Assessment of Uncertainties in Energetic and Exergetic Performances of a Flat Plate Solar Water Heater

'is paper aims to quantify sensitivities of energy and exergy performances of Flat Plate Solar Water Heaters (FPSWHs) with respect to measurement parameters. For that purpose, a computational tool is developed and validated by using outdoor conditions according to the test standard EN 12975. First of all, numerical simulations are compared with experimental results and available data in the literature, and the comparison shows a good agreement. 'en, we apply the proposed model to the quantification of uncertainties associated with transient simulation. Results show that ambient temperature is the main relevant factor in operating conditions, and its effect reaches 13.7% and 3.89% on energy and exergy efficiencies, respectively, when the deviation in the sensor measurement is about ±1°C. When 0.15 v% multiwall carbon nanotubes (MWCNT)-Ethylene-Glycol (EG) (30 : 70) nanofluid is used as working fluid, results show that a suitable choice of nanofluid properties achieves 84.7% of the thermal efficiency during the zero reduced temperature conditions compared to 75.4% when the collector works with E-G. Using common empirical correlations affects substantially the accuracy of the fitting parameters, and the deviation in exergy efficiency reaches 1.18%.


Introduction
ere is evidence that the investment in solar water heater installations is reversed by gaining electricity for the first 4 to 5 years, and then the hot water becomes free during the life time of equipment over 15 years. In addition, governments provide subsidized loans to finance equipment costs. For our country, Tunisia, the solar water heater market has grown significantly in recent years thanks to the subsidy granted by the program to promote the use of the solar water heater (35% of the purchase price is refunded), which is among the projects eligible for the Clean Development Mechanism (CDM). is mechanism represents one of the three flexibility mechanisms provided in Article 12 of the Kyoto Protocol of Climate Change Convention signed by Tunisia [1]. To reach this goal, prior to the sale of a Solar Water Heater (CES), each supplier must submit to the National Agency of Energy Management (ANME) a specific eligibility file for each proposed model. Several studies have been carried out in the direction of perfecting solar water heaters, and currently, there are several international standards that set the test methods for solar water heaters in order to predict their performance. Among the global testing methods, we mention the DST test procedure that was developed in Germany, and that has been registered as ISO 9459-5. Moreover, it is for this reason that about hundreds of technical papers and reports of Performance Test according to the DST procedure have been elaborated on request in Research and Technology Centre of Energy (CRTEn). e second test procedure (Input/output) does not require a study of the whole system, i.e., without the storage tank.
In fact, all test procedures aim to evaluate the thermal efficiency of solar collectors; a few investigations [2] are found in the literatures that are interested in studying their exergy efficiencies. Kalogirou et al. [3] reviewed the exergy analysis of solar thermal collectors. ey gave a clear methodology of exergy analysis of solar collectors. ey highlighted that exergy is more representative of performance evaluation. Farahat et al. [4] carried out a numerical study to optimize the exergy performance of flat plate solar collectors by using design parameters. ey validated their code experimentally under given design and operating conditions. eir results showed that there exist optimum values of the maximum exergy efficiency for a specific mass flow rate and absorber area.
