Energy and Exergy Analyses of a Flat Plate Solar Collector Using Various Nanoﬂuids: An Analytical Approach

: Energy and exergy (EnE) efﬁciencies are considered the most important parameters to compare the performance of various thermal systems. In this paper, an analysis was carried out for EnE efﬁciencies of a ﬂat plate solar collector (FPSC) using four different kinds of nanoﬂuids as ﬂowing mediums, namely, Al 2 O 3 /water, MgO/water, TiO 2 /water, and CuO/water, and compared with water as a ﬂowing medium (traditional base ﬂuid). The analysis considered nanoﬂuids made of nanomaterials’ volume fractions of 1–4% with water. The volume ﬂow rates of nanoﬂuids and water were 1 to 4 L/min. The solar collector (cid:48) s highest EnE efﬁciency values were obtained for CuO/water nanoﬂuid among the four types of nanoﬂuids mentioned above. The EnE efﬁciencies of the CuO nanoﬂuid-operated solar collector were 38.21% and 34.06%, respectively, which is signiﬁcantly higher than that of water-operated solar collectors. For the same volume ﬂow rate, the mass ﬂow rate was found to be 15.95% higher than water for the CuO nanoﬂuid. The EnE efﬁciency of FPSC can also be increased by increasing the density and reducing the speciﬁc heat of the ﬂowing medium.


Introduction
Nowadays, solar thermal collectors and photovoltaics are primary devices for collecting solar energy [1]. The solar thermal collector is a device to absorb heat from sunlight. It is designed in such a way that it helps to convert solar irradiation into more functional energy forms, such as hot water, molten salt, or applied to produce steam to generate electricity. Flat plate solar collector (FPSC) is considered as the better performing solar heater within other types of solar heaters. These FPSC are considered to have a fixed area for collecting the incoming radiation and are typically stationary throughout the day [2,3]. They are also considered to have a fixed area for collecting incoming radiation. The efficiency of these systems is explained as the extent to which the thermal energy impacts on the collector surface, and heat is transferred to the working fluid. Improved collector efficiency will be achieved by increasing solar absorption while reducing outgoing heat losses, a process known as energy transfer optimisation [4]. The collector s efficiency can be improved by substituting the absorbing medium with high thermal conductivity (TC) fluids. The TC of a fluid can be increased by mixing a small quantity of nanoparticles with base fluid. For example, a 1 vol % of multi-wall carbon nanotube with base fluid could enhance TC by 40% [5].
The particle size of nanomaterials (nm range) is an important parameter. Although nanofluid increases heat transfer performance, large nanoparticles may cause instability of suspensions, flow resistance, clogging, and abrasion problems [6,7]. Nanofluids are colloidal suspensions of nanomaterials, including metals, metal oxides, non-metal, and oxides and carbon nanotubes in solutions or base fluids that are commonly employed as carrier fluids [8]. The nanofluids provide a larger surface area, higher thermal conductivity, better stability, and minimal clogging. In addition, nanoparticles are capable of changing the fluid characteristics [9].
The thermal performance of a FPSC can be significantly enhanced by employing an absorbing medium called nanofluids. Natarajan and Sathish [10] studied the impact of carbon nanotube (CNT) on the TC over the base fluid. They demonstrated that using these fluids as an agent medium in a standard solar water heater would improve performance. In a directive solar collector (DAC), Tyagi et al. [11] used aluminium nanoparticles with water as an absorbent medium. They reported that the DAC had 10% more efficiency than a standard flat plat solar collector [12]. Otanicar and Golden [13] studied the improvement of solar collector efficiency compared to a conventional collector under the environmental and economic influence and suggested some possible improvements. The effect of nanofluids on the performance of microthermal collectors has been experimentally studied by Otanicar et al. [14]. When the nanofluid is used as an absorbing medium, they observed an improvement in the efficiency of about 5% as compared to water.
