Simulated Performance Analysis of a Hybrid Water-Cooled Photovoltaic/Parabolic Dish Concentrator Coupled with Conical Cavity Receiver

Abstract

The present research discloses a novel hybrid water-cooled Photovoltaic/Parabolic Dish Concentrator coupled with conical cavity receiver and spectral beam splitter (PV/PDC-CCR-BSF). In effect, a compact co-generating solar-concentrating PV system involving a subsequent optical interface has been fully developed and numerically tested. The optical performance of the proposed hybrid solar-concentrating system was modeled and assessed using the RT 3D-4R method while the thermal yield of the system was examined using the Finite Element Method. In addition to that, different configurations of serpentine-shape embedded water-cooling pipes (rectangle, semicircle, semi-ellipse and triangle) have been tested and optimized for maximum heat collection and minimum operating cell temperature. The performance of all the tested serpentine-shape embedded water-cooling pipes was evaluated with respect to conventional serpentine-shape water-cooling pipes. The outcomes indicated that the triangular cross-section outperforms other shapes in terms of heat dissipation capabilities, with about −446 W and maximum useful thermal power in the medium of the heat transfer fluid of 11.834 kW.

1. Introduction

Hybrid concentrating photovoltaic/thermal (CPV/T) solar systems are often used to further boost the useful work potential and the cost effectiveness of CPV technology [1,2,3]. The advantage of CPV/T’s usability is that it simultaneously produces heat and electrical output power from the incoming sun irradiation focused on the receivers at high power densities. The key component of this process is the Beam Splitter Filter (BSF) [4], which separates the reflected solar rays according to their wavelengths; solar rays with wavelengths in the bandpass zone are transmitted, otherwise they are reflected. To ensure the splitting of all reflected solar rays, the BSF’s position should be carefully optimized to ensure an acceptable optical efficiency. Several surveys on CPV/T systems coupled to a BSF have been examined in the literature [5,6,7,8,9,10,11,12]. Furthermore, solar hybrid systems using beam-splitter technology have attracted particular interest, as revealed through many investigations [4,5,6]. In addition, the design of cooling systems has been optimized with the aim of ensuring maximum thermos-electrical efficiencies and a longer lifespan for the entire system. In fact, micro-channel-based heat sinks (stepwise, manifold etc.) are the most investigated and employed cooling designs to ensure uniform temperature distribution across the solar cells. However, the excessive pressure drop should be considered and carefully optimized to maintain high exegetic efficiency. The micro-channel-based heat sink can be hybridized with impinging jets to further cool down the hot spots of the HCPV cells, but this solution is difficult to manufacture. The topic of advanced cooling technology has received significant attention from numerous researchers, such as Tyagi et al. [13], Sultan et al. [14], Lupu et al. [15] and Papis-Fraczek [16], who have extensively reviewed and analyzed it. Moreover, Akbarzadeh et al. [17] developed a theoretical study of a solar-concentrating parabolic trough system, and the results showed that the potential improvements that might be achieved by the suggested concentrating solar system were dependent on the optical properties of the reflector and the effectiveness of the cooling system. Mittelman et al. [18] analyzed solar cooling with CPV/T systems by combining heat and power to simultaneously cogenerate heat and electricity. The authors coupled the CPV/T with a single-effect absorption chiller. Moreover, Mittelman et al. [19] studied desalination units driven by CPV/T systems. According to the results, solar CPV/T combined with desalination systems has a promising potential that can compete with the traditional reverse osmosis (R.O.) desalination. Othman et al. [20] tested the proof of concept of a hybrid solar CPV/T consisting of a double-pass photovoltaic thermal (PV/T) solar air concentrator with a compound parabolic collector (CPC). The study aimed not only to increase the sunspot power density focused on the solar cells, but also to enhance the rate of heat transfer to the flowing air. Hedayatizadeh et al. [21] solved numerically the output power of a hybrid PV/T system based on CPC. The accuracy of the obtained outcomes was experimentally validated through data collection and analysis. He et al. [22] proposed an analytic model to study the CPV/T system’s characteristics and assess its overall performance. Meng et al. [23] suggested a novel design of CPV/T technology, aiming to achieve a high effective concentration ratio and a more uniform distribution of the solar cells across the receiver. Koronaki et al. [24] investigated a hybrid solar collector with a flat-plate receiver. The gold goal of the work was to estimate and enhance the overall performance of the proposed collector. The outcomes proved that the investigated solar collector could work efficiently throughout the year and record a seasonal capacity of generated power production of 2.2 kW in summer, 2.8 kW in spring and 2.6 kW in autumn. Wang et al. [25] studied the thermos-economic performance of a solar combined heat and power collector (SCHP) for the supply of electrical and heat loads. The outcomes showed that the selective spectrally coating filters employed with the hybrid PV/T collectors could significantly reduce the solar cells’ temperatures. Furthermore, they had the capability to yield thermal outputs at high temperatures for steam and hot water supply. Felsberger et al. [26] studied the pre-feasibility of an industrial thermal Parabolic Trough Collector (PTC). The findings revealed that the designed collector allowed a safe range of cell temperatures below 86 °C, which corresponds to an electrical cell efficiency of above 26.5% and Heat Transfer Fluid (HTF) temperatures of about 65 °C. Renno [27] investigated the performance of a line-focus CPV/T system applied to a residential user. In this study, the cooling fluid temperatures of the CPV/T system, the temperature stratification in the tank in winter and in summer, and the temperatures of the user loads were solved and calculated. The results demonstrated that the CPV/T system could supply a significant fraction of the annual heat loads. In addition, Renno et al. [28] developed a theoretical model of an Organic Rankine Cycle (ORC) system driven by a CPV/T collector. The results proved that the proposed CPV/T-ORC combined system could be an efficient alternative for power cogeneration. Nasseriyan et al. [29] conducted an experimental and numerical study to investigate the output power of the Solarus asymmetric CPC. The findings showed that maintaining a temperature difference with respect to the ambient temperature of around 23 °C can result in a rise in the electrical and thermal output power by 25% and 3%, respectively. Gakkhar et al. [30] reported and analyzed the cogenerated power of an experimental and numerical investigation of a hybrid CPV/T collector. The obtained results demonstrated that the mean overall efficiency of the system was 61.42%, 64.61% and 66.36% for a flow rate in the inner tube of 0.075 kg/s, 0.083 kg/s and 0.091 kg/s, respectively, for an annulus flow rate of 0.008 kg/s. Ustaoglu et al. [31] developed a new design for an enhanced CPV system, a compound hyperbolic concentrator-trumpet photovoltaic thermal system (CHCT-PV/T). The study reported that the non-imaging-light CHCT-PV/T can efficiently substitute CPV/T collectors. Lin et al. [32] explored the potential breakthroughs that can be achieved successfully by combining a CPV/T collector and a variable effect absorption chiller. Gomaa et al. [33] carried out a trade-off analysis to analyze the overall performance of a CPV/T collector to explore and define the operating boundaries within which the maximum output power can be produced. The outcomes revealed that the threshold electrical and thermal power of 170 W and 580 W, respectively, can be generated at a geometric concentration ratio of 3× and a water mass flow rate of 1 kg/min. Cabral et al. [34] analyzed an experimental study of a PV/T collector, and it was found that the proposed design was able to resolve the issue of the cells overheating when the sun was overhead, in otherwise normal incidence irradiation. Han et al. [35] presented a novel hybrid PV/T system based on a Silicon on Glass (SOG) Fresnel and beam splitting filter. Using a ray-tracing method, the optical efficiency of the proposed model was evaluated. The results depicted that beam filter coating characteristics have a consequential influence on the overall system performance and behavior. Gorouh et al. [36] developed a zero-dimensional thermal model to investigate a CPV/T collector at low optical concentration levels. In fact, the thermal and electrical peak efficiencies of the proposed collector were found to be 69.6% and 6.1%, respectively. According to the literature, many research studies have investigated the cooling mechanisms of the solar cells integrated into hybrid concentrated solar systems using numerical simulations tools or by conducting experimental studies, and then analyzing the collected data. The main purpose was to boost and optimize the thermos-optical efficiencies of such systems. However, there is a lack of sufficient research exploring through trade-off analysis the impact of the design of the active cooling-based embedded serpentine pipes on the cogenerated energies of such systems. In the present work, the effectiveness of a novel hybrid PV/PDC-CCR-BSF system is investigated. The examination revolves around understanding and contrasting the thermos-optical characteristics of the optimal cooling pipe designs with the conventional one using a numerical tool. The objective is first to model the opto-thermal behavior of the entire system, which has never been executed before. Then, the proposed hybrid system aims to identify the most effective and convenient alternative solution of conventional active cooling-based serpentine pipes. The performance of all the tested serpentine-shape embedded water-cooling pipes (rectangular, semicircular, semi-elliptical and triangular) was evaluated with respect to conventional serpentine-shape water-cooling pipes. The comparison of the performance of the tested serpentine-shape embedded water-cooling pipes indicated that the triangular cross-section outperformed other shapes in terms of heat dissipation capabilities and maximum useful thermal power in the medium of the heat transfer fluid.

