# An Integration Model for Flux Density Distribution Formed by a Heliostat
## 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 [PERSON]'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. Comparing the elliptic Gaussian model with the experimental data, the results of the elliptic Gaussian model are better than SoITrace 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%. * 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. * 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. * 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 [PERSON]'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. Conceptualization, Y.S. ; methodology, Y.S., W.H. and L.Y. ; software, Y.S. ; validation, 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. This research received no external funding. Not applicable. The authors declare no conflict of interest. List of Symbols