Optical modeling of a cylindrical-hemispherical receiver for parabolic dish concentrator

Among all sub-systems of a solar thermal energy system, the receiver plays a significant role when it gets heat energy from the concentrator. The reliability of such systems depends on the amount of solar energy that the receiver collects and other optical parameters like focal length, aperture diameter, surface absorptivity, and slope error. The present paper discusses the optical analysis of a cylindrical-hemispherical receiver coupled with a parabolic dish concentrator having 3-m diameter. The study has been carried out using the SolTrace software by varying the parameters like receiver aperture diameter (Da) ranging from 0.125 to 0.162 m, surface error of the concentrator varying from 1.7453 to 34.907 mrad, and also surface absorptivity (α) changing from 75 to 95% for different receiver distances (H) ranging from 1.7 to 1.95 m. The simulation results show that the optical efficiency is maximum when the receiver with 0.150-m aperture diameter is placed at a distance of 1.85 m from the concentrator. An increase in slope errors from 1.7453 to 17.453 mrad decreases the average optical efficiency by almost 50% for all receiver diameters. It is also noticed that uniform heat flux distribution can be achieved when the receiver’s position is maintained at H = 1.85 m from the concentrator for 0.150-m receiver diameter and 95% absorptivity of the receiver surface. The simulated results of heat flux intensity on the receiver surface are then compared and validated by the experimental results available in the literature. The simulated optical efficiency of the present receiver is found to be 8% higher when compared with a conventional cylindrical receiver with similar dimensions.