2. Methodology

To design the entire system, the geometry modeling equations of the parabolic dish, conical cavity receiver and beam splitting filter were first formulated. The second step was dedicated to numerically investigating the spectral optical behavior of the hybrid system and then the heating load and the useful gain energy that would be removed within the domain of the working fluid for different cooling pipe cross-section shapes.

2.1. Geometry Modeling

The schematic of the investigated system is depicted in Figure 1 and Figure 2. Keeping the same focal point and the symmetric axis, the BSF is positioned above the parabolic dish collector (PDC) to concentrate almost all of the reflected solar rays. In the present study, the BSF is considered as the key element of the proposed hybrid solar system since it has a significant impact on the thermos-optical performance and responsiveness of the whole system. The principal characteristic of the BSF is the wide spectral selective coating which divides the concentrated solar rays according to their wavelengths. In fact, the spectral reflectivity (ρ) of the BSF selective coating is expressed as the following [5,7,8]:

$$\left\{\begin{array}{c}\rho \left(\lambda \right)=0 for \lambda \ge \hslash c/{\mathcal{E}}_g\\ \rho \left(\lambda \right)=1 for \lambda <\hslash c/{\mathcal{E}}_g\end{array}\right.$$

(1)

where the solar cell’s bandgap is denoted by ${\mathcal{E}}_g$, the Planck’s constant is ℏ, and c denotes the speed of light in vacuum.

Equation (1) illustrates the splitting process of the reflected rays intercepted by the BSF surface. Solar rays having wavelengths below that of the semiconductor energy bandgap will be transmitted throughout the splitter filter placed at the focal plane of the concentrator. The other sun rays will be reflected back and projected on the plane of the solar cells, positioned at the reference zero-height of the considered vertical datum.

The parabolic dish concentrator is considered as a full tracker solar system to enable the exclusion of any further optical losses rather than that of the BSF.

