| This paper presents an analytical solution for calculation of flux density distribution on the image plane of a round spherical heliostat which is deduced from the convolution-based integration method.The two-dimensional integration for flux density calculation is converted to a one-dimensional integration and solved as a function,thereby reducing the amount of calculation and obtaining fast computation.The flux density on the receiver plane can be calculated through projection even if the distance between the heliostat and receiver is not equal to the focal length of the heliostat.The ray tracing and numerical method are applied to validate the accuracy of the analytical method which shows the analytical method can be applied to most heliostats for solar tower system with a faster calculation speed than SolTrace code,and the error source is analyzed,which is inversely proportional to the cube of the focal length f and is proportional to the biquadrate of the heliostat radius R0.It is proved that a radially symmetric solar flux density distribution on the image plane is produced by a round spherical heliostat,which is so far the only radially symmetric solar flux density distribution formed by the heliostat when the incident angle is not 0.In this paper,based on the convolution integral method,the Gaussian flux density function at the image plane is deduced after simulating the image function of the heliostat with Gaussian distribution function.The flux density distribution of various round or rectangular focusing heliostats at the receiver plane can be calculated through projection.The simulation results are compared with SolTrace analysis and experimental data.Comparison of the flux density distributions between this method and SolTrace show excellent agreement especially when the optical error is equivalent to or larger than the Gaussian function parameter of heliostat image which fits to most of heliostats in a solar tower system.The average prediction error of the elliptical Gaussian function from the experimental data is about 2.24%,which is less than that of SolTrace in most cases as SolTrace applies the circular Gaussian distribution for optical error.The added error source comes from simulating the image of the heliostat with Gaussian distribution.In this paper,two methods for calculating the flux density are proposed,which can deepen people’s understanding of the concentrating performance of heliostats and can be applied to the simulation and optimization of solar systems. |