Ziyadanogullari et al. [15] experimented with Al 2 O 3 /water and CuO/water to investigate the performance efficiency of FPSC. Their result showed CuO/water nanofluid achieved maximum efficiency compared to the other two nanofluids. The reason for these results may be because of differences in TC coefficients of the nanoparticles. Said et al. [16,17] experimentally and theoretically investigated the impact of utilising different sized Al 2 O 3 nanoparticles (20 nm and 13 nm) suspended into a base fluid (water) on the EnE efficiencies of the FPSC. They performed their experiments with nanofluid of 0.3 vol % and 0.1 vol %, varying the mass flow rate from 0.5 to 1.8 kg/min. They reported that the nanoparticle size has an impact on the performance of FPSC. They found a 3% higher thermal performance enhancement when the size varies from 20 nm to 13 nm. Additionally, they recorded 83.5% theoretical thermal efficiency at 0.3 vol % and 1.8 kg/min of a mass flow rate. An overview of the literature analysis is presented in Table 1. Table 1 provides an overview of FPSC performance improvement using metallic oxide nanofluid. It specifically indicates that the nanofluid was used in FPSC as heat transfer fluid, with a broad range of nanoparticles varying from 0.05 to 3 vol % and mass flow rate varying from 0.96 to 7.5 kg/min. It shows the optimum nanoparticle volume fraction (vol %) corresponding to FPSC maximum efficiency enhancement in nanofluid in the perspective of mass flow rate in laminar and turbulent flow conditions. Table 1 further indicates that employing Al 2 O 3 /water as a heat transfer fluid inside the FPSC improved the effectiveness of FPSC by 2-31.6% when nanoparticle concentrations of 0.1 to 3 wt.% and mass flow rates of 0.5 to 7.5 kg/min were maintained. In addition, Cu/water and CuO/water nanofluids improve the efficiency of 6.3 to 27.3% when the experiments were employed at a volume fraction that varies from 0.025 to 0.5 vol % and the mass flow rate changes from 1 to 8.8 kg/min, when compared to traditional fluids. The energy equation alone is insufficient to determine the efficiency of a FPSC in thermodynamics analysis. The second law, also known as exergy analysis, is more successful in determining the source and amount of irreversibility and may be utilised to soptimise system efficiency. Exergy is the maximum output that may be obtained in relation to the difference in temperature [29]. Saidur et al. [30] performed an exergy study on a variety of solar energy applications and FPSCs' [31]. Alim et al. [32], and Zhong et al. [33] conducted a thorough study of entropy generation in a FPSC utilising various metal oxide nanofluids. Most of the published research [34][35][36][37] focus on the thermal performance of FPSC utilising different working fluids at a constant flow rate which has a limited application in reality. Therefore, a more detailed and accurate study of their performance requires more and more up-to-date research. The aim of our paper is to analyse the EnE efficiency of a FPSCs' using the different absorbing medium as nanofluids. Furthermore, the effect of mass flow rate, volume fraction, specific heat, density, and outlet temperature on EnE efficiency for a given operating condition are investigated. The effect of fraction factor and pumping power in variable flow rate can be evaluated in this study. A previous study was done for a single/given particle, whereas, for variable flow rate, we can determine which particle is performing better compared to many others. The nanofluids made of Al 2 O 3 , MgO, TiO 2 , and CuO nanoparticles in water were considered in this study. The first law of thermodynamics was used to calculate energy efficiency, while the second law of thermodynamics was used to determine exergy efficiency.

The First Law of Thermodynamics for Energy Efficiency
The first law of thermodynamics states that the energy is conserved, and the overall amount of conserved energy remains constant. However, different types of energies (such as potential, kinetic, and internal) can be changed from one form to another. An energy balance can be expressed using the law as follows [38]: The absorbing mediums' potential heat gain (Q u ) may be expressed as, The following formulas determine the heat capacity and density of nanofluid. [39,40] C p,n f = φC p,np The Hottel-Whillier Equation is another equation for calculating the potential heat gain rate . Q u of a FPSC. The Equation that accounts for heat losses between the solar collector and atmosphere as follows [41].
The amount of absorbed irradiation per unit area of solar collector absorber plate (S) is calculated as follows, S = I T (τα) (6) where (τα) is an optical efficiency (η o ) [42].
An energy balance equation of the absorber plate in the rating form can be expressed for a steady-state condition as [42], In this study, the absorber plate s average temperature, absorbed irradiation flux per unit area, overall loss, and absorber plates' area were all considered to remain constant. The instantaneous collector efficiency is calculated by Equation (9) or Equation (10).
