Effects of critical geometric parameters on the optical performance of a conical cavity receiver

Abstract

The optical performance of a receiver has a great influence on the efficiency and stability of a solar thermal power system. Most of the literature focuses on the optical performance of receivers with different geometric shapes, but less research is conducted on the effects of critical geometric parameters. In this paper, the commercial software TracePro was used to investigate the effects of some factors on a conical cavity receiver, such as the conical angle, the number of loops of the helical tube, and the distance between the focal point of the collector and the aperture. These factors affect the optical efficiency, the maximum heat flux density, and the light distribution in the conical cavity. The optical performance of the conical receiver was studied and analyzed using the Monte Carlo ray tracing method. To make a reliable simulation, the helical tube was attached to the inner wall of the cavity in the proposed model. The results showed that the amount of light rays reaching the helical tube increases with the increasing of the conical angle, while the optical efficiency decreases and the maximum heat flux density increases. The increase in the number of loops contributed to an increase in the optical efficiency and a uniform light distribution. The conical cavity receiver had an optimal optical performance when the focal point of the collector was near the aperture.

Keywords

parabolic collector / conical cavity receiver / critical geometric parameters / optical performance

Introduction

With the growing consumption of fossil energy and increasing environmental pollution, renewable energy such as solar energy attracts more and more attention. Compared with other forms of solar thermal utilization, the solar parabolic dish system has the highest thermal efficiency, a concentration ratio of more than 2000, and an operating temperature of over 900 K [¹,²]. As a core component of the system, the cavity receiver takes on the role of converting solar energy into thermal energy for the working fluid. The optical and thermal performance of the receiver has a great influence on the efficiency and stability of the whole system.

The boundary conditions of the inner surface of the cavity have often been simplified in different ways by many researchers. Flesch et al. [³] and Xiao et al. [⁴] did not consider the helical tube in the cavity when exploring the effects of wind on the thermal performance of the receiver. Their models were greatly simplified. Wu et al. [⁵,⁶] studied the convection and radiation loss of a fully open cylindrical cavity receiver. The inner surface of the cavity was regarded as a wall with a constant heat flux. In the numerical simulation of a hemispherical cavity receiver, Cui et al. [⁷] assumed that the wall temperature of the helical tube was constant. This assumption was based on the fact that the helical tube material has a good thermal conductivity. Compared to the work of Cui et al. [⁷], it was more advisable to simplify similar models, as done by Reddy et al. [⁸] and Vikram & Reddy [⁹]. He divided the surface of the helical tube into several parts, and the temperature of each part was set to different values. Thus the simulated situation was closer to the actual temperature distribution.

In the above studies, some rules concerning the thermal performance of receivers are revealed, but there are improvements to be made. When assuming the boundary conditions of the inner surface of the cavity, the constant wall temperature condition and the constant heat flux condition are systematically incomplete. The overall performance of the receiver includes the optical and thermal performance. Ideally, the light distribution obtained from the optical part can be introduced into the thermal part for calculation. Therefore, some researchers have studied the optical performance of receivers. Based on some reasonable assumptions, the theoretical analysis methods were used to derive models for the heat flux distribution and optical loss of receivers [¹⁰,¹¹]. In addition, some researchers used ray tracing software or code based on the Monte Carlo method to simulate receiver models. This latter method can obtain more accurate results and richer information [¹¹].

Xie et al. [¹²] performed optical simulations of cylindrical, conical and spherical cavity receivers. The results showed that the conical receiver had the highest optical efficiency and the most uniform heat flux distribution. In the subsequent study, the conical receiver with a conical angle of 100° was considered to have the highest thermal efficiency [¹³]. For similar problems, Daabo et al. [¹⁴] concluded that the conical receiver had the highest optical efficiency, while the cylindrical receiver had a most uniform heat flux distribution. In subsequent work, it was found that the thermal efficiency of the conical receiver was the highest, while that of the cylindrical receiver was the lowest [¹⁵]. Li et al. [¹⁶] studied the effects of some geometric parameters of the cylindrical receiver on the optical efficiency and the heat flux distribution. The results showed that the optical efficiency decreased with increasing diameter ratio. However, the increase in the height ratio made the heat flux distribution more uniform. To solve the problem of low optical efficiency caused by the absence of tubes covering the bottom surface of cylindrical receivers, Wang et al. [¹⁷] proposed a new receiver model with a convex bottom surface. The investigation results showed that the optical efficiency was significantly improved. Shuai et al. [¹⁸] studied the effects of sun shape and surface slope error on receivers of different geometries. Then an upside-down pear cavity receiver was designed, which showed a more uniform heat flux distribution compared to receivers of traditional geometries. Some researchers are more concerned about the influence of the concentrator on the optical performance of the system, such as the focal length of the collector [¹⁹] and the optical error [²⁰]. Current research on the optical performance of cavity receivers is still at an early stage and the theory is not yet mature. Therefore, when similar models are investigated in the literature, the conclusions are sometimes inconsistent or even contradictory. In regard to optical simulation, researchers do not always consider the helical tube inside the cavity to simplify models. Therefore, it is very meaningful to perform optical simulation of a cavity receiver whose inner wall is covered by a helical tube.

