A Detailed Numerical Study of a Nanoﬂuid-Based Photovoltaic/ THERMAL Hybrid System under Non-Uniform Solar Flux Distribution

: The concentrated photovoltaic/thermal system (CPVT) adopting spectral beam splitting is a promising ﬁeld of solar energy research. However, the thermo-electric properties of ﬂuid-based CPVT collectors, which depend strongly on the non-uniform concentrated energy ﬂux, remain unclear. This study aims to ﬁll the gap and explore the thermo-electric properties of ﬂuid-based CPVT collectors under non-uniform energy ﬂux based on the ﬁnite volume method (FVM) with the Monte Carlo Ray-Trace (MCRT) method. The actual solar ﬂux distribution on the receiver surface is obtained using Tracepro software. Then, the realistic non-uniform energy ﬂux was employed in ANSYS Workbench 2022R1 software as a boundary condition to increase the accuracy of the CFD modeling of the system. The model is validated by comparing the results of the reference data. Moreover, the impact of uniform and non-uniform energy ﬂux on the PV cell temperature is analyzed. In addition, the effects of mass ﬂow rate on the electrical and thermal performance of the system are investigated. The results show that the PVT hybrid system has high conversion efﬁciency, with a total efﬁciency of more than 50%. Notably, the extreme non-uniformity of the solar-concentrated energy ﬂux can result in local overheating of the PV cell, which may lead to irreversible damage.


Introduction
Against the depletion of traditional energy sources, humanity is constantly looking for alternatives to conventional fossil energy sources to alleviate the energy crisis. Under these circumstances, various new energy sources, such as solar, wind, tidal, and geothermal, begin to emerge. The development and utilization of renewable energy have become the focus of attention today. Among the numerous renewable energy sources, solar energy is considered one of the most promising energy sources due to its abundant reserves, wide distribution, and clean and pollution-free characteristics. There are 3.85 × 10 24 J of radiation quantities to the earth's surface per year, and only the portions of 0.5 h could provide the needs of the global energy demand for a whole year. Accordingly, the efficient utilization technology of solar energy has a broad development perspective [1,2].
The impending demand for solar energy development is to improve utilization efficiency. One of the significant trends is the photovoltaic/thermal (PVT) hybrid utilization technology [3,4]. The current application of PVT technology has mainly relied on flat-plate PV modules due to their large-scale commercial application. However, the total efficiency of this hybrid approach is quite limited, and the grade of the heat source obtained could be much higher. Based on this, more and more attention has been paid to composite technology using concentrated technology, known as CPVT. The CPVT hybrid system achieves higher total utilization efficiency and grade heat [5][6][7]. Going one step further, when high-grade thermal energy has the potential to be used directly for power generation, aims to fill the gap and explore the thermo-electric properties of fluid-based CPVT collectors under non-uniform energy flux. A low-concentrating CPVT collector based on splitting NF, built by our team, was selected, and a comprehensive 3-D coupled opticalthermal model with non-uniform concentrated energy flux was developed. Furthermore, the influence of the thermo-electric properties of the PVT hybrid system by the variation of the NF volume flow rate was investigated.

Physical Model of CPVT System
An NF-based PV/T hybrid system with dual channels was selected as the physical model, as shown in Figure 1. The selected PVT system's splitting and cooling channels were arranged in series. To improve the receiver's thermo-electric performance, an air gap was employed to thermally decouple the PV module and the filter, allowing the splitting NF to operate at a higher temperature range than the PV module. In addition, the splitting NF's thermodynamic properties are defined as a water parameter reference. The main parameters and material properties of the PVT system are given in Table 1.    The ideal optical filter for the m-Si PV cell is characterized by Equation (1) [34]. In the present work, the optical properties of the splitting liquid were assumed to be that of the stated ideal optical filter. The spectral absorptivity (α λ ) should be 0 for the spectral response range of 380-1100 nm and 1 for the rest band. It was designed to absorb the underutilized spectrum of the PV module.
We adopted a formal expression for the optical characterization of splitting NF to simplify our calculations in this study. The MCRT approach is a ray-tracing methodology for analyzing the optical interaction between physical components. For realizing the MCRT, a commercial ray-tracing software, Tracepro, was used.

