# An Integration Model for Flux Density Distribution Formed by a Heliostat
###### 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 [PERSON]'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. Gaussian distribution; direct integration method; [PERSON]'s equations; solar flux density; optical error +
Footnote †: journal: Appl. Sci. **Citation:** [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] An Integration Model for Flux Density Distribution Formed by a Heliostat. _Appl. Sci._**2022**, 12, 10191. [[https://doi.org/10.3390/appl22010191](https://doi.org/10.3390/appl22010191)]([https://doi.org/10.3390/appl22010191](https://doi.org/10.3390/appl22010191))  
Academic Editor: [PERSON]  
Received: 29 July 2022  
Accepted: 28 September 2022  
Published: 11 October 2022  
**Published:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. **[PERSON]; [PERSON]; [PERSON]. 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]. [PERSON] [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.