Introduction
Solar energy is considered the best alternative to fossil fuels, which mainly cause global warming. Concentrating solar energy collectors is one of the emerging ways to harness solar thermal energy. According to Kumar et al. (2022) and Jaffe (1983), the parabolic dish concentrator (PDC) is in the top position due to its high concentration ratios. In such systems, the concentrator helps to collect and concentrate the solar rays at the focal point. Compared with other solar concentrating systems, the PDC system can generate a higher temperature of the fluid flowing through the receiver placed at the focal point. The receiver, acting like a heat exchanger, is placed at the focal point of the concentrator, absorbs the heat energy from the concentrated rays, and releases heat to the heat transfer fluid. The exchange of heat energy to the heat transfer fluid to the receiver determines the overall efficiency of the PDC system. This is why proper optical modeling is essential for the system. Optical modeling helps to reduce experimental expenditure and time. In recent times, Sagade (2013) has done the optical analysis of a parabolic dish solar water geyser system using the SolTrace software where the distribution of heat flux on the individual coils has been shown for the conical-shaped receiver. Le Roux et al. (2014) used the ray tracing and receiver modeling method to find the optimum area ratio of a receiver to the concentrator having a rim angle of 45° with an optical error of 10 mrad and a tracking error of 1°. The rectangle cavity receiver in a small-scale solar thermal Brayton cycle is used for this study. And the optimum receiver-to-concentrator area ratio is obtained as 0.0035. Reddy et al. (2015) used the SolTrace software to study the focal image characteristics of a fuzzy focal solar dish collector. This study's results help design a suitable receiver for the dish collector system. Craig et al. (2016) also studied tubular receivers used in the Brayton cycle using combined CFD and SolTrace to evaluate absorbed solar radiation on the surface of the tubes. They explained the approach of importing complex geometries generated in CFD to the Monte Carlo raytracing method (MCRTM). Li et al. (2016) studied the optical performance of a PDC and cavity receiver system using the MCRTM. Effects of geometrical and surface properties like diameter ratio, height ratio, and absorptivity of the cavity receiver are analyzed concerning its optical performance. The results show that the optical efficiency of simulated results and results obtained from correlations complement each other. Zou et al. (2017) studied the effect of geometric parameters on the thermal performance of a cylindrical cavity receiver. The heat flux distribution on the receiver surface was shown using the Soltrace software, and the heat losses were calculated using ANSYS. Pavlovic et al. (2017) investigated the PDC system with a spiral receiver numerically and experimentally. They used the Optis-Works ray-tracing software to analyze solar ray distribution on the receiver surface. The developed thermal model is solved using the engineering equation solver (EES). In other research, Pavlovic et al. (2018) analyzed spiral and conical cavity receivers' optical, thermal, and exergetic performance. An optical tool was used to simulate the PDC, and they combined it with the thermal modeling and then validated them with the experimental results. The conical receiver's optical efficiency is 1.38% more than the spiral receivers. Soltani et al. (2019) have combined the computational fluid dynamics (CFD) and ray-tracing method for estimating the cylindrical receiver's optical and thermal modeling with helically baffled annular space. They used ANSYS for the CFD modeling and SolTrace as the primary software for ray tracing. They have shown that the thermal performance of the receiver was increased up to 65% concerning change in aperture diameter and focal length. Cherif et al. (2019) conducted a parametric study of a PDC system using the SolTrace software to find the best configuration for achieving optimal performance. Craig et al. (2020) used the Sol-Trace software to analyze the total heat flux distribution on a tubular cavity receiver and also found out the amount of heat absorbed by the receiver walls at various inclination angles. It is observed that the optical efficiency of the receiver was around 70%. Sasidharan and Dutta (2021) have also characterized flux distribution as a focal point for a shuffler-type concentrator. They showed that the average flux density generated from the experimental setup is matched with the flux value spatially resolved from numerical analysis. Leiva Butti et al. (2021) modeled a solar biomass gasifier using MCRTM, where the heat flux distribution on the cylindrical cavity receiver was evaluated using the SolTrace software. The effect of variables like solar absorptance, the receiver's reflection type, and tracking error on the flux distribution is explained.
Few of previous research works have also focused on the variation of optical efficiency concerning the receiver geometry. Using an integral relationship to evaluate energy distribution on the receiver, Sharma et al. (1983) computed solar intensity distribution on both flat-and cylindricalshaped receivers. Johnston (1995) conducted flux mapping of a 400-m 2 parabolic dish concentrator to characterize the flux distribution at the focal point. Furthermore, the author compared the results with the ray-tracing outcomes for a dish with a surface error of 6 mrad. The geometric optics method, which includes accounting for surface errors, point offset errors, and finite sun shape, was utilized by Jones and Wang (1995) to estimate the flux distribution on the cylindrical receiver of a point-focusing parabolic dish collector (PDC). Johnston (1998) did an experimental and theoretical analysis of a 20-m 2 PDC to characterize the focal image of the system. Experimentally measured flux distribution is compared with the fluxes generated by the ray-tracing algorithm for different slope errors. Kaushika and Reddy (2000) did the inadequacy of a conventional receiver in a fuzzy focal dish was concluded after investigating the thermal performance and optimization of cavity receivers for a low-cost solar parabolic dish. Sendhil Kumar and Reddy (2008) carried out a numerical investigation to assess the impact of receiver types, angle of inclination, and area ratio on the performance of PDC with the cavity receiver, semi-cavity receiver, and modified cavity. Using MCRT, Wang et al. (2013) analyzed the optics of a cylindrical cavity featuring a convex bottom surface and explored how dead space affects its optical efficiency. Daabo et al. (2016a, b) compared the receiver's thermal and optical behavior of three different geometries. Optic-Works software was used for ray-tracing analysis, and CFD was used for thermal modeling. The research was done based on two optical parameters, like the shape of the receiver and absorption ratio, which affected the focal point region of the concentrator. Daabo et al. (2017), in other work, used similarly shaped receiver geometries used in previous work and analyzed optical efficiency using ray tracing and CFD. In this work, the optical parameters considered were the pitch of the tube coil used in the receiver and the tube diameter. Zou et al. (2017) used MCRTM to solve heat flux distribution and absorptance of a cylindrical cavity receiver. The analysis has been done on three critical properties of the cavity receiver: aperture diameter, focal length, and number of coil loops in the receiver. Si-Quan et al. (2019) analyzed a spherical cavity receiver using MCRTM for the optical performance of the receiver. Reflected ray losses and optical efficiency concerning the focal length region have been analyzed with the ray-tracing analysis. Xiao et al. (2019) had done optical efficiency of a conical receiver using the TracePro software. Effects of geometrical parameters like the cone angle of the receiver, the number of loops in the spiral tube, and the focal point of the concentrator were studied. Zhang et al. (2020) optimized a conical receiver's performance using optical and thermal modeling. The optical analysis is done using the TracePro software and then coupled with ANSYS for CFD analysis. Parameters like the receiver's cone angle, insulation thickness, and the number of loops influencing optical efficacy are analyzed in this work. From the data, it is observed that increased cone angle number of loops of the receiver, the optical efficiency is decreased by around 1%. Madadi Avargani et al. (2020) did the thermal analysis of a cylindrical cavity receiver using CFD and ray-tracing methods. The influence of optical parameters like slope error on the heat flux distribution on the receiver surface is explained in this work. The increase in the slope error of the concentrator from 10-to 35-mrad heat flux distribution is reduced by 60%. In a recent study, Cisneros-Cárdenas et al. (2021) utilized experimental and ray-tracing methods to examine the radiation flux distribution in a PDC. The authors employed the MCRT technique in the Tonatiuh software for ray tracing and estimated that the maximum and minimum flux values were 10 MW/m 2 and 4.5 MW/m 2 , respectively. Rajan and Reddy (2022) investigated the optical performance of a corrugation cavity receiver used for 100-m 2 PDC. They used the ASAP software to study the heat flux distribution and internal reflection of the rays at different optical parameters like focal length, aperture diameter, absorptivity, and the tapered angle of the corrugation cavity receiver. The maximum optical efficiency of the receiver is observed as 82.93% at a specific receiver position.
The available literature shows how the receiver's geometrical and optical parameters determine the receiver's efficiency in the PDC system. Most of these studies are focused on the parameters like shape, height, and absorptance of receivers. However, few papers focus on hemispherical-cylindrical receivers, and the available information is insufficient. Apart from this, the slope error of the concentrators, an important parameter affecting the heat flux uniformity, has not been discussed much in previous literature.
In the present work, the focus is given to studying the effect of parameters like aperture diameter (D a ), receiver distance from the concentrator (H), surface absorptivity (α), and slope error (θ s ) on the optical efficiency of a cavity receiver having cylindrical-hemispherical-type shape. In this type of receiver, the upper part is hemispherical, and the lower portion is cylindrical. Since the present study focuses on the optical efficiency of the receiver, the work primarily highlights the helical coil tube without considering insulation on its outer surface. The optical efficiencies of such cylindrical-hemispherical receivers are then compared with that of conventional cylindrical receivers having similar dimensions. In addition, the slope error, an essential characteristic of the concentrator surface and less discussed in the previous research has also been considered to determine its effect on the optical efficiency.

