface

When the new absorptivityσca is set to 0.80, 0.83, 0.86, 0.89, 0.92, and 0.95, respectively. The results of the receiver’s optical efficiency in six cases are shown in Table 11 and Figure 12. When the absorptivity decreases, the optical efficiency η will decrease from 89.64 to 86.94. This finding is mainly caused by the fact that when the absorptivity decreases, the reflectivity increases, and the solar radiation that enters the receiver cavity will escape more, thus reducing the optical efficiency of the receiver. In addition, although the decrease of absorptivity will lead to the decrease of the nonuniformity coefficient Vσ and PCR, the decrease also comes at the expense of optical efficiency; therefore, the coating of the receiver should be overhauled regularly in real application to ensure its good performance.Table 11
Optical performance of the receiver when the absorptivity of the cavity receiver surface is changed.
Parameters
σca=0.8
σca=0.83
σca=0.86
σca=0.89
σca=0.92
σca=0.95
η (%)
86.94
87.46
88.00
88.49
88.92
89.64
PCR
760.30
771.03
778.01
789.30
796.38
806.82
Vσ
0.28
0.28
0.29
0.30
0.31
0.31Figure 12
LCR distribution of the receiver when the absorptivity of the receiver surface is changed.


#### 4.2.4. Influence of Focal Distance of Parabolic Dish Concentrator

When the radiusRd of the parabolic dish concentrator was kept constant, and the focal distance f of the parabolic dish was changed to 8.99 m, it was found from Table 12 and Figure 13 that the PCR of the receiver with concave quartz window was decreased from 1396.6 to 831.57, and the Vσ was decreased from 0.70 to 0.34 compared with that of the receiver with plane quartz window. When the focal distance f was 9.99 m, the PCR of the receiver with concave quartz window was decreased from 1237.30 to 788.69, and the Vσ was decreased from 0.66 to 0.33 compared with that of the receiver with plane quartz window. Therefore, for the parabolic dish concentrator with different focal distance, the concave quartz window can also improve the uniformity of the flux distribution of the receiver cylindrical cavity. On the other hand, the parabolic dish with proper focal distance should be used in the SDCR system.Table 12
Optical performance parameters of the receiver of parabolic dish concentrator at different focal distances.
f (m)
f=8.99
f=8.99
f=9.99
f=9.99
Quartz window
Plane
Concave
Plane
Concave
η (%)
90.40
89.71
90.37
89.66
PCR
1396.60
831.57
1237.30
788.69
Vσ
0.70
0.34
0.66
0.33Figure 13
LCR distribution of the receiver of parabolic dish concentrator at different focal distances.


## 4.1. Flux Distribution of the Receiver with a Quartz Window

At present, planar quartz windows are widely used in cavity receivers. In this paper, the optical performance of a cavity receiver with a planar quartz window is studied. The diameter of the planar quartz window is 240 mm, and the thickness is 8 mm. The following conclusions are obtained by simulation with the OptisWorks software. The optical efficiency of the cavity receiver is 90.80%. The LCR distribution on the sidewall of the receiver is shown in Figure5(b). The nonuniformity coefficient of the flux distribution on the sidewall of the receiver is 0.68, and the peak concentration ratio (PCR) on the sidewall of the receiver is 1320.21. The results show that the flux distribution of the receiver is highly nonuniform and the PCR is relatively large.Figure 5
Optical performance of the cylindrical cavity receiver with a conical quartz window. (a) Optical efficiency. (b) LCR. (c) Nonuniformity coefficient. (d) PCR.
(a)
(b)
(c)
(d)


### 4.1.1. Conical Quartz Window

When a conical quartz window is used in a cavity receiver, there are 36 different structural schemes of the quartz window according to the dimensionsHqw and Dqw. The values of dimension Hqw and Dqw of the quartz window refer to Table 1 (the same as below). The results of the optical performance of the receiver obtained by the simulation calculation are shown in Figure 5 and Table 5. As shown in Figure 5(a), the optical efficiency of the receiver using the various quartz window schemes mentioned above is between 87.08% and 90.08%. The optical efficiency of some of the receivers is greatly reduced due to the total reflection of the solar radiation when it passes through the quartz window. Considering that the optical efficiency is an important factor to measure the optical performance of the receiver, this paper discusses only the optical performance of a quartz window scheme whose optical efficiency is equal to or higher than 89% (the same as below). Therefore, the optical efficiency of the other quartz window receivers is 89.01-90.10%, which is close to the optical efficiency of the 8 mm planar quartz window receiver. Figure 5(b) shows the LCR distribution of the receiver’s sidewall when a planar quartz window and a conical concave quartz window are used. Considering the large amount of data, only the LCR distribution curve of four extreme working conditions is thickened (the same as in the following sections). It can be seen from the curve in Figure 5(b) that the PCR of the receiver with a conical concave quartz window is much smaller than that with a planar quartz window, and Figure 5(c) shows the nonuniformity coefficient distribution of the receiver sidewall when the conical quartz window is used. It can be seen from Figure 5(c) that the nonuniformity coefficient Vσ is between 0.35 and 0.53; while keeping Hqw constant, the nonuniformity coefficient Vσ monotonically increases, and it is in the range of 22.06-48.53% and lower than that of the receiver with the 8 mm planar quartz window receiver. Figure 5(d) shows the PCR on the sidewall of the receiver when the conical quartz window is used. It can be seen from the figure that the value range of the PCR on the receiver is 838.90-997.68. While keeping Hqw constant, the PCR increased monotonically, and it decreased by 24.43-36.46% compared with the PCR with the 8 mm planar quartz window receiver. Therefore, in terms of the overall optical performances of the receiver, the conical quartz window is the best when Hqw is 26 mm and Dqw is 160 mm.Table 5
Optical performance of the cylindrical cavity receiver with a conical quartz window.
Hqw, mm
Parameters
Dqw, mm
140
160
180
200
220
240
18
η (%)
89.60
89.77
89.90
89.96
90.01
90.10
PCR
887.64
894.03
909.89
941.54
971.94
997.68
Vσ
0.46
0.47
0.49
0.51
0.52
0.53
20
η (%)
89.41
89.59
89.72
89.82
89.93
89.96
PCR
872.76
885.35
892.65
896.25
923.06
948.15
Vσ
0.43
0.44
0.47
0.48
0.50
0.51
22
η (%)
89.21
89.44
89.59
89.71
89.78
89.89
PCR
856.22
870.52
879.89
890.62
894.74
914.51
Vσ
0.40
0.42
0.44
0.46
0.48
0.49
24
η (%)
89.01
89.26
89.43
89.56
89.67
89.76
PCR
838.80
854.42
869.97
880.01
886.83
892.79
Vσ
0.37
0.39
0.41
0.44
0.46
0.47
26
η (%)
88.54
89.09
89.28
89.43
89.54
89.66
PCR
821.33
841.31
859.92
867.25
877.88
887.27
Vσ
0.33
0.35
0.39
0.41
0.43
0.45
28
η (%)
87.08
88.88
89.12
89.28
89.42
89.53
PCR
806.83
825.80
845.64
858.37
869.27
878.43
Vσ
0.32
0.31
0.36
0.39
0.41
0.43