Table 1. Geometrical parameters.

	Item	Symbol	Values [m]
Parabolic reflector	Aperture radius	wc	2.5
	Focal length	f	3
Conical cavity receiver	Upper radius	rmax	0.1
	Lower radius	rmin	0.05
Beam splitter	Aperture radius	wsf	0.1
	Focal length	fs	0.05
PV cells	Aperture width	wsc	1.4

shows the main components and dimensions of the proposed hybrid system.

2.1.1. Parabolic Dish Collector

Several optical and geometric parameters can influence the processes for concentrating sunlight, such as the size of the reflectors and receivers, the spectral response of the parabolic collector and receiver materials, and the accuracy and resolution of the solar tracking system [37,38,39,40,41]. In fact, the parabolic collector equation is as follows [42]:

$$z=\frac{x^2+y^2}{4f}$$

(2)

where f is the focal length of the parabolic collector.

Thus, the reflector depth (d_c) is deduced and can be written as follows [42]:

$$d_c=\frac{w_c^2}{4f}$$

(3)

where w_c is the reflector aperture radius.

The PDC system’s effectiveness is contingent upon the material and design selection of the receiver, which has an enormous influence on the yield of the whole system. The selection of the optimal design should consider the main purpose or application of the developed system [41]. The ratio of the intake aperture area to the exit receiver area is known as the geometric concentration ratio (GC) [42]:

$$GC=\frac{A_{inlet}}{A_{outlet }}$$

(4)

where $A_{inlet} \mathrm{and} A_{outlet}$ are the inlet aperture area and the outlet receiving area, respectively.

As presented by Kalogirou [43], for higher conversion efficiencies, in systems converting the solar radiation into heat by absorption, the GC should range between 600 and 2000×. Thus, the design of the receiver should take into account the geometrical concentration ratio and the optical properties of the involved materials [44].

2.1.2. Conical Cavity Receiver (CCR)

The performance of the parabolic solar concentrator system was studied and evaluated for different shapes of receivers [45,46,47,48,49,50,51,52,53]. According to the literature, these shapes can be categorized into two groups depending on their design: either a simple shape such as a cylindrical cavity [45], conical cavity [48,49,50], spherical cavity and flat disk [54], or a more intricate shape [51,52]. In fact, a comparison between these different shapes in terms of optical efficiency was conducted to determine the most efficient geometry [46,47]. This article focuses on the geometry of the conical receiver as the most efficient of all other simple receiver geometries [46,47].

The CCR equation In the Cartesian system coordinates is the following [54]:

$$\mathrm{z}=z_{cr}+[r_{max}-\left({\left(x_{cr}-x\right)}^2+{\left(y_{cr}-y\right)}^2\right)\cot ({\theta }_{op})]$$

(5)

where $x_{cr},y_{cr} \mathrm{and} z_{cr}$ are the center’s coordinates, r_max and r_min are the radius of the conical upper and lower bases, and ${\theta }_{op}$ is the CCR’s half opening angle, which is described as:

$${\theta }_{op}={\tan }^{-1}(\frac{r_{max}-r_{min}}{h_r})$$

(6)

where h_r is the conical cavity height.

The formula of the geometric concentration ratio is then:

$$GC=\frac{w_c^2}{\left(r_{max}+r_{min}\right)*\sqrt{{\left(h_r\right)}^2+(r_{max}-r_{min}{)}^2}}$$

(7)

The collector aperture area $A_c$ is determined from the relation:

$$A_c={\pi w}_c^2$$

(8)

while the receiver surface area $A_r$ is expressed as follows:

$$A_r=\pi \left(r_{max}+r_{min}\right)\sqrt{{\left(h_r\right)}^2+(r_{max}-r_{min}{)}^2}$$

(9)

The main objective of this article is to numerically model the opto-thermal behavior of the entire system (water-cooled PV/PDC-CCR-BSF) and to demonstrate the upper hand and superior performance of such novel technology to pave the avenues for future experimental prototyping and validation. A helical water tube is installed across the innermost portion of the conical cavity receiver. The conical height serves to carry out the circulation of cooling water from bottom to top.

2.1.3. PV Cells

A small-sized PV panel serves as a second receiver in the design of the proposed hybrid PV/PDC-CCR-BSF system. The objective is to generate electricity while simultaneously producing heat [55,56,57,58,59]. In fact, single-junction solar cells are placed at the vertical datum origin height, which is at the vertex of the parabolic dish. Since it is well known that solar cells are very sensitive to external influences (environmental and climatic conditions), the solar-cell panels are made up of various layers (Figure 1). These layers are as follows, from top to bottom: (1) the topmost layer of the solar panel is a glass layer, which is a protective layer made up of a highly transparent medium that is resistant to several conditions; (2) the solar cells are then covered with one ethylene vinyl acetate (EVA) layer (3) and then sandwiched between other two EVA layers, to protect the solar cells’ circuits from soft vibrations and shocks and to prevent the penetration of dirt and moisture; (4) the panel back sheet is a thin and opaque film that is composed of Thiophene-benzothiazole (TBT) to reduce the effects of the ultraviolet exposures from the bottom. For the PV/T panel, a cooling pipe (5) is embedded in the back sheet of the PV panel. In fact, once the reflected solar rays from the BSF reach the hybrid PV/T panel, a major fraction of the heat dissipated by the solar cells will be transferred to the water flow circulating inside the cooling pipe. The conventional cooling pipe PV system, carrying water, is a serpentine cylindrical shape [60].