The current study was conducted under the usual incidence condition; F R (τα), F R , and U 1 were constant at the tested temperatures [5]. However, it should be remembered that different types of energy have additional chances of generating work. As a result, efficiency specification was limited to a comparison of quantities that were metrically equivalent but not conceptually equivalent.

The Second Law of Thermodynamics for Exergy Efficiency
Energy efficiency is not enough to describe a thermodynamic process because the energy equation does not encounter internal losses. The second law of thermodynamics provides more information about the optimal operating stage, inefficiencies, corresponding magnitudes and traces [31,43]. It assumes that actual processes are not reversible and that the processes accumulate entropy. Molecular diffusion, friction, and hysteresis are examples of typical irreversible processes. The second law, according to the Clausius argument, can be written as [38] The irreversibility is not considered in the analysis of the first law of thermodynamics. On the other hand, the second law addresses irreversibility without using the word work. The first and second laws are merged to gather further details. The Gouy-Stodola equation [44] can be obtained by combining Equations (1) and (11).
Exergy is described as the work that is obstructed to its dead state. When the environment and the system are in balance, there is no more work to be done. The process is said to be in a dead state at this stage. So, the Equation (12) can be rewritten in terms of exergy for a control volume as shown below: Therefore, where the exergy of work E . m can be defined as follows: .
The difference in measured exergy at the inlet and outlet of the control volume will then be used to quantify irreversibility. In steady state, the exergy balance per unit area of the solar collector is given by [45]: The loss of exergy because the pressure drop of the fluid is assumed to be negligible. Alternatively, Equation (14) can also be represented as [46], In steady state, the exergy collection rate is the amount of energy gained by the heat transfer fluid as the temperature of the fluid rises from T f in at the inlet to T f out at the outlet. The exergy collection rate can be expressed without considering the mechanical exergy if the fluid is incompressible by using the following Equation: .
When considering the usable exergy ratio for solar radiation, two crucial points should be kept in mind. On earth, the radiating solar flux can always be assumed to be a steady state condition but not in an equilibrium condition. The other is that the Sun s radiation is an open system, which implies it is not possible to recover photons, as opposed to the equilibrium system. Carnot of (1 − Ta/Ts) is deemed appropriate from these facts for exergy with the solar radiation, which has the same form as the result of Jeter [47]. The exergy flow from the sun can be defined from the above information as: The absorber plate is absorbed solar radiation from the sun, , and to heat is then transferred to working fluid. As shown in Figure 1, exergy destruction occurs during these two processes, including the flowing sections, as illustrated [45].
Absorption exergy loss (radiation ¬ plate): when solar light at Ts is absorbed by the absorber at Tc, an exergy annihilation event occurs.
Leakage exergy loss (plate ¬ ambient): a process of exergy loss followed by heat leaking from the absorber into its environs.
Conduction exergy loss (plate ¬ fluid): heat conduction between the absorber and the heat transfer fluid causes exergy annihilation. Absorption exergy loss (radiation ⮕ plate): when solar light at Ts is absorbed by the absorber at Tc, an exergy annihilation event occurs.
Leakage exergy loss (plate ⮕ ambient): a process of exergy loss followed by heat leaking from the absorber into its environs.
Conduction exergy loss (plate ⮕ fluid): heat conduction between the absorber and the heat transfer fluid causes exergy annihilation.
Through Gouy-theorem Stodola′s [48], the aforementioned three types of exergy loss processes are strongly connected to the entropy generating rates. The following thermodynamical concepts may be used to derive these three entropy production rates: Even though Equations (18)-(20) cannot be integrated without knowing local absorber temperature (T1) and the heat transfer coefficient distribution; these equations may be approximated using the mean absorber temperature as shown below: The first right term in Equation (23) is an entropy flow from the absorber fluid, and the second one is an entropy of energy gathered from the absorber. The difference between the two variables becomes the entropy production rate while the heat is transferred from the absorber into the fluid. The phrase of exergy loss Equation (14) can be seen from Equations (21)-(23) using the Gouy-Stodola′s theorem as, Through Gouy-theorem Stodola s [48], the aforementioned three types of exergy loss processes are strongly connected to the entropy generating rates. The following thermodynamical concepts may be used to derive these three entropy production rates: ∆ .