When investigating the optical performance of the cavity receiver, most researchers pay attention to the differences between the receivers of different geometries. They try to find an optimal geometric structure, such as cylindrical, conical, spherical or nonconventional-shaped receivers [¹²,¹⁴,¹⁷,¹⁸]. However, there are not many further studies on the effects of critical geometric parameters of a receiver. For example, the influencing mechanism is not clear of the cavity depth, the size of the aperture, the conical angle, and the receiver’s position affecting the optical performance. To obtain the optimal optical performance, these critical parameters should be considered. Comparing several traditional geometric receivers, cylindrical receivers and conical receivers are found to have a lower manufacturing difficulty and lower cost. Then, referring to the conclusion that the optical performance of conical receivers is better [¹²,¹⁴], it is necessary to study the optical performance of the conical cavity receiver.

In this paper, the effects of the critical geometric parameters on the optical performance of conical cavity receivers are investigated, such as the conical angle, the number of loops of the helical tube, and the distance between the focal point of the collector and the aperture. These parameters affect the optical efficiency, the maximum heat flux density, and the light distribution in the cavity receiver. To get closer to the actual situation, this study assumes that the inner wall of the cavity is covered by a helical tube in the proposed model. This assumption makes the propagation of light rays in the cavity more complicated.

Physical model

As shown in Fig. 1, the parabolic collector model and the cavity receiver model are established. A virtual circle is created as the light source plane, which emits light rays. Most of the dimensions are based on the experimental platform designed by our research group, as shown in Fig. 2. The parabolic collector is an ideal paraboloid with a diameter of 5060 mm and a focal length of 3200 mm.

Figures 3(a) and 3(b) illustrate the cross section and the external view of the cavity receiver respectively. The thickness (s) of the thermal insulation layer is 75 mm. The maximum diameter (d_cav) of the cavity is 460 mm. At the same time, the diameter (d_ap) of the aperture is 200 mm by reference to Refs. [²¹,²²] about the aperture size of a similar receiver model. The conical angle (b) and the length (L) of the cavity are varied. For the proposed model, different cavity lengths correspond to different numbers (n) of loops of the helical tube. In addition, the diameter (d_t) of the helical tube is 42 mm and the wall thickness (d_t) is 3 mm. In view of manufacturing errors, the pitch is set to 47 mm, leaving a gap between each loop.

For convenience, the cylindrical receiver is regarded as a conical receiver with a conical angle of 0°. To investigate the influence of the conical angle, receivers with 6 loops are studied as the conical angle varies from 0° to 15°. To investigate the influence of the number of loops, receivers with conical angles of 15° are studied as the number of loops varies from 5 to 8. When the above two parameters are discussed, the position of the receiver is changed. That is, the distance (d) between the focal point of the collector and the aperture varies from -120 mm to 150 mm. A distance of less than 0 mm means that the focal point is inside the cavity. A distance of more than 0 mm means that the focal point is outside the cavity, as shown in Fig. 3(a).

When the conical angle is 15° and the number of loops is 8 in the proposed model, the minimum winding radius of the helical tube is approximately 90 mm. As a comparison, the diameter of the tube is 42 mm. Referring to the experiment made by Prakash et al. [²³,²⁴], a small winding radius leads to a large manufacturing stress and residual stress. Then, it increases the difficulty of winding. To avoid an overly small winding radius, a larger conical angle and a larger number of loops are avoided. When the number of loops is less than 5, the working fluid in the tube may not be sufficiently heated. Therefore, this option is not considered. When the distance between the focus point and the aperture varies from -120 mm to 150 mm, it is ensured that most of the light rays enter the cavity. This distance range is mainly focused on.

Optical theory and analysis

In this paper, the commercial software TracePro7.3.4 is used to explore the optical performance of the conical cavity receiver. In previous studies, this software has been frequently used for optical analysis [¹²,²⁵–²⁹]. In TracePro, users can trace the random rays emitted from the light source. The position and direction of these random rays, as well as the reflection, refraction, and scattering processes generated during the propagation are simulated based on the Monte Carlo ray tracing method. Each ray is traced and counted until the energy of the ray reaches a negligible level or the ray escapes the system.