In general, numerical tools are used in order to validate the experimental results. e latter often shows several uncertainties that directly affect the collector's performances. e physics community usually defined an interval of "probable" values of the quantity to be measured in which we have a 95% chance of finding the "true value. " We are talking about 95% confidence interval. Several authors have been interested in uncertainty tests for solar water heaters. Tang et al. [5] used a test standard published by the National Technology Supervision Bureau of China in 1997 to assess the uncertainty of heat loss coefficient, U L , of a glass evacuated solar collector tube. ey found that the choice of the test procedure is the most influencing parameter on the evaluation of U L . eir results also showed that measuring errors of instruments contributed little to U L . Kicsiny [6] validated a model based on Multiple Linear Regression (MLR) for solar collectors. ey showed that the MLR-based model has a lot of precision than models working on regression in the literature. eir results showed that the proposal based on the MLR model is considered to be the simplest black box with a precision greater than 5%. ey also showed that the MLR-based model gives collector output temperature results with higher accuracy than the physically based model. Facão and Oliveira [7] presented a new method in order to evaluate heat pipe solar collector efficiency. ey analyzed the uncertainty by separating systematic and random uncertainties. eir results showed that the uncertainty of efficiency is higher for smaller values of (T fm − T a )/G, whereT fm , T a , and G are the fluid mean temperature, the ambient temperature, and the solar incident irradiation, respectively. Sowmy et al. [8] evaluated the uncertainty in the efficiency tests of conventional flat plate solar collectors by using a solar simulator. ey performed indoor tests for the uncertainty estimation associated with the determination of efficiency. ey evaluated the contribution of the errors in the instrumentation, the test conditions, and the statistical regression. eir results showed that using an artificial solar simulator is primordial in order to obtain good accuracy.
eir study also showed that regression uncertainties also have a significant effect as measurement uncertainty on the results. Mathioulakis et al. [9] performed a systematic analysis of the contribution of all components of uncertainty based on the ISO 9806-1 procedure test to determine the final uncertainty in the parameters of the characteristic equation and the instantaneous efficiency of the collector. ey proposed a step-by-step methodology that uses specific statistical tools to assess the reliability of the test procedure and also to quantify the quality of the adjustment.
eir results showed that the stepwise methodology is a reliable and efficient tool to predict the performance of the collector by considering the efficiency of the collector a key parameter. Unlike the ISO 9806-1 standard which does not provide the uncertainty of the parameters of the efficiency curve, Sabatelli et al. [10] conducted an experimental study in order to find the uncertainty of curve fitting parameters. In their study, the authors proposed a methodology to solve this problem and to evaluate not only the parameters and their uncertainties, but also the reliability of the test procedure and its effectiveness aptitude. ey performed a sensitivity analysis to evaluate the effects of measurement errors on the uncertainty values in the estimates settings.
Regression Models and Sensitivity Analysis for the ermal Performance of solar collectors were also studied by Rehman et al. [11]. e authors used the water, 20% glycol-water, and 40% glycol-water as working fluids. ey found that the coefficients of determination exceed 97% for the efficiency models and 93% for the fluid outlet temperature.
ey also performed a sensitivity analysis to determine the most influential design, operational and natural parameters on the collector's efficiency. Karwa and Baghel [12] used a mathematical model to study the effect of measurement uncertainties on the thermohydraulic performance of solar plate heaters. ey showed that, based on thermohydraulic consideration, the driving height is 10 mm is preferable.
eir results also showed that uncertainty in the wind heat transfer coefficient could affect the thermal efficiency by 2.6%. e efficiency varies by 1.3% when the sky temperature varies by 5 K. is efficiency was affected by approximately −1.5 to 1.3% by varying the solar irradiation from 500 to 1000 Wm −2 .
eir study also showed that by increasing the flow, these effects decrease. ey are lower for the collector with a selective coating on the surface of the absorbing plate.
As described above, the determination of the influence input parameters on the efficiency of solar collectors attracted the intention of several studies in recent years. e determination of tolerances and the effect of uncertainties measurement on the results is essential for designing solar collectors and influences the manufacturing costs. is also helps in evaluating and comparing possible configurations of these systems. e studies carried out in this area are limited and require further investigation essentially in terms of exergy performances. e aim of this study is to quantify some pertinent parameter sensitivities on both energy and exergy performances of a FPSWH. We focus on better understanding the origin of different sources of deviation in energy and exergy efficiencies estimation during solar collector test motivated by applications that aim to improve solar water heaters. In the following section, we describe the solar collector tested in the Research and Technology Centre of Energy (CRTEn), the mathematical model, and the solution procedure taken to analyze the effect of uncertainties of mentioned parameters on thermal behavior and exergy performance of the system.