Table 2. Optical and thermophysical properties of the different PV panel layers.

	Glass	EVA	Cell	Backsheet
Material	Silica glass	Elvax 250 (28% VA, 25 MI)	polycrystalline cell	Thiophene-Benzothiazole
Width (m)	1.4	1.4	0.2	1.4
Length (m)	1.4	1.4	0.2	1.4
Thickness (m)	0.0032	0.0017	0.0002	0.02
Refractive index	1.45	1.49 [61]	4.052	1.46 [62]
Thermal conductivity (W/m K)	1.83	0.15	60	0.15
Specific heat capacity (J/Kg K)	730	2090	320	1250
Density (Kg/m3)	2203	950	5323	1170

recapitulates the main optical and thermophysical properties of the different PV panel layers.

The aim of the serpentine-shaped water flow embedded in the back sheet of the PV panel is to ensure optimal operating conditions for the solar cells and to avoid any extreme heat loads and temperatures for optimum electrical efficiencies. While cooling pipes have long been used to remove heat from panels, research needs to be carried out on the ideal cross-section configuration of the tubing in relation to the rear sheet enabling efficient heat dispersal. In this regard, the computational fluid dynamics code ANSYS FLUENT was utilized in order to simulate and expose the serpentine-shaped water flow integrated into the heat sink, depicted in Figure 2, to perform evaluation with four distinct cross-sections.

2.1.4. Beam Splitter Filter

The paraboloidal equation is defined as follows [42]:

$$z=\frac{x^2+y^2}{4f_s}+z_{fs}$$

(10)

where $z_{fs}$ is the elevation of the filter over the dish reflector and $f_s$ is the focal length.

As mentioned before, the BSF splits the sunlight according to the wavelengths. Over the BSF area, a spectrally selective coating is disposed. The bandpass section of the spectral width is selected when the wavelength is above 1100 nm [8,63].

2.2. Simulation Methods

2.2.1. Optical RT 3D-4R Method

The investigation of the optical performance of the hybrid PV/PDC-CCR-BSF is carried out with numerical modeling based on the ray-tracing technique (RT 3D-4R) [54,64,65,66,67]. This method is developed to compute the optical path length and ray intensities for unpolarized electromagnetic waves. The solar disk theory is assumed, and the sun subtend angle ${\sigma }_s/2$ is set equal to 16 and 32 arcminutes [68]. The dependence of the refraction indexes on the temperatures is neglected.

Each integration node over the dish reflector area can be coordinated as the following:

$$\left\{\begin{array}{c}{x}_c\left({i_c,j}_c\right)=-r_c\left(i_c\right) cos\left({\phi }_c\right(j_c\left)\frac{\pi }{180}\right)\\ y_c\left({i_c,j}_c\right)=-r_c\left(i_c\right) sin\left({\phi }_c\right(j_c\left)\frac{\pi }{180}\right)\\ z_c\left({i_c,j}_c\right)=\frac{{{x_c\left({i_c,j}_c\right)}^2+y_c\left({i_c,j}_c\right)}^2}{4 f}\end{array}\right.$$

(11)

where $\left\{\begin{array}{c}{r}_c\left(i_c\right)=(i_c-1){dr}_c\\ {\phi }_c\left(j_c\right)=(j_c-1){d\phi }_c\end{array}\right.,\mathrm{a}\mathrm{n}\mathrm{d} {dr}_c,$ and ${d\phi }_c$ are the two reflector spatial steps.

The conical cavity receiver coordinates $x_r(i_r,j_r)$,$y_r(i_r,j_r)$ and $z_r(i_r,j_r)$ could be expressed as follows:

$$\left\{\begin{array}{c}{x}_r\left(i_r,j_r\right)=x_{cr}-r_r\left(i_r\right) cos\left({\phi }_r\right(j_r\left)\frac{\pi }{180}\right)\\ {\mathrm{y}}_{\mathrm{r}}\left(i_r,j_r\right)=y_{cr}-r_r\left(i_r\right) sin\left({\phi }_r\right(j_r\left)\frac{\pi }{180}\right)\\ {\mathrm{z}}_{\mathrm{r}}\left(i_r,j_r\right)=z_{cr}+(r_{max}-r_r(i_r\left)\right)\ast \cot (\alpha )\end{array}\right.$$

(12)

where $\left\{\begin{array}{c}{r}_r\left(i_r\right)=(i_r-1){dr}_r\\ {\phi }_r\left(j_r\right)=(j_r-1){d\phi }_r\end{array}\right.,$ and ${dr}_r$ and ${d\phi }_r$ are, respectively, the two reflector space increments.

The block diagram of the developed RT3D-4R algorithm is described in Figure 3.

2.2.2. Heat Transfer Modeling of the PV and Conical Cavity Receivers

The numerical model and simulation of the proposed solar system is based on Finite Element Method analysis. Figure 4 shows the generated mesh over both the conical cavity receiver and the PV solar cells. The physics-controlled mesh is adapted according to the physics settings for any boundary or domain condition and the constraints applied to the developed model geometry.

Heat Transfer on Sinusoidal Coiled Tube of the PV Cells Panels

The 4 different embedded serpentine-shaped water flow configurations were numerically modeled in the ANSYS FLUENT module. The front side of the PV panel was subjected to a constant heat flux of $6400 \mathrm{W}/{\mathrm{m}}^2$ while the sides and the back of the panel were exposed to surrounding air at $T_{sur}=298 \mathrm{K} \mathrm{and} h=5 \mathrm{W}/{\mathrm{m}}^2·\mathrm{K}$, as shown in Figure 5. An emissivity of $\epsilon =0.9$ was considered for all surfaces exposed to surrounding air.