Methodology
In this study, the solar thermal system consists of a parabolic dish concentrator (PDC) with a cavity receiver of cylindrical-hemispherical geometry, as shown in Fig. 1. Such a cylindrical-hemispherical type receiver has a cylindrical body with a hemispherical top. This type of cavity receiver is unique in shape compared to other receiver geometries described in published literature.
The aperture diameter of the concentrator is taken as 3 m, and the rim angle is considered as 45° to achieve maximum efficiency (Daabo et al. 2016a). The direct normal irradiation (DNI) is taken as 1000 W/m 2 . The receiver's height is taken as 0.152 m, and the receiver aperture diameter varies from 0.125 to 0.162 m, as shown in Fig. 2. The receiver has 12 turns of coils with inner and outer coil diameters of 10 mm and 11 mm, respectively. Since the focal length of the present system is 1.8 m, the receiver distance from the concentrator has been maintained from 1.7 to 1.95 m.
The surface reflectivity of the concentrator is set at 96%, and three different slope errors, 1.7453, 17.453, and 34.907 mrad (Madadi Avargani et al. 2020) for the concentrator have been considered in the optical analysis. Other optical parameters, like the surface absorptivity of the receiver, have been considered ranging from 75 to 95%. For this work, the concentrator errors like specular, sun shape, and tracking error have not been considered as per Daabo et al. 2016a, b(b). The design parameters of the present solar thermal system are given in Table 1.