Mathematical Model
As described previously, the aim of this work is to quantify the effects of several operating parameters on the energy and exergy efficiencies of a FPSC. For this reason, it is essential to develop an efficient numerical tool allowing quantifying the uncertainties in the FPC's performances. For this purpose, a validation step is necessary. e validation of the code helps to determine if the mathematical model is able to investigate the heat transfer and control the exchange of entropy in the system. e test of the solar collector is carried out according to EN 12975 procedure. Solar collector testing determines the reliability and efficiency characteristics of a collector. Knowledge about efficiency makes it possible to optimize the cost/benefit ratio and to classify the performance of collectors in the productive sector. As a result of the test, it is possible to compute the energy that the collector will produce in a specific climate region.
e model used in this study is the one-dimensional mathematical model previously used by Zima and Dziewa [13], Saleh [14], Saleh et al. [15], and Cadafalch [16]. is model consists of five nodes and thus a system of five equations. It has been shown that this model is useful for analyzing the influence of the mass flow rate on the performance of the collector and also the analysis of the determination of the real time collector's performances. is model can be applied to the collectors in a serpentine or parallel tube arrangement configuration with single or double covers [13]. e energy balance equation for the glass cover is written as follows: where C g is the specific heat of the glass cover, ρ g and V g are its density and volume, respectively. T am , T g , T ab and T a are the ambient, the glass, the absorber, and the air gap temperature, respectively. α is the absorption coefficient. e radiation heat transfer coefficient between the outer surface of the glass cover and the ambient is computed as follows: where F ab−g is the view factor between two surfaces of absorber and glass. e radiation transfer coefficient between the glass and the ambient is computed as the following: h c g−a � Nu a λ a /δ a is the convective heat transfer coefficient between the absorber and the air gap, where Nu a is the average Nusselt number for the air gap computed as follows [17]: c is the collector inclination angle and Ra � gβΔTL 3 /]α is the Rayleigh number. α, λ a , δ a , A ab , ε g and A g are the absorption coefficient, the thermal conductivity, the air gap thickness, the absorber area, and the emissivity of the cover glass and its area, respectively. h c g−am � Nu am λ am /δ is the convective heat transfer coefficient between the glass and the ambient. Nu am is the average Nusselt number for the ambient defined as follows [18]: Re am is the ambient Reynolds number, Pr am is the Prandtl number of the ambient air. G is the heat flux of solar radiation, p is the tube pitch, Δz is the spatial size of control volume, σ is the Stefan-Boltzmann constant, ε g is the emissivity of the glass, and T sky � 0.0552T 1.5 am [19] is the sky temperature. e energy balance on an elemental volume of the air gap gives the following: where C a , ρ a , V a , T a , T ab are, respectively, the specific heat of the air, its density, its volume, its temperature, and the absorber temperature. e energy balance equation for the absorber is written as follows: where h c ab−a � h c g−a and h f � Nu f λ f /d in is a transfer on the internal surface of the collector tube. e empirical formula of Heaton et al. [20] using the assumption that the flow is fully developed and is used as follows: λ f is the fluid thermal conductivity, d in is the inner diameter of the collector tube, L is the length of the analyzed collector, λ is is the thermal conductivity of insulation, δ is is its thickness, σ is Stefan-Boltzmann constant, ε ab is the emissivity on absorber, and ε g is the glass emissivity. e energy balance equation for the insulation: where C is is the specific heat of the insulation, ρ is is its density, V is is its volume, and T is is the insulation temperature. e radiation transfer coefficient between the insulation and the ambient is computed as follows: e energy balance equation for working fluid (water): where p, Δz, d in and h f � Nu f λ f /d in are the tube pitch, the spatial size of the control volume, the inner diameter of the collector tube, and the heat transfer on the internal surface of the collector tube, respectively. Based on the above equations, we can calculate the thermal efficiency of the collector as follows: where _ Q u is the useful heat rate to the fluid. e exergy efficiency of the solar collector measures the deviation of the real process from the ideal one. It is given as follows [3]: where g and f are computed using the Farahat's proposal [4], which assumes zero pressure drop in the collector. In equation (13), the solar exergy is obtained using the Jeter expression [21] and T s � 5770 K is the apparent black body temperature of the sun, where T out is the output temperature, T in is the input temperature, and T am is the ambient one and where the two functions f � _ Ex s / _ Q s and g � ( _ Ex out, f − _ Ex in, f )/ _ Q u express the ratio of the solar radiation exergy rate to the solar radiation heat rate on the collector and the ratio of the exergy increase of the fluid to the useful fluid heat rate, respectively. _ Ex s is the solar radiation exergy and _ Q s is the solar radiation heat rate on the collector.