Even though Equations (18)-(20) cannot be integrated without knowing local absorber temperature (T 1 ) and the heat transfer coefficient distribution; these equations may be approximated using the mean absorber temperature as shown below: The first right term in Equation (23) is an entropy flow from the absorber fluid, and the second one is an entropy of energy gathered from the absorber. The difference between the two variables becomes the entropy production rate while the heat is transferred from the absorber into the fluid. The phrase of exergy loss Equation (14) can be seen from Equations (21)-(23) using the Gouy-Stodola s theorem as, Energies 2021, 14, 4305 As a result, the steady-state exergy-balance-equation of a solar collector may be obtained by inserting Equations (16), (17), and (24) into Equation (14). After a few modifications, it becomes: .
The energy balance of a solar collector can be derived by rearranging Equation (25) From Equation (25), the exergetic efficiency can be expressed by Equation (27) as follows: In Equations (27) and (28), the words in brackets indicate loss of exergy and have the following physical significance: e opt : Optical loss percent solar energy absorbed owing to glazing transmissiveness and amorphous absorption. e rp : When solar radiation at Ts is absorbed by the absorber at Tc, there is a loss fraction. (Absorption at low temperatures degrades the high-quality energy). e pa : A portion of the exergy lost by the absorber to the environment. e pf : The heat transfer from the absorber to the fluid is accompanied by a heatconduction loss percentage.
The terms (1 − η=) and U l (Tc − Ta)/IT in the well-known formulation of energy efficiency correspond to two of the aforementioned loss fractions, e opt and e pa ; it is worth mentioning that the word e pf is quite close to the collector efficiency factor, which stands for loss of heat conduction. The following link is discovered using the temperature distribution correlations in the collector [45]: In order to remove the component of output fluid temperature from Equation (27), the collector exergy efficiency correlation is recast as follows [49]:

Input Data
Some assumptions were made to simplify the analysis in this study. Other heat transfer coefficients were assumed to be constant, such as the absorbing fluid characteristics, exergy flow rate, Solar flux, total heat loss, and heat transfer coefficients. Furthermore, the temperature of the fluid entering the system and the temperature of the surrounding environment were both considered to be constant and equal. Tables 2-4  on the characteristics of nanoparticles, the solar collectors' parameter, and the analytic condition, respectively.

Methodology
The excel equation solver was used to extract the results. The input parameters from Tables 2-4 and, the Equations from 1-30 have been used to obtain analytical results. The Equations (3) and (4) have been used to estimate the specific heat, and density of a given nanofluids. The Equations (9), (10), and (30) have been used to calculate the EnE analysis of this analysis.

Results and Discussions
The chosen environmental and analytical conditions of the solar collector and constant parameters for the analytical approach are shown in Table 5. Collector efficiency was calculated using Equations (9), (10), and (30) and Tables 2-4. The EnE efficiency values are presented in Table 5. The maximum enhancement results of EnE analysis are represented in Table 5. It is showed that the highest level of EnE enhancement was found for CuO/water nanofluids. The results are similar to available literature [32]. Energetic and exergetic efficiencies of different nanofluids such as SiO 2 /water, TiO 2 /water, Al 2 O 3 /water, CuO/water, and Gr/water, were experimentally investigated by Verma et al. [53]. They found the CuO/water nanofluids achieved maximum efficiencies. A group of nanofluids (SiO 2 /water, TiO 2 /water, Al 2 O 3 /water, GNP/water, and SWCNT/water) were also examined and compared their efiiciencied by Elcioglu et al. [54]. Additionally, they verified that, the increase in volume fraction of nanofluids enhanced the absorber's efficiency for all nanofluids investigated.
