Design and Principle of Novel Linear Solar Concentrator With Asymmetric Parabolic Reflector and Independently Movable Receiver Open Access

The linear solar concentrator, as a typical optical component for harnessing solar thermal energy, has a wide application in power generation and industrial heat supply [1]. Parabolic trough collector (PTC) and linear Fresnel reflector (LFR) designs are the two types that have been commercialized in concentrating solar power (CSP) plants [2,3]. The literature has relevant studies on optimizing the efficiency of PTC and LFR, analyzing key behaviors, trends, and exploring new designs to enhance optical performance [4–8]. However, neither has been able to completely replace traditional fossil fuel-based boilers due to their shortcomings in terms of stability and cost-effectiveness. The conventional solar concentrator contributes a significant portion of the overall cost, accounting for 30–44% of the budget required for constructing a CSP plant [9,10]. Given its operational characteristics, the solar concentrator, which consists of an array of mirrors, is driven by a motor-controlled device to continuously track incident sunlight. Therefore, the additional costs associated with operation and maintenance are even higher compared to other solar energy application technologies, such as photovoltaics [11–13]. Besides, a certain distance must be maintained between two adjacent arrays of mirrors to prevent mutual occlusion, which may further reduce the land use ratio [14]. For example, a PTC with north-south axial orientation can lose over 30% of incident solar radiation due to the gap arrays, while for an LFR, the loss is around 16% [15–17]. The previous studies [18–25] have reported specific approaches to address these two kinds of drawbacks. Pujol Nadal and Martínez Moll [18–23] introduced the concept of a fixed-mirror solar concentrator and conducted a series of studies on its optical and thermal performance, prototype characterization, and control system. The concentrator employs arc-distributed stepped mirrors to achieve concentration with fixed reflectors and a moving receiver, which thereby enhances the land use ratio and simplifies the tracking mechanism. However, manufacturing such stepped mirrors presents challenges, and mutual shading among the mirrors causes optical losses. Subsequently, they proposed another design, a single-mirror and tracking absorber system. The parabolic trough mirrors are fixed horizontally on the ground, while the receiver moves along a circular path [24,25]. This system has a simple structure that allows integration into lightweight building rooftops. While the shading issue is resolved, the use of a symmetrical parabolic reflector makes the concentration performance highly sensitive to the solar incidence angle. The light spot expands rapidly when increasing the transversal incidence angle. In general, low overall concentration limits the system to low-latitude regions and makes it suitable only for low-temperature applications of 100 °C–200 °C.

This work presents a novel conceptual design for the linear solar concentrator to address the challenges mentioned earlier. A key feature of the new concentrator is the use of a stationary reflector with an asymmetrical parabolic mirror and a movable receiver based on the nonimaging concentrating principle [26,27]. The design offers several advantages: (1) the laydown configuration of the new reflector eliminates the gap arrays in traditional PTC/LFR systems, achieving land use ratio of 100% under ideal conditions; (2) the system does not require complex mechanical components to move the reflector, which enhances system reliability and reduces costs; and (3) most importantly, the asymmetrical parabolic shape reduces the impact of solar incidence angles on concentration performance, which extends the system's application range across latitudes.

The conceptual design of the novel solar concentrator is first introduced in detail in Sec. 2. Section 3 presents the mathematical model of solar incidence angle and deviation angle based on geometrical-optic principles, and the optical performance analysis method based on the Monte Carlo ray tracing method [28,29]. We utilize the developed model to calculate the annual evolution of incidence angles and deviation angles in the reference location, Nanjing, China (32°N, 119°E), and then via ray tracing simulation to analyze the optical performance of the novel solar concentrator, such as the local/overall concentration ratio and the optical efficiency, in Sec. 4. The main conclusions are provided in Sec. 5.