In order to evaluate the effects of the uncertainty of the experimental measurement equipment on the precision of the energy and exergy efficiencies, the theoretical uncertainty of the functioning parameters, F, was evaluated assuming that it depends on the operating parameters x 1 , x 2 , . . . , x n and according to the following relationship: where u x 1 , u x 2 , u x n are the independent uncertainties of parameters x 1 , x 2 , . . . , x n . e derivative of F is computed by using first-order finite difference approximation as follows: where Δ is the standard deviation of the value of the operating parameter x i .

Validation with Existing Literature Data.
Numerical simulation enables a better understanding of the physical phenomena involved in heating the solar collectors' working fluid and also assists in the proper sizing of thermal systems.
It also provides an advantage over experimental testing as it is a lower cost tool as there is no need for sophisticated instruments and prototypes. Saleh [15] developed a mathematical model for simulating the transient behavior of a flat plate solar collector. e mathematical model was solved numerically by using an implicit scheme. He validated his model experimentally and found that the calculation scheme is useful for transient simulations of energy systems. Hilmer et al. [22] developed a model of first-order partial differential equations in order to calculate the short-term dynamic behavior of solar collectors for various mass flow rates. is model is applied for a large unglazed collector for heating a public outdoor pool in Marburg (Germany). eir results showed that the simple single-capacity model gives a high accuracy even for very wide flow rates by using average hourly values of the useful energy gain. ey showed that in a steady state with a constant flow rate, the model gives better results. Results of our computational tool were compared to the values collected from both the numerical and experimental results of Saleh [14] according to the operating parameters, specifically for the steady state tests. e flowchart of the code solves the mathematical model equations ((1)-(5)) of the solar collector for each operating configuration involved in the heat transfer process. Each component of the system is simulated using the same weather data as the experimental test. e parameters of the experimental system, as well as initial conditions, are entered as input in the simulation program. In the first step, the preprocessing provides a database with the physical properties of the water and air relevant to the problem, such as thermal conductivity, specific mass, and other parameters. e implementation also involves physical heat transfer and turbulence models. erefore, the input data includes the total water flow, the solar radiation, and the ambient temperature. After the stabilisation of system temperatures , the simulation program provides a zero solar radiation flow over a period of time to simulate the cooling of the experimental model of reservoirs disconnected from solar collectors. Numerical results of Saleh [14] were found by using an adapted numerical model for a flat solar collector and performed by comparing the input and output temperatures with experimental results. Data collection was recorded every minute for inlet, outlet, and ambient temperatures. In the case of solar radiation, an average was recorded every 15 minutes. To verify the sensitivity of the mathematical model, when subjected to heating and then thereafter, the same input conditions were considered, such as dimensions and materials of the collector. Operating conditions were as follows: Solar Radiation: 800 W/m 2 constant for 1 hour; then zero radiation, ambient temperature is 20°C, the collector inclination is 45°, the time interval is 120 minutes and the volumetric flow is taken 1.5 GPM.
