# An Integration Model for Flux Density Distribution Formed by a Heliostat
## 2 Methodology
### 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. \[F_{total}=\sum_{i=1}^{3}\delta_{i}\times F_{i}(i=1,2,3) \tag{12}\]  
where \(F_{i}\) 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 \(S_{1}\), the receiver plane is \(S_{2}\), \(n_{1}\) and \(n_{2}\) are normals of the cell surfaces on the heliostat and the receiver, \(O_{1}\) and \(O_{2}\) are the points on the cell surfaces on the heliostat and the receiver, and \(\beta_{1}\) and \(\beta_{2}\) are the included angles between the line \(O_{1}O_{2}\) and normals of each cell surface. For a single part, the analytic formula of the flux density at point \(O_{2}\) in the receiver surface is [15]:  
\[F=\iint_{S_{1}}\frac{IBcos\beta_{1}cos\beta_{2}}{r^{2}}dS_{1} \tag{13}\]  
where \(r=|\,O_{1}O_{2}\,|\) is the distance between the reflection point and the receiver point, \(I\) is 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 \(\sigma_{sun}\) can be determined by direct solar irradiation \(I_{D}\)[8]:  
\[\sigma_{sun}=\frac{[3.7648-3.8413(I_{D}-1)+1.5923\times 10^{2}(I_{D}-1)^{2}]}{ \sqrt{2}} \tag{14}\]  
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 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. For the first part of the reflected ray \(I_{1}\), the upper surface of the glass directly reflects it, so the average optical errors were calculated as follows:  
\[\sigma_{sx}^{2}=4\sigma_{slopex}^{2}+(\frac{2 tan^{2}\lambda cos\beta sin\beta}{1 +tan^{2}\lambda cos^{2}\beta}\sigma_{slopey})^{2} \tag{15}\]  
\[\sigma_{sy}^{2}=4\sigma_{slopex}^{2}+(\frac{2 tan^{2}\lambda cos\beta sin\beta}{1 +tan^{2}\lambda cos^{2}\beta}\sigma_{slopex})^{2} \tag{16}\]  
where \(\sigma_{slopex}\) and \(\sigma_{slopey}\) are slope errors in the X direction and Y direction, respectively. For the second part of the reflected ray \(I_{2}\), 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.