Figure 1 depicts the end view schematic diagram of the novel linear solar concentrator. It incorporates an asymmetric parabolic reflector and an independently movable receiver. The solar incidence angle is defined as the angle between the sun's ray and the vertical line, and the deviation angle is the angle between the sun's ray and the axis of the parabola. The reflector section is truncated from a parabola and oriented along an east-west axis (perpendicular to the paper surface) with its two ends fixed at the same height. Given the nature of nonimaging concentration, the sunlight reflected by the reflector forms focal zones instead of focuses, except when the incident sunlight aligns with the parabola's symmetry axis. To maximize the capture of reflected sunlight, the receiver needs to be installed on a receiver trace that crosses these focal zones based on the sun's positions. Interestingly, calculations reveal that the receiver trace resembles a circular shape. Hence, an arc approximation is adopted here for the optimal solution. The geometry of the concentrator is determined by the parameters of the truncated width (W), the parabolic the focal length (f), the tilt angle (θ_p), the center coordinates and radius of the receiver trace (C and r, respectively), and the diameter of the tubular receiver (d). The values can be found in Sec. 4.2. Due to the scope of this article, the detailed theoretical methodologies concerning these geometric parameters are not included here and will be presented in future publications.

Figure 2 shows the operational diagram of two schemes, i.e., (1) the sliding scheme and (2) the rotating scheme. The cross section of the reflector is horizontally oriented upward, with its base fixed on the ground. Although the reflector remains stationary during the day, its tilt angle is designed to be adjustable periodically to accommodate different seasons.² The tubular receiver is positioned on an arc trace or held in place between two parallel rotating rods. Both schemes enable the receiver to move along the preset trace to capture sunlight concentrated from the reflector throughout the day. The sliding scheme allows for movement along customized curved paths, whereas the rotating scheme only allows for movement along the arc trace. Because of the difference in power mechanism, the sliding scheme offers a slight advantage in terms of weight and cost. In contrast, the rotating scheme provides improved accuracy and stability. The concentrator system employs a similar layout as PTC systems that feature arrays of novel linear concentrator units connected in series and arranged in parallel. Within each array, the receivers are joined end to end. To facilitate the seamless interconnection between arrays, stainless steel flexible hoses serve as the flexible joints connecting the movable absorber tubes with the fixed headers anchored at the rotating center points. The interconnection can accommodate both sun-tracking movements and thermal expansion and contraction caused by heat transfer fluids [30,31]. Since the thermomechanical analysis falls outside the scope of this article, it will not be discussed here.

In this section, we introduce the mathematical model for calculating the solar incidence angle and the deviation angle, and the optical simulation using the Monte Carlo ray tracing method for assessing optical performance of the novel concentrator.

First, the geometric relationship between solar incidence angle, solar elevation angle, and solar azimuth angle is analyzed. As shown in Fig. 3(a), when the axial direction of the linear concentrating collector aligns with the east-west orientation, the solar incidence angle $(π2−θr)$ and the north-south elevation angle θ_s are complementary. Conversely, when the axial direction aligns with the north-south orientation, the solar incidence angle and the east-west elevation angle θ_e are complementary [32]. Therefore, the mathematical model of north-south elevation angle and east-west elevation angle should be established first.

The solar incidence angle can be obtained when the concentrator is placed on the north-south axis and the east-west axis by Eqs. (4) and (5).

Equation (8) indicates that when the parabolic tilt angle equals half of the interval of solar incidence angle variation, the maximum absolute value of deviation angle is minimized, which equals half of the solar incidence angle variation.

It is interesting to note that the deviation angle is solely determined by the declination angle and the hour angle, regardless of the geographical location (latitude).

According to the definition of a parabola, when the deviation angle is zero, the incident ray is perfectly reflected on the parabolic focus. The absolute value of the deviation angle represents the degree of deviation from the “ideal focus.” In cases where deviation occurs and the realistic sun shape is considered, the path of light becomes more intricate. Thus, the Monte Carlo ray tracing method is employed to achieve the optical simulation [34,35]. The computer-aided design models for the reflector and receiver, the related optical properties, as well as the specifications for the solar light source, are imported and configured within the optical simulation software tracepro. As this article focuses on track accuracy rather than energy loss, both the mirror and absorber are considered ideal optical elements with a reflectivity and absorptivity of 1. The incident source utilizes the solar beam that has an angular profile equal to that measured for the sun [36]. Data for the solar profile are taken from Astrophysical Quantities [37]. Using this setup, the trajectory of the sunray and the size of the spot is then simulated. It enables a comprehensive analysis of the relationship between concentrating performance and the deviation angle.