Thermal and Electrical Model
The numerical heat transfer computation was implemented using the ANSYS Workbench 2022R1 platform with the concentrated non-uniform solar flux distribution to simulate the PVT hybrid system. The following assumptions are involved in the model: The solar intensity can be considered as steady and the constant state study is conducted; b.
3-D fluid flow and heat transfer; c.
The NF flow is laminar, uniform, incompressible, and single phase; d.
The thermal properties of NFs and used materials are temperature independent; e.
The effects of thermal contact resistance and gravitational force are ignored; f.
The top surface of the primary glass of components is exposed to free convection, while the bottom surface and side walls of the silicon layer are adiabatic.
By applying the above assumptions, the continuity, momentum, and energy governing equations can be given as follows [29]. Continuity: Momentum: Energy in the NF flow: Energy in the solid parts and silicon substrate: where Sh(R) and Sh(el) mean the volumetric source of absorbed radiation at each substrate and output electric power, respectively. In the analysis, the thermal η th and photoelectric η el conversion efficiencies can be applied to assess the NF-CPV/T system performance, and they are determined by the following Equations (2) and (3), respectively: where E th is the absorbed thermal energy by the NF; c, ρ, Q, are the specific heat capacity, the density, and the flow rate, respectively; T out and T in are the outlet temperature and inlet temperature of the splitting NF, respectively; C is the concentration ratio; A c equals to the area of the PV cell; and I sun is the solar intensity.
where E el is the electric output of the PV module at the temperature T cell ; δ is the temperature coefficient of PV efficiency (assuming the temperature coefficient of mono-silicon cell is equal to 0.43%); FF is the fill factor which is defined as the ratio of the maximum power that can be delivered by a cell to the product of its short-circuit photocurrent J sc and open-circuit photovoltage V oc ; and η ref is the photoelectric conversion efficiency of PV cell at the reference temperature T ref . The total energy efficiency of the hybrid system is obtained by: Sustainability 2023, 15, 4377

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As the values of heat and electricity are different and depend strongly on the application, a total effective exergy efficiency can be defined as: where w is a weight coefficient that converts the thermal energy to an equivalent amount of electricity and reflects the worth of thermal energy relative to that of electricity. The relative value of heat and electricity in a hybrid system depends on the temperature of the heat output. The coefficient changes in different applications. However, in this paper, we applied a simplified evaluation model and discussed only the output characteristics of the system with respect to the operating flow rate. Herein, w is assumed to be 0.3 [29]. The ambient temperature of the simulation was 300 K and the flow rates of the NF were 17.9712 L/h, 21.3408 L/h, 28.08 L/h, 33.696 L/h, and 40.435 L/h, respectively. The SIMPLE algorithm was used for the pressure-velocity coupling, and the discretization was implemented with a second-order upwind scheme. Furthermore, the convergence criteria were selected to be <10 −3 for continuity and momentum equations and <10 −6 for energy equations. A grid-independence study was conducted to make the solution independent of the grid size ( Table 2). The number of elements varied between 52,380-511,212, and the pressure drop and outlet temperature of splitting NF were calculated. Therefore, 511,212 elements were selected for further analysis. The swept meshing method is employed to generate the model grid as presented in Figure 2.  The concentrated non-uniform solar flux distribution on the PV module was simulated by the MCRT method. The flow and heat transfer of NF and the heat conduction through the solid part were simulated by the FVM. It is worth mentioning that the nonuniform solar flux distribution provided by the MCRT is delivered to FVM, serving as an important boundary condition for the flow and heat transfer simulation, which is realized by using UDFs (user-defined functions) in the ANSYS Workbench 2022R1 platform.  The concentrated non-uniform solar flux distribution on the PV module was simulated by the MCRT method. The flow and heat transfer of NF and the heat conduction through the solid part were simulated by the FVM. It is worth mentioning that the non-uniform solar flux distribution provided by the MCRT is delivered to FVM, serving as an important boundary condition for the flow and heat transfer simulation, which is realized by using UDFs (user-defined functions) in the ANSYS Workbench 2022R1 platform.