### 4.1.2. Spherical Quartz Window

When the spherical quartz window is used in the cavity receiver, the results of the optical performance of the receiver obtained by the simulation are shown in Figure6 and Table 6. As shown in Figure 6(a), the scheme with the optical efficiency lower than 89% is first removed. Figure 6(b) shows the LCR distribution of the receiver’s sidewall when the planar quartz window and the spherical concave quartz window are used. According to the curve in Figure 6(b), it can be found that the PCR of the receiver with the spherical concave quartz window is much smaller than that of the receiver with the planar quartz window. Figure 6(c) shows the nonuniformity coefficient distribution on the sidewall of the receiver when the spherical quartz window is used. It can be seen from the figure that the nonuniformity coefficient Vσ is between 0.36 and 0.55. While keeping Hqw constant, the nonuniformity coefficient Vσ increases monotonically with Dqw, and it decreases by 19.12-47.06% compared with the nonuniformity coefficient of the receiver with the 8 mm planar quartz window receiver. Figure 6(d) shows the PCR on the sidewall of the receiver when the spherical quartz window is used. It can be seen from the figure that the value range of the receiver PCR is 856.42-1070. While keeping Hqw constant, the PCR increases monotonically with Dqw, and it decreases by 18.95-35.13% compared with the PCR using the 8 mm planar quartz window receiver. Therefore, in terms of the overall optical performance of the receiver, the spherical quartz window is the best when Hqw is 22 mm and Dqw1 is 160 mm.Figure 6
Optical performance of the cylindrical cavity receiver with the spherical quartz window. (a) Optical efficiency. (b) LCR. (c) Nonuniformity coefficient. (d) PCR.
(a)
(b)
(c)
(d)Table 6
Optical performance of the cylindrical cavity receiver with a spherical quartz window.
Hqw, mm
Parameters
Dqw, mm
140
160
180
200
220
240
18
η (%)
89.62
90.15
90.28
90.36
90.44
90.48
PCR
857.19
910.50
957.90
1000.00
1040.00
1070.00
Vσ
0.37
0.42
0.47
0.51
0.54
0.55
20
η (%)
88.66
89.86
90.20
90.32
90.36
90.41
PCR
825.84
880.92
926.45
971.53
1010.00
1040.00
Vσ
0.35
0.38
0.44
0.48
0.51
0.54
22
η (%)
87.45
89.25
90.10
90.25
90.32
90.37
PCR
801.34
853.23
900.65
942.97
981.63
1010.00
Vσ
0.35
0.36
0.40
0.45
0.49
0.52
24
η (%)
86.13
88.43
89.81
90.16
90.28
90.34
PCR
774.57
831.48
878.94
919.50
958.04
995.78
Vσ
0.34
0.35
0.38
0.43
0.47
0.50
26
η (%)
84.74
87.52
89.30
90.08
90.23
90.29
PCR
752.89
807.73
856.42
898.27
937.46
976.56
Vσ
0.34
0.35
0.36
0.40
0.45
0.48
28
η (%)
83.38
86.52
88.72
89.85
90.17
90.28
PCR
733.21
787.74
838.66
881.20
919.11
953.81
Vσ
0.34
0.35
0.36
0.38
0.42
0.46


