Linear Fresnel reflectors (LFRs) have gained increasing attention due to their advantages of simplified construction, reduced wind loads, cost-effectiveness, and optimal land area utilization. Mirrors are the focusing components of LFRs, including flat, parabolic, and cylindrical shapes. Flat mirrors have limited focusing ability, with a reflected light spot width not smaller than its width. The use of slightly curved cylindrical or parabolic mirrors can improve the focusing ability. Existing research on the optimization of cylindrical mirrors in LFRs is mostly limited to unaltered curvature radii. Although the production is simple and cost-effective, the individual optimization of the curvature of each mirror (half mirror field) can improve the optical performance of the system more effectively. We aim to investigate the optimization design problem of cylindrical mirrors in LFRs and propose an optimized calculation method for the curvature radius of cylindrical mirrors. A general calculation model is established, which only requires considering its distance from the center of the field and the transversal incidence angle of the sun during effective sunrise to obtain the optimal value.

Firstly, based on the characteristic that the reflected rays passing through the two endpoints of the cylindrical mirror of LFRs always deviate towards the direction closer to the center when the mirror deviates from the reference position, an optimized calculation method for the curvature radius of cylindrical mirrors is proposed, and the calculation formula is derived. Secondly, using the method of polynomial surface fitting, the curvature radii calculated for cylindrical mirrors at different widths, distances from the center of the field, and the transversal incidence angles of the sun during effective sunrise are processed to obtain a general calculation model. The accuracy of the model is validated using numerically precise calculation results. Finally, the optical performance of the optimized cylindrical mirrors is analyzed using a ray tracing-based optical model. The focusing characteristics of the system are analyzed using an optical model based on SolTrace, with an LFR optimized in existing literature as an example.

It is assumed that a transversal incidence angle of the sun during effective sunrise is 30°, and the curvature radii for cylindrical mirrors with different widths are nearly identical at the same distance from the center of the field (Fig. 4). By taking the transversal incidence angle of the sun during effective sunrise from 20° to 40° with an interval of 1°, the curvature radii for cylindrical mirrors with a relative width of 0.09 are calculated at different distances from the center of the field. A good fit is achieved with a polynomial surface fitting order of 3 for the distances from the center and an order of 1 for the transversal incidence angle of the sun during effective sunrise (Fig. 5). After ignoring the influence of mirror slope error and tracking error, it can be observed that as the distance from the center increases, the lateral offset of reflected rays from the cylindrical mirror exhibits a linear increasing trend. Under the same distance from the center, the maximum lateral drift of reflected rays from the cylindrical mirror increases with wider mirror widths (Fig. 7). At the center of the field, the lateral offset of reflected rays from the cylindrical mirror in response to the transversal incidence angle demonstrates a symmetrical relationship around 90°. Overall, a decreasing trend is followed by an increasing trend. When the cylindrical mirror deviates from the center of the field, the lateral offset of reflected rays from the cylindrical mirror first decreases and then increases across the entire range of transversal incidence angles, with the minimum value determined by the distance from the center (Fig. 8). With an increase in the transversal incidence angle of the sun during effective sunrise, the optical efficiency of the system and the concentrated solar flux on the absorber surface continue to rise, while the uniformity shows a decreasing trend followed by slight fluctuations within a narrower range. The concentrated solar flux primarily concentrates on the lower half of the absorber tube (Figs. 9 and 10).

The optimal curvature radius of the cylindrical mirrors has little correlation with its width but rather depends on the distance from the mirrors to the center of the field and the transversal incidence angle of the sun during effective sunrise. The results obtained from the general calculation model closely match the numerically precise calculations, with a maximum deviation of 1.26% and an average deviation of 0.38%. By considering the slope error, tracking error, and curvature radius error of the cylindrical mirrors, the real-time optical efficiency remains above 59.46% when the transversal incidence angle exceeds 45°. Within a small range on the aperture (relative distance -0.05-0.05), the concentrated solar flux density is high and exhibits good uniformity, making it suitable for concentrating photovoltaic systems.