Optical Analysis
The purpose of the optical study is to determine the realistic energy flux distribution that is received by the PV module. In the present work, the reflector surfaces are specified to have 100% internal reflectance, and the PV module is assumed to be a perfect receiver. Additionally, to evaluate the non-uniformity of the focused light onto the upper surface of the receiver, direct solar beams usually fall into the domain at 1000 W/m 2 . The design parameters of the CPC concentrator are summarized in Table 3. The traced rays with a sun source with 1000 W/m 2 DNI and a 0 • incidence angle are depicted in Figure 3a (i.e., rays are normal to the aperture of the CPC concentrator). However, for better visualization, just 1000 rays are displayed. Figure 3b illustrates the corresponding energy flux map (b) on the receiver surface. As can be seen from the figure, the concentrated solar energy flux shows a fairly uniform distribution trend along the receiver length (z). In contrast, the opposite is true along the receiver width (x). It should be emphasized that to improve the computational efficiency, we did not simulate the energy flux distribution of the proposed system in full size, shown in Table 1, but only a partial component is completed. In the subsequent calculation, only the energy flow distribution on the X-axis section of the PV module is applied.
corresponding energy flux map (b) on the receiver surface. As can be seen from the figur the concentrated solar energy flux shows a fairly uniform distribution trend along th receiver length (z). In contrast, the opposite is true along the receiver width (x). It shoul be emphasized that to improve the computational efficiency, we did not simulate th energy flux distribution of the proposed system in full size, shown in Table 1, but only partial component is completed. In the subsequent calculation, only the energy flo distribution on the X-axis section of the PV module is applied.
(a) (b) A ray independence analysis is performed using 50,000, 100,000, and 500,000 rays t ensure that the local flux intensity at the receiver does not exactly fluctuate with th number of rays traced. The fluctuation in local flux intensity over the receiver width whe tracing 50,000, 100,000, and 500,000 rays are depicted in Figure 4. As can be observed, th flux intensity fluctuates slightly while tracing 50,000 rays as compared to that whi tracing 100,000 rays. However, the local flux intensity remains relatively constant whe the number of rays is increased from 100,000 to 500,000. Accordingly, 500,000 rays a chosen and set for later research.  A ray independence analysis is performed using 50,000, 100,000, and 500,000 rays to ensure that the local flux intensity at the receiver does not exactly fluctuate with the number of rays traced. The fluctuation in local flux intensity over the receiver width when tracing 50,000, 100,000, and 500,000 rays are depicted in Figure 4. As can be observed, the flux intensity fluctuates slightly while tracing 50,000 rays as compared to that while tracing 100,000 rays. However, the local flux intensity remains relatively constant when the number of rays is increased from 100,000 to 500,000. Accordingly, 500,000 rays are chosen and set for later research.
tracing 50,000, 100,000, and 500,000 rays are depicted in Figure 4. As can flux intensity fluctuates slightly while tracing 50,000 rays as compare tracing 100,000 rays. However, the local flux intensity remains relativel the number of rays is increased from 100,000 to 500,000. Accordingly, chosen and set for later research.   The distribution of concentrated solar energy flux is affected by the relative position of the concentrator and receiver. Herein, three different cases for the relative positions of the underside of the concentrator and the upper surface of the receiver were considered: (1) case 1: 0 mm, (2) case 2: 12 mm, (3) case 3: 19 mm. In addition, the uniform concentrated solar energy flux is selected as the comparative one, marked as case 4. The analyses are carried out separately for the four designed cases. Figure 5 illustrates the influences of the relative positions of the underside of the concentrator and the upper surface of the receiver on the non-uniformity of the concentrated light on the surface of the receiver. In general, the total energy flux presents as an M-shaped intensity distribution. For case 1, the concentrated energy flux is focused in a narrower region at the center of the receiver and has a higher flux intensity compared with case 2 and case 3. As the relative distance between the receiver and the concentrator increases, the distribution of the concentrated energy flux on the receiver surface becomes more expansive, and the energy flux intensity at the central region gradually decreases. For case 3, the concentrated energy flux distribution covers the entire receiver surface with good uniformity, and the average energy flux intensity is approximately equal to 3990 W/m 2 .