In this section, the concentrating characteristics and feasibility of the novel concentrator are analyzed based on the aforementioned model and method. The variation of the deviation angle over a whole year and its impact on the concentrating performance are discussed. Demonstrating that the collector consistently meets the concentrating performance requirements across varying deviation angles throughout its operational duration would provide evidence of the feasibility of the novel solar concentrator from an optical perspective.

According to the model of the solar incidence angle, taking Nanjing as an example, the variation of the solar incidence angle of the collector in different horizontal axial directions during the work time of a year is analyzed.

After calculation, the variation curve of solar incidence angle θ_r for collector with different axial declination angles during the daytime is obtained and shown in Fig. 4. Because of the symmetrical distribution of data in the first and second half of the year, Fig. 4 only shows the data of half a year. The definition of annual solar incidence angle variation is the difference between the maximum solar incidence angle and the minimum solar incidence angle in a year during the daytime.

Figure 4 shows the solar incidence angle as a function of daytime and dates for different axial declination angles (from 0 deg to 90 deg) in the Nanjing area. When the deviation angle is 0 deg, the solar incidence angle in the morning and afternoon is symmetrical. During the period from spring equinox to autumn equinox, the solar incidence angle at 12:00 is the largest, and that at 8:00 and 16:00 is the smallest. During the autumnal to vernal equinoxes, the solar incidence angle at 12:00 is the smallest, and that at 8:00 and 16:00 is the largest. With the increase of the axial declination angle, the maximum solar incidence angle of a year is always at 8:00 of winter solstice and increases from 72.9 deg to 77.4 deg, and the minimum solar incidence angle of the year is at 16:00 of the summer solstice (deviation angle < 47.5 deg) or at 16:00 of the winter solstice (deviation angle > 47.5 deg) and decreases from −8.9 deg to −77.4 deg. This demonstrates that the lowest annual variation in the solar incidence angle occurs when the axial declination angle is set to zero, which corresponds to the east-west orientation arrangement.

Figure 5 further shows the results of the solar incidence angle variation throughout the year and the 4 standard days: winter solstice, summer solstice, spring equinox, and autumn equinox, with different axial declination angles for the Nanjing area during the daytime. With an increasing axial declination angle, these solar incidence angle variations all increase, among which, the annual solar incidence angle variation expands from 81.9 deg to 154.7 deg, and the winter solstice is the day with the largest increase in the range of incidence angle, from 17.5 deg to 154.7 deg. The smallest values for the solar incidence angle variation occur when the axial declination angle is 0 deg or the collector is orientated in the east-west direction. Conversely, the largest values for solar incidence angle variation occur when the axial declination angle is 90 deg or the collector is orientated in the north-south direction.

Equation (9) proves that for regions of the Earth, although the parabolic tilt angle and the distribution of the solar incidence angle are different, the distribution of the deviation angle is consistent with that in the Nanjing area, which is only related to time. Therefore, it is only necessary to analyze the relationship between the deviation angle and time.

When the axial direction of the concentrator is east-west and the parabolic tilt angle is the local latitude, the distribution diagram of the deviation angle can be calculated according to Eq. (9), as shown in Fig. 6. The absolute value of the deviation angle in spring and autumn (mainly distributed within 0–28 deg) is smaller than that in summer and winter (mainly distributed within 16–40 deg). Compared with morning and evening, the absolute value of the deviation angle is smaller at noon, especially in winter and summer.

According to the calculated results of the deviation angle, during the daytime, the average of the absolute values of the deviation angle in winter/summer, spring/autumn, and the whole year are 26.35 deg, 11.02 deg, and 18.37 deg, respectively.