In order to verify if results of the numerical model of the proposed model behaved according to results of the reference, the same collector was considered, that is, the flat type with glass cover, for water heating applications, with an external area of 2 m 2 . e collector is lined with aluminum frame and 5″ thick rock wool insulation material at the bottom. Looking at Figure 1, we can see a comparison between the inlet and outlet temperature curves of numerical and experimental results obtained by Saleh [14] as well astemperature curves of numerical results obtained by the author. It is possible to verify a strong correlation between results.

Validation with Tested Model in the LPT Laboratory.
e second step in the validation process is the comparison of our results with those obtained experimentally in our laboratory. e FPC under investigation was tested in the Research and Technology Centre of Energy (CRTEn), Borj-Cedria. It is, as displayed in Figure 2, detailed as follows: e collector has overall dimensions of 2.004 × 1.004 × 0.95 m 3 , a surface area of 2.12 m 2 and the capacity of the absorber is 2.85 l. e painting is highly selective with an absorption coefficient 95.3% and an emission factor 4.7%. e solar collector was fixed so that the angle of incidence of the radiation is 45°. e solar collector test bench consists of the following: (1) Heat pump to set the solar collector water inlet temperature. (2) An anemometer to measure the wind speed. During the test, the global solar radiation is measured by the calibrated first class pyranometer (Kipp & Zonen) and the ambient temperature so that the probe was placed in a protected and ventilated cage placed at the height of 1.25 m from the surface of the trial. Finally, PT100 probes for measuring different temperatures (input, output, and ambient) were used. A multichannel digital data acquisition unit (type Agilent HP34970 A) was connected to the FPSC device to collect the data obtained during measurements. For the numerical investigation, ASHRAE data is used to obtain thermophysical properties of water.
In order to show the precision of the experimental investigation, we summarize in Table 1 measurement errors attributed to the uncertainly of measurements. In fact, uncertainties are mainly attributed to errors occurring during measurement and the errors of the measurement apparatus, the sensitiveness of the data acquisition system, and Pt100 temperature sensors. Figure 3 is a representation of the instantaneous thermal efficiency curve of the solar collector η en as a function of the reduced temperature T r � (T m − T a )/G where T m � (T out + T in )/2 and T in is the inlet temperature. As it is predicted by a linear equation η en � η 0 − U L T r , where U L is the heat loss coefficient, η en is almost linear, and slopes of curves are about −5.44 and -5.06 obtained numerically and experimentally, respectively. If we take into consideration the sensitivity of the equipment and the error occurring during the experimental investigation presented by error bars in Figure 2, we can affirm the accuracy of the proposed numerical model. Hence, the simulation program could reproduce with acceptable accuracy the real behavior of the FPC system. Table 2 summarizes results of the measured ambient T a , the solar radiation G, the inlet temperature, and the rate of temperature increase ΔT � T out − T in . e table also shows a comparison between computed values of energy and exergy efficiencies η en and η ex obtained numerically and experimentally. In general, results are encouraging and the      proposed program was shown to be a reliable method for evaluating both energy and exergy performances of the FPC. e slight discrepancy between η en and η ex obtained by using the experimental set-up and simulation program can be explained by errors due to the misplacing of the temperature sensor or a faulty connection wire is suspected. Other reasons for the discrepancy are the used 1D mathematical model to compute the temperatures at each node and the simplified theoretical formula of the exergy efficiency, which does not take into account the pressure drops in the collector. e table shows that by increasing ΔT, the energy efficiency decreases while the exergetic efficiency increases. When ΔT is high, the useful exergy rate decreases quickly. Figure 4 shows the ±50 W/m 2 of uncertainty in solar irradiation effects on the thermal efficiency of the tested collector. Note that 100 W is the