Thermal Analysis
The thermal analysis involved the determination of PV cell temperature as well as the hermos-electric properties of the PVT hybrid system as a function of NF volume flow. The analysis has been carried out under uniform and non-uniform solar flux distributions.
For validating the present thermal model, the obtained cell temperature is compared with the experimental findings of our team, which is obtained through the experiment rig as shown in Figure 6. The PV/T system consisted of two parts: a CPC concentrator and a PV module integrating the splitting channel. The CPC was fixedly placed above the PV module. Therein, the geometric concentration ratio (CR) was designed as 4 and the concentrating efficiency was approximately 87.5%. The present thermal results are compared with the experimental results [35] in Table 4, which justifies the employed thermal simulation procedure. concentrated energy flux is focused in a narrower region at the center of the receiver and has a higher flux intensity compared with case 2 and case 3. As the relative distance between the receiver and the concentrator increases, the distribution of the concentrated energy flux on the receiver surface becomes more expansive, and the energy flux intensity at the central region gradually decreases. For case 3, the concentrated energy flux distribution covers the entire receiver surface with good uniformity, and the average energy flux intensity is approximately equal to 3990 W/m 2 .

Thermal Analysis
The thermal analysis involved the determination of PV cell temperature as well as the thermo-electric properties of the PVT hybrid system as a function of NF volume flow. The analysis has been carried out under uniform and non-uniform solar flux distributions.
For validating the present thermal model, the obtained cell temperature is compared with the experimental findings of our team, which is obtained through the experiment rig as shown in Figure 6. The PV/T system consisted of two parts: a CPC concentrator and a PV module integrating the splitting channel. The CPC was fixedly placed above the PV module. Therein, the geometric concentration ratio (CR) was designed as 4 and the concentrating efficiency was approximately 87.5%. The present thermal results are compared with the experimental results [35] in Table 4, which justifies the employed thermal simulation procedure.   Figure 7 shows the comparison results of temperature contours of the PV module under three cases of non-uniform concentrated energy flux and one case of a uniform one. Apparently, the non-uniform distribution of the solar concentrated energy flux results in significant differences in the internal temperature distribution of the PV modules. With the increasing non-uniformity of the solar-concentrated energy flux, the temperature difference inside the PV module is gradually augmented. At this time, the hightemperature region is mainly focused at the center of the PV module. For case 1, the maximum temperature inside the PV module can be up to 369.99 K, which is much higher than the normal average heating temperature of the PV cell. It is worth pointing out that this situation can lead to irreversible high-temperature damage to the PV module for the actual operating PV experiment duel. On the other hand, as the uniformity of the concentrated energy flux on the PV module surface is improved, the temperature difference of the PV module in the direction perpendicular to the NF flow gradually becomes smaller. Comparing case 3 and case 4, a similar temperature distribution is presented on the surface of the PV module. The two cases have approximately equal maximum temperatures of 347.81 K and 347.69 K, respectively. Contrasting four cases, the average temperatures of the PV modules are almost equivalent, as well as the NF output temperature. Accordingly, when ignoring the high-temperature damage of the PV Figure 6. Experiment rig of the PVT hybrid system in the outdoor environment [35].  Figure 7 shows the comparison results of temperature contours of the PV module under three cases of non-uniform concentrated energy flux and one case of a uniform one. Apparently, the non-uniform distribution of the solar concentrated energy flux results in significant differences in the internal temperature distribution of the PV modules. With the increasing non-uniformity of the solar-concentrated energy flux, the temperature difference inside the PV module is gradually augmented. At this time, the high-temperature region is mainly focused at the center of the PV module. For case 1, the maximum temperature inside the PV module can be up to 369.99 K, which is much higher than the normal average heating temperature of the PV cell. It is worth pointing out that this situation can lead to irreversible high-temperature damage to the PV module for the actual operating PV experiment duel. On the other hand, as the uniformity of the concentrated energy flux on the PV module surface is improved, the temperature difference of the PV module in the direction perpendicular to the NF flow gradually becomes smaller. Comparing case 3 and case 4, a similar temperature distribution is presented on the surface of the PV module. The two cases have approximately equal maximum temperatures of 347.81 K and 347.69 K, respectively. Contrasting