Solar energy has the most significant amount of energy on earth. It spreads all over the planet’s surface in the form of solar radiation with total radiation of 3,400,000 EJ per year, which can cover the yearly global energy consumption [1]. It is a clean, eco-friendly, and widely available energy on earth that can fulfill the growing global energy demands. It can be harnessed using different technologies, such as photovoltaic systems applications, which generate electricity from solar radiation and concentrated solar power applications, like linear Fresnel reflectors, solar dishes, parabolic and compound parabolic trough collectors (PTC), and power towers [2]. It has been proven that the optimum design of PTC components such as the collector structure, tracking system, and reflector can result in good PTC designs with minimal system capital cost of up to $75–$100/m², which can be considered highly cost-effective solar concentration systems [3].

There are three important aspects in PTC analysis and design: (1) PTC configurations, (2) Methods of solution, and (3) Key behaviours and trends. PTC configuration concerns the different shapes, designs, analyses, and applications of PTCs. Gharat et al. [3] discussed various configurations and components of PTCs, showing how they may have evolved to simplify PTCs' manufacture, installation, and maintenance and enhance their performance. The system's capital cost can be decreased by enlarging the parabolic trough's aperture. Hydraulic rotary actuators may be preferable to the traditional step-reduction gearbox to achieve superior tracking precision. Kasem [4] showed how to modify PTC for air conditioning and desalination in addition to heating. PTCs can also provide energy that can be utilized to run stirring engines and be stored in hydraulic accumulator systems. Jamali [5] focused on the significance of solar mirror reflectance and how it affects the thermal performance of parabolic trough solar collectors. Aluminium and silver mirrors are the most popular and often used in mirrors for PTC reflectors. However, we have to weigh the benefits of mirrors against their drawbacks in light of their immediate needs.

According to Yilmaz et al. [6], the PTC's optical and thermal characteristics are crucial for enhancing the PTC's general performance. They argued that reducing the PTC cost, increasing optical precision, and enhancing reliability are important. Better system controllability techniques can be attained using dynamic simulation of Direct Steam Generation (DSG) and solar plants. They also showed that PTCs thermal performance can be enhanced by adding turbulators, changing the absorber tube, and using nanofluids [6]. For optical analysis, Malan et al. [7] found that the flux distribution analysis is important for improving the PTC optical performance. The history of flux distribution at the focal plane with a flat receiver began in 1957 and continued into the 1980s when a cylindrical receiver was added. After 2010, most of the research used the Monte Carlo Ray Tracing (MCRT) approach for PTCs optical analysis [7].

Accurate mathematical models are critical for optimizing PTC performance, so various models have been developed to predict solar radiation absorption, heat transfer fluid flow, and heat loss mechanisms. Analytical, numerical, and experimental models have been established for PTC analysis, with each offering different advantages and limitations [8,9]. The creation of precise and effective models can aid in the optimization of PTC system design and operation, resulting in a greater uptake of this renewable energy technology [10].

Many of the studies found in the literature concentrate on PTCs configurations, analysis, and design. The present work investigates the development of efficient PTCs structure configurations and solution methods and compares them. Three methods are developed and compared regarding their accuracy, finally leading to a new method that is found to be more accurate than the traditional methods since it maximizes the positives and minimizes the negatives among the methods available in the literature. Some key behaviors and trends regarding PTC analysis and design are summarized, and some conclusions are obtained.

The solar PTC consists of a reflector with a parabolic cross-section trough shape, receiver tube, supporting structure and solar tracking system, as shown in Fig. 1. The receiver, which is the heat collection element (HCE), is a black pipe located at the trough's focal line that receives the concentrated solar energy and converts it into heat, as depicted in Fig. 1. When a solar beam falls on the trough, it is reflected to the receiver at which the heat transfer fluid is located. The heat produced by the radiation is used to warm the fluid [3,11]. This section reviews the development of PTCs configurations over the years, including the solar tracking system, the receiver, and the working fluid.

Several mathematical models and solution methods are used to obtain the optical, thermal, structural, and fluid dynamic performance of a PTC. These models are summarised in Fig. 4.

Fors the receiver to be structurally stable, the study of the absorber tube wall overheating caused by partial or total dry-out is essential. The glass cover might break due to the significant stress that deflects the absorber tube in the transverse direction. Thermal studies have revealed that the absorbers' temperature distribution is Y-axisymmetric. The little inlet and exit surfaces experience the highest thermal stress. The displacement in the Z-direction steadily rises from the inlet to the outlet side. The large temperature gradient caused by the irregular distribution of solar radiation is to blame for this. Adopting an eccentric receiver tube and a 90° orientation angle may decrease the thermal stress by about 41.4%. The internal surface centre of the receiver moves higher, but the exterior surface centre remains the same [28].

The addition of porous medium at the absorber tube's inner surface speeds up heat transfer because of the following reasons: (a) disruption of the boundary layer lowers thermal resistance; (b) an increase in turbulence intensity speeds up fluid mixing; and (c) an increase in effective thermal conductivity because of the porous medium high thermal conductivity and high fluid area density [23].

Also, increasing the wet surface area of the absorber tube's inner surface, including micro-grooves, speeds up the rate at which heat is transferred from the absorber to the working fluid. Capillary-driven flow minimises the size of the dry region in stratified and stratified-wavy flow by pushing liquid water into micro-grooves. It aids in accelerating heat transmission and lowering the thermal gradient along the circumference of the absorber wall. In addition, convective boiling is accelerated by microcircumferential grooves on the inner wall of the PTC absorber tube, increasing the heat transfer rate [29]. As the wall thickness is increased, the pressure-induced stress decreases whereas the thermal stress increases. Because steam has a poor heat transfer coefficient, the overheated part becomes critical [30].