standard uncertainty of the major calibrated first class pyranometer which is required by the EN 12975. As displayed in the figure, the uncertainty in the irradiance has an effect on both reduced temperature's results and thermal efficiency's values. More importantly, a more noticeable influence on the reduced temperature than on the efficiency of the collector is observed. More precisely, an uncertainty of ±50 W/m 2 in G affects the reduced temperature about ±0.004 Km 2 /W. is uncertainty produces a deviation in efficiency results of about 5.5%. Figure 5 shows the effects of ±1°C uncertainty in thermocouples measuring the ambient temperature on the thermal performance of the collector. It is clear from the figure that when the reduced temperature is small, i.e., the collector is at near-ambient temperature, the 2°C of uncertainty in the ambient temperature has no significant effect on the thermal efficiency. When the reduced temperature increases, the effect of the uncertainty appears on both Tr and η en . e figure also shows that when the collector is at far away-ambient temperature, uncertainty in the ambient temperature has an important effect on the result of the reduced temperature which decreases substantially. Since the reduced temperature gives rise to the difference between the mean collector and the ambient temperature, one can explain the accentuated effect of the ambient temperature on T r . e deviation reaches 0.033 Km 2 /W and 9.9% in T r and η en , respectively. erefore, the collector efficiency is significantly influenced by the ambient temperature. is trend of the results shows that both uncertainties measurements in the ambient temperature must be taken into account as for regression parameters [11]. Figure 6 shows the effect of ±1°C of uncertainty when using Pt100 ermometer at the inlet of the collector on its thermal performance. As shown in the figure, the 2°C of uncertainty in the inlet temperature has a slight effect on the result of the reduced temperature but has no effect on the thermal efficiency. In fact, this result is expected because the inlet temperature has no requirement in the test standard. More importantly, the uncertainty of ±1°C in T in affects the reduced temperature by ±0.001 Km 2 /W. e flow rate uncertainty effect of ±10% on thermal efficiency is shown in Figure 7. As displayed in the figure, the gap between measured and mean flow rate values has no effect on the η en and it has a slight effect on the reduced temperature. When the deviation in mass flow rate is about ±10%, the uncertainty in T r decreases about 0.002 Km 2 /W. Figure 8 shows the effect of a deviation of about ±10% in flow rate on the collector performance for a test sample. According to the figure, the uncertainty in the flow rate has a small effect on the temperatures of the absorber, the air gap, and the glass. Importantly, these deviations are about 0.9°C, 0.8°C, and 0.6°C for T ab , T a, and T g , respectively. A remarkable effect of uncertainty is observed in outlet temperature, which deviates by 1.5°C. e effect is then accentuated in the power output of the collector which deviates by 6.8 W.

Results and Discussion
It is important to note that apart from uncertainties in operating parameters, regression uncertainties should be taken into account and should not be neglected, as confirmed by Facão and Oliveira [7] and Sowmy et al. [8]. Table 3 reports the effect of measurement errors in several operating parameters on regression coefficients η 0 , a 1 and a 2 . e table reveals that the estimation in conversion factor mean value η 0 (efficiency for T m � T am ) is affected by 1%, 8.2%, and 0.4% when there is an uncertainty in G, T am , and T in . e heat loss coefficient a 1 is shifted by ±84.79% when the uncertainty of measurement is about ±1°C. is trend of results means that other operating parameters do not have the same order of relevance as ambient temperature. With the same uncertainty, the heat loss coefficient dependent on temperature a 2 is moved by 0.022 Wm −2 K −2 . Mean uncertainties for linear analysis procedure were, respectively, 4.1% and 2.6 Wm −2 K −2 for η 0 and a 1 . is trend of the results means that regression uncertainties do not have the same order of relevance as ambient temperature uncertainty. Figure 9 reports the variation of the exergy efficiency versus the inlet temperature. e figure shows that the exergy efficiency increases with the inlet temperature. When the water is heated