All cross-sectional areas, the pipe’s overall length, its shape (serpentine), its entry temperature, its velocity and the mass flow rate with which water passed through the pipe were maintained uniform for homogeneity across the board. The dimensional features of the tubing sectional areas are listed in

Table 3. Channel/pipe cross-section geometry and dimensions for the same cross-sectional area.

No	Geometry	Dimensions	Width w (mm)	Perimeter * p (mm)	Depth d (mm)	Hydraulic Diameter Dh (mm)
1.			45.14	141.8	22.57	22.6
2.			50.00	132.0	16.00	24.2
3.			63.66	158.3	16.00	20.2
4.			80.00	169.4	20.00	18.9

* Perimeter p is the sum of length of the sides of the channel minus the top side or width w.

, and

Table 4. Boundary conditions common to all cases.

Parameter	Value
Pipe wall material	Copper
Pipe wall thickness (mm)	0.5
Pipe inlet/outlet areas (m2)	0.0008
Pipe length (m)	12.637
Inlet temperature (K)	298.0
Inlet flow velocity (m/s)	0.125
Inlet mass flow rate (kg/s)	0.10
Air temperature (K)	298.0
Heat flux (W/m2)	6400.0
Material surface emissivity	0.9

lists the boundary conditions common to all cases, as shown in Figure 2.

The photovoltaic module and tubing configuration computing domain was modeled in the Design Modeler module of ANSYS. The domain was then discretized using unstructured mesh made up of tetrahedral cells. To properly resolve the fluid boundary layer adjacent to the solid walls, the first height, the rate of increase and number of layers were set, to ensure that the wall y+ < 1. For the solution, the RANS equations representing the mass, momentum and energy conservation equations, and the steady pressure-based model employing the Spalart–Allmaras model were used.

The mass, momentum and energy conservation equations for a steady, three-dimensional flow are given by:

$$\mathrm{Mass}: \nabla ·\left(\rho \stackrel{⃑}{V}\right)=0$$

(13)

$$\mathrm{Momentum}: \nabla ·\left(\rho \stackrel{⃑}{V}\stackrel{⃑}{V}\right)=-\nabla p+\nabla ·\mu \left(\nabla \stackrel{⃑}{V}+\nabla {\stackrel{⃑}{V}}^T-\frac{2}{3}\nabla ·\stackrel{⃑}{V}I\right)+\rho \stackrel{⃑}{g}$$

(14)

$$\mathrm{Energy}: \nabla ·\left\{\stackrel{⃑}{V}\left(\rho E+p\right)\right\}=\nabla ·\left(k_{eff}\nabla T-{\sum }_j{h}_j{\stackrel{⃑}{J}}_j+{\stackrel{̿}{\tau }}_{eff}·\stackrel{⃑}{V}\right)$$

(15)

where transfers of energy via conduction, diffusion and dissipation through viscosity are represented by the initial three terms on the right-hand side of Equation (15).

A second-order upwind approach is employed to discretize Equations (13)–(15), which are subsequently resolved using a pressure–velocity coupling mechanism. The criteria for solution convergence are based on residual values below 1 × 10⁻⁵ and net heat flux and net mass flow rate values below 1 × 10⁻².

3. Results of Simulation Methods

3.1. Validation of the RT3D-4R Technique

To validate the developed in-house optical model, the obtained results are compared against those obtained using other modeling and simulation tools such as the MATLAB program (R2020a) [53], COMPREC code [53] and experimental measurements by Jeter [69]. The same material and geometrical properties have been assumed to deliver accurate and fair validation:

• Focal distance: $f$ = 1 m;
• Rim angle: 45° and 60°;
• Sun subtend disk angle: ${\theta }_s$ = 16′ and 32′;
• Solar irradiance: $I_r=1000 \mathrm{W}/{\mathrm{m}}^2$.

Figure 6 depicts the comparison between the experimental and numerical results in terms of LCR as a function of the normalized sunspot coordinates for two reference sun subtend angle values. The normalized sunspot coordinate represents the position of a point in the flat-disk receiver, normalized with respect to the focal distance of the parabolic dish collector. It is evident from Figure 6 that the optical approach currently being used yields a pattern that exhibits good agreement with both numerical and experimental results.

3.2. Hybrid Parabolic Dish Collector Using Spectral Beam Splitter Technology

3.2.1. Beam Splitter Filter and Concentrated Solar Rays

After the incident rays have been reflected by the paraboloidal mirrors of the PDC, they reach the boundaries of the BSF, and after that they are filtered to be either transmitted to the outer walls of the cavity receiver or reflected onto the solar cells array. The bending rate of the transmitted rays depends on the refractive index of the BSF and the angular displacement of the incident rays away from the normal to the element boundaries of the BSF. The reflected rays from the BSF are considered as a new set of incident rays and are projected onto the solar cells’ surface area with minimum interception angle. In fact, the BSF technology splits the impinged rays as function of their wavelengths. Almost 85% of the spectrum light density, which is reflected from the BSF, is concentrated on the PV/T panel. The rest of the spectrum light density is transmitted and reaches the CCR [8].

