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Applied Sciences
, Volume 12 (20) – Oct 11, 2022

/lp/multidisciplinary-digital-publishing-institute/an-integration-model-for-flux-density-distribution-formed-by-a-z6O3TdfRk3

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applied sciences Article An Integration Model for Flux Density Distribution Formed by a Heliostat 1 2 2 3 3, Chenggang Zong , Yemao Shi , Liang Yu , Bowen Liu and Weidong Huang * Asset Management Co., University of Science and Technology of China, 96 Jinzhai Road, Hefei 230026, China School of Earth and Space Science, University of Science and Technology of China, 96 Jinzhai Road, Hefei 230026, China Department of Environmental Science and Engineering, University of Science and Technology of China, 96 Jinzhai Road, Hefei 230026, China * Correspondence: huangwd@ustc.edu.cn; Tel.: +86-551-63606631; Fax: +86-551-63607386 Abstract: An accurate ﬂux density calculation is essential for optimizing and designing solar tower systems. Most of the existing methods introduce multiple assumptions, and the accuracy and scope of the application are limited. This paper proposes an integration model used to calculate the ﬂux density distribution after only applying the Gaussian model for solar brightness distribution. It is the ﬁrst time that multiple reﬂections and the inﬂuence of the optical error transferred from different planes of the glass mirror are considered in order to build an optical model for the ﬂux density of a heliostat. The reﬂection from two surfaces of the glass mirror used to form three main parts of beams was considered in the present model, and Fresnel’s equations were applied to calculate the energy of the three parts of reﬂected rays. An elliptic Gaussian model was applied for the optical error distribution of the heliostat. The model error was evaluated using the experimental data of ten heliostats, and the applicability and accuracy of the model were veriﬁed through ﬂux distribution and an intercept factor. The average relative prediction error of the present model from the experimental data was only 2.83%, which is less than SolTrace and other models. Citation: Zong, C.; Shi, Y.; Yu, L.; Liu, B.; Huang, W. An Integration Model Keywords: Gaussian distribution; direct integration method; Fresnel’s equations; solar ﬂux density; for Flux Density Distribution Formed optical error by a Heliostat. Appl. Sci. 2022, 12, 10191. https://doi.org/10.3390/ app122010191 1. Introduction Academic Editor: Antonio Di Bartolomeo The accurate prediction of a solar ﬂux density distribution formed by a heliostat has attracted widespread attention. It can avoid an excessive solar energy density and calculate Received: 29 July 2022 the receiver ’s intercept factor and optical efﬁciency to the focusing solar spot, which can be Accepted: 28 September 2022 used to design and optimize the heliostat ﬁeld [1,2]. Published: 11 October 2022 The simplest calculation method uses functions to describe the ﬂux density distribution Publisher’s Note: MDPI stays neutral on the receiver plane directly. The UNIZAR function proposed by Collado et al. [3] and the with regard to jurisdictional claims in HFCAL function proposed by Michael Kiera [4] is simple and fast in its calculation. Still, published maps and institutional afﬁl- there is a signiﬁcant error in calculation results compared with the experimental data. iations. Current methods for calculating the heliostat ﬂux density mainly include the convo- lution integration method and ray-tracing method [5]. The ray-tracing method is based on the law of light reﬂection used to track the path of incident rays in the system. The Gaussian model is mainly used to describe the optical error of a heliostat in order to bring Copyright: © 2022 by the authors. errors. However, it still requires extensive calculation and is not suitable for optimization. Licensee MDPI, Basel, Switzerland. The convolution integration method is an indirect integration method that regards the This article is an open access article ﬂux density on the image plane as the convolution of the solar brightness distribution and distributed under the terms and image function of the heliostat. It then projects the ﬂux density distribution on the image conditions of the Creative Commons plane onto the receiver plane. The convolution integration is the approximation of the Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ actual physical image. At the same time, the projection process also introduces more errors 4.0/). as it assumes that the solar ray is entirely from the center of the heliostat. Although the Appl. Sci. 2022, 12, 10191. https://doi.org/10.3390/app122010191 https://www.mdpi.com/journal/applsci Appl. Sci. 2022, 12, 10191 2 of 25 calculation speed is faster than the ray-tracing method, it is sometimes not accurate enough to calculate the ﬂux density distribution of a single heliostat. However, it is more accurate than assuming that all incident radiation has been intercepted by the receiver [6]. Lipps [7] comprehensively considered the solar brightness distribution and the elliptic Gaussian optical error distribution and solved the ﬂux density distribution by using the convolution method. The HELIOS [8] system uses two-dimensional Fourier transform to calculate the error of the reﬂected ray generated by the ellipse Gaussian distribution of the surface error in the face of different solar intensity distribution functions. The optical error distribution function of an ellipse Gaussian has been proven to be more consistent with the actual data and applicable to a wider range of situations to a certain extent. However, the above methods do not consider the transmission of the optical error from the heliostat to reﬂected light, which is different from the actual situation. Previous studies have shown that even if the optical error is the same at different parts of the heliostat, the optical error transferred to the reﬂected ray is different [9], resulting in a difference from the actual situation. Before, we applied the convolution method to deduce an equation used to simulate the solar ﬂux of a heliostat [10]. Its calculation speed is fast, and its calculation data are relatively accurate. However, it introduces the Gaussian function for a solar brightness distribution and an image of the heliostat and convolution methods, and the scope of the application is limited. The convolution method assumes that the image of the heliostat is similar to its shape. Suppose that the ratio of the heliostat size to the distance from the receiver or the incident angle is too large. In this case, it will bring many errors and the prediction result of the solar ﬂux will be inaccurate. Moreover, the distance between the heliostat and receiver plane is also required to be large enough. Furthermore, the projection method must be applied to bring many more errors. The majority of the heliostats in the current tower solar system apply the mirror with a reﬂecting surface on the back of the glass. Experiments show that the reﬂected rays are mainly divided into three parts: the rays directly reﬂected from the glass surface, the main rays reﬂected from the metal reﬂecting surface [11], and part of the main rays reﬂected off the upper glass surface and then reﬂected back off the metal reﬂecting surface. In current studies of solar tower systems, they are all treated as the same reﬂected beam, which is different from the actual situation. In this paper, under the only assumption of an approximate description of the sun shape, a direct integration method was applied to deduce an integration equation to compute the solar ﬂux reﬂected from the heliostat at any point on the receiver plane, considering the inﬂuence of the optical error of the heliostat and multiple reﬂections by the two planes. An elliptic Gaussian model was used to describe the optical error of the heliostat. A numerical code was developed to calculate a rectangular spherical heliostat’s ﬂux density distribution and intercept factor based on this model. Compared with the experimental data of ten heliostats, the applicability and accuracy of the model were veriﬁed. 