Mathematical formulation
This section describes the mathematical model's basics and the PDC system's optical efficiency coupled with a cylindrical-hemispherical receiver. The parabolic dish concentrates the solar irradiation at a point where the receiver is placed to collect that radiation. The distance between the concentrator base and the focal point, known as focal length, is an important parameter to determine the overall system's efficiency (Kumar et al. 2022). Concerning the aperture diameter of the concentrator (D c ) and the rim angle (φ), the focal length (f) of the concentrator can be presented as per Eq. 1.
The receiver, the system's core, plays a significant role in determining the overall efficiency. Equation 2 (Kumar et al. 2022) gives the relation of receiver aperture diameter (Da) with acceptance angle (θ), focal length (f), and rim angle (φ), as shown below.
Again, the amount of heat absorbed by the fluid while flowing through the annular space of the receiver concerning the total solar radiation heat that concentrates on the receiver surface defines the efficiency of the receiver. This energy conversion helps to find out the optical efficiency ( optical ) of the receiver calculated (Rajan and Reddy 2022) using Eq. 3 as stated below.
where Q absorber and Q total are the total solar irradiation absorbed by the receiver surface and solar energy input from the concentrator. The total heat flux absorbed by the receiver using direct and indirect solar radiation concentrating on its surface is represented by Eq. 4 (Rajan and Reddy 2022).
where Q d and Q ref are the solar irradiation absorbed by the receiver directly and indirectly. Since the receiver surface is not a perfect absorber, there are other characteristics, like the reflection phenomenon on the surface. As a result, the reflected radiation is measured on the surface for single and multiple reflections using Eq. 5 (Rajan and Reddy 2022), as given below.
where Q 1.ref and Q n,ref are the reflections of the indirect radiation for the first and multiple times on the receiver surface.

Optical analysis
In this work, the SolTrace ray-tracing software, for its high accuracy and low computational cost, has been used for the optical analysis of the receiver. This SolTrace software uses the MCRT method to perform the analysis shown in Fig. 3 (Craig et al. 2020).
This advantageous method involves tracing the vectors through the space, where it calculates the ray direction until it hits the surface and is absorbed by it (Craig et al. 2016). Pillbox sun-shape distribution has been considered in this study for distributing and analyzing solar irradiation. Since the construction of complex geometries like circular shape concentrators using the SolTrace software is not easy; MAT-LAB or Python code is used to convert such complex geometries into a finite number of elements (Craig et al. 2016).  The process flow chart for modeling optical efficiency is shown in Fig. 4. The CAD model of the receiver is exported to ANSYS to generate a mesh file, and the generated data is reinterpreted using MATLAB before exporting it into the SolTrace software which works on MCRTM. The surface properties of the receiver and the concentrator are considered per the model's requirement. While carrying out the optical modeling, it is essential to determine the number of rays interacting with the receiver surface. Figure 5a shows the ray sensitivity analysis of the receiver geometry. It is observed that divergence is less than 0.5% for the absorbed heat flux by using 0.6 million rays on the further increase in the ray count.
So, the number of rays considered for the present analysis is 0.6 million. In Fig. 5b it is shown the absorbed heat flux with varying ray count. During the study, the data file of ray interactions is generated using the SolTrace software which is further converted into a heat flux data file of individual rays using MATLAB script. The heat flux ray data is then exported to ANSYS fluent, giving the total heat flux absorbed on the receiver surface.

