# An Integration Model for Flux Density Distribution Formed by a Heliostat
## 3 Model Validation
### Verification with Experimental Data  
During July, [PERSON] 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\({}^{2}\). 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. 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  
Figure 5: (**a**) Flux density distribution computed by the circular Gaussian model and SolTrace (\(\mathrm{kW/m^{2}}\)), (**b**) intercepts computed by the circular Gaussian model and SolTrace. Figure 6: Distribution of ten heliostats and receiver. 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 \(I_{1}\), \(I_{2}\), and \(I_{3}\) were approximately 5%, 91.5%, and 3.5%, respectively. 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. 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.