2. Methodology 2.1. The Energy Density Ratios of the Reﬂected Ray Solar rays from various positions will reach different positions on the receiver surface after being reﬂected by a certain point of the heliostat. Optical errors at different points of the heliostat will cause the reﬂected ray to deviate from the ideal direction, as shown in Figure 1. We calculated the ﬂux density contribution of the reﬂected solar ray to a certain point on the receiver surface by integrating all of the reﬂection points on the heliostat [9]. Therefore, the distribution of the ﬂux density on the whole receiver surface could be computed. The rays reﬂected by the heliostat were divided into three main parts [11]: I (4%), I (89.5%), and I (3.5%), as shown in Figure 2. The incident rays ﬁrst reach the upper 2 3 Appl. Sci. 2022, 12, 10191 3 of 25 surface of the glass, and part I is directly reﬂected, accounting for approximately 5%. The rest of the rays are transmitted through glass, reﬂected by the metal reﬂecting surface, and transmitted out of the upper surface. These are the main reﬂected rays, namely part I , accounting for approximately 91.5% when the incidence angle is small. The remaining part is reﬂected again from the upper surface, and again from the reﬂective metal surface. When it reaches the upper surface, it transmits out part I , accounting for approximately 3%. The rest of the rays still travel inside the glass, but the fraction is negligible. Figure 1. Slope error s and reﬂection cone q. slo pe Figure 2. Composition of reﬂected rays. The focuses of the three parts of the beams are different, but the part I mainly determines the distribution of the ﬂux density on the receiver plane, so, in practice, the rays of part I are aligned with the center of the receiver plane. When the distance between the receiver plane and the heliostat is large enough, the difference in the focal positions of the three parts can be ignored. Meanwhile, the contribution of part I and part I to the ﬂux density is not tiny. When a more accurate ﬂux density distribution is required, the optical errors of these two parts should be taken into account, and the ﬂux density distribution is different from that of part I . In particular, part I and I are not taken into account 2 1 3 when calculating the intercept factor, and the curvature of the intercept curve will differ signiﬁcantly from the actual situation, especially when the intercept factor reaches 90%. The proportion of I , I , and I is related to the incident angle. There are also polar- 1 2 3 ization changes in reﬂection and refraction on the glass surface. We ﬁrst decomposed the amplitude vector of the ray into two directions. The one perpendicular to the vibration of the incident plane is called the S component, and the one parallel to the vibration of the incident plane is called the P component. They have identical parts in solar rays. Appl. Sci. 2022, 12, 10191 4 of 25 The incident angle is i and the ratios of the refractive index of the two sides of the reﬂected glass interface are n and n , respectively. According to the refraction law, the 1 2 refraction angle inside the glass can be calculated as i . According to Fresnel’s equations [12], the reﬂectivity of the reﬂected ray in the P and S components can be calculated: tan (i i ) 2 1 R = jr j = (1) p p tan (i + i ) 1 2 sin (i i ) 1 2 R = jr j = (2) s s sin (i + i ) 1 2 Similarly, the transmissivity of the refracted ray in the P and S components can be calculated: n cosi sini cosi 2sini cosi 2 2 1 2 2 1 T = jt j = ( ) (3) p p n cosi sini cosi sin(i + i )cos(i i ) 1 1 2 1 1 2 1 2 n cosi sini cosi 2sini cosi 2 2 1 2 2 1 T = jt j = ( ) (4) s s n cosi sini cosi sin(i + i ) 1 1 2 1 1 2 As light passes through the metal reﬂecting surface, namely the silver plating layer, it is refracted and reﬂected as it reacts with the silver. Part of the refraction into the silver plating layer is equivalent to loss: 2 2 2 (n cosi ) + n k s s silver R = (5) 2 2 2 (n + cosi ) + n k s s 1 2 2 2 (n ) + n k cosi silver s R = (6) 2 2 2 (n + ) + n k cosi silver silver R and R are the reﬂectivity of light on the silver surface in the P and S s p components. i is the incident angle reaching the silver surface. n is the ratio of the 3 3 refractive index of the silver plating layer relative to the glass, and, here, it was equal to 0.051585. k is the dielectric constant, and, here, it was equal to 3.9046. The wavelength of the incident ray was 587.6 nm [13]. As light passes through the glass, some of it is absorbed by the glass. Heliostats usually use ultra-clear glass. According to the experimental results, approximately 1% [14] of light is absorbed when it passes through the glass perpendicularly, mainly absorbed by impurities in the mirror. The incident ray is natural light, and the light intensity is I , which has the same amplitude in the P and S components. Thus, the light intensity in the P and S components is equal. By solving the energy density of the reﬂected and refracted ray at the interface bound- ary each time, the energy density of the three parts of rays reﬂected off the glass surface can be obtained. Finally, the proportion of three parts of the reﬂected ray can be calculated. I = I (T R ) (7) 1 0 p s 0 silver 0 silver 2 I = I (T T R + T T R ) (1 a) (8) 2 0 p s p p s p 0 0 silver 2 0 0 silver 2 4 I = I (T R T (R ) + T R T (R ) ) (1 a) (9) 3 0 p s p p p s s p where R and R are the reﬂectivity of the ﬁrst reﬂections in the P and S components, p s 0 0 respectively. R and R are the reﬂectivity of the second reﬂections in the P and S compo- p s nents, respectively. T and T are the transmissivities of the transmission into the glass in p s Appl. Sci. 2022, 12, 10191 5 of 25 0 0 the P and S components, respectively. T and T are the transmissivities of the transmission p s out of the glass in the P and S components, respectively. a is the absorption ratio of glass to light, which is calculated by: a = 1 ex p(# ) (10) cosi where d is the thickness of the glass, and # is the absorption coefﬁcient, which should be determined from the test