Thermal efficiency, pressure drop, and heat transfer decrease with the increase in HTF inlet temperatures and increase with the addition of nanoparticles to synthetic oil to increase the Reynold number. Complex impacts of the Darcy number might be seen. As the Darcy number rises, thermal efficiency and the heat transfer coefficient also decrease. However, the pressure drop is inversely proportional to the Darcy number decreases. Installing a porous structure within the absorber tube improves thermal efficiency at lower Reynolds numbers (5*10⁵–15*10⁵). However, porous structures with lower Darcy numbers (Da = 0.07) have no significant effect on improving the thermal efficiency of PTCs at high Reynolds numbers [31].

The drag and friction coefficients decrease with the increase in Reynold's number, but the heat transfer coefficient increases. Speed drops off away from the receiver pipe's output line along the centre line axis. Additionally, the dispersion of velocity drops in a laminar flow is more significant than in a turbulent flow. Pressure drop is more considerable for laminar flow than the turbulent flow and decreases with increasing distance along the centre line towards the outlet line of the receiver pipe. In both laminar and turbulent flows, the temperature distribution along the centreline receiver pipe is constant. With increasing heat flow, the temperature of the parabolic collector's thermal output at the absorber tube wall rises.

In the following, we investigate the accuracy of the above-described three models in addition to the one developed by the author. The comparison between these models and the experimental data provided by Bellos et al. [10] is presented in Table 3. The experimental data provides the values for various parameters, including T_out, ηo and ηth, under pre-estimated conditions of GB,Tam,Tin and V.

Model I revealed a discrepancy in the forecasted optical efficiency, measuring at 0.08%. Conversely, the error in thermal efficiency has a positive correlation with the input temperature. This suggests that the methodology employed to estimate the output temperature may possess inherent limitations.

The results of Model II show a good agreement with the experimental data. It gives the lowest error for the thermal efficiency of 3% for T_in = 470.65 K.

Model III also shows a good agreement with the experimental data, with a maximum error in the output temperature of 4% for T_in = 470.65 K. However, this depends on the good choice of a compensation factor in the iterative technique, which is not usually easy.

Finally, the proposed model shows the best agreement with the experimental data, with a maximum error in the output temperature of 4.7% for T_in = 375.35 K, as shown in Fig. 8. Table 3 below compares the different models used in thermal analysis. The results are also depicted in Fig. 9 for a velocity of 0.0008 m³/s.

Fig. 8 shows variation of the outlet temperature predicted by the various models with the inlet temperature for V = 0.0008 m³/s.

In the present work, we investigate different PTC configurations and methods of solution found in literature and compare their accuracies with a newly developed method. We start by discussing the PTCs’ main components and their evolution with time. Then, we investigate the different mathematical models available for solving PTC’s optical, thermal, and structural behaviors in addition to the key solution techniques. Some key behaviors, and trends regarding PTC analysis and design are discussed.

Regarding the mechanical design of the reflector structure, the space tube design is preferred. For optimal reflectance, polymer-based reflective sheets are comparable to glass mirrors, but their resilience, corrosion resistance, and reflectivity may require more testing under humidity and cyclical conditions. Solar tracking systems commonly utilize the shadow effect idea, but it has been demonstrated that artificial intelligence-powered image processing may achieve higher accuracy. In addition, hydraulic rotary actuators are preferable to traditional step reduction gearboxes for superior tracking precision in significant solar fields. Ultimately, optimising the collector structure, tracking system, and reflector may lead to the PTC design with optimal performance, and it may reduce the system's capital cost by up to $75–$100/m².

When creating an adequate parabolic trough solar collector, the flux distribution analysis is one of the most critical factors. The primary method currently used for the optical analysis is MCRT, which has replaced older methods like limb darkening studies. Some researchers have investigated methods to reduce the computational time of MCRT using algorithms or two-dimensional methods. For thermal analysis, TFM is the best method to use for PTCs in direct steam generation, with different fluid phases considered. For structural analysis of PTCs for home applications, FEM is the most cost-effective and suitable method. Regarding fluid dynamics analysis, most studies use software such as ANSYS FLUENT, Ansys CFX, Thermo-fluids, Solidworks, and COMSOL Multiphysics. General fluid dynamics software can also be used for PTC simulations, such as Sol Trace, MCRT-code/Fluent, STAR-CCM+, TracePro, or MSC Nastran.

In the present work, the optical efficiency of parabolic trough solar collectors is accurately predicted by validating the proposed models using experimental data from Bellos. Three models are examined, each resulting in different thermal efficiency. However, the models encountered contradicting challenges when confronted with the experimental data and faced limitations in estimating the output temperature. After that, a new model that combines the optical model developed in [26] with the solar energy calculations based on location and date in [25] and the output temperature has resulted in the highest thermal efficiency and the best matching with the experimental results.

For future work, there is a need to improve the mathematical model for the analysis and design of multistage PTCs. This model can design, analyze, and perform parametric studies on multistage PTCs, leading to better performance and cost savings. There are also other forms of solar concentrators that need better mathematical modelling and formulation, such as Flat Solar Concentrators.