Study on the Optical Properties of Triangular Cavity Absorber for Parabolic Trough Solar Concentrator

A theoretical analytical method for optical properties of cavity absorber was proposed in this paper and the optical design software TracePro was used to analyze the optical properties of triangular cavity absorber. It was found that the optimal optical properties could be achieved with appropriate aperture width, depth-to-width ratio, and offset distance from focus of triangular cavity absorber. Based on the results of orthogonal experiment, the optimized triangular cavity absorber was designed. Results showed that the standard deviation of irradiance and optical efficiency of optimized designed cavity absorber were 30528W/m and 89.23%, respectively. Therefore, this study could offer some valuable references for designing the parabolic trough solar concentrator in the future.


Introduction
Concentrating solar power (CSP) system mainly consists of parabolic trough solar concentrator, linear Fresnel solar concentrator, compound parabolic solar concentrator, and solar tower.Sait et al. [1] investigated the optical and thermal analysis of different blocks of the solar power plant and they compared Fresnel arrays with parabolic trough solar concentrators.They found that the optical performance of Fresnel array was very close to that of parabolic trough solar concentrator, with similar values of maximum flux intensities.However, parabolic trough solar concentrator, as the most mature technology, has been intensively researched and its cost gradually reduced [2][3][4][5][6][7][8][9][10][11][12][13].
In addition, parabolic trough solar concentrator has been put into commercial operations in many countries [14].Practically with the regulation of the tracking system, the collecting mirror of parabolic trough solar concentrator concentrates the solar rays to the absorber at the focal line and then the radiation energy converts into heat energy which transmits to the internal working medium of the absorber.Subsequently, it was confirmed that the absorber is the carrier of solar thermal conversion and its optical properties directly influenced the performance of the entire system [15].
There are two main types of absorbers for parabolic trough solar concentrator, namely, vacuum tube and cavity.Vacuum tube absorber has numerous advantages such as high efficiency and small heat loss.Nevertheless, it has some deficiencies such as technical difficulty, high cost, and gas leakage upon long-term running, while the cavity absorber could make up those shortages of vacuum tube absorber [16][17][18].Therefore, a group of scholars [19][20][21][22][23][24] have done a lot of studies on the structure of cavity absorbers in order to improve the cavity absorber properties, and they reported that the cavity absorbers with triangular structure had a high optical efficiency.
Based on the existing researches, a new theoretical analytical method for cavity absorber optical properties was proposed by using Monte Carlo Ray Tracing (MCRT) [25,26] method.Furthermore, the distribution of solar rays on absorbing surface of triangular cavity absorber was carried out in a simulation experiment, and the optimized triangular cavity absorber was designed to provide high optical properties.

Heat Loss Characteristics of Solar Energy Absorber Surface
Assuming that there is a flat plate absorber as shown in Figure 1, the total area of the absorber is  and the total radiation energy from the sun is .When the irradiance of the absorber surface is uniform,  can be estimated as where  is the direct irradiance.When the irradiance of the absorber surface changed to a nonuniform distribution,  can be calculated as shown in where the whole absorber area is divided into  equal areas   and   is irradiance of each   , so Obviously, the absorber surface has more irradiation, so the high surface temperature can be estimated as where  and   are the surface temperatures of  and   . is the proportionality coefficient, which is the function of absorber material, ambient environment, sky parameters, and so on.
According to the Stefan-Boltzmann law, where  unif and  non are the heat losses of the absorber surface irradiance in uniform and nonuniform phases, respectively,  is Stefan-Boltzmann constant, and  is the emissivity of surface.The relationship value between  unif and  non is analyzed as follows.According to the relationship between the arithmetic average and the root-mean-square value, when the equality held up as follows: By assuming that where   is the proportionality coefficient, from formulae (( 5)-( 6)), And then Therefore, by comparison between (7), (8), and (20), it can be found that the irradiant heat loss of nonuniform distribution is higher than that of uniform distribution.