four cases, the average temperatures of the PV modules are almost equivalent, as well as the NF output temperature. Accordingly, when ignoring the high-temperature damage of the PV modules, the thermal output of the PVT hybrid system will barely be affected by the non-uniform solar concentrated energy flux. Above the analysis shows that rational optimization of solar concentrated energy flux distribution is of great significance to the electrical power output of PV modules. than the normal average heating temperature of the PV cell. It is worth pointing out that this situation can lead to irreversible high-temperature damage to the PV module for the actual operating PV experiment duel. On the other hand, as the uniformity of the concentrated energy flux on the PV module surface is improved, the temperature difference of the PV module in the direction perpendicular to the NF flow gradually becomes smaller. Comparing case 3 and case 4, a similar temperature distribution is presented on the surface of the PV module. The two cases have approximately equal maximum temperatures of 347.81 K and 347.69 K, respectively. Contrasting four cases, the average temperatures of the PV modules are almost equivalent, as well as the NF output temperature. Accordingly, when ignoring the high-temperature damage of the PV modules, the thermal output of the PVT hybrid system will barely be affected by the nonuniform solar concentrated energy flux. Above the analysis shows that rational optimization of solar concentrated energy flux distribution is of great significance to the electrical power output of PV modules. Based on the numerical model of the PVT hybrid system established in the paper, we further analyzed the effect of the volume flow rate on the thermo-electric performance characteristics of the system. The results are shown in Figure 8 below. From the results, it can be seen that the total thermal efficiency of the system increases significantly with the increase in the volume flow rate. In addition, the electrical efficiency is continuously improved as the PV temperature gradually decreases with the rise in the NF volume flow rate. In general, the total energy efficiency of the PVT hybrid system is above 55% and progressively improves with increasing NF volume flow rate. The overall exergy efficiency of the PVT hybrid system exceeds 30%. It can be seen that the concentrating PVT system based on an NF filter has high solar-energy conversion efficiency. Based on the numerical model of the PVT hybrid system established in the paper, we further analyzed the effect of the volume flow rate on the thermo-electric performance characteristics of the system. The results are shown in Figure 8 below. From the results, it can be seen that the total thermal efficiency of the system increases significantly with the increase in the volume flow rate. In addition, the electrical efficiency is continuously improved as the PV temperature gradually decreases with the rise in the NF volume flow rate. In general, the total energy efficiency of the PVT hybrid system is above 55% and progressively improves with increasing NF volume flow rate. The overall exergy efficiency of the PVT hybrid system exceeds 30%. It can be seen that the concentrating PVT system based on an NF filter has high solar-energy conversion efficiency.
can be seen that the total thermal efficiency of the system increases significantly with the increase in the volume flow rate. In addition, the electrical efficiency is continuously improved as the PV temperature gradually decreases with the rise in the NF volume flow rate. In general, the total energy efficiency of the PVT hybrid system is above 55% and progressively improves with increasing NF volume flow rate. The overall exergy efficiency of the PVT hybrid system exceeds 30%. It can be seen that the concentrating PVT system based on an NF filter has high solar-energy conversion efficiency.

Conclusions
In this study, a detailed 3-D numerical simulation of a splitting NF-based PVT hybrid system under non-uniform concentrated energy flux was carried out. The optical simulation based on MCRT was employed to obtain the realistic energy flux distribution on the receiver. A thermal analysis based on FVM is carried out to determine the thermal output and the cell temperature distribution. The following key conclusions are summarized for the present work: (1) The distribution of concentrated solar energy flux is dramatically affected by the relative position between the CPC concentrator and receiver. A relatively uniform concentrated solar radiation distribution is obtained in case 3; (2) The non-uniform concentrated solar energy flux can result in non-uniform temperature profiles of the PV module, which may lead to local overheating. The temperature distribution of the PV module corresponding to case 3 is similar to that of the uniform energy flux; (3) The PVT hybrid system has high conversion efficiency, with a total conversion efficiency of more than 50%. Data Availability Statement: Not applicable.

Conflicts of Interest:
The authors declare no conflict of interest.

Nomenclature
A C solar receiving area of the PVT system (m 2 ) CR concentration ratio C p specific heat capacity (J/(kg·K)) I sun direct solar intensity (W/m 2 ) m mass flow rate of the fluid filter (kg/s)