data [14] for the speciﬁc mirror. Finally, the energy density ratios of the three parts can be calculated: d = (i = 1, 2, 3) (11) I + I + I 1 2 3 2.2. Solution of Flux Density on Receiver Plane Although the energy density distribution formed by the three parts of the reﬂected ray is different, the calculation process of the ﬂux density of each part is similar. It only needs to calculate the ﬂux density on the receiver plane of a single part and then multiply it by the energy density ratios to obtain the ﬂux density of the three parts. Finally, we can calculate the ﬂux density distribution on the whole receiver plane. F = d F (i = 1, 2, 3) (12) total å i i i=1 where F is the ﬂux density of a single part. For a solar tower system, it is assumed that a receiver plane is a square plane and the heliostat is a rectangular spherical mirror, as shown in Figure 3. Suppose that the heliostat reﬂector is S , the receiver plane is S , n and n are normals of the cell surfaces on the 1 2 1 2 heliostat and the receiver, O and O are the points on the cell surfaces on the heliostat and 1 2 the receiver, and b and b are the included angles between the line O O and normals of 1 2 1 2 each cell surface. For a single part, the analytic formula of the ﬂux density at point O in the receiver surface is [15]: ZZ I Bcosb cosb 1 2 F = dS (13) where r = |O O | is the distance between the reﬂection point and the receiver point, I is 1 2 the total intensity of the reﬂected ray of a single part, and B is the intensity distribution of reﬂected rays, obtained from the convolution calculation of the sun shape and optical error. To calculate the intensity distribution of the reﬂected ray, we need to determine the solar intensity distribution ﬁrstly, which is described by a circular Gaussian function. An approximate description of the intensity of sunlight is the only assumption in this paper. Due to many factors, such as sun position and atmospheric absorption, the solar intensity changes from time to time. However, the standard deviation of the solar intensity distribution s can be determined by direct solar irradiation I [8]: sun D 2 2 [3.7648 3.8413( I 1) + 1.5923 10 ( I 1) ] D D s = p (14) sun The optical error of the reﬂected ray must then be determined. This paper adopted the elliptic Gaussian function to describe the slope error on a heliostat. It is necessary to consider that rays will be reﬂected by the upper and lower surfaces of the mirror. The equations for calculating optical errors are also different for each part of the reﬂected ray. The optical error of each point on the heliostat is different [9,16]. To improve the calculation speed, we can integrate the whole heliostat and replace the optical error of each point with the average optical error. To compare the two calculation results, we calculated the difference with ten heliostats in Section 3.2. From the results, the average difference Appl. Sci. 2022, 12, 10191 6 of 25 between the two is less than 0.05%. Therefore, in the subsequent calculation of the optical error, the optical error of each point was no longer considered separately but was only replaced by the average slope error of the surface. Figure 3. X axis points due east, Y axis points due north, and Z axis points zenith. For the ﬁrst part of the reﬂected ray I , the upper surface of the glass directly reﬂects it, so the average optical errors were calculated as follows: 2tan lcosbsinb 2 2 2 s = 4s + ( s ) (15) sx slo pey slo pex 2 2 1 + tan lcos b 2tan lcosbsinb 2 2 2 s = 4s + ( s ) (16) sy slo pex slo pex 2 2 1 + tan lcos b where s and s are slope errors in the X direction and Y direction, respectively. slo pex slo pey For the second part of the reﬂected ray I , it will be refracted by the lower surface and reﬂected by the upper surface. Given mirrors’ relatively mature manufacturing technology, the slope errors on the upper and lower surfaces can be considered equal. Appl. Sci. 2022, 12, 10191 7 of 25 Therefore, the standard deviation of the average error in two directions was calculated as follows: 2 2 2n sin l cosl 2 2 2 0 2 2 q q s = (1 ) [s + ( 1)s ] + 4s + sx slo pex slo pey slo pex 2 2 2 2 n sin l 2n n sin l 0 0 (17) 2 2 2 2 n sin l 2 4n 2sin l (1 cosl) 0 2 2 2 2 ( q 1)s + (1 ) (s + s ) slo pey slo pex slo pey cosl 2cosl 2 2 n n sin l 2 2 cosl 2n sin l 2 2 2 0 2 2 q q s = (1 ) [s + ( 1)s ] + 4s + sy slo pey slo pex slo pey 2 2 2 2 n sin l 2n n sin l 0 0 (18) 2 2 2 2 n sin l 2 4n 2sin l (1 cosl) 0 2 2 2 2 ( q 1)s + (1 ) (s + s ) slo pex slo pey slo pex cosl 2cosl n n sin l where n is the refractive index ratio of glass to air, constant at 1.51 in general, and l is the incident angle of the heliostat reﬂection point, respectively. For the third part of the ray I , the ray passes through the glass again, and its optical error is two more reﬂection errors than that of the ray I : 2 2 cosl 2n sin l 2 2 2 0 2 2 q q s = (1 ) [s + ( 1)s ] + 12s + sx slo pex slo pey slo pex 2 2 2 2 n sin l 2n n sin l 0 0 (19) 2 2 2 2 n sin l 2 4n 2sin l (1 cosl) 0 2 2 2 2 ( q 1)s + (1 ) (s + s ) slo pey slo pex slo pey cosl 2cosl n n sin l 2 2 cosl 2n sin l 2 2 2 0 2 2 q q s = (1 ) [s + ( 1)s ] + 12s + sy slo pey slo pex slo pey 2 2 2 2 n sin l 2n n sin l 0 0 (20) 2 2 2 2 n sin l 2 4n 2sin l (1 cosl) 0 2 2 2 2 ( q 1)s + (1 ) (s + s ) slo pex slo pey slo pex cosl 2cosl n n sin l The total error of the reﬂected ray at each point of the heliostat was calculated by convolution of three Gaussian function errors as follows: 2 2 2 2 s = s + s + s (21) x sun sx trx 2 2 2 2 s = s + s + s (22) y sun sy try where s and s are the tracking errors in the X direction and Y direction. The tracking trx try errors of all models used in this paper were 0. In the elliptic Gaussian model, the distribution of reﬂected light intensity can be calculated as: 1 1 q y B(q , q ) = exp[ ( + )] (23) x y 2 2 2ps s 2 s s x y x y where q is the angle between O O in the X direction and the reﬂected ray from the center x 1 2 of each cell on the heliostat, and q is the angle between O O in the Y direction and the 1 2 reﬂected ray from the center of each cell on the heliostat. s and s are the total errors in x y the X direction and Y direction, respectively. In this paper, based on the elliptic Gaussian model, we simpliﬁed the calculation process of an optical error and built a circular Gaussian model, which was applied to verify the model’s accuracy by comparing it with SolTrace [17]. It can also be applied to quickly Appl. Sci. 2022, 12, 10191 8 of 25 compute a ﬂux density distribution when the accuracy requirement is not high. In the circular Gaussian model, the reﬂected ray was no longer divided into three parts and was only calculated as the same reﬂected ray. The slope error s was simpliﬁed to radial, slo pe and the tracking error s was taken into account, so the total error was calculated as [18]: tr 2 2 2 2 s = s + s + s (24) sun slo pe tr In the later model veriﬁcation, the optical error in SolTrace will be consistent with that in the circular Gaussian model and calculated by the least square method. Therefore, the intensity distribution function of the reﬂected ray in the circular