Investigating the outputs of Luminosu et al. [43] for the solar collector s open circuit mode and the computer simulation of the solar collector developed by Farahat et al. [31] enables the verification of the findings acquired through the analytic approach. Table 6 compares the analysed exergy efficiency with an experimental work [43], and analytical analysis [32] and the computer simulation [31] performed in Iran. These findings are in excellent accordance with the literature data of Reference [32]. Our results vary only about 9% with Reference [32], which is well acceptable. The deviation was higher for the others two data of references [31,43] due to their numerical and experimental parameters were different compared to the present analysis. For example, the solar radiation and the temperature difference are different from our analysis. It was realised that the solar radiation and ambient temperature had an impact on the exergy efficiency, particularly at the high flow rates [21]. The results are also in good agreement with the recently published experimental study [55]. They reported the maximum exergy enhancement was about 3.33% which is very close to this study (3.35%) analysis. The following observations can be noted.
As nanofluids replace water as an absorbing medium, the nanofluids' viscosity, density, and TC increase, but its specific heat decreases compared to the base fluid (water). Interestingly, these findings are in agreement with those of Pandey et al. [50].
To gain exergy, the entropy generation number is typically expected to decrease. The entropy generation number decreases when nanofluids are used as agent fluids [51]. With an increase in particle concentration, the heat transfer rate is increased [56]. Figure 2 illustrates the changes of the energy efficiency of a FPSC as a function of the volume flow rate of 1 to 3.8 L/min. It can be seen from Figure 2 that the collector s energy efficiency increases linearly and steadily with the increase of volume flow rate. Equation (9) and input data were used to calculate energy efficiency, considering output temperature difference constant and 2 vol % nanoparticles. It can be found from Figure 2 that the collector efficiency increases by 38.21% for CuO/water nanofluid and 34.17%, 33.21, and 32.91% for Al 2 O 3 /water MgO/water and TiO 2 /water nanofluids compared to the base fluid. These results satisfactorily agree with the results of Yousefi et al. [12] and Tyagi et al. [11].
 To gain exergy, the entropy generation number is typically expected to decrease. The entropy generation number decreases when nanofluids are used as agent fluids [51].  With an increase in particle concentration, the heat transfer rate is increased [56]. Figure 2 illustrates the changes of the energy efficiency of a FPSC as a function of the volume flow rate of 1 to 3.8 L/min. It can be seen from Figure 2 that the collector′s energy efficiency increases linearly and steadily with the increase of volume flow rate. Equation (9) and input data were used to calculate energy efficiency, considering output temperature difference constant and 2 vol % nanoparticles. It can be found from Figure 2 that the collector efficiency increases by 38.21% for CuO/water nanofluid and 34.17%, 33.21, and 32.91% for Al2O3/water MgO/water and TiO2/water nanofluids compared to the base fluid. These results satisfactorily agree with the results of Yousefi et al. [12] and Tyagi et al. [11].  Figure 3 shows the effect of output temperature on energy efficiency. Energy efficiency was calculated using Equation (9). It can be seen from Figure 3 that the energy efficiency increases significantly with increasing output temperature of all fluids. Solar collectors that utilise nanofluids have a higher output temperature, which produces nanoparticles in the basis fluid, and therefore have greater efficiency. The relevant definition is: "The heat needed to increase the temperature of a unit mass of a material by one unit of temperature". As a result, at a similar heat flow rate, lower specific heat capacity material should offer higher temperature, as stated in the description.  Figure 3 shows the effect of output temperature on energy efficiency. Energy efficiency was calculated using Equation (9). It can be seen from Figure 3 that the energy efficiency increases significantly with increasing output temperature of all fluids. Solar collectors that utilise nanofluids have a higher output temperature, which produces nanoparticles in the basis fluid, and therefore have greater efficiency. The relevant definition is: "The heat needed to increase the temperature of a unit mass of a material by one unit of temperature". As a result, at a similar heat flow rate, lower specific heat capacity material should offer higher temperature, as stated in the description. The specific heat of nanofluids as a function of nanoparticle volume fraction is shown in Figure 4. When the nanoparticle volume fraction in the base fluid is higher, the specific heat is reduced. Due to these reasons, the overall output temperature and efficiency both increase. Figure 4 demonstrated the CuO nanofluid has a lower specific heat than Al2O3, The specific heat of nanofluids as a function of nanoparticle volume fraction is shown in Figure 4. When the nanoparticle volume fraction in the base fluid is higher, the specific heat is reduced. Due to these reasons, the overall output temperature and efficiency both increase. Figure 4 demonstrated the CuO nanofluid has a lower specific heat than Al 2 O 3 , MgO and TiO 2 nanofluids for a given volume fraction. The relative