### 4.1.3. Sinusoidal Quartz Window

When the sinusoidal quartz window is used in the cavity receiver, the results of the optical performance of the receiver obtained by the simulation are shown in Figure7 and Table 7. As shown in Figure 7(a), the scheme with an optical efficiency lower than 89% is first removed. Figure 7(b) shows the LCR distribution of the receiver’s sidewall when the planar quartz window and the sinusoidal concave quartz window are used. According to the curve shown in Figure 7(b), it can be found that the PCR of the receiver with the sinusoidal concave quartz window is much smaller than that of the receiver with the planar quartz window. Figure 7(c) shows the nonuniformity coefficient distribution on the sidewall of the receiver when the sinusoidal quartz window is used. It can be seen from the figure that the nonuniformity coefficient Vσ is between 0.31 and 0.47, and it decreases by 30.88-54.41% compared with the nonuniformity coefficient of the receiver with the 8 mm planar quartz window receiver. Figure 7(d) shows the PCR on the sidewall of the receiver when the spherical quartz window is used. It can be seen from the figure that the value range of the receiver PCR is 798.09-923.42. While keeping Hqwconstant, the PCR decreases first and then increases with Dqw, and it decreases by 30.06-39.55% compared with the PCR using the 8 mm planar quartz window receiver. Therefore, in terms of the overall optical performance of the receiver, the sinusoidal quartz window is the best when Hqw is 28 mm and Dqw1 is 240 mm.Figure 7
Optical performance of the cylindrical cavity receiver with the sinusoidal quartz window. (a) Optical efficiency. (b) LCR. (c) Nonuniformity coefficient. (d) PCR.
(a)
(b)
(c)
(d)Table 7
Optical performance of the cylindrical cavity receiver with a sinusoidal quartz window.
Hqw, mm
Parameters
Dqw, mm
140
160
180
200
220
240
18
η (%)
89.62
89.74
89.89
90.04
90.10
90.19
PCR
1020.00
842.89
848.35
872.79
895.15
923.42
Vσ
0.45
0.43
0.43
0.45
0.46
0.47
20
η (%)
89.07
89.56
89.82
89.90
90.02
90.09
PCR
1050.00
864.82
824.05
843.68
866.86
890.32
Vσ
0.41
0.39
0.39
0.40
0.42
0.44
22
η (%)
88.00
89.03
89.59
89.77
89.91
90.02
PCR
1070.00
898.06
800.60
818.04
840.02
868.37
Vσ
0.39
0.36
0.35
0.37
0.39
0.41
24
η (%)
86.49
87.69
88.64
89.60
89.77
89.91
PCR
1090.00
903.88
782.94
798.09
823.02
846.22
Vσ
0.38
0.3
0.32
0.32
0.35
0.38
26
η (%)
84.54
86.13
87.69
88.87
89.62
89.81
PCR
1060.00
876.82
759.26
780.13
801.78
823.77
Vσ
0.38
0.32
0.30
0.30
0.31
0.35
28
η (%)
82.53
84.25
86.47
87.58
88.84
89.64
PCR
968.15
851.38
753.77
762.26
784.89
806.82
Vσ
0.35
0.31
0.29
0.28
0.29
0.31


### 4.1.4. Hyperbolic Tangent Quartz Window

When the hyperbolic tangent quartz window is used in the cavity receiver, the results of the optical performance of the receiver obtained by the simulation are shown in Figure8 and Table 8 As shown in Figure 8(a), the scheme with an optical efficiency of lower than 89% is first removed. Figure 8(b) shows the LCR distribution of the receiver’s sidewall when the planar quartz window and the hyperbolic tangent concave quartz window are used. According to the curve in Figure 8(b), it can be found that the PCR of the receiver with the hyperbolic tangent concave quartz window is much smaller than that of the receiver with the planar quartz window. Figure 8(c) shows the nonuniformity coefficient distribution on the sidewall of the receiver when the hyperbolic tangent quartz window is used. It can be seen from the figure that the nonuniformity coefficient Vσ is between 0.39 and 0.53, and it decreases by 22.06-42.65% compared with the nonuniformity coefficient of the receiver with an 8 mm planar quartz window receiver. Figure 8(d) shows the PCR on the sidewall of the receiver when the spherical quartz window is used. It can be seen from the figure that the value range of the receiver PCR is 875.89-983.03, and it decreases by 25.54-33.66% compared with the PCR using the 8 mm planar quartz window receiver. Therefore, in terms of the overall optical performance of the receiver, the hyperbolic tangent quartz window is the best when Hqw is 28 mm and Dqw1 is 240 mm.Figure 8
Optical performance of the cylindrical cavity receiver with the hyperbolic tangent quartz window. (a) Optical efficiency. (b) LCR. (c) Nonuniformity coefficient. (d) PCR.
(a)
(b)
(c)
(d)Table 8
Optical performance of the cylindrical cavity receiver with a hyperbolic tangent quartz window.
Hqw, mm
Parameters
Dqw, mm
140
160
180
200
220
240
18
η (%)
89.55
89.68
89.80
89.87
89.94
90.01
PCR
935.45
929.29
923.92
944.31
964.25
983.03
Vσ
0.48
0.49
0.50
0.51
0.52
0.53
20
η (%)
89.34
89.51
89.62
89.70
89.80
89.87
PCR
902.86
899.55
900.57
906.09
917.23
935.43
Vσ
0.46
0.46
0.47
0.48
0.49
0.50
22
η (%)
89.12
89.32
89.47
89.57
89.66
89.73
PCR
897.65
890.99
894.18
894.15
900.04
902.07
Vσ
0.43
0.44
0.45
0.46
0.47
0.48
24
η (%)
88.76
89.11
89.27
89.39
89.48
89.61
PCR
915.02
887.99
889.89
888.34
891.25
898.07
Vσ
0.41
0.42
0.43
0.44
0.45
0.46
26
η (%)
88.02
88.73
89.07
89.24
89.32
89.44
PCR
936.02
880.12
881.76
884.29
885.88
885.70
Vσ
0.39
0.39
0.40
0.41
0.43
0.44
28
η (%)
86.97
88.05
88.72
89.04
89.18
89.30
PCR
941.77
871.41
870.93
875.89
879.66
884.90
Vσ
0.37
0.37
0.38
0.39
0.41
0.42