Gaus- sian model can be expressed as: 1 q B(q) = exp( ) (25) 2 2 2ps 2s q is the angle between O O and the reﬂected ray from the center of each cell on the 1 2 heliostat, and s is the total error. Therefore, for a single part of a reﬂected ray, the ﬂux density distribution function that the heliostat devotes to a speciﬁc point on the receiver plane can be transformed into: Circular Gaussian model: ZZ q I cosb cosb exp( ) 1 2 2 p 2s F = dS (0 b , b ) (26) 1 1 2 2 2 2ps r 2 Elliptic Gaussian model: q y ZZ I cosb cosb exp[ ( + )] 1 2 2 2 s s p x y F = dS (0 b , b ) (27) 1 1 2 2ps s r 2 S x y 2.3. Flux Density Computation Method When considering the intensity of the reﬂected solar ray, we ﬁrst divided the heliostat into smaller grids, and the center point of each grid was used to represent the grids. Then, we solved the ﬂux density contribution of all reﬂected rays to a certain point on the receiver plane. Finally, we integrated these contributions to compute the ﬂux density value of a speciﬁc point on the receiver plane [19]. The actual calculation process of a speciﬁc point is mainly divided into four steps: i Location determination. ii Meshing process and coordinate system rotation. iii Calculation of geometrical optics. iv Solution of the ﬂux density at a speciﬁc point. The calculation details are shown in Figure 4. Further, the ﬂux density of other grid points was calculated according to the above method. As shown in Appendix A, the actual calculation process is very complicated, so we developed a numerical code to help us complete the analysis and calculation. To verify the reliability of the proposed method, we calculated the root mean square error (RMSE) of the ﬂux density and intercept factor between the models and experimental data [19]: u v [F (u, v) F (u, v)] å å 1 2 1 1 R MSE = (28) uv 1 where F (u, v) and F (u, v) are the ﬂux density calculated by the model and measured 1 2 data, respectively. This formula can also calculate the root mean square error between two different models. Appl. Sci. 2022, 12, 10191 9 of 25 Figure 4. Solution of the ﬂux density at a speciﬁc point. 3. Model Validation 3.1. Comparison with SolTrace SolTrace [17] is a software that simulates the real optical action process based on the traditional ray-tracing method, developed by The National Renewable Energy Laboratory (NREL). In SolTrace, the ﬂux density value calculated from ﬁve million random rays is accurate enough, so we set the number of rays to ﬁve million in the subsequent calculation using SolTrace. SolTrace can only be used to calculate the solar intensity radial distribution model. Therefore, we can compare the results of SolTrace with that of the circular Gaussian model, which were used to verify the calculation results of the circular Gaussian model. The origin of the coordinate system is located in the center of the bottom of the tower. The positive direction of the Y-axis is due north, the positive direction of the X-axis is due east, and the positive direction of the Z-axis is the zenith. The tower height is 120 m, the receiver is placed perpendicular to the ground, the solar altitude angle is 45 , the solar azimuth angle is 0 , and the solar light intensity is 1 kW/m . All parameters of the heliostat are shown in Table 1. The slope error in the circular Gaussian model is 1.7 mrad, consistent with the slope error setting in SolTrace. After setting each parameter, we calculated the flux density distribution of the rectangular heliostat using the circular Gaussian model and SolTrace, respectively. Table 1. Heliostat parameters. X (m) Y (m) Z (m) f (m) Width Length cosl s (mrad) s (mrad) 0 sun tr 300 500 0 595.3 6 9 0.8022 2.51 0 Figure 5a shows the distribution of the ﬂux density on the receiver surface computed by SolTrace and the circular Gaussian model. As SolTrace can only apply the circular Gaussian model, here, we also used it in our model for comparison. Since the reﬂected Appl. Sci. 2022, 12, 10191 10 of 25 ray is at an acute large incident angle to the receiver surface, the distribution of the ﬂux density on the receiver surface is elliptical, and the average absolute difference computed by the two methods is 0.64%. Figure 5b shows the intercept factor calculated by SolTrace and the circular Gaussian model. On the whole, the intercept factor computed by SolTrace is slightly larger than that. The ﬁgures show that the results computed by the circular Gaussian model and SolTrace have a good consistency. Figure 5. (a) Flux density distribution computed by the circular Gaussian model and SolTrace (kW/m ), (b) intercepts computed by the circular Gaussian model and SolTrace. 3.2. Veriﬁcation with Experimental Data During July, Collado carried out an experiment in the tower heliostat system [20]. He recorded the experimental data, which included the measured solar ﬂux contours and intercept factors of the heliostat in ten different positions on the receiver plane. The ten heliostats measure 6.6778 m in width and 6.819 m in length. The spherical heliostat’s total mirror area is 39.9126 m . Figure 6 depicts the position distribution of heliostats. The experimental results of the ﬂux density distribution and intercept factor were compared to SolTrace, the elliptic Gaussian, and the circular Gaussian models. The slope errors of all models were calculated using the least square method. Meanwhile, the reﬂected light intensity distribution in SolTrace is consistent with a circular Gaussian model. Because the circular Gaussian model results are not signiﬁcantly different from those of SolTrace, they were only recorded as errors in the tables. Figure 6. Distribution of ten heliostats and receiver. The elliptic Gaussian model must first calculate the energy density ratios of ten heliostats’ reflected rays. Given the incident angles of the ten heliostats, and after accounting for reflection Appl. Sci. 2022, 12, 10191 11 of 25 and refraction at the glass interface and light energy loss at the silver plating layer, the energy density ratios of the reflected rays of each heliostat were calculated, as shown in Table 2. The energy density ratios of I , I , and I were approximately 5%, 91.5%, and 3.5%, respectively. 