heat of Al 2 O 3 and TiO 2 nanofluid was nearly identical. Even though the specific heat of these nanofluids varied, they were all above the specific heat of the base fluid. These findings are consistent with those of Kamyar et al. [57] and Sohel et al. [58]. The specific heat of nanofluids as a function of nanoparticle volume fraction is shown in Figure 4. When the nanoparticle volume fraction in the base fluid is higher, the specific heat is reduced. Due to these reasons, the overall output temperature and efficiency both increase. Figure 4 demonstrated the CuO nanofluid has a lower specific heat than Al2O3, MgO and TiO2 nanofluids for a given volume fraction. The relative heat of Al2O3 and TiO2 nanofluid was nearly identical. Even though the specific heat of these nanofluids varied, they were all above the specific heat of the base fluid. These findings are consistent with those of Kamyar et al. [57] and Sohel et al. [58]. The effect of nanoparticle volume fraction on exergy efficiency is illustrated in Figure  5. Equations (16), (17), (27) and (30) was used to calculate the exergy efficiency. The efficiency of solar collector increases linearly with the volume fraction. For example, for given nanoparticles of 3.2 vol %, the efficiency increases by 11.45% for CuO nanofluid, 9.7% for The effect of nanoparticle volume fraction on exergy efficiency is illustrated in Figure 5. Equations (16), (17), (27) and (30) was used to calculate the exergy efficiency. The efficiency of solar collector increases linearly with the volume fraction. For example, for given nanoparticles of 3.2 vol %, the efficiency increases by 11.45% for CuO nanofluid, 9.7% for TiO 2 nanofluid, and 9.35% for Al 2 O 3 and 9.53% for MgO nanofluid compared to base fluid (water). TiO2 nanofluid, and 9.35% for Al2O3 and 9.53% for MgO nanofluid compared to base fluid (water). Exergy efficiency as the function of fluid volume flow rate is represented in Figure 6. The efficiency of exergy was determined using Equation (30). The analysis indicates that the exergy efficiency of a water operated collector is minimum; therefore, a good amount  Exergy efficiency as the function of fluid volume flow rate is represented in Figure 6. The efficiency of exergy was determined using Equation (30). The analysis indicates that the exergy efficiency of a water operated collector is minimum; therefore, a good amount of irreversibility is present in the solar collector with water as the flowing medium. However, the figure shows that the exergy efficiency can be increased by using nanofluids as the flowing medium. As an absorbent medium, CuO nanofluid may be the right choice because the exergetic efficiency is better compared to the others nanofluids, and water. The TC of a nanofluid is an increasing function of nanoparticle volume fraction, according to the Hamilton and Crosser model [59]. It may be explained by increasing the effective surface area for heat transmission by increasing the volume fraction of nanoparticles. Additionally, the nanoparticles inherent the higher TC that improves the TC of nanofluids. This phenomenon may account for an increase in exergy efficiency. Exergy efficiency as the function of fluid volume flow rate is represented in Figure 6. The efficiency of exergy was determined using Equation (30). The analysis indicates that the exergy efficiency of a water operated collector is minimum; therefore, a good amount of irreversibility is present in the solar collector with water as the flowing medium. However, the figure shows that the exergy efficiency can be increased by using nanofluids as the flowing medium. As an absorbent medium, CuO nanofluid may be the right choice because the exergetic efficiency is better compared to the others nanofluids, and water. The TC of a nanofluid is an increasing function of nanoparticle volume fraction, according to the Hamilton and Crosser model [59]. It may be explained by increasing the effective surface area for heat transmission by increasing the volume fraction of nanoparticles. Additionally, the nanoparticles′ inherent the higher TC that improves the TC of nanofluids. This phenomenon may account for an increase in exergy efficiency.   Figure 7 represents the variation of analytically determined exergy efficiency of solar collectors with nanoparticles volume fraction. A solar collector using CuO/water nanofluid has the greatest exergy efficiency for the same flow rate as the others nanofluids. The CuO/water nanofluid along with solar collector, for example, has a 14.12% greater exergy efficiency than a typical solar collector operating on the water at a 2.4 L/s volume flow rate. Al 2 O 3 and MgO show approximately equal exergy efficiency but higher than water. On the other hand, TiO 2 provides slightly better exergy efficiency compared to base fluid, Al 2 O 3 and MgO nanofluids. Thus, the analytical results indicate that the maximum exergy efficiency can be achieved with collectors using nanofluids as an agent medium. This improvement was most likely aided by the following factors: At a fixed Reynolds number, (I) the suspension TC increases as the volume fraction of nanoparticles, and (II) the nanofluids' convective heat transfer coefficient is higher than that of the base fluid. Those similar conclusions were reported by Duangthongsuk and Wongwises [60], Xuan and Li [61], and He et al. [62]. When the collector absorbent surface area is identical, the mass flow rates and the specific heat may have a considerable effect on the exergetic efficiency of solar collectors, according to the exergy efficiency Equation (30).