## 4.1.1. Conical Quartz Window

When a conical quartz window is used in a cavity receiver, there are 36 different structural schemes of the quartz window according to the dimensionsHqw and Dqw. The values of dimension Hqw and Dqw of the quartz window refer to Table 1 (the same as below). The results of the optical performance of the receiver obtained by the simulation calculation are shown in Figure 5 and Table 5. As shown in Figure 5(a), the optical efficiency of the receiver using the various quartz window schemes mentioned above is between 87.08% and 90.08%. The optical efficiency of some of the receivers is greatly reduced due to the total reflection of the solar radiation when it passes through the quartz window. Considering that the optical efficiency is an important factor to measure the optical performance of the receiver, this paper discusses only the optical performance of a quartz window scheme whose optical efficiency is equal to or higher than 89% (the same as below). Therefore, the optical efficiency of the other quartz window receivers is 89.01-90.10%, which is close to the optical efficiency of the 8 mm planar quartz window receiver. Figure 5(b) shows the LCR distribution of the receiver’s sidewall when a planar quartz window and a conical concave quartz window are used. Considering the large amount of data, only the LCR distribution curve of four extreme working conditions is thickened (the same as in the following sections). It can be seen from the curve in Figure 5(b) that the PCR of the receiver with a conical concave quartz window is much smaller than that with a planar quartz window, and Figure 5(c) shows the nonuniformity coefficient distribution of the receiver sidewall when the conical quartz window is used. It can be seen from Figure 5(c) that the nonuniformity coefficient Vσ is between 0.35 and 0.53; while keeping Hqw constant, the nonuniformity coefficient Vσ monotonically increases, and it is in the range of 22.06-48.53% and lower than that of the receiver with the 8 mm planar quartz window receiver. Figure 5(d) shows the PCR on the sidewall of the receiver when the conical quartz window is used. It can be seen from the figure that the value range of the PCR on the receiver is 838.90-997.68. While keeping Hqw constant, the PCR increased monotonically, and it decreased by 24.43-36.46% compared with the PCR with the 8 mm planar quartz window receiver. Therefore, in terms of the overall optical performances of the receiver, the conical quartz window is the best when Hqw is 26 mm and Dqw is 160 mm.Table 5
Optical performance of the cylindrical cavity receiver with a conical quartz window.
Hqw, mm
Parameters
Dqw, mm
140
160
180
200
220
240
18
η (%)
89.60
89.77
89.90
89.96
90.01
90.10
PCR
887.64
894.03
909.89
941.54
971.94
997.68
Vσ
0.46
0.47
0.49
0.51
0.52
0.53
20
η (%)
89.41
89.59
89.72
89.82
89.93
89.96
PCR
872.76
885.35
892.65
896.25
923.06
948.15
Vσ
0.43
0.44
0.47
0.48
0.50
0.51
22
η (%)
89.21
89.44
89.59
89.71
89.78
89.89
PCR
856.22
870.52
879.89
890.62
894.74
914.51
Vσ
0.40
0.42
0.44
0.46
0.48
0.49
24
η (%)
89.01
89.26
89.43
89.56
89.67
89.76
PCR
838.80
854.42
869.97
880.01
886.83
892.79
Vσ
0.37
0.39
0.41
0.44
0.46
0.47
26
η (%)
88.54
89.09
89.28
89.43
89.54
89.66
PCR
821.33
841.31
859.92
867.25
877.88
887.27
Vσ
0.33
0.35
0.39
0.41
0.43
0.45
28
η (%)
87.08
88.88
89.12
89.28
89.42
89.53
PCR
806.83
825.80
845.64
858.37
869.27
878.43
Vσ
0.32
0.31
0.36
0.39
0.41
0.43


## 4.1.2. Spherical Quartz Window

When the spherical quartz window is used in the cavity receiver, the results of the optical performance of the receiver obtained by the simulation are shown in Figure6 and Table 6. As shown in Figure 6(a), the scheme with the optical efficiency lower than 89% is first removed. Figure 6(b) shows the LCR distribution of the receiver’s sidewall when the planar quartz window and the spherical concave quartz window are used. According to the curve in Figure 6(b), it can be found that the PCR of the receiver with the spherical concave quartz window is much smaller than that of the receiver with the planar quartz window. Figure 6(c) shows the nonuniformity coefficient distribution on the sidewall of the receiver when the spherical quartz window is used. It can be seen from the figure that the nonuniformity coefficient Vσ is between 0.36 and 0.55. While keeping Hqw constant, the nonuniformity coefficient Vσ increases monotonically with Dqw, and it decreases by 19.12-47.06% compared with the nonuniformity coefficient of the receiver with the 8 mm planar quartz window receiver. Figure 6(d) shows the PCR on the sidewall of the receiver when the spherical quartz window is used. It can be seen from the figure that the value range of the receiver PCR is 856.42-1070. While keeping Hqw constant, the PCR increases monotonically with Dqw, and it decreases by 18.95-35.13% compared with the PCR using the 8 mm planar quartz window receiver. Therefore, in terms of the overall optical performance of the receiver, the spherical quartz window is the best when Hqw is 22 mm and Dqw1 is 160 mm.Figure 6
Optical performance of the cylindrical cavity receiver with the spherical quartz window. (a) Optical efficiency. (b) LCR. (c) Nonuniformity coefficient. (d) PCR.
(a)
(b)
(c)
(d)Table 6
Optical performance of the cylindrical cavity receiver with a spherical quartz window.
Hqw, mm
Parameters
Dqw, mm
140
160
180
200
220
240
18
η (%)
89.62
90.15
90.28
90.36
90.44
90.48
PCR
857.19
910.50
957.90
1000.00
1040.00
1070.00
Vσ
0.37
0.42
0.47
0.51
0.54
0.55
20
η (%)
88.66
89.86
90.20
90.32
90.36
90.41
PCR
825.84
880.92
926.45
971.53
1010.00
1040.00
Vσ
0.35
0.38
0.44
0.48
0.51
0.54
22
η (%)
87.45
89.25
90.10
90.25
90.32
90.37
PCR
801.34
853.23
900.65
942.97
981.63
1010.00
Vσ
0.35
0.36
0.40
0.45
0.49
0.52
24
η (%)
86.13
88.43
89.81
90.16
90.28
90.34
PCR
774.57
831.48
878.94
919.50
958.04
995.78
Vσ
0.34
0.35
0.38
0.43
0.47
0.50
26
η (%)
84.74
87.52
89.30
90.08
90.23
90.29
PCR
752.89
807.73
856.42
898.27
937.46
976.56
Vσ
0.34
0.35
0.36
0.40
0.45
0.48
28
η (%)
83.38
86.52
88.72
89.85
90.17
90.28
PCR
733.21
787.74
838.66
881.20
919.11
953.81
Vσ
0.34
0.35
0.36
0.38
0.42
0.46