Table 1 presents the distribution proportion of the deviation angle during a year. If the working time of winter and summer is adjusted to 09:00–15:00, the proportion of time that the deviation angle is in the range of (−30 deg, 30 deg) will increase from 90.8% to 99.5%, which indicate that the absolute value of the deviation angle is larger in the morning and evening of winter/summer.

This case study focuses on the Nanjing region of China. On the basis of theoretical solutions, we have established a set of moderate design parameters for the concentrator as annotated in Fig. 1. The width of the reflector is W = 850 mm. The tubular receiver has a diameter of d = 60 mm. The parabolic collector is oriented along the east-west axial direction with a focal length of f = 978 mm and a tilt angle of θ_p = 32 deg. The rotating center of the receiver trace is located at C (−404.6 mm, 747.7 mm), with a radius of r = 811.7 mm. A sensitivity study was first conducted to determine the ray's number for the optical simulation. Table 2 presents the predicted results of total absorbed fluxes and corresponding computational times for various ray counts ranging from 350,000 to 4,300,000. The relative error was calculated by comparing against the result for the highest ray count (4,300,000 rays). One can see that, as the number of rays increases to 1.4 million, the relative error has diminished to the level of 10⁻⁴ over a computational time of 48 s. Thus, a consistent ray count of 1,400,000 was adopted for all simulations in this study.

By changing the solar incidence angle, the concentrating performance of sunlight reflected by the reflector is simulated. Figure 7 shows the trajectory of sunlight when the solar incidence angles are 2 deg, 12 deg, 22 deg, 32 deg, 42 deg, 52 deg, and 62 deg (corresponding deviation angles are −30 deg, −20 deg, −10 deg, 0 deg, 10 deg, 20 deg, and 30 deg). The trajectory of sunlight reveals that the locations of concentrating points for deviation angles varying from −30 deg to 30 deg adhere a circular pattern. When the deviation angle deviates from zero, the reflected sunlight creates an approximate triangular convergence area on the surface of the receiver. The distribution of reflected sunlight is uneven with an increase in solar concentration closer to the symmetry axis of the parabola.

A larger absolute value of deviation angle results in a more dispersed solar concentration. For the design case, when the absolute value of the deviation angle ranges from 0 deg to 10 deg, the concentrating performance of the novel concentrator is comparable to that of PTC. Within the 10 deg to 20 deg range, the spots enlarge, yet the concentrator maintains good performance. However, when the absolute value of the deviation angle falls between 20 deg and 30 deg, the spots are anticipated to expand further, which thereby limits the concentration ratio.

Figure 8 shows the detailed contour maps of local concentration ratio on the surface of the cylindrical receiver for different deviation angles. Due to the nature of nonimaging concentration, the map exhibits a tidal-like pattern when the deviation angle is nonzero. Specifically, the distribution along the axial direction (Z/L) remains relatively uniform, whereas in the circumferential direction (ϕ), an asymmetric distribution emerges with a peak local concentration ratio squeezed to one side and gradually decaying toward the other side forming a “tail” reminiscent of the ebbing tide. For the deviation angle of −30 deg, the peak in the local concentration ratio is observed on the right side at ϕ = π/2, followed by a long “tail” extending leftward with the local concentration ratio gradually decreasing. As ϕ decreases to 3π/2, the local concentration ratio diminishes to around 2. The length of “tail” shortens and eventually disappears as the deviation angle increases from a negative value toward zero. When the deviation angle reaches 0 deg, the map of local concentration ratio exhibits a symmetric distribution with a minimized width of 1.5 rad that spans from ϕ = 3.0 rad to 4.5 rad, leading to the best-concentrating performance. In addition, under the same given conditions, the local concentration ratio for positive deviation angles is lower than that for negative deviation angles. For example, at a deviation angle of −20 deg, the peaks of the local concentration ratio emerge at ϕ = 3π/4, achieving a maximum value of 42, which is higher than the value of 29 correspondingly observed at a deviation angle of 20 deg.