This improvement was most likely aided by the following factors: At a fixed Reynolds number, (I) the suspension TC increases as the volume fraction of nanoparticles, and (II) the nanofluids' convective heat transfer coefficient is higher than that of the base fluid. Those similar conclusions were reported by Duangthongsuk and Wongwises [60], Xuan and Li [61], and He et al. [62]. When the collector absorbent surface area is identical, the mass flow rates and the specific heat may have a considerable effect on the exergetic efficiency of solar collectors, according to the exergy efficiency Equation (30).   Figure 8, the mass flow rate of nanofluid rises as the volume percentage of nanoparticles increases; however, specific heat has the opposite effect. The following formula was used to determine the mass flow rate: relation, ṁ = Vρ Where, V represents a volume flow rate in L/min. The nanoparticles' density and volume fraction of given geometrical dimensions are responsible for the variation in mass flow rate.
For instance, considering the volume flow rate of 1 lit/min, the mass flow rate was found to be 0.01667, 0.01824, 0.01840, 0.01825, and 0.01933 for water, Al2O3, TiO2, MgO, and CuO nanofluids, respectively for 3.2% nanoparticles volume concentration. This indicates that about an 11.45% increase in the mass flow rate is possible for CuO nanofluid than water. These findings are in agreement with the results published by Sohel et al. [58].   Figure 8, the mass flow rate of nanofluid rises as the volume percentage of nanoparticles increases; however, specific heat has the opposite effect. The following formula was used to determine the mass flow rate: relation,ṁ = Vρ Where, V represents a volume flow rate in L/min. The nanoparticles' density and volume fraction of given geometrical dimensions are responsible for the variation in mass flow rate. Although its outlet temperature has a more significant impact on energy efficiency, it raises the absorber plate temperature, potentially causing exergy loss. According to the literature, the leading cause of exergy loss in the collector is the temperature difference between the sun and absorber plate. If the temperature of the absorber plate rises, this disparity increases as well, resulting in the decrease of collector exergy loss. According to Jafarkazemi et al. [46], if the flow rate is increased by 0.01 kg/s it results in a significant reduction of temperature on the absorber plate. The heat loss coefficient reduces with the For instance, considering the volume flow rate of 1 lit/min, the mass flow rate was found to be 0.01667, 0.01824, 0.01840, 0.01825, and 0.01933 for water, Al 2 O 3 , TiO 2 , MgO, and CuO nanofluids, respectively for 3.2% nanoparticles volume concentration. This indicates that about an 11.45% increase in the mass flow rate is possible for CuO nanofluid than water. These findings are in agreement with the results published by Sohel et al. [58].