In the simulation, the continuous light ray propagation is replaced by the discrete light ray propagation. The discrete light rays are randomly sampled based on probability density. Tracing rays in this way requires a large and complex program. By simulating large number of light rays, reliable simulation results can be obtained.

The direct normal irradiance of sunlight reaching the ground is represented by I₀. In the software, many rays of the same energy are used to approximately simulate the process. The number of rays is represented by N in the proposed model. The correct number of rays ensures a short calculation time and high calculation accuracy. According to past research experience, the number of rays that is suitable is between 10⁶ and 10⁷. When the light rays reach the parabolic collector, the energy carried by each reflected ray (E₀) can be determined using Eq. (1).

(1)E_{\mathrm{0}}={\displaystyle \frac{\mathrm{\pi }{R}^{\mathrm{2}}{I}_{\mathrm{0}}\rho }{N}},

where R is the radius of the light source and r is the reflectivity of the collector.

Then, the intersection of each ray with the surface is counted and the current reflection mode is determined. If the reflected rays reach the surface of the receiver, the reflection mode is diffuse reflection. In this case, the reflection model is based on a bidirectional reflection distribution function (BRDF). If the reflected rays reach the collector, the reflection mode is specular reflection. The angle of reflection is equal to the angle of incidence. If the reflected rays escape from the system, they are not counted. Therefore, the energy carried by each ray after the first reflection (E₁) in the cavity can be determined using Eq. (2).

(2)E_{\mathrm{1}}=E_{\mathrm{0}}(\mathrm{1}-\alpha ),

where α is the absorptivity of the material covering the receiver and the tube. Each reflected ray whose energy is less than 0.1% of the initial energy is no longer traced. In this paper, the wall of the cavity and the wall of the heliacal tube are made of the same material. Therefore, the reflectivity of each reflection surface in the cavity is the same value. If the number of reflections of a ray in the cavity is represented by m, the energy carried (E_m) by the ray can be determined using Eq. (3).

(3)E_m=E_{\mathrm{0}}{(\mathrm{1}-\alpha )}^m,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}m=\mathrm{0},1,2,3,\mathrm{...},m_{\max },

where m_max is the maximum number of reflections of a ray. Most of the rays reflected by the collector can enter the cavity. Then a small portion of the rays escapes through the aperture and it is almost impossible for them to re-enter the cavity. After the surfaces of the receiver model are partitioned, the light rays and energy of each area are counted. Then, the energy in each area of the receiver (Q_n) can be determined using Eq. (4).

(4)Q_n=\alpha {\displaystyle {\sum }_{m=\mathrm{0}}^{m_{\max }}}{E}_m{N}_{m,n},

where N_m,n is the number of rays reaching area n when the number of reflections is m. The value of N_m,n in different areas is different according to the statistical results of the ray tracing. Then, the energy flux density in each area of the receiver (I_n) can be determined using Eq. (5).

(5)I_n={\displaystyle \frac{Q_n}{S_n}},

where S_n is the area of the corresponding part. After counting the energy flux density in each area, the maximum heat flux density (f_max) on the receiver can be obtained.

In the proposed model, the reflectivity of the collector material is a constant. To directly assess the utilization of light energy, the optical efficiency of the cavity receiver is defined by reference to the work performed by Li et al. [¹⁶]. The optical efficiency is the ratio of the total energy absorbed by the inner surface of the cavity to the solar energy emitted on the collector. The optical efficiency (h_opt) can be determined using Eq. (6).

(6){\eta }_{\mathrm{opt}}={\displaystyle \frac{Q_{\mathrm{rec}}}{Q_{\mathrm{col}}}},

where Q_rec is the total energy absorbed by the inner surface of the cavity and Q_col is the solar energy emitted on the collector. Obviously, the newly defined optical efficiency is lower than the regular optical efficiency of the receiver. However, the optical performance of the cavity receiver can be directly evaluated. Q_rec is further divided into several parts, which can be calculated using Eq. (7).

(7)Q_{\mathrm{rec}}=Q_{\mathrm{n}}+Q_{\mathrm{f}}+Q_{\mathrm{b}}+Q_{\mathrm{s}}+Q_{\mathrm{t}},

where Q_n, Q_f, Q_b, Q_s, and Q_t represent the energy absorbed by the neck, the front, the bottom and the side of the cavity and the tube wall, respectively, as shown in Fig. 4.

To quantitatively evaluate the light distribution in the cavity, the energy absorption proportion of the tube (P_t) is presented in this paper. It is the ratio of the energy absorbed by the heliacal tube to the total energy absorbed by the inner surface of the cavity, which is defined in Eq. (8).