3.2.2. Concentration Ratio and Temperature Distribution on the CCR

In this section, the concentrated solar density (CSD) distribution on the conical cavity receiver is discussed. With the same case studies investigated in previous work [54], the simulation details are used with characteristics: w_c = 2.5 m, f = 3 m, r_max = 0.1 m, r_min = 0.05 m, θ_S = 16′ and E_S = 1 kW/m². The simulation outcomes are illustrated in Figure 7. It is evident that the CSD distribution exhibits a non-uniformity aspect along the cavity’s height. Furthermore, the highest CSD value is around 40 kW/m² situated at 0.1 m from the conical cavity bottom base. Compared with that found by Daabo et al. [46,47], this finding proves the reliability of the presented simulation method.

The optical efficiency is the ratio of the average local concentration ratio to the geometric concentration ratio [70]. The optimization of the hybrid PV/PDC-CCR-BSF system in terms of optical efficiency is presented in [54].

Figure 7b shows the temperature distribution on the CCR where an increase in the temperature with height is observed. The top part of the CCR is found to have the maximum surface temperature where maximum temperature values reach more than 170 °C. This temperature trend is influenced by the convective heat transfer rates of the water flow inside the helical tube, which are faster than the conduction heat transfer flow over the cavity inner walls exposed to the transmitted radiative sun power intensities. The cold-water inlet is fed in at the bottom base of the cavity, with the water temperature increasing from the inlet value of 20 °C to around 104 °C at the top of the CCR before leaving the helical tube.

3.2.3. Heat Load and Temperature Distribution of the Solar Cells with Conventional Coiled Pipe

Approximately 66% of the concentrated solar energy received by the BSF device is reflected toward the PV cells. As stated earlier, the BSF device is positioned in such a way that its focal point matches the primary parabolic reflector. Consequently, the solar rays bouncing off the BSF maintain a path parallel with the parabolic reflector’s symmetry axis and are then finished by being absorbed by the PV cell positioned on the parabolic reflector’s center. As a result, the CSD distribution over the PV cells area is spread out uniformly, reaching a maximum value of 8 kW/m² (Figure 8). At the edges, the CSD has decreased since the sunspots are extending over a smaller area than that of the PV cells.

Figure 8b,c depict the temperature distribution of the cooled solar cells using conventional water-cooled pipe. For a full pipe flow, the maximum solar cell temperature has been decreased from 90.5 °C to 49.6 °C.

3.2.4. Heat Load and Temperature Distribution of the Solar Cells with Different Coiled Pipe Forms

This article aims to optimize the geometry of the serpentine-shaped water flow embedded in the heat sink placed underneath the solar cells of the PV panel. The conventional cooling pipe, characterized by a cylindrical shape, is in direct contact with the back sheet of the panel. Thus, the main idea is to examine the effect of different cooling pipe cross-sections with the same flow area but different contact areas that can lead to higher heat transfer to the cooling pipe. In this regard, four different cross-section area shapes are analyzed. The areas and pipe length are fixed at 0.0008 m² and 12.637 m, respectively.

The distribution of heat flux on the Interface surface along the coolant conduit and panel rear sheet is depicted in Figure 9 for the four different cooling pipe cross-section cases. Since the contact sides of the cooling pipe channel have different widths, the heat transfer is directly proportional to the contact areas. However, a small difference is observed since the difference between the contact areas of the four different cross-sections is small and is listed in Table 4 as the width parameter for comparison.

Figure 10 depicts the temperature distribution on the serpentine shape of the different serpentine-shape embedded cooling pipes. The inlet water temperature is fixed at 298 °K for all cases. However, one can observe that the outlet temperature values differ for the four cases, see

Table 5. Temperature and thermal transfer rate description for the top, bottom, sides, and underneath of the sheet of the channel.

Geometry	${\mathit{T}}_{\mathit{o}\mathit{u}\mathit{t}}$ (K)	$∆\mathit{T}$ (°C)	${\mathit{T}}_{\mathit{c}\mathit{h},\mathit{t}\mathit{o}\mathit{p}}$ (K)	${\mathit{T}}_{\mathit{c}\mathit{h},\mathit{s}\mathit{i}\mathit{d}\mathit{e}\mathit{s}}$ (K)	${\mathit{T}}_{\mathit{b}\mathit{a}\mathit{c}\mathit{k}}$ (K)	${\dot{\mathit{q}}}_{\mathit{c}\mathit{h},\mathit{t}\mathit{o}\mathit{p}}$ (kW)	${\dot{\mathit{q}}}_{\mathit{c}\mathit{h},\mathit{s}\mathit{i}\mathit{d}\mathit{e}\mathit{s}}$ (W)	${\dot{\mathit{q}}}_{\mathit{b}\mathit{a}\mathit{c}\mathit{k}}$ (W)	${\dot{\mathit{q}}}_{\mathit{o}\mathit{u}\mathit{t}}$ (kW)
	326.2	28.2	341.28	311.93	344.76	11.690	−136	−855	11.491
	326.6	28.6	339.35	311.72	342.70	11.765	−154	−779	11.548
	327.9	29.9	334.65	312.10	337.60	11.941	−149	−603	11.730
	328.3	30.3	330.29	313.82	333.10	12.098	−195	−446	11.834

. In the case study of the conventional cooling pipe, the variation in temperature across the water at the point of entry and the outflow is about 19 °C, which is lower than that found for the other geometries. In fact, the temperature difference is about 28.6 °C for the rectangular cooling pipe cross-section, 28.2 °C for the semicircular cooling pipe case, 29.9 °C for the semi-elliptical cooling pipe case and 30.3 °C for the triangular cooling pipe case. Thus, the largest is found for the triangular cross-section cooling pipe by virtue of its larger width, see Table 3.