Although its outlet temperature has a more significant impact on energy efficiency, it raises the absorber plate temperature, potentially causing exergy loss. According to the literature, the leading cause of exergy loss in the collector is the temperature difference between the sun and absorber plate. If the temperature of the absorber plate rises, this disparity increases as well, resulting in the decrease of collector exergy loss. According to Jafarkazemi et al. [46], if the flow rate is increased by 0.01 kg/s it results in a significant reduction of temperature on the absorber plate. The heat loss coefficient reduces with the reduction of temperature between the absorber plate and the atmosphere, resulting in an improvement of the collector s thermal efficiency. The argument is consistently endorsed by Figure 9. The mass flow rate was substantially improved by increasing the nanoparticle volume concentration. Although its outlet temperature has a more significant impact on energy efficiency, it raises the absorber plate temperature, potentially causing exergy loss. According to the literature, the leading cause of exergy loss in the collector is the temperature difference between the sun and absorber plate. If the temperature of the absorber plate rises, this disparity increases as well, resulting in the decrease of collector exergy loss. According to Jafarkazemi et al. [46], if the flow rate is increased by 0.01 kg/s it results in a significant reduction of temperature on the absorber plate. The heat loss coefficient reduces with the reduction of temperature between the absorber plate and the atmosphere, resulting in an improvement of the collector′s thermal efficiency. The argument is consistently endorsed by Figure 9. The mass flow rate was substantially improved by increasing the nanoparticle volume concentration.  The conventional working fluid and nanofluids can be compared for their contribution to total heat gain. After determining the thermal efficiency of a solar collector, the possibility for reducing the collector s area can be calculated by [63]: Figure 10 illustrated the size reduction of FPSC with a different working medium. The calculation is considered at nanoparticle of 0.032 vol % and volume flow rate of 2 L/min. The water-CuO working fluids shows the maximum decrease of FPSC size compared to others working medium. The difference of FPSC size reduction of water-CuO nanofluids with other nanofluids due to the higher EnE efficiency of water-CuO nanofluids. Figure 10 illustrated the size reduction of FPSC with a different working medium. The calculation is considered at nanoparticle of 0.032 vol % and volume flow rate of 2 L/min. The water-CuO working fluids shows the maximum decrease of FPSC size compared to others working medium. The difference of FPSC size reduction of water-CuO nanofluids with other nanofluids due to the higher EnE efficiency of water-CuO nanofluids.

Conclusions
The benefits of utilising various nanofluids in a FPSC were investigated in this research, considering the impacts of volume flow rate, nanoparticles volume fraction, mass flow rate, density, and specific heat on the solar collector′s EnE efficiency. Based on the results, the following main conclusions can be drawn.
1. The analytical result shows CuO nanofluid increased the EnE efficiency of a FPSC by 38.21% and 14.86%, respectively. 2. Additionally, the study demonstrated that increasing volume fraction, mass flow rate, and density could improve EnE efficiency. Whenever the volume flow rate remains constant, the mass flow rate can be increased by adding nanoparticles into the base fluid, which has higher efficiency. Specific heat is one of the essential parameters for efficiency improvement. By reducing the specific heat of a fluid, the efficiency of a FPSC can be improved. It is simple to do so by suspending a small number of nanoparticles. 3. CuO nanofluid is a better option for increasing both the EnE efficiency of FPSC.
To reduce the complexity of the analysis, the overall heat loss coefficient, nanoparticle properties, inlet temperature, pressure drop, area of the absorber plate on the performance of the collector, heat removal factor, etc., were considered constant. This emphasises that more study is required for thoroughly investigating the heat transfer performance and efficiency of FPSC with nanofluids.

Conclusions
The benefits of utilising various nanofluids in a FPSC were investigated in this research, considering the impacts of volume flow rate, nanoparticles volume fraction, mass flow rate, density, and specific heat on the solar collector s EnE efficiency. Based on the results, the following main conclusions can be drawn.

1.
The analytical result shows CuO nanofluid increased the EnE efficiency of a FPSC by 38.21% and 14.86%, respectively.

2.
Additionally, the study demonstrated that increasing volume fraction, mass flow rate, and density could improve EnE efficiency. Whenever the volume flow rate remains constant, the mass flow rate can be increased by adding nanoparticles into the base fluid, which has higher efficiency. Specific heat is one of the essential parameters for efficiency improvement. By reducing the specific heat of a fluid, the efficiency of a FPSC can be improved. It is simple to do so by suspending a small number of nanoparticles.

3.
CuO nanofluid is a better option for increasing both the EnE efficiency of FPSC.
To reduce the complexity of the analysis, the overall heat loss coefficient, nanoparticle properties, inlet temperature, pressure drop, area of the absorber plate on the performance of the collector, heat removal factor, etc., were considered constant. This emphasises that more study is required for thoroughly investigating the heat transfer performance and efficiency of FPSC with nanofluids.