1 2 3 Table 2. Parameters of ten heliostats. Heliostats Incidence (Rad) d d d 1 2 3 1 0.8188 5.08% 91.50% 3.42% 2 0.8477 4.96% 91.65% 3.39% 3 0.8253 5.05% 91.53% 3.42% 4 0.8510 4.95% 91.67% 3.38% 5 0.8327 5.02% 91.58% 3.40% 6 0.8421 4.98% 91.63% 3.39% 7 0.8922 4.83% 91.82% 3.35% 8 0.8740 4.88% 91.76% 3.36% 9 0.9484 4.75% 91.92% 3.33% 10 0.9513 4.74% 91.93% 3.33% The slope errors of all models were computed by the least square method. Table 3 shows the optical errors of the three parts of the elliptic Gaussian model, SolTrace, and circular Gaussian models. Meanwhile, the optical errors of SolTrace and the circular Gaussian model refer to the average slope error of the mirror surface. Table 3. Optical errors of ten heliostats calculated by different models (unit: mrad). s s s s s s s SolTrace Heliostats Elliptical-I Elliptical-I Elliptical-I Elliptical-I Elliptical-I Elliptical-I and Circular 1 1 2 2 3 3 x y x y x y Optical 1 1.93 1.56 2.27 2.01 3.72 3.28 2.10 2 1.79 2.58 2.39 2.94 3.92 4.85 2.73 3 2.37 0.26 2.56 1.36 4.22 2.18 2.07 4 2.85 1.40 3.15 2.18 5.20 3.55 2.58 5 2.56 1.30 2.85 2.01 4.70 3.26 2.46 6 1.61 1.76 1.99 2.01 3.27 3.45 2.36 7 0.67 3.62 2.09 3.87 3.42 6.43 3.26 8 1.76 2.11 2.20 2.45 3.63 4.05 2.62 9 6.49 3.65 7.09 5.16 11.86 8.57 3.84 10 7.07 3.67 3.68 5.37 12.85 8.93 5.54 Figures 7–16 show the ﬂux density distribution and intercept factor of ten heliostats measured by the experiment and computed using the circular Gaussian model and SolTrace. The elliptic Gaussian model in this paper is superior to SolTrace in ﬂux density distribution and intercept factor calculation. The absolute average difference in distribution of the ﬂux density and intercept factor between the elliptic Gaussian model and experimental data are shown in Tables 4 and 5. We compared the ﬂux density and intercept factor calculated by multiple models, and the calculation results of SolTrace and the circular Gaussian model are also shown in the tables. In terms of the ﬂux density distribution, the result indicates that the average absolute difference between the elliptic Gaussian and measured data is 2.83%. The average absolute difference between the circular Gaussian and measured data is 4.40%, and the average absolute difference between SolTrace and measured data is 3.50%. Therefore, the average absolute difference between the elliptic Gaussian model and measured SolTrace data is the smallest. The calculation result of heliostat 10 is relatively the best, with a 1.91% smaller average difference than that of SolTrace. The ﬂux density distribution calculated by the elliptic Gaussian model and SolTrace was compared using Heliostat #10. The elliptic Gaussian model simulates the experimental data’s ﬂux density contour well, as shown in Figure 16a, and the long and short axes are consistent. However, the SolTrace-calculated results deviated from the measured contour Appl. Sci. 2022, 12, 10191 12 of 25 and could not adequately ﬁt the ﬂux density distribution. In terms of data, the average absolute difference in ﬂux density distribution between the elliptic Gaussian model and the experimental data is 3.38%, which is signiﬁcantly smaller than the 4.43% of SolTrace and the 5.97% of circular Gaussian. Figure 7. (a) Contours of the measured and computed ﬂux distribution (kW/m ): Heliostat #1, (b) measured and computed intercepts vs. the side length of a square receiver: Heliostat #1 . Figure 8. (a) Contours of the measured and computed ﬂux distribution (kW/m ): Heliostat #2, (b) measured and computed intercepts vs. the side length of a square receiver: Heliostat #2 . Figure 9. (a) Contours of the measured and computed ﬂux distribution (kW/m ): Heliostat #3, (b) measured and computed intercepts vs. the side length of a square receiver: Heliostat #3 . Appl. Sci. 2022, 12, 10191 13 of 25 Figure 10. (a) Contours of the measured and computed ﬂux distribution (kW/m ): Heliostat #4, (b) measured and computed intercepts vs. the side length of a square receiver: Heliostat #4 . Figure 11. (a) Contours of the measured and computed ﬂux distribution (kW/m ): Heliostat #5, (b) measured and computed intercepts vs. the side length of a square receiver: Heliostat #5 . Figure 12. (a) Contours of the measured and computed ﬂux distribution (kW/m ): Heliostat #6, (b) measured and computed intercepts vs. the side length of a square receiver: Heliostat #6 . In terms of the intercept, the elliptic Gaussian model and circular Gaussian model proposed in this paper are consistent with the intercept factor calculated by SolTrace. The result of intercept factors suggests that the absolute average difference in the elliptic Gaussian and measured data is 1.13%, the absolute average difference in the circular Appl. Sci. 2022, 12, 10191 14 of 25 Gaussian and measured data is 2.28%, and the absolute average difference in SolTrace and measured data is 2.36%. In most cases, the intercept factor of the elliptic Gaussian model is much better than that of SolTrace. Figure 13. (a) Contours of the measured and computed ﬂux distribution (kW/m ): Heliostat #7, (b) measured and computed intercepts vs. the side length of a square receiver: Heliostat #7 . Figure 14. (a) Contours of the measured and computed ﬂux distribution (kW/m ): Heliostat #8, (b) measured and computed intercepts vs. the side length of a square receiver: Heliostat #8 . Figure 15. (a) Contours of the measured and computed ﬂux distribution (kW/m ): Heliostat #9, (b) measured and computed intercepts vs. the side length of a square receiver: Heliostat #9 . Appl. Sci. 2022, 12, 10191 15 of 25 Figure 16. (a) Contours of the measured and computed ﬂux distribution (kW/m ): Heliostat #10, (b) measured and computed intercepts vs. the side length of a square receiver: Heliostat #10 . Table 4. The absolute average difference in ﬂux density of ten heliostats. Heliostats Elliptical SolTrace Circular 1 3.32 3.87 4.04 2 2.86 3.38 4.57 3 3.64 4.41 5.26 4 2.40 2.87 3.34 5 2.33 2.89 3.13 6 2.60 3.72 5.20 7 2.87 3.21 3.50 8 2.40 2.88 3.72 9 2.50 3.30 5.28 10 3.38 4.43 5.97 Table 5. The absolute average difference in intercept factor of ten heliostats. Heliostats Elliptical SolTrace Circular 1 1.27 2.64 2.50 2 1.26 2.09 2.22 3 0.46 1.76 1.64 4 1.21 2.54 2.35 5 0.49 1.90 1.78 6 1.26 2.19 2.07 7 1.75 2.36 2.27 8 0.94 2.25 2.30 9 1.30 2.90 2.82 10 1.38 2.95 2.82 Taking Heliostat #5 as an example, the intercept factor obtained by experimental data, the elliptic Gaussian model, circular Gaussian model, and SolTrace are shown in Figure 11b. The absolute average difference between the intercept factor calculated by the elliptic Gaussian model and the experimental data is only 0.49%. The absolute average difference between the intercept factor calculated by the circular Gaussian model and the experimental data is 3.12%. The absolute average difference in the intercept factor calculated by SolTrace and the experimental data is 3.25%. The intercept factor calculated by the elliptic Gaussian model is, to a large extent, close to the actual situation. Compared with the circular Gaussian model and SolTrace, the method of the reﬂected ray being divided into I , I , and I can well explain the curvature changes in the intercept curve. The 1 2 3 intercept factors of SolTrace and the circular Gaussian model are both close to 100% when the side length of the receiver plane reaches 6 m, whereas the intercept factor of the elliptic Appl. Sci. 2022, 12, 10191 16 of 25 Gaussian model and experiment data is close to 98%. This is mainly due to the large optical error of reﬂected rays in part I , which prevents it from being completely intercepted when it reaches the receiver plane. The circular Gaussian model and SolTrace do not take this situation into account, so the calculated result of the intercept factor is markedly different from the actual situation. 