## 4.1.3. Sinusoidal Quartz Window

When the sinusoidal quartz window is used in the cavity receiver, the results of the optical performance of the receiver obtained by the simulation are shown in Figure7 and Table 7. As shown in Figure 7(a), the scheme with an optical efficiency lower than 89% is first removed. Figure 7(b) shows the LCR distribution of the receiver’s sidewall when the planar quartz window and the sinusoidal concave quartz window are used. According to the curve shown in Figure 7(b), it can be found that the PCR of the receiver with the sinusoidal concave quartz window is much smaller than that of the receiver with the planar quartz window. Figure 7(c) shows the nonuniformity coefficient distribution on the sidewall of the receiver when the sinusoidal quartz window is used. It can be seen from the figure that the nonuniformity coefficient Vσ is between 0.31 and 0.47, and it decreases by 30.88-54.41% compared with the nonuniformity coefficient of the receiver with the 8 mm planar quartz window receiver. Figure 7(d) shows the PCR on the sidewall of the receiver when the spherical quartz window is used. It can be seen from the figure that the value range of the receiver PCR is 798.09-923.42. While keeping Hqwconstant, the PCR decreases first and then increases with Dqw, and it decreases by 30.06-39.55% compared with the PCR using the 8 mm planar quartz window receiver. Therefore, in terms of the overall optical performance of the receiver, the sinusoidal quartz window is the best when Hqw is 28 mm and Dqw1 is 240 mm.Figure 7
Optical performance of the cylindrical cavity receiver with the sinusoidal quartz window. (a) Optical efficiency. (b) LCR. (c) Nonuniformity coefficient. (d) PCR.
(a)
(b)
(c)
(d)Table 7
Optical performance of the cylindrical cavity receiver with a sinusoidal quartz window.
Hqw, mm
Parameters
Dqw, mm
140
160
180
200
220
240
18
η (%)
89.62
89.74
89.89
90.04
90.10
90.19
PCR
1020.00
842.89
848.35
872.79
895.15
923.42
Vσ
0.45
0.43
0.43
0.45
0.46
0.47
20
η (%)
89.07
89.56
89.82
89.90
90.02
90.09
PCR
1050.00
864.82
824.05
843.68
866.86
890.32
Vσ
0.41
0.39
0.39
0.40
0.42
0.44
22
η (%)
88.00
89.03
89.59
89.77
89.91
90.02
PCR
1070.00
898.06
800.60
818.04
840.02
868.37
Vσ
0.39
0.36
0.35
0.37
0.39
0.41
24
η (%)
86.49
87.69
88.64
89.60
89.77
89.91
PCR
1090.00
903.88
782.94
798.09
823.02
846.22
Vσ
0.38
0.3
0.32
0.32
0.35
0.38
26
η (%)
84.54
86.13
87.69
88.87
89.62
89.81
PCR
1060.00
876.82
759.26
780.13
801.78
823.77
Vσ
0.38
0.32
0.30
0.30
0.31
0.35
28
η (%)
82.53
84.25
86.47
87.58
88.84
89.64
PCR
968.15
851.38
753.77
762.26
784.89
806.82
Vσ
0.35
0.31
0.29
0.28
0.29
0.31