(8)P_{\mathrm{t}}={\displaystyle \frac{Q_{\mathrm{t}}}{Q_{\mathrm{rec}}}},

where Q_t is the energy absorbed by the helical tube. A higher energy absorption proportion of the tube means a better light distribution in the cavity.

Assumptions and software settings

The purpose of this paper is to study the effects of the critical geometric parameters of a conical cavity receiver on the optical performance. Therefore, the machining error and tracking error of the collector are not considered. When using the receiver, phenomena occur such as oxidation at high temperature and adhesion of dust on the inner surface of the cavity. Correspondingly, the following assumptions are made in the simulation: the dish collector is an ideal paraboloid; air is a noninvolved medium for radiation; and the reflection mode of rays in the cavity is diffuse reflection.

Taking into account the material limitations of the actual process, stainless steel is used to make helical tubes and cover the surfaces of the thermal insulation layer. According to the setting in Ref. [³⁰], the direct normal irradiance of sunlight is assumed to be 1000 W/m². The main settings in the software are as follows:

1) The reflectivity of the collector material is 0.9. The inner wall material of the cavity receiver is the same as that of the helical tube, with a reflectance of 0.15 and an absorptivity of 0.85.

2) The light source is a circle with a radius of 2500 mm. The number of rays is 10⁶. The direct normal irradiance is 1000 W/m².

3) The light flux threshold is 0.001, below which light rays are not taken into consideration.

Results and analysis

In this paper, there are three main indicators for evaluating the optical performance of the conical cavity receiver: optical efficiency (h_opt), maximum heat flux density (f_max), and the energy absorption proportion (P_t) of the tube. A greater optical efficiency means that more light rays enter the cavity. A small maximum heat flux density ensures that the receiver will not be overheated locally. A higher energy absorption proportion indicates that the light distribution is better. Combining these three indicators, a more accurate analysis of the optical performance can be made.

Optical efficiency analysis

Figures 5 and 6 reflect the variation of optical efficiency for different conical angles and different numbers of loops when the distance between the focus point and the aperture varies, respectively. It is observed that the optical efficiency first rises sharply as the distance increases. Then, it remains relatively stable, and finally drops sharply. The corresponding three situations are demonstrated in Fig. 7. Figure 7(a) illustrates the case where the distance is less than 0 mm, and the focus point is inside the cavity. At this time, the optical efficiency is low because a large number of rays are blocked by the front cover of the receiver. In Fig. 7(b), the focal point is near the aperture, and most rays can enter the cavity. Thus, the optical efficiency is high. Figure 7(c) shows the case where the distance is greater than 0 mm, and the focal point is outside the cavity. At this time, the optical efficiency is low because a portion of the rays is blocked again. The maximum optical efficiency occurs when the distance is -60 mm.

A comparison of the four curves in Fig. 5 indicates that the optical efficiency decreases with increasing conical angle. This phenomenon can be explained by the size of the absorption area in the cavity. When the conical angle is small, the surface area of the cavity is large. After light rays enter the cavity, they reflect more between the surfaces in the cavity. Therefore, the energy can be fully absorbed and the optical efficiency is high. The maximum optical efficiency occurs when the conical angle is 0°.

The four curves in Fig. 6 exhibits that the optical efficiency increases slightly as the number of loops increases. This result can also be attributed to the size of the absorption area in the cavity. Figure 8 presents a cross-sectional view of the receiver for different numbers of loops. The larger number of loops means greater lengths and greater surface area. Light energy can be more fully absorbed in the cavity, resulting in an increase in optical efficiency. In addition, when the number of loops is sufficiently large, its continued increase does not effectively increase the optical efficiency. Thus, the curves for 7 and 8 loops are almost coincident in Figs. 6(a) and 6(b). The maximum optical efficiency occurs when the number of loops is 8.

A comparison of the curves in Figs. 5 and 6 suggests that the maximum difference in optical efficiency for the different conical angles is approximately 0.5% and approximately 0.1% for different numbers of loops. It is concluded that the effect of conical angle on optical efficiency is significantly greater than the effect of the number of loops.

Maximum heat flux density analysis

Figures 9 and 10 reflect the variation of maximum heat flux density at different conical angles and different numbers of loops when the distance between the focus point and the aperture varies, respectively. All curves first decrease and then increase because the location of the maximum heat flux density varies with distance.