from 17.48°C to 26.37°C, the exergy increases by 2.8%. e highest exergy observed is equal to 8.3% for T in � 75.61°C. It is important to note that an optimum in exergy rate is observed by Ge et al. [23] and Farahat et al. [4]. Figure 10 shows the contribution of several operating parameters on the deviation in instantaneous energy efficiency values. e effect of uncertainty in _ m and T in is around 1.09% and 1.42%, respectively. However, the ambient temperature was found to be the most influencing parameter, and its effect reaches 13.7%. e second important parameter is the solar irradiance, and its value may affect the energy efficiency by 5.47%. Figure 11 shows the contribution of several operating parameters on the deviation in exergy efficiency. e effect of uncertainty in _ m and T in is around 0.34% and 0.53%, respectively. However, the ambient temperature was found to be the most influencing parameter and its effect reaches 3.89%. e second important parameter is the solar irradiance and its value may affect the exergy efficiency by 1.11%. e aim of this section is the analysis of the effect of a deviation in thermophysical properties of the working fluid on energetic and exergetic efficiencies of the collector. erefore, we consider a new class of heat transfer fluids composed of metal nanometer-sized conductive particles dispersed in the base fluid often called nanofluid. e majority of published works highlight that classical correlations are not able to predict the superior characteristics of convective heat transfer of nanofluids. Figure 12 shows the efficiency of the collector with ethylene-glycol mixture (E-G) (30 : 70) with and without MCNTnanoparticles at 0.15v% versus the reduced temperature parameters. Experimental conditions are kept the same as the previous section and thermophysical properties data of E-G + MCNT nanofluid are given by Kumaresan and Velraj [24]. ese properties depend on both the volume fraction and the temperature. e figure clearly shows that the efficiency of the collector using E-G + MCNTnanofluid is highest and the η 0 values when using pure E-G are the lowest. is result agrees well with previous investigations found in the literature [25,26]. e η 0 values for E-G with and without MCNT nanoparticles are 84.7% and 75.4%, respectively. erefore adding nanoparticles enhances energy parameter η 0 value by about 9.3%.   In order to reveal the effect and the importance of the determination of thermophysical properties of the working fluid on the thermal performance of the collector, numerical investigations were carried out as follows: (1) Investigation I: e working fluid is the Ethylene-Glycol. (2) Investigation II: e working fluid is at 0.15v% Carbon Nanotubes + Ethylene-Glycol (30 : 70). ermophysical properties are extracted from data of Kumaresan and Velraj [24]. Other investigations are performed using the same nanofluid as follows: (i) Investigation II-a: Only λ f is computed by using Maxwell's model as follows [27]: where φ = 0.15% is volume fraction, λ s and λ EG are the thermal conductivities of the MWCNT nanoparticles and the E-G fluid, respectively. (ii) Investigation II-b: Only μ f is computed by using Brinkman's model [28] by using the following equation: where μ EG is the E-G dynamic viscosity. (iii) Investigation II-c: C f is computed by using Xuan and Roetzel's model which is claimed to be the fittest for getting specific heat of nanofluid [29,30]: (iv) Investigation II-d: μ f , λ f, and C p are computed from prescribed theoretical correlations and combined together. Figure 13 presents the effect of correlation determining the viscosity μ f , the thermal conductivity λ f, and the specific heat C p of nanofluid on the thermal efficiency of the collector. Correlations used are Maxwell's model, Brinkman's model, and Xuan and Roetzel's model for μ f , λ f, and C p , respectively. Reference characteristic parameters of the flat plate solar collector use thermophysical properties of E-G + MCNT nanofluid published by Kumaresan and Velraj [24]. e figure indicates that when we use Brinkman's model for the viscosity, η 0 increases about 6.3%. By using Maxwell's model for the thermal conductivity, η 0 increases about 1.4% and the specific heat formulae of the nanofluid is trend of results reveals the importance of the determination of thermophysical properties of the nanofluid (and hence the preparation step of the nanofluid) for the accuracy of fitting parameters.
As expected, the specific