## 4.1.4. Hyperbolic Tangent Quartz Window

When the hyperbolic tangent quartz window is used in the cavity receiver, the results of the optical performance of the receiver obtained by the simulation are shown in Figure8 and Table 8 As shown in Figure 8(a), the scheme with an optical efficiency of lower than 89% is first removed. Figure 8(b) shows the LCR distribution of the receiver’s sidewall when the planar quartz window and the hyperbolic tangent concave quartz window are used. According to the curve in Figure 8(b), it can be found that the PCR of the receiver with the hyperbolic tangent concave quartz window is much smaller than that of the receiver with the planar quartz window. Figure 8(c) shows the nonuniformity coefficient distribution on the sidewall of the receiver when the hyperbolic tangent quartz window is used. It can be seen from the figure that the nonuniformity coefficient Vσ is between 0.39 and 0.53, and it decreases by 22.06-42.65% compared with the nonuniformity coefficient of the receiver with an 8 mm planar quartz window receiver. Figure 8(d) shows the PCR on the sidewall of the receiver when the spherical quartz window is used. It can be seen from the figure that the value range of the receiver PCR is 875.89-983.03, and it decreases by 25.54-33.66% compared with the PCR using the 8 mm planar quartz window receiver. Therefore, in terms of the overall optical performance of the receiver, the hyperbolic tangent quartz window is the best when Hqw is 28 mm and Dqw1 is 240 mm.Figure 8
Optical performance of the cylindrical cavity receiver with the hyperbolic tangent quartz window. (a) Optical efficiency. (b) LCR. (c) Nonuniformity coefficient. (d) PCR.
(a)
(b)
(c)
(d)Table 8
Optical performance of the cylindrical cavity receiver with a hyperbolic tangent quartz window.
Hqw, mm
Parameters
Dqw, mm
140
160
180
200
220
240
18
η (%)
89.55
89.68
89.80
89.87
89.94
90.01
PCR
935.45
929.29
923.92
944.31
964.25
983.03
Vσ
0.48
0.49
0.50
0.51
0.52
0.53
20
η (%)
89.34
89.51
89.62
89.70
89.80
89.87
PCR
902.86
899.55
900.57
906.09
917.23
935.43
Vσ
0.46
0.46
0.47
0.48
0.49
0.50
22
η (%)
89.12
89.32
89.47
89.57
89.66
89.73
PCR
897.65
890.99
894.18
894.15
900.04
902.07
Vσ
0.43
0.44
0.45
0.46
0.47
0.48
24
η (%)
88.76
89.11
89.27
89.39
89.48
89.61
PCR
915.02
887.99
889.89
888.34
891.25
898.07
Vσ
0.41
0.42
0.43
0.44
0.45
0.46
26
η (%)
88.02
88.73
89.07
89.24
89.32
89.44
PCR
936.02
880.12
881.76
884.29
885.88
885.70
Vσ
0.39
0.39
0.40
0.41
0.43
0.44
28
η (%)
86.97
88.05
88.72
89.04
89.18
89.30
PCR
941.77
871.41
870.93
875.89
879.66
884.90
Vσ
0.37
0.37
0.38
0.39
0.41
0.42


## 4.2. Comprehensive Analysis

### 4.2.1. Comparison of the Different Designs

According to the calculation results in Section4.1, when the cavity receiver uses four types of quartz windows with different structures, i.e., conic, spherical, sinusoidal, and hyperbolic tangent, its optical performance is evaluated. The optimal structure schemes of the four types of quartz windows are shown in Table 9, Figure 9 shows the LCR distribution of the receiver sidewall under the four quartz window structure schemes, and Figure 10 shows the LCR distribution of the receiver sidewall under the four quartz window structure schemes. According to Table 9 and Figure 9, further analysis of the quartz windows of the four structural schemes shows that the optical efficiency of all of the structural schemes is basically the same. From the comparison of the nonuniformity coefficient and the peak value of the concentration ratio, it can be found that the optimal optical properties of the four types of quartz windows are sinusoidal, conical, spherical, and hyperbolic tangent quartz windows. The nonuniformity coefficient of the receiver with the sinusoidal quartz window is 0.31 and the PCR is 806.82.Table 9
Optical performance parameters of a cavity receiver with a quartz window with four optimal structure schemes.
Name
Hqw (mm)
Dqw (mm)
η (%)
Vσ
PCR
Receiver with sinusoidal quartz window
28
240
89.64
0.31
806.82
Receiver with conical window
26
160
89.09
0.35
841.31
Receiver with spherical quartz window
22
160
89.25
0.36
853.23
Receiver with hyperbolic tangent quartz window
28
200
89.04
0.39
875.89Figure 9
LCR distribution of a receiver with a concave quartz window with four optimal structure schemes.Figure 10
LCR contours of a receiver with a concave quartz window with four optimal structure schemes: (a) sinusoidal, (b) conical, (c) spherical, and (d) hyperbolic tangent.
(a)
(b)
(c)
(d)


### 4.2.2. Influence of the Installation Position of the Cavity Receiver

According to the analysis results in Section4.2.1, the sinusoidal quartz window receiver with the optimal structure scheme is further discussed, and the quartz window’s dimensions are Hqw=28mm and Dqw=240mm when the distance L between the apex of the dish concentrator and the point Oqw on the receiver is changed (shown in Figure 1), in other words, when L=f+0.06m, L=f+0.012m, L=f−0.006m, and L=f−0.012m, respectively. The optical performance of the receiver in these four cases is simulated and calculated, and the results are shown in Table 10 and Figure 11. The results show that the optical efficiency basically remains unchanged when L increases, the nonuniformity coefficient increases, and the PCR increases. When L is equal to f−0.006m, the optical efficiency of the receiver basically remains unchanged, but the nonuniformity coefficient drops to 0.31, and the PCR drops to 766.93. The optical performance of the receiver has been further improved.Table 10
Optical performance parameters of the receiver when changing the installation position of the sinusoidal quartz window (Hqw=28 mm, Dqw=240 mm).
Parameters
L=f
L=f+0.06
L=f+0.012
L=f−0.006
L=f−0.012
η (%)
89.64
89.60
89.59
89.62
89.64
PCR
806.82
854.75
909.41
766.93
737.93
Vσ
0.31
0.38
0.34
0.31
0.33Figure 11
LCR distribution of the receiver when changing the installation position of the sinusoidal quartz window (Hqw=28mm, Dqw=240mm).