4. Analysis and Discussion In summary, the average difference, minimum difference, and maximum difference in ﬂux density distribution between the elliptic Gaussian model and experimental data are 2.83%, 2.33%, and 3.64%, and the average difference, minimum difference, and maximum difference in intercept factor are 1.13%, 0.46%, and 1.75%. The error of the model arises from the following aspects: i Each point on the heliostat is affected by gravity, temperature, and wind. The slope er- ror will change to different degrees, resulting in the irregular ﬂux density distribution of experimental results and errors with model calculation results. ii The solar intensity distribution uses the circular Gaussian function, which is different from the actual solar intensity distribution and will bring some errors. This model considers rays reﬂected by two planes of the glass mirror, and the focused light spot formed on the receiver plane is divided into three parts, which has been observed experimentally [11]. This is due to the ray reﬂected by the upper and lower surfaces of the glass. In this paper, when calculating the ﬂux density of the tower heliostat, its inﬂuence was also considered, signiﬁcantly impacting the calculation result. Although the focused spot position of the three reﬂections differs little, the error transfer effect of the three reﬂections needs to be calculated separately due to the difference in the number of reﬂections. The intensity distribution of reﬂected rays is very different in three parts, resulting in a different ﬂux density distribution and larger light spot dispersion, which is more consistent with the actual situation and introduces fewer calculation errors. Conventional ray-tracing methods consider the single beam and use the circular Gaus- sian model to describe the transfer effect of the surface error distribution. The comparison results in this paper show that the average difference between SolTrace and experimental data is 3.50%, and the maximum difference is 4.43%. In addition, even if the Monte Carlo method [21] is used to reduce the amount of calculation, it is still necessary to simulate as many reﬂected rays of the heliostat as possible to obtain high-precision results. Therefore, the speed is relatively slow, which is not suitable for optimizing ﬂux density calculation. When using an i7-4790 CPU 3.6 GHz personal computer, SolTrace needs 7.125 s to calculate the ﬂux density distribution reﬂected by a single heliostat. In comparison, the elliptic Gaussian model in this paper only needs 2.211 s to calculate. Collado et al. [3] used a circular Gaussian function to describe the ﬂux density formed by a heliostat on the image plane. When calculating the intercept factor based on this method, the error can reach up to 9.3%. The convolution integration method is another common method. A circular Gaussian distribution was ﬁrst used to describe the surface error of a heliostat, but the difference is relatively large compared with experimental data. In the 1980s, Lipps [7] proposed the method of convolution of the solar brightness distribution and elliptic Gaussian error to calculate and solve the ﬂux density distribution, which requires an interpolation calculation of the data table to be established in advance. In Lipps’ paper, HCOEF and KGEN methods were proposed. HCOEF has a fast calculation speed, and the calculation time is approximately 0.27 s. However, when the slope error of the heliostat is small, or the heliostat is very close to the receiver plane, unacceptable errors will occur, and the peak value of the calculated ﬂux density is four to ﬁve times the actual value. However, although the calculation results of the KGEN method can ﬁt the actual situation to a certain extent, the calculation speed is very slow, and the calculation time is 31.3 s. The HELIOS [8] system uses the two-dimensional fast Fourier transform method to replace convolution with function multiplication. It calculates the ellipse Gaussian Appl. Sci. 2022, 12, 10191 17 of 25 error distribution of a reﬂected ray generated by the ellipse Gaussian distribution of a surface error, which is suitable for more general cases. However, it consumes many computing resources. The calculated results of the HELIOS system were compared with the experimental data of a single heliostat in New Mexico [22], and the central proﬁles of the calculated results and the experimental data were presented. The difference between the actual data and the calculated results was approximately 1.5%. From the central proﬁles, the calculation error of the HELIOS system is not signiﬁcant. However, the comparison of the ﬂux density distribution involves interpolation in two directions. In the worst case, interpolation will bring about an error of approximately 2%. There is still a signiﬁcant error between the calculated results of the HELIOS system and the actual data. The numerical method proposed by the HELIOS system and Lipps has an essential assumption as a convolution method. It is necessary to assume that all reﬂected rays are reﬂected from the center of the heliostat and projected onto the receiver plane to calculate the ﬂux density of the receiver plane. A large calculation error will occur when the distance between the receiver plane and the image plane is large. Only when the distance between the heliostat and the receiver plane is close to the focal length is the calculation result of the convolution method accurate enough. However, the direct integration method does not need to introduce this assumption in the calculation, and the error is smaller than the indirect integration. The application range of direct integration is wide. Even if the distance between the heliostat and receiver plane is unequal to the focal length, the ﬂux density distribution can be calculated accurately. For the convolution method we proposed before [10], we calculated the intercept factor of ten heliostats mentioned in Section 3.2. The calculated results of ten heliostats are superior to that of the UNIZAR function [3] and the HFCAL function [4]. The indirect integration’s average, minimum, and maximum differences are 2.24%, 1.3%, and 2.7%. The indirect method simulates well when the intercept factor is lower than 95%, so, in most cases, it outperforms the UNIZAR and HFCAL functions. When the intercept factor is lower than 95%, the elliptic Gaussian model proposed in this paper is consistent with the indirect integration method. When the intercept factor is higher than 95%, we can see the limitations of the convolution, and the elliptic Gaussian model performs better. As a result, the average absolute difference in the elliptic Gaussian model is 1.11% smaller than the indirect integration method. Furthermore, in Heliostat #9 and #10, where the indirect integration method performs the worst, the absolute difference is only 1.3%. This result embodies the advantage of the direct integration method. This paper has only one assumption about the sun shape, which is as close to the actual physical image as possible. 