Table 5 summarizes the simulation results of the four different serpentine-shape embedded water flow cross-sections.

In Table 5, it is clear that the triangle cross-section appears to be the best choice in comparison to the others in terms of extracting heat from the rear of the sheet in that it results in the maximum thermal energy rate to the cooling water (${\dot{q}}_{out}=11.834$ kW), which is the thermal power of the liquid out of the pipe, and the lowest back-sheet temperature (T_back = 333.1 K) and heat flux (${\dot{q}}_{back}=$ −446 W), meaning more cooling capability than the other cross-section cooling pipes.

4. Comparison of Previous Similar Studies

This article aims to optimize the geometry of the serpentine-shaped water flow embedded in the heat sink placed underneath the solar cells of the PV panel. The optical performance of the proposed hybrid solar-concentrating system was modeled and assessed using the RT 3D-4R method, while the thermal yield of the system was examined using the Finite Element Method. Different configurations of serpentine-shape embedded water-cooling pipes (rectangle, semicircle, semi-ellipse and triangle) have been tested and optimized for maximum heat collection and minimum operating cell temperature. The performance of all the tested serpentine-shape embedded water-cooling pipes (rectangle, semicircle, semi-ellipse and triangle) was evaluated with respect to conventional serpentine-shape water-cooling pipes. The outcomes indicated that the triangular cross-section outperforms other shapes in terms of heat dissipation capabilities and maximum useful thermal power in the medium of the heat transfer fluid.

Table 6. A comparative overview of previous studies.

Authors	Numerical/Experimental	Concentrator Solar System Device	Type of Cooling System	Optical Concentration Ratio	Heat Transfer Fluid	ΔT (°C)	Efficiencies/Electrical or Thermal Power	Main Outcomes/Milestones
Akbarzadeh et al. [17]	Numerical & experimental	One-axis tracked east-west parabolic trough concentrator	Thermosyphon external to the structure	20 suns	Water & R-11	21	Electrical power = 20.6 W	The performance improvement of the proposed concentrating solar system is caused by enhancement of the primary optical stage properties by the effectiveness of the cooling mechanism.
Othman et al. [20]	Numerical & experimental	Parabolic concentrator	Fins peripheral to the structure	1.95 suns	Air	18	Thermal efficiency = 70% Electrical efficiency = 8%	The surge in the cogeneration effectiveness of the system.
Hedayatiza-deh et al. [21]	Numerical	Compound parabolic concentrator	Duct external to the system	2 suns	Water	7.7	Thermal efficiency = 51.4584% Electrical efficiency = 9.6113%	The investigation of the thermal and electrical performance of the hybrid PV/T-system-based compound parabolic concentrator.
He et al. [22]	Experimental	Non-imaging-based-diffuse-reflection PV/T concentrator solar system	Channel exterior to the arrangement	≤2 suns	Water	41	Average output electrical power = 22.9 W	The proposed optical design boosted the productivity of the CPV/T system.
Koronaki et al. [24]	Numerical & experimental	Compound parabolic concentrator	Duct external to the system	1.414 suns	Water	7.8	Useful energy = 2278 W Efficiency = 30%	Enhanced optical design of the hybrid solar collectors with flat-plate receiver. Long-term collected data and performance analysis.
Wang et al. [25]	Experimental	Parabolic trough concentrator	Channel external to the system	1.25 suns	Water	0–30	Fraction of solar energy = 60% of annual hot water load Net electrical output = 60% of the annual electric loads	Lower cell temperature by the implementation of a spectral beam splitter, higher steam production, and more significant hot water supply.
Felsberger et al. [26]	Numerical & experimental	Parabolic trough concentrator	Channel	900	Water/glycol	20.5	Electrical power = 3.87 W	The potential to combine the simultaneous generation of both heat and electricity by retrofitting traditional parabolic collector which is typically only employed in thermal systems with multijunction solar cells.
Renno et al. [27,28]	Numerical & experimental	Parabolic trough concentrator	Channel	100	Water	28–56	Thermal energy = 23 kWh–63 kWh Electrical energy = 33 kWh–50 kWh	When considering energy options, an integrated CPV/T-ORC system proves to be the best choice for a commercial consumer.
Nasserian et al. [29]	Numerical & experimental	Compound parabolic concentrator	Channel	1.52	Water	35.1	Electrical power = 142.2 W Thermal power = 511 W	The improvement of the thermal and electrical production of a Solarus Power Collector.
Gakkhar et al. [30]	Numerical & experimental	Parabolic trough concentrator	Pipe	6	Water	28.7	Thermal efficiency = 49.48% Electrical efficiency = 12%	The evaluation of the performance of hybrid system in terms of respective efficiencies.
Ustaoglu et al. [31]	Numerical	Compound hyperbolic trumpet concentrator	Pipe	1.94	Water	1.2	Power output = 42.9%	Offer the CPV solution’s benefits in order to attain a more affordable design and superior performance.
Lin et al. [32]	Numerical	Parabolic dish concentrator	Channel	10	Water	30	Exergy efficiency = 32–33%	The suggested system provides variable cogeneration of electricity and cooling.
Gomaa et al. [33]	Numerical & experimental	Fresnel flat mirror concentrator	Pipe	3 suns	Water	8–13.5	Electric power = 170 W Thermal power = 580 W	Both electrical and thermal efficiency was increased by raising the cooling water capacity in both setups. This maximized the heat recuperation from the photovoltaic module (PV).
Gabral et al. [34]	Experimental	Parabolic trough concentrator	Channel	2	Water	25	Electrical efficiency = 8% Thermal efficiency = 58.3%	Moving the highest power of energy at normal incidence angles to more extreme incidence angles.
Hen et al. [35]	Mathematical	Flat mirror concentrator	Channel exterior to the system	24 suns	Liquid	23.9	Optical efficiency = 84.82% Overall efficiency = 46.77%	Enhancement of the optical behavior and overall system performance due to the optimized beam filter coating characteristics.
Gorouh et al. [36]	Numerical & experimental	Parabolic trough concentrator	Channel	2.7	Water	40	Electrical efficiency = 6.1% Thermal efficiency = 69.6%	Finding the essential CPVT collector design parameters for additional improvements.
Present work	Numerical	Hybrid photovoltaic/parabolic dish concentrator	Variously sized and positioned channels that are incorporated onto the PV rear sheet	6.4 suns	Water	29–33	Optical efficiency = 67% Thermal power = 11.834 kW	PV module back sheet with 3D-printed and embedded channels of dissimilar cross-sections and positions with respect to the heat transfer source.