Validation of the present optical model
The results obtained by the present optical simulation, which needs to be validated, have been compared with the data available in the literature (Johnston 1998). To do this, the geometry of the present model, having a 3-m concentrator diameter, was modified to suit the dimensions of the experimental prototype. The modified diameter of the concentrator was 5 m keeping the constant receiver distance at 1.8 m. Moreover, the receiver of the present model was also changed to a circular copper plate having a 0.5-m diameter to match the dimensions of the experimental setup, as mentioned in Johnston 1998. The modified geometry of the present model thus becomes similar to Johnston's experimental model, except that the current study's concentrator is considered a single reflector, unlike facetted mirrors used in the experimental model. The value of DNI is taken as 1000 W/m 2 in both cases. The simulated results from the present work have been compared with the experimental data (Johnston 1998), as shown in Fig. 6a.
This figure also shows the comparison of present simulation results with results from published literature (Rajan and Reddy 2022) which are found to be good in agreement. Figure 6b shows that the solar heat flux is higher at the center and gradually decreases toward the periphery. The values of total heat flux for the experiment, literature, and the present study are 14.8 kW, 14.781 kW, and 14.768 kW, respectively. The numerical model is validated using Eqs. (6-8) for absolute error, percentage error, and coefficient of determination (Habchi et al. 2021(Habchi et al. , 2023. where y a and y b are the deviations from the base value and the value of the base case, respectively, and m is the number of numerical data. The data in Table 2 indicates that this study's absolute and percentage errors are reasonably low compared to the literature results. Additionally, the maximum coefficient of determination (R 2 ) is close to 0.999995, consistent with the literature results. Based on these observations, it is evident that the current study's validation has good agreement with the literature results. The minor deviations may be due to errors in the experimental setup like tracking error, slope error, and limb darkening effect error. Wind conditions and environmental effects could be the other reasons for such deviations. Therefore, it may be concluded that the present model results are validated with experimental data.

Results and discussion
The present work has been carried out to analyze the effects of receiver aperture diameter, slope error, and surface absorptivity on the optical efficiency of the receiver concerning receiver distance from the concentrator. The details of the study are discussed below.

Influence of receiver aperture diameter
The proper mounting and appropriate position of the receiver enhance its optical efficiency. As the distance from the concentrator increases, the heat flux intensity increases up to a certain distance and gradually decreases. Figure 7 shows the heat flux distributions on the receiver with varying receiver distances from the concentrator ranging from 1.7 to 1.95 m. In Fig. 8, the variations of optical efficiency have been highlighted with varying receiver distances from 1.7 to 1.95 m when the receiver aperture diameters are 0.162, 0.150, 0.138, and 0.125 m. The figure shows the maximum optical efficiency for a 0.150-m receiver diameter at a receiver distance of 1.85. It is seen that the larger receiver diameter captures more solar irradiation with high heat loss from the receiver. Similarly, a decreased aperture diameter also results in less absorption of solar irradiation with significant heat loss. Maximum optical efficacy is 82.1% for the receiver with an aperture diameter of 0.150 m compared with all the other cases.

Influence of slope error of concentrator surface
One of the critical parameters affecting the optical efficiency is the slope error of the concentrator surface. The concentrator with an irregular surface causes non-uniformity of the rays on the receiver surface. Rays are escaped from the receiver to outer space, resulting in non-uniformity in the heat flux distribution. Surfaces with a slope error of 0 mrad are also ideal surfaces where the solar rays hit the concentrator surface and reflect perfectly to a point on the receiver surface. Similarly, surfaces with slope errors, called real surfaces, irregularly reflect the solar rays toward the receiver surface. Heat flux distributions of an ideal surface along with a real surface with slope errors 1. 7453, 17.453, and 34.907 mrad are analyzed; these slope errors are used by Madadi Avargani et al. (2020) for optical analysis of helically baffled cavity receiver and shown in Fig. 9 for a given receiver distance of 1.7 m. Figure 10a-d shows how the optical efficiencies vary with different receiver distances for slope errors of 0, 1. 7453, 17.453, and 34.907 mrad and receiver aperture diameters of 0.125, 0.138, 0.150, and 0.162 m. It is observed from Fig. 10c that the concentrator with an ideal surface gives the highest optical efficiency of 82.1% at receiver distance H = 1.85 m for the receiver aperture diameter of 0.150 m. On the other hand, from Fig. 10d, it is evident that for a real surface with a slope error of 1.7453 mrad, the receiver's highest optical efficiency is 63.94% at the height of 1.8 m for a receiver aperture diameter of 0.162 m. It is also observed that, at this point, the optical efficiency of a real surface is 2% higher than the ideal surface because of increased ray interactions. It is further noticed that there is a significant drop in the optical efficiencies of the receiver with an increase in slope errors. Figure 10d shows that, for 1.8-m receiver distance, the peak and the lowest optical efficiencies are found to be 63.94% and 4.7% for real surfaces with slope