Figure 9 presents the profiles of the average local concentration ratio (C_avg) along the circumferential direction of the receiver surface (ϕ) for different deviation angles. As the deviation angle changes from 30 deg to −30 deg, the peak of C_avg increases first and subsequently declines. When increasing the deviation angle within the range of −30 deg to −10 deg, the peak of C_avg increases from 20 to 50. This is attributed to the improved concentrating effect and the shortened optical path between the reflector and the receiver. It is interesting to note that the maximum C_avg is achieved at a deviation angle of −10 deg, even though perfect imaging concentration theoretically corresponds to a zero-deviation condition. This is because that as deviation angle reaches 0 deg, the solar incident angle exceeds 30 deg, which leads to a detrimental cosine effect on the reflector surface. The calculation shows that the total solar flux reaching the reflector aperture is reduced to merely 85% of that under vertical incidence. As the deviation angle continues to increase from 0 deg to 30 deg, the concentrating performance deteriorates and the peak of C_avg drops back down to 20. The optical efficiency remains above 96% across the entire range of deviation angles, peaking at 99% when the deviation angle reaches 30 deg. Simulations indicate that the optical loss occurs primarily at both ends of the receiver, which can be mitigated by utilizing a sufficiently long receiver in practical applications. Note that in real-world conditions, one can expect a slightly lower optical efficiency due to the unaccounted realistic reflective properties of the mirror, e.g., reflectivity and slope error. Nevertheless, this discrepancy would not be significant.

In addition, the relationship between the overall concentration ratio (C_t) and the maximum norm of deviation angle $(|αd|max)$ is analyzed. As shown in Table 3, the overall concentration ratio increases significantly with the reduction of maximum deviation angle. As discussed in Sec. 4.1, the deviation angle almost falls within the range of ±30 deg during the adjusted working time (as given in Table 1) throughout a year. Therefore, for this design case, the overall concentration ratio is of 15. The concentration performance can be further enhanced by additional measures such as implementing a secondary concentrator or employing a seasonally adjustable reflector.

Combining the aforementioned analysis, the following conclusions can be drawn: the smaller the absolute value of the deviation angle, the better the concentrating performance of the novel concentrator. In addition, for regions with different latitudes, the deviation angle of the novel concentrator has the same distribution characteristics, such as when the novel concentrator is placed on the east-west axis and the parabolic tilt angle is the local latitude, the absolute value of the deviation angle in a year is the smallest, except in the morning and evening of winter/summer, the deviation angle of the rest time is within the range of −30 deg–30 deg. And when the deviation angle is in the range of −30 deg–30 deg, the novel concentrator has a good concentrating performance. Therefore, it can be concluded that the novel concentrator with the stationary reflector is suitable for concentrating solar energy.

This article introduces a novel design for a linear solar concentrator that consists of a stationary reflector and movable receiver, along with an innovative nonimaging concentrating method. Through mathematical modeling and optical simulation, it demonstrates that the deviation angle is solely a function of time and is independent of location. The novel concentrator achieves the best optical performance by aligning the concentrator in the east-west axial direction and adjusting the tilt angle to the local latitude. An overall concentration ratio of 15 is obtained when the deviation angle is particularly within the range of −30 deg to 30 deg. By using ideal mirrors, the optical efficiency remains above 96% across the entire range of deviation angles and reaches a peak of 99% when the deviation angle is 30 deg. These findings suggest that the novel concentrator design is globally applicable and offers consistently good performance.

This modeling study preliminarily validated the feasibility of the new approach, which shows advantages in land use enhancement and the simplification of tracking components compared to traditional optics such as parabolic troughs and linear Fresnels. Future endeavors could concentrate on two aspects: (1) evaluating the solar-to-thermal performance of this new solar concentrator configuration through coupled modeling of fluid flow and heat transfer; (2) exploring the prospects for cost reduction associated with this new design to further validate its economic benefits.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.