### 4.2.3. Influence of the Absorptivity of the Cavity Receiver Surface

When the new absorptivityσca is set to 0.80, 0.83, 0.86, 0.89, 0.92, and 0.95, respectively. The results of the receiver’s optical efficiency in six cases are shown in Table 11 and Figure 12. When the absorptivity decreases, the optical efficiency η will decrease from 89.64 to 86.94. This finding is mainly caused by the fact that when the absorptivity decreases, the reflectivity increases, and the solar radiation that enters the receiver cavity will escape more, thus reducing the optical efficiency of the receiver. In addition, although the decrease of absorptivity will lead to the decrease of the nonuniformity coefficient Vσ and PCR, the decrease also comes at the expense of optical efficiency; therefore, the coating of the receiver should be overhauled regularly in real application to ensure its good performance.Table 11
Optical performance of the receiver when the absorptivity of the cavity receiver surface is changed.
Parameters
σca=0.8
σca=0.83
σca=0.86
σca=0.89
σca=0.92
σca=0.95
η (%)
86.94
87.46
88.00
88.49
88.92
89.64
PCR
760.30
771.03
778.01
789.30
796.38
806.82
Vσ
0.28
0.28
0.29
0.30
0.31
0.31Figure 12
LCR distribution of the receiver when the absorptivity of the receiver surface is changed.


### 4.2.4. Influence of Focal Distance of Parabolic Dish Concentrator

When the radiusRd of the parabolic dish concentrator was kept constant, and the focal distance f of the parabolic dish was changed to 8.99 m, it was found from Table 12 and Figure 13 that the PCR of the receiver with concave quartz window was decreased from 1396.6 to 831.57, and the Vσ was decreased from 0.70 to 0.34 compared with that of the receiver with plane quartz window. When the focal distance f was 9.99 m, the PCR of the receiver with concave quartz window was decreased from 1237.30 to 788.69, and the Vσ was decreased from 0.66 to 0.33 compared with that of the receiver with plane quartz window. Therefore, for the parabolic dish concentrator with different focal distance, the concave quartz window can also improve the uniformity of the flux distribution of the receiver cylindrical cavity. On the other hand, the parabolic dish with proper focal distance should be used in the SDCR system.Table 12
Optical performance parameters of the receiver of parabolic dish concentrator at different focal distances.
f (m)
f=8.99
f=8.99
f=9.99
f=9.99
Quartz window
Plane
Concave
Plane
Concave
η (%)
90.40
89.71
90.37
89.66
PCR
1396.60
831.57
1237.30
788.69
Vσ
0.70
0.34
0.66
0.33Figure 13
LCR distribution of the receiver of parabolic dish concentrator at different focal distances.


## 4.2.1. Comparison of the Different Designs

According to the calculation results in Section4.1, when the cavity receiver uses four types of quartz windows with different structures, i.e., conic, spherical, sinusoidal, and hyperbolic tangent, its optical performance is evaluated. The optimal structure schemes of the four types of quartz windows are shown in Table 9, Figure 9 shows the LCR distribution of the receiver sidewall under the four quartz window structure schemes, and Figure 10 shows the LCR distribution of the receiver sidewall under the four quartz window structure schemes. According to Table 9 and Figure 9, further analysis of the quartz windows of the four structural schemes shows that the optical efficiency of all of the structural schemes is basically the same. From the comparison of the nonuniformity coefficient and the peak value of the concentration ratio, it can be found that the optimal optical properties of the four types of quartz windows are sinusoidal, conical, spherical, and hyperbolic tangent quartz windows. The nonuniformity coefficient of the receiver with the sinusoidal quartz window is 0.31 and the PCR is 806.82.Table 9
Optical performance parameters of a cavity receiver with a quartz window with four optimal structure schemes.
Name
Hqw (mm)
Dqw (mm)
η (%)
Vσ
PCR
Receiver with sinusoidal quartz window
28
240
89.64
0.31
806.82
Receiver with conical window
26
160
89.09
0.35
841.31
Receiver with spherical quartz window
22
160
89.25
0.36
853.23
Receiver with hyperbolic tangent quartz window
28
200
89.04
0.39
875.89Figure 9
LCR distribution of a receiver with a concave quartz window with four optimal structure schemes.Figure 10
LCR contours of a receiver with a concave quartz window with four optimal structure schemes: (a) sinusoidal, (b) conical, (c) spherical, and (d) hyperbolic tangent.
(a)
(b)
(c)
(d)


## 4.2.2. Influence of the Installation Position of the Cavity Receiver

According to the analysis results in Section4.2.1, the sinusoidal quartz window receiver with the optimal structure scheme is further discussed, and the quartz window’s dimensions are Hqw=28mm and Dqw=240mm when the distance L between the apex of the dish concentrator and the point Oqw on the receiver is changed (shown in Figure 1), in other words, when L=f+0.06m, L=f+0.012m, L=f−0.006m, and L=f−0.012m, respectively. The optical performance of the receiver in these four cases is simulated and calculated, and the results are shown in Table 10 and Figure 11. The results show that the optical efficiency basically remains unchanged when L increases, the nonuniformity coefficient increases, and the PCR increases. When L is equal to f−0.006m, the optical efficiency of the receiver basically remains unchanged, but the nonuniformity coefficient drops to 0.31, and the PCR drops to 766.93. The optical performance of the receiver has been further improved.Table 10
Optical performance parameters of the receiver when changing the installation position of the sinusoidal quartz window (Hqw=28 mm, Dqw=240 mm).
Parameters
L=f
L=f+0.06
L=f+0.012
L=f−0.006
L=f−0.012
η (%)
89.64
89.60
89.59
89.62
89.64
PCR
806.82
854.75
909.41
766.93
737.93
Vσ
0.31
0.38
0.34
0.31
0.33Figure 11
LCR distribution of the receiver when changing the installation position of the sinusoidal quartz window (Hqw=28mm, Dqw=240mm).