5. Conclusions This paper proposes a new model for calculating the ﬂux density distribution by a heliostat. Compared with the convolution integration method, this model uses the direct integration method with only one assumption, so it has a wide range of applicability and a high accuracy. We reconstructed the actual optical process, where reﬂected rays are divided into three parts, and applied Fresnel’s equations to calculate the energy distribution ratios of reﬂected rays. The slope errors of reﬂected rays in the three parts are also different, which needed to be calculated separately, and were then superimposed to compute the distribution of the ﬂux density on the receiver plane. i Comparing the elliptic Gaussian model with the experimental data, the results of the elliptic Gaussian model are better than SolTrace and indirect integration. The distribution of the ﬂux density is consistent with the experimental data, and the curvature variation in the intercept factor is closer to the experimental data. The average difference in ﬂux density distribution is 2.83%, the minimum difference is 2.33%, and the maximum difference is 3.64%; the average difference in the intercept factor is 1.30%, the minimum difference is 0.46%, and the maximum difference is 2.30%. Appl. Sci. 2022, 12, 10191 18 of 25 ii The circular Gaussian model simpliﬁed by the elliptic Gaussian model was compared with SolTrace. Under the condition that the reﬂected light intensity distribution was consistent, the ﬂux density distribution and intercept factor obtained by the circular Gaussian model were consistent with SolTrace. The absolute differences were only 0.64% and 0.58%. However, the present model has a lower prediction ability than the SolTrace. iii The integration model proposed in this paper was proven to be more accurate and applicable than convolution and function methods. The function methods are mainly based on experience and have a signiﬁcant error in calculation results. Meanwhile, convolution methods themselves introduce too many assumptions, resulting in many limitations in the calculation. iv This is the ﬁrst time that multiple reﬂections and the inﬂuence of an optical error transferred from different planes of the glass mirror were considered in order to build an optical model for the ﬂux density of a heliostat. The reﬂection from two surfaces of the glass mirror to form three main parts of beams was considered in the present model, and Fresnel’s equations were applied to calculate the energy of the three parts of reﬂected rays. This may be the main reason for why more accurate results with the present model were obtained, as more reﬂections will add more optical errors to the reﬂection ray to diffuse the solar ray. As the present model has a greater prediction precision, it may be applied for the optimization of the optical ﬁeld design to obtain a better design. The model applies the Gaussian model for solar brightness distribution. It is a good approximation if the optical error of the heliostat is much larger than the solar brightness distribution. However, the optical error of the heliostat gradually becomes smaller, so the model may not give a good prediction to the heliostat with a lower optical error. Author Contributions: Conceptualization, Y.S.; methodology, Y.S., W.H. and L.Y.; software, Y.S.; vali- dation, Y.S.; investigation, Y.S.; resources, Y.S.; data curation, Y.S.; writing—original draft preparation, Y.S.; writing—review and editing, B.L.; supervision, W.H. and C.Z. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conﬂicts of Interest: The authors declare no conﬂict of interest. List of Symbols B intensity distribution function of reﬂected ray d the thickness of the glass E the total ﬂux density reﬂected by the heliostat RMES the root mean square error of the ﬂux density and intercept factor between the models and experimental data f intercept factor int f focal length F solar ﬂux, (kW/m ) i ,i the reﬂection angle or refraction angle at air–glass interface 1 2 i the incident angle reaching the silver surface I total intensity of reﬂected ray I intensity of incidence ray I , I , I the energy density of reﬂected ray of the ﬁrst, second, and third part 1 2 3 I direct solar irradiation, (kW/m ) k dielectric constant Appl. Sci. 2022, 12, 10191 19 of 25 n the ratio of refractive index of glass n the ratio of refractive index of the silver plating layer relative to the glass O grid points on the heliostat O grid points on the receiver r distance between the reﬂection point and the receiver point, (m) R reﬂectivity of reﬂected ray in the S component R reﬂectivity of reﬂected ray in the P component S the scope of integration on the heliostat, (m ) S the scope of integration on the receiver, (m ) T transmission of reﬂected ray in the P component T transmission of reﬂected ray in the S component Greek symbols: l the incident angle of the solar ray to the heliostat, (rad) b the azimuth angle of the mirror reﬂection point of the heliostat, (rad) b the included angle between O O and heliostat’s normal 1 1 2 b the included angle between O O and receiver ’s normal 2 1 2 a the absorption ratio of glass to light, q the angle variable, (mrad) d the energy density ratios # the absorption coefﬁcient d the standard deviation of the solar intensity distribution sun d the standard deviation of the slope errors at the optical surface in slo pex transverse(x) direction, (mrad) d the standard deviation of the slope errors at the optical surface in slo pey longitudinal (y) direction, (mrad) d the standard deviation of the optical error in transverse(x) direction sx d the standard deviation of the optical error in the Y direction sy d the standard deviation of the optical error distribution in the X direction, (mrad) d the standard deviation of the optical error distribution in the Y direction, (mrad) d the standard deviation of the average error Subscripts and superscripts: i three parts of reﬂected ray parameters p,s the direction that is perpendicular or parallel to the vibration of the incident plane silver on the silver surface sun sun shape tr heliostat tracking error Appendix A Appendix A.1. Location Determination In a solar tower system, the given quantities are the altitude angle a , the azimuth rn angle g of the receiver plane, and the normal vector of the receiver plane (u , v , w ). rn rn rn rn Firstly, we need to calculate the coordinates of the incident ray (u , v , w ) from sun sun sun the solar altitude angle s and azimuth angle g : s s (u , v , w ) = (cosa sing ,cosa sing , sina ) (A1) sun sun sun s s s s s Then, the altitude angle a and azimuth angle g of the central reﬂected ray can be r r calculated by the position of the heliostat and the receiver plane: Z Z receiver heliostat a = atan(p ) (A2) 2 2 (x x ) + (y y ) receiver heliostat receiver heliostat x x receiver heliostat g = atan( ) (A3) y y receiver heliostat Appl. Sci. 2022, 12, 10191 20 of 25 where (x , y , z ) and (x , y , z ) represent the heliostat cen- receiver receiver receiver heliostat heliostat heliostat ter coordinates and receiver center coordinates in the global coordinate system, respectively. The altitude angle a and azimuth angle g of the heliostat normal vector should also n n be calculated: sina + sina s r a = atan( ) (A4) 2 2 cos a + cos a + 2cosa cosa cos(g g ) s r s r r s cosa sing + cosa sing r r s s g = atan( ) (A5) cosa cosg + cosa sing r r s s Appendix A.2. Meshing Process and