provides a comparative overview in terms of improving both thermal and electrical efficiency. Table 6 provides various information like whether the analysis was conducted in various studies experimentally or numerically, the concentrator solar system used in these studies, the type of cooling system used, the optical concentration ratio, the heat transfer fluid, change in temperature, various efficiencies and the main outcomes of these studies.

5. Discussion

A crucial factor in ensuring optimal performance of the hybrid PV/PDC-CCR-BSF system is the integration of a sophisticated cooling pipe system to avoid cell overheating, which may result in an output voltage drop and a short lifespan of the solar cells. In this study, the design optimization, a serpentine-shape embedded water flow cooling pipe, was carried out by trade-off analyses using the optical ray-tracing technique and the Finite Element Methods, using ANSYS FLUENT software (version 2019R3) to simulate the relevance of the key designing parameters of the cooling system configurations.

To achieve this goal, four different serpentine-shape water-cooling pipes embedded in the plate of the underneath heat sink were investigated, while maintaining the same surface area; the four tested geometries were semicircular, semi-elliptical, rectangular and triangular pipe. Simulation was carried out to assess the thermo-optical efficiency of the selected serpentine-shape embedded cooling pipe configurations. A comparison between the proposed four different embedded water flow pipes’ cross-sections and the traditional circular cross-section cooling pipe was assessed. The heat exchange area between the panel and the pipe changed from one form to another.

The simulation’s findings indicated that the amount of heat dissipated is contingent upon the contact area utilized for transfer. Hence, the triangular cross-section shape showcases the utmost performance capability:

• The triangular cross-section shape resulted in the largest increment in temperature, in other words the maximum useful gain energy, recording the highest increment in temperature of 30.3 °C. The triangular cross-section shape recorded the fastest rate at which heat was transferred to the water flow in the serpentine-shape embedded cooling pipes.
• The same cooling pipe design, i.e., the triangular cross-section shape, recorded the lowest heat sink temperature resulting in the highest heat transfer rate to the cooling water flow of 11.834 kW.

6. Conclusions

This current work discloses a novel hybrid water-cooled Photovoltaic/Parabolic Dish Concentrator coupled with a conical cavity receiver and spectral beam splitter (PV/PDC-CCR-BSF). The optical modeling has been developed based on the ray-tracing method (RT3D-4R) and the thermal analysis has been elaborated using Finite Element Method software (COMSOL Multiphysics, version 5.2). In the current study, different configurations of serpentine-shape embedded water-cooling pipes (rectangle, semicircle, semi-ellipse and triangle) have been tested and optimized for maximum heat collection and minimum operating cell temperature. The performance of all the tested serpentine-shape embedded water-cooling pipes was evaluated with respect to the conventional serpentine-shape water-cooling pipes. The findings of this study lead to the following conclusions and suggestions:

• First, the optical validity of the RT3D-4R method has been approved to simulate the performance of the parabolic dish concentrator for two case studies (flat receiver and conical receiver).
• Second, the thermo-optical characteristics of the proposed PV/PDC-CCR-BSF system have been optimized and performed.
• Third, the thermo-optical performance of the proposed system for four different cooling pipe cross-sections of serpentine-shape embedded water-cooling pipes have been numerically simulated and the results compared against those of the conventional circular cross-section cooling pipe placed underneath the PV panel rear sheet. The outcomes showed that:

  ○

  The heat sinks and the water-cooling pipe’s interface width determine the heat flux pattern and transfer rates in a comparative manner.

  ○

  The water temperature increment between the inlet and outlet ports of the embedded serpentine water-cooling pipe is equal to 28.6 °C for the rectangular cross-section cooling pipe, 28.2 °C for the semicircular cross-section cooling pipe, 29.9 °C for the semi-elliptical cross-section cooling pipe and 30.3 °C for the triangular cross-section cooling pipe.

  ○

  Amongst the four different cross-sections, the triangle cross-section is found to have a greater heat extraction capability than the other cross-sections. It gives the highest heat rate to the cooling water, the lowest back-sheet temperature and heat flux (${\dot{q}}_{back}=$ −446 W), yielding the highest amount of removed thermal power in the water flow medium.

  ○

  The outcomes demonstrated that the traditional cooling pipe, which is also placed underneath the solar cells, has poorer cooling performance in comparison with the four serpentine-shape embedded water-cooling configurations.

As an avenue of the present research, further analysis and enhancement of the serpentine-shape embedded water-cooling configurations is carried out in another concurrent study.