Influence of absorptivity of the receiver surface
The phenomena of absorptivity of a cavity receiver are assessed by the amount of received, reflected, and absorbed rays by its surface. With the help of simulation, the distribution pattern of heat flux on the receiver surface can not only be studied, but it becomes easier to find out the high and dead intensity areas of heat flux on the receiver. The high and dead intensity areas should be eliminated from the receiver to improve optical efficiency. Due to the high heat flux concentration, the tube material used in the receiver sometimes gets damaged, which finally affects the overall performance. Consequently, it becomes necessary to find out the optimal position of the receiver to avoid such phenomena. The optical efficiencies of the receiver have been evaluated for different receiver distances (H) with receiver diameters (D a ) varying from 0.125 to 0.162 m and also with absorptivity (α) ranging from 75%, 85%, and 95%. From Fig. 11a-c, it is observed Fig. 10 a-d Optical efficiency vs receiver distance for different slope errors of concentrator and for different receiver aperture diameters that the nature of simulated results is all similar, and the maximum optical efficiency in each case is found to be at H = 1.85 m for 0.150-m receiver diameter. The simulated optical efficiencies are given in Table 3.

Efficiency comparison of the present receiver with a conventional cylindrical receiver
In the present study, the geometrical parameters of a cylindrical-hemispherical-type receiver have been evaluated to find its optimized design parameters. These parameters are compared with a conventional cylindrical receiver with similar overall dimensions. The optical efficiencies of these two types of receivers having 0.150-m diameters have been simulated, compared, and represented in Fig. 12 for different receiver distances varying from 1.7 to 1.95 m. The simulation results show that the efficiencies of the present cylindrical-hemispherical receiver and Fig. 11 a-c Optical efficiency vs receiver distance for varying absorptivity of the receiver at different receiver aperture diameters This enhancement in the efficiency of a cylindricalhemispherical receiver could be due to a more significant number of internal reflections of the rays with higher absorption of the heat flux. In contrast, in the case of a cylindrical receiver, the internal reflections of the rays are less because of its open/hollow space at the top, leading to a loss of optical efficiency. Therefore, it may be concluded that the cylindrical-hemispherical receiver could be a better alternative than the conventional one. Table 4, however, summarizes the optical efficiencies of other types of receivers found in available literature along with the present receiver.

Conclusions
The study performed optical modeling of a cylindrical-hemispherical receiver used in solar thermal systems using the SolTrace software. The MCRT method was used to evaluate heat flux distribution on the receiver surface. Three parameters, aperture diameter, surface absorptivity, and slope error of the concentrator, were considered to investigate optical efficiency at different receiver distances. Results showed a maximum optical efficiency of 82.1% at a distance of 1.85 m with an aperture diameter of 0.150 m. The slope errors of the concentrator greatly affected heat flux distribution on the receiver surface. The concentrator with a 1.7453-mrad slope error had the highest optical efficacy of 63.94% with a 0.150-m receiver aperture diameter at a distance of 1.85 m. However, an increase in slope errors from 1.7453 to 17.453 mrad resulted in a 50% decrease in optical efficiency for all receiver diameters. A uniform heat flux distribution was achieved at a distance of 1.85 m for a 0.150-m receiver diameter and 95% surface absorptivity. Results were compared and validated with experimental data, showing a 0.216% deviation in the percentage error. The performance of the cylindricalhemispherical receiver was also compared to a conventional cylindrical receiver, resulting in an 8% increase in efficiency. These findings may be helpful in further evaluation of the cylindrical-hemispherical receiver.

Acknowledgements
The authors thankfully acknowledge Director, CSIR-CMERI, for his kind support and encouragement. CSIR-HRDG, New Delhi, may also be acknowledged for granting the CSIR-SRF (2021) fellowship.
Author contribution Kolli Harish Kumar: conceptualization and writing-original draft preparation.
Desireddy Shashidhar Reddy: writing-review and editing. Malay Karmakar: writing-review and editing. K H K has analyzed the data regarding optical efficiency of the cylindrical-hemispherical receiver which is the major contribution in writing the manuscript. D S R and M K have reviewed and done the constructive editing for the manuscript. All authors have read and approved the final manuscript.

Data availability
The data supporting this study's findings are available in the supplementary material.

Declarations
Ethics approval Not applicable.