Coordinate System Rotation The next step is to grid the heliostat and the receiver plane into a 100-by-100 grid, and the center point of each grid, namely the grid point, is used to represent the grid. The unit grid of a heliostat is expressed by Dhg, and the unit grid of a heliostat is expressed by Drg. Then, we calculated the coordinates of the heliostat and the receiver plane in the coordinate system. In the receiver plane coordinate system, the coordinate of the receiver plane is (x , y , z ) , where x and y were obtained by meshing the length and width of the r r r r r receiver receiver plane. Since the receiver plane is ﬂat, the z coordinate is always 0. In the coordinate system of the heliostat, the coordinate of the heliostat is (x , y , z ) , r r r heliostat where x and y were obtained by meshing the length and width of the heliostat. Since the h h heliostat is spherical, the z coordinate is: 2 2 2 z = 4 f x y (A6) h h After completing the meshwork of the heliostat and receiver plane, the next step is to transform the heliostat coordinate system and receiver plane coordinate system into the global coordinate system. The total rotation matrix from the global coordinate system to the heliostat coordinate system is: 2 3 cosg sing 0 n n 4 5 A H = sing sina cosg sina cosa (A7) n n n n n sing cosa cosg cosa sina n n n n n Thus, the coordinate of the heliostat grid point in the global coordinate system is: (x , y , z ) = (x , x , x ) + (x , y , z ) A H (A8) h h h global heliostat heliostat heliostat global h h h heliostat Similarly, the rotation matrix from the global coordinate system to the receiver plane coordinate system is: 2 3 cosg sing 0 rn rn 4 5 AR = sing sina cosg sina cosa (A9) rn rn rn rn rn sing cosa cosg cosa sina rn rn rn rn rn After gridding the heliostat, we can further determine the coordinates of the heliostat spherical center: x = x + 2 f (cosa sing ) (A10) n n s pherical heliostat 0 y = y + 2 f (cosa cosg ) (A11) 0 n n s pherical heliostat z = z + 2 f sina (A12) 0 n s pherical heliostat Further determining the normal vector of each point on the heliostat: s phericalx u = (A13) hn so Appl. Sci. 2022, 12, 10191 21 of 25 y y s pherical h v = (A14) hn so z z s pherical h w = (A15) hn so where l is the distance between the heliostat grid point and the heliostat spherical center, so calculated as follows: 2 2 2 l = sqrt(x x ) + (y ) + (z z ) (A16) so s pherical h s phericaly s pherical h Appendix A.3. Calculation of Geometrical Optics Then, the reflected ray can be calculated according to the incident vector (u , v , w ) sun sun sun and the normal vector of the heliostat grid point (u , v , w ): hn hn hn u = 2(u u + v v + w w )u u (A17) ray sun sun sun sun hn hn hn hn v = 2(u u + v v + w w )v v (A18) ray sun sun sun sun hn hn hn hn w = 2(u u + v v + w w )w w (A19) ray sun sun sun sun hn hn hn hn The receiver point, which coordinates in the global coordinate system, is formed when a reﬂected ray hits the receiver plane: z w + y v + x u u x v y w z receiver rn receiver rn receiver rn rn rn rn h h h t = , (A20) u u + v v + w w rn ray rn ray rn ray x = x + u t (A21) t f h ray y = y + v t (A22) t f h ray z = z + w t (A23) ray t f h The receiver point O on the receiver is converted from the global coordinate system to the receiver plane coordinate system: (x , y , z ) = (x x , y y , z z ) AR. (A24) r f r f r f receiver r f receiver r f receiver r f receiver global Heliostat grid point coordinates are converted from the global coordinate system to the receiver plane coordinates: (x , y , z ) = (x x , y y , z z ) AR (A25) h h h receiver h receiver h receiver h receiver global The O O vector is: 1 2 u = (x x ) (A26) O O r 1 2 v = (y y ) (A27) O O r h 1 2 w = (z z ) (A28) O O r h 1 2 The angle between O O and the heliostat normal vector is: 1 2 u u + v v + w w O O hn O O hn O O hn 1 2 1 2 1 2 b = acos( ) (A29) 2 2 2 u + v + w O O O O O O 1 2 1 2 1 2 The angle between O O and the receiver normal vector is: 1 2 u u + v v + w w rn rn rn O O O O O O 1 2 1 2 1 2 b = acos( q ) (A30) 2 2 2 u + v + w O O O O O O 1 2 1 2 1 2 Appl. Sci. 2022, 12, 10191 22 of 25 Firstly, we calculated the distances between points in the coordinate system of the receiver plane: (1) The distance between receiver point O and heliostat grid point O is: 3 1 2 2 2 l = (x x ) + (y y ) + (z z ) (A31) hr f h r f h r f h r f and l in the X direction and the Y direction are, respectively: hr f 2 2 l = (x x ) + (z z ) (A32) hr f x h r f h r f 2 2 l = (y y ) + (z z ) (A33) hr f y h r f h r f (2) The distance between receiver point O and receiver plane grid point O is: 3 2 2 2 2 l = (x x ) + (y y ) + (z z ) (A34) r r r r f r r f r f r f and l in the X direction and the Y direction are, respectively: r f r 2 2 l = (x x ) + (z z ) (A35) r r r f rx r f r f 2 2 l = (y y ) + (z z ) (A36) r f ry r r f r r f (3) The distance between heliostat grid point O and receiver plane grid point O is: 1 2 2 2 2 l = (x x ) + (y y ) + (z z ) (A37) r r r rh h h h and l in the X direction and the Y direction are, respectively: rh 2 2 l = (x x ) + (z z ) (A38) rhx r h r h 2 2 l = (y y ) + (z z ) (A39) r r rhy h h Then, the included angle between the reﬂected ray and O O of each grid can be 1 2 calculated, as shown in Figures A1–A3: 2 2 2 l + l l rhx hr f x r f rx q = acos( ) (A40) 2l l rhx hr f x 2 2 2 l + l l rhy hr f y r f ry q = acos( ) (A41) 2l l rhy hr f y 2 2 2 l + l l rh hr f r f r q = acos( ) (A42) 2l l rh hr f Appl. Sci. 2022, 12, 10191 23 of 25 Figure A1. The included angle between the reﬂected ray and O O of each grid under the circular 1 2 Gaussian model. Figure A2. The included angle between the reﬂected ray of each grid and O O in the X-direction 1 2 under the elliptic Gaussian model. Appl. Sci. 2022, 12, 10191 24 of 25 Figure A3. The included angle between the reﬂected ray of each grid and O O in the Y-direction under the elliptic Gaussian model. Appendix A.4. Solution of the Flux Density at a Speciﬁc Point Finally, considering the sun shape and optical errors, the distribution function of reﬂected rays on the plane of the receiver was computed: Circular Gaussian model: 1 q B(q) = exp( ) (A43) 2 2 2ps 2s Elliptic Gaussian model: 1 1 q B(q , q ) = exp[ ( + )] (A44) x y 2 2 2ps s 2 s s x y x y Therefore, the contribution of the heliostat to the ﬂux density of the grid point on the receiver plane is: Circular Gaussian model: I cosb cosb B(q) Dhg 1 2 F = (A45) u,v Elliptic Gaussian model: I cosb cosb B(q , q ) Dhg 1 2 x y F = (A46) u,v where F is the contribution of grid points in row u and column v on the heliostat to the u,v ﬂux density value of a point on the receiver plane. Finally, the ﬂux density value of a point on the receiver plane can be obtained by integrating all grid points on the heliostat. References 1. Boukelia, T.E.; Arslan, O.; Bouraoui, A. 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Applied Sciences – Multidisciplinary Digital Publishing Institute

**Published: ** Oct 11, 2022

**Keywords: **Gaussian distribution; direct integration method; Fresnel’s equations; solar flux density; optical error

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