As the distance increases, there are three different heat flux distributions as displayed in Fig. 11. In Fig. 11(a), the location of the maximum heat flux density appears at the upper part of the helical tube. Few light rays reach the lower part of the helical tube. The heat flux distribution is extremely nonuniform and the corresponding distances are -90 mm to -30 mm. As the distance increases, the location of the maximum heat flux density changes to the lower part of the helical tube, as shown in Fig. 11(b). The area of the loop near the aperture is larger when the conical angle is not 0°. This effect results in a decrease in the maximum heat flux density. The heat flux distribution becomes relatively uniform and the corresponding distances are 0 mm to 30 mm. As the distance increases further, the location of the maximum heat flux density appears at the neck of the receiver, as shown in Fig. 11(c). Thus, the maximum heat flux density changes little when the conical angle and the number of loops are varied. The heat flux density distribution is not ideal and the corresponding distances are 60 mm to 150 mm. The maximum heat flux density is relatively small when the distance is 30 mm.

Figure 9 shows that the maximum heat flux density increases with increasing conical angle when the distance varies from -90 mm to 30 mm. As shown in Fig. 12, the four light distributions correspond to four different conical angles. As the conical angle increases, the area of the helical tube decreases. Therefore, more rays can reach the tube, which inevitably leads to an increase in the local heat flux density. When the conical angle is 0°, the maximum heat flux density is relatively small.

There is a slight difference among the four curves in Fig. 10. In most cases, the curves are nearly coincident. When the distance is between -90 mm and -30 mm, the maximum heat flux density increases slowly as the number of loops increases. Similar to the rule for the optical efficiency, the number of loops has a small effect on the maximum heat flux density that is less than the effect of the conical angle.

Energy absorption proportion analysis

In a cavity receiver, heat is transmitted to the working fluid in the tube by heat conduction, convection, and radiation. The best way is that the light rays reach the tube directly and heat the working fluid by heat conduction. Therefore, the energy absorption proportion of the tube proposed can quantitatively reflect the light distribution in the cavity. When the energy absorption proportion is greater, the light distribution is better.

Figures 13 and 14 depict the variation of energy absorption proportion for different conical angles and different numbers of loops when the distance between the focal point and the aperture varies, respectively. All curves in Figs. 13 and 14 decrease first and then increase. Figure 6(a) shows the case where the focal point is inside the cavity. Many rays entering the cavity reach the bottom of the cavity and cannot reach the tube. When the focal point is near the aperture, most light rays can enter the cavity and reach the tubes, as shown in Fig. 7. Therefore, the energy absorption proportion is high. Figure 7(c) shows the case where the focal point is outside the cavity. The energy absorption proportion is low again because many rays reach the neck of the cavity. The maximum energy absorption proportion occurs when the distance is -30 mm.

Figure 13 shows that the energy absorption proportion of the tube increases as the conical angle increases. This result means that the light distribution in the cavity is better when the conical angle is larger. As shown in Fig. 12, the bottom area of the cavity is greatly reduced when the conical angle increases. Therefore, more light rays reach the tubes, resulting in a high-energy absorption proportion. The maximum energy absorption proportion occurs when the conical angle is 15°.

As shown in Fig. 14, the energy absorption proportion of the tube increases as the number of loops increases. This result indicates that the light distribution in the cavity is better when the number of loops is larger. Figure 8 shows that when the number of loops increases, the bottom area of the conical cavity decreases. In contrast, the area of the helical tube increases, more light rays reach the tube and reflect fully. The maximum energy absorption proportion occurs when the number of loops is 8.

The conical angle and the number of loops have effects on the optical efficiency of less than 0.5%, but the effects on the energy absorption proportion of the tube can reach 20%. It is concluded that the conical cavity receiver with a larger conical angle and more loops has a better optical performance. In addition, the receiver has a higher optical efficiency, a lower maximum heat flux density, and a higher energy absorption proportion when the focal point is near the aperture. Finally, when the focus point is located near the aperture, a cavity receiver with 8 loops and a conical angle of 15° has the best optical performance in the proposed model.

Conclusions

In this paper, the ray tracing method is used to simulate the optical performance of a conical cavity receiver. Three factors affecting the optical performance are investigated. They are the number of loops, the conical angle, and the distance between the focal point and the aperture when the diameter of the aperture, the incident light conditions and the material are constant. Based on the results, the following conclusions can be made:

For a conical cavity receiver, the optical efficiency increases with the decrease of the conical angle and the increase of the number of loops.

The maximum heat flux density decreases as the conical angle decreases and the focal point is near the aperture.

The increase of the conical angle and the number of loops contribute to the increase of light rays reaching the helical tubes.

In the proposed model, an optimal optical performance can be achieved when the focal point is located near the aperture, the cavity receiver has 8 loops, and the conical angle is 15°.

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