## 4.2.3. Influence of the Absorptivity of the Cavity Receiver Surface

When the new absorptivityσca is set to 0.80, 0.83, 0.86, 0.89, 0.92, and 0.95, respectively. The results of the receiver’s optical efficiency in six cases are shown in Table 11 and Figure 12. When the absorptivity decreases, the optical efficiency η will decrease from 89.64 to 86.94. This finding is mainly caused by the fact that when the absorptivity decreases, the reflectivity increases, and the solar radiation that enters the receiver cavity will escape more, thus reducing the optical efficiency of the receiver. In addition, although the decrease of absorptivity will lead to the decrease of the nonuniformity coefficient Vσ and PCR, the decrease also comes at the expense of optical efficiency; therefore, the coating of the receiver should be overhauled regularly in real application to ensure its good performance.Table 11
Optical performance of the receiver when the absorptivity of the cavity receiver surface is changed.
Parameters
σca=0.8
σca=0.83
σca=0.86
σca=0.89
σca=0.92
σca=0.95
η (%)
86.94
87.46
88.00
88.49
88.92
89.64
PCR
760.30
771.03
778.01
789.30
796.38
806.82
Vσ
0.28
0.28
0.29
0.30
0.31
0.31Figure 12
LCR distribution of the receiver when the absorptivity of the receiver surface is changed.


## 4.2.4. Influence of Focal Distance of Parabolic Dish Concentrator

When the radiusRd of the parabolic dish concentrator was kept constant, and the focal distance f of the parabolic dish was changed to 8.99 m, it was found from Table 12 and Figure 13 that the PCR of the receiver with concave quartz window was decreased from 1396.6 to 831.57, and the Vσ was decreased from 0.70 to 0.34 compared with that of the receiver with plane quartz window. When the focal distance f was 9.99 m, the PCR of the receiver with concave quartz window was decreased from 1237.30 to 788.69, and the Vσ was decreased from 0.66 to 0.33 compared with that of the receiver with plane quartz window. Therefore, for the parabolic dish concentrator with different focal distance, the concave quartz window can also improve the uniformity of the flux distribution of the receiver cylindrical cavity. On the other hand, the parabolic dish with proper focal distance should be used in the SDCR system.Table 12
Optical performance parameters of the receiver of parabolic dish concentrator at different focal distances.
f (m)
f=8.99
f=8.99
f=9.99
f=9.99
Quartz window
Plane
Concave
Plane
Concave
η (%)
90.40
89.71
90.37
89.66
PCR
1396.60
831.57
1237.30
788.69
Vσ
0.70
0.34
0.66
0.33Figure 13
LCR distribution of the receiver of parabolic dish concentrator at different focal distances.


## 5. Conclusions

Four types of concave quartz windows, including conical, spherical, sinusoidal, and hyperbolic tangent, are proposed to be used in the cylindrical cavity receiver. For each concave quartz window, 36 structural schemes are selected. The research shows that the concave quartz window can improve the flux uniformity of the receiver significantly. The specific conclusions of this paper are as follows:(i)
The optical efficiency of the conventional planar quartz window is 90.80%, and the nonuniformity coefficient is 0.68, while the PCR on the sidewall of the receiver is 1320.21
(ii)
The optical properties of the receiver with four different concave quartz windows are evaluated. WhenHqw is 28 mm and Dqw is 240 mm, the sinusoidal quartz window is the best scheme. The optical efficiency of the receiver is 89.04%, which is basically the same as that of the flat quartz window receiver, but the nonuniformity coefficient is reduced to 0.31, and the PCR is reduced to 806.82
(iii)
The other three optimal schemes of the concave quartz windows are the following: whenHqw is 26 mm and Dqw is 160 mm for the conical quartz window, the optical efficiency of the receiver is 89.09%, the nonuniformity coefficient is 0.35, and the value of the PCR is 841.31. When Hqw is 22 mm and Dqw is 160 mm for the spherical quartz window, the optical efficiency of the receiver is 89.25%, the nonuniformity coefficient is 0.36, and the value of the PCR is 853.23. When Hqw is 28 mm and Dqw is 200 mm for the hyperbolic tangent quartz window, the optical efficiency of the receiver is 89.04%, the nonuniformity coefficient is 0.39, and the value of the PCR is 875.89
(iv)
The receiver of the sinusoidal quartz window (Hqw=28mm, Dqw=240mm) with the optimal scheme is further discussed and found. When L=f−0.006m, the optical efficiency of the receiver basically remains unchanged, but the nonuniformity coefficient drops to 0.31, and the PCR drops to 766.93. The optical performance of the receiver has been further improved. When the absorptivity of the receiver decreases, the optical efficiency η will decrease; therefore, the coating of the receiver should be overhauled regularly in real application to ensure its good performance. For the parabolic dish concentrator with different focal distance, the concave quartz window can also improve the uniformity of the flux distribution of the cylindrical cavity receiver



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*Source: 2421391-2020-04-09.xml*