Simulation Model of Optical Properties
Based on the MCRT, TracePro simulation experiment was carried out for the optical properties of triangular cavity absorber as shown in Figure 2. TracePro software was developed for system's optical design based on solid model drawing software of ACIS as the core.The TracePro has strong optical analysis and information transmission capacity and was widely applied in solar system, optics device design field, autolighting system, infrared optical imaging, indoor lighting system, optical thin film, and imaging system.
During the TracePro simulation process, the reflection, refraction, absorption, and transmission behaviors of solar rays were tracked and recorded one by one until the lights overflow the system or totally adsorbed and the solar ray tracing would be stopped at this time as shown in Figure 3.The solar rays from the sun of the parabolic trough solar concentrator were assumed as parallel.The surface of concentrating mirrors in the PTC was ideal without any mechanical or manufacturing error and the one-dimensional automatic tracking device in the PTC has no error or lag.

Optical Properties of Absorber
In order to analyze the properties of cavity absorber properly, optical efficiency and standard deviation of irradiance should be investigated.The optical efficiency demonstrates the ratio between the radiant energy received by cavity absorber and the direct radiant energy received by parabolic trough solar concentrator.Moreover, it is a key parameter for analyzing the optical properties of cavity absorber.Thereby, the optical efficiency can be estimated as shown in where  is the optical efficiency of cavity absorber;  is the area of absorption surface of cavity absorber; (, ) is the irradiance distribution function of absorption surface of cavity absorber;  is the included angle between the axis and the aperture of the triangular cavity absorber;  is aperture area of parabolic trough solar concentrator.The standard deviation of irradiance () is used to measure the evenness of energy distribution inside the cavity absorber.The cavity absorber at high standard deviation of irradiance has high fluctuation gradient and high heat loss.Furthermore, it has an important role for researching the heat loss of cavity absorber and () can be calculated as shown in where () is probability density of (, ).() is average of irradiance as shown in It is very difficult to determine the irradiance distribution function (, ) and the probability density ().Therefore, the irradiance of absorption surface of cavity absorber is a dispersed calculation in TracePro simulation experiment.The calculation formulas for the optical efficiency, average of irradiance, and deviation of irradiance are shown in where  and  are the -axis and -axis direction of absorption surface, respectively,  and  are the total step number of axis and -axis, respectively,   is the absorption area unit of absorption surface of cavity absorber,   is the direct solar irradiance, and   is the irradiance of   .

Optical Properties Simulation
The impacts on the optical properties caused by aperture width, depth-to-width ratio, and offset distance from focus of triangular cavity absorber were simulated.In the simulation experiment [12], the focal distance, the aperture width, and the aim angle of parabolic trough solar concentrator were 100 cm, 240 cm, and 45 ∘ , respectively.The absorptivity, reflectance, and transmittance of the concentrating mirror in the parabolic trough solar concentrator were 6.5%, 93.0%, and 0.5%, respectively.The solar direct irradiance was 1000 W/m 2 .Cavity absorber has axial symmetry.Therefore, only half-cavity absorber has been analyzed for researching its optical properties.The parameters of the single factor simulation experiment are shown in Table 1.

Optical Properties
Caused by the Aperture Width.The irradiance distribution of absorbing surface inside the cavity at different aperture width is shown in Figure 4.It can be seen that the absorbing surface irradiance of the triangular cavity absorber firstly increased and then decreased with the increasing of the distance away from the symmetry axis.In addition, the maximum irradiance gradually decreased with the increment of the aperture width from the symmetry axis.
The phenomenon was very interesting and it was explained in latter paragraphs.The aperture widths of triangular cavity absorber were 30 mm, 45 mm, 60 mm, 75 mm, and 90 mm whereas the maximum irradiances were 83066 W/m  Figure 5 shows the effects of the aperture width of triangular cavity absorber on the optical properties.It was observed that the absorber's optical efficiency and the standard deviation of irradiance were decreasing with the increment of the aperture width.When the aperture widths of cavity absorber were 30 mm and 90 mm, the optical efficiencies were 86.63% and 86.29%, respectively.Meanwhile, the standard deviations of irradiances were 29037 W/m 2 and 11529 W/m 2 , respectively, that is, because of the increment of the aperture width increasing the absorption area of cavity and reducing the radiation energy per area unit on absorbing surface.Therefore, the aperture width should not be very large.The large aperture width reduces the optical concentration ratio and increases the angle factor between absorbing surface and the ambient.In addition, it increases the heat losses of irradiance from the cavity absorber.However, actual parabolic trough solar concentrator has some errors such as the machining errors and tracing accuracy errors.The actual focal line width was bigger than that of ideal condition.Thereby, the width of the cavity absorber should not be very small.

Optical Properties Caused by the Depth-to-Width Ratio.
The irradiance distribution of absorbing surface inside the cavity at different depth-to-width ratios is shown in Figure 6.It can be observed that the maximum irradiance decreased firstly and then increased with the increment of the depthto-width ratio of cavity absorber.
Figure 7 shows the effect of the depth-to-width ratios of triangular cavity absorber on the optical properties.It was found that the absorber's optical efficiency increases with the increment of the depth-to-width ratio.In contrast, the standard deviation of irradiance decreases with the increment of the depth-to-width ratio.The depth-to-width ratios were 0.2 : 1.0 and 1.0 : 1.0, the optical efficiencies were 79.72% and 89.03%, and the standard deviations of irradiances were 33747 W/m 2 and 19502 W/m 2 , respectively.
The solar rays can easily enter the inner of the cavity absorber with the increasing of cavity's depth-to-width ratios.Furthermore, the aperture of the cavity can detain the solar rays and it will be difficult to escape.Therefore, this was very beneficial for the irradiance uniformity of inner surface to improve the optical efficiency and decrease the standard deviation of the irradiance.

Optical Properties Caused by the Offset Distance from
Focus. Figure 8 illustrates the irradiance distribution of absorbing surface inside the cavity at different offset distance from focus of the cavity (offset distance from focus was a negative value when the absorber was close to the trough system and a positive value on the contrary).It can be seen that the irradiance of absorbing surface increased firstly and then decreased with the increment of the distance from the symmetry axis.Moreover, the maximum irradiance decreased with increasing the distance from the symmetry axis and with the increment of the offset distance from focus.The optical properties of the cavity could be changed at different offset distances from the focus as shown in Figure 9.It was clear that the absorber's standard deviation of irradiance decreased with the increment of the offset distances from the focus, while the optical efficiency increased firstly and then decreased with the increment of the offset distances from the focus.The offset distances from focus were −20 mm, −10 mm, and 20 mm, the optical efficiencies were 87.65%, 88.13%, and 83.05%, and the standard deviations of the irradiances were 43324, 31058, and 6741, respectively.The parabolic trough solar concentrator concentrates the solar rays near its focal line and then the concentrated solar rays near the focal line can be absorbed and reflected by reaching the absorbing surface or it can be diffused.Therefore, by increasing the diffusing paths of concentrated rays, the solar energy will be more uniformity and the standard deviation of the irradiance will be decreased.It was also observed that when the offset distance from focus of cavity became more than zero, the diffused concentrated rays will be decreased, and it cannot reach the absorbing surface.Thereby, the optical efficiency and the standard deviation of the irradiance will be both decreased.In contrast, when the offset distance from focus of cavity became less than zero, the aperture of the cavity can detain the solar rays and it will be difficult to escape from the inner of cavity, and this resulted in improving the optical efficiency.In other words, when the offset distance from focus was too small, some solar rays could not reach the absorbing surface and the optical efficiency decreased.Subsequently, the cavity absorber was properly shifted downward from system's focal line position.

Optimization
In order to get the optimal optical properties of the cavity absorber, the structure and installation position of cavity absorber were properly designed according to the experimental results of each single factor on the optical properties of cavity absorber.The three-factors-three-levels orthogonal experiments L9 (3 3 ) were designed and the parameters of the orthogonal experiments are shown in Table 2.
Table 3 shows the optical efficiency of orthogonal experiment.It was found that the depth-to-width ratio of cavity absorber was the most sensitive parameter on the optical properties.The range of optical efficiency was 0.336%.The installation position of cavity absorber was relatively weak on the optical properties (0.160%).Furthermore, the optical efficiency of the optimized cavity absorber was 89.33% when its aperture width, depth-to-width ratio, and offset distance were 70 mm, 0.85 : 1.00, and −15 mm, respectively.
The standard deviation of irradiance is shown in Table 4, and it can be seen that the standard deviation of irradiance was the most sensitive on the installation position.The range of the standard deviation of irradiance reached 5392.Moreover, the standard deviation of irradiance of optimized cavity absorber was 25257 when its aperture width, depth-towidth ratio, and offset distance were 72 mm, 0.75 : 1.00, and −5 mm, respectively.
The results of the structure parameters and installation position on cavity absorber by the orthogonal experiment on the optical efficiency and standard deviation of irradiance were different, which can be discussed as follows.
The influence of aperture width on the optical efficiency and standard deviation of irradiance were between depth-towidth ratio and offset distance, which can be seen from the primary and secondary factors as shown in Tables 3 and 4. Therefore, the aperture width can be set as  2 for considering the properly optical efficiency.
The depth-to-width ratio of cavity absorber was the most sensitive on the optical efficiency.The larger depth-to-width ratio of cavity absorber was important to improve the optical efficiency.On the other hand, the larger depth-to-width ratio increased the cavity absorber's center of gravity and decreased the instability.Therefore,  2 can be selected as the value of cavity absorber's depth-to-width ratio.
The solar rays could not be absorbed by cavity absorber for only one time but it could be absorbed by the inner of cavity absorber several times.It was found that the deep of cavity assisted the solar rays to enter the inner of cavity absorber, and the solar rays were detained by the aperture of the cavity absorber, which was important for the thermal performance of the cavity absorber.Consequently,  1 can be selected as the value of cavity absorber's offset distance.
According to the above analysis, the optimized results were  2  2  1 at the aperture width of 70 mm, depth-to-width ratio of 0.80 : 1.00, and offset distance of −15 mm, respectively.
Results revealed that the standard deviation of irradiance and optical efficiency of optimized designed cavity absorber were 30528 and 89.23%, respectively.The optical properties of the optimized designed cavity absorber are shown in Figure 9.It can be seen that the solar rays are concentrated inside the optimized designed cavity absorber, and then the concentrated solar rays are divergent and reach the absorbing surface of the cavity absorber.Furthermore, the reflected solar rays from the absorbing surface could reach the absorbing surface.Meanwhile, the solar rays could be absorbed by the absorbing surface of the cavity absorber several times, which is important to improve the photothermal performance of the optimized designed cavity absorber.
We can find in Figure 10 that the concentrated solar rays were concentrated at the aperture of cavity absorber reflect mirror.The concentrated solar rays met the absorber surface at the aperture which was absorbed by the cavity absorber;  otherwise the concentrated solar rays would be divergence and then divergence concentrated solar rays were arriving at the inner absorbing surface of cavity absorber.So the maximum irradiance achieved positions were not in the axis of symmetry of the cavity absorber in Figure 10.

Conclusion
The optical properties of triangular cavity absorber for parabolic trough solar concentrator were simulated and optimized.Furthermore, the optimized triangular cavity absorber was designed from the orthogonal experiment.Results illustrated that the standard deviation of irradiance and optical efficiency for the optimized designed cavity absorber were 30528 and 89.23%, respectively.Results also revealed that the triangular cavity absorber's aperture width, depth-to-width ratio, and offset distance from focus under different conditions of appropriate values were beneficial to improve the optical properties.Therefore, it is recommended to select the aperture width of 70 mm, depth-to-width ratio of 0.80 : 1.00, and offset distance of −15 mm, respectively.

Figure 4 :Figure 5 :
Figure 4: The irradiance distribution of absorbing surface at different aperture width.

Figure 6 :Figure 7 :
Figure 6: The irradiance distribution of absorbing surface at different depth-to-width ratio.

Figure 8 :Figure 9 :
Figure 8: The irradiance distribution of absorbing surface for cavity at different focus offset distance.

Figure 10 :
Figure 10: The optical properties of optimized designed cavity absorber.

Table 1 :
Simulation parameters for the optical properties of the cavity absorber.

Table 2 :
The parameters of the orthogonal experiments.

Table 3 :
The orthogonal experimental results of optical efficiency on cavity absorber.

Table 4 :
The orthogonal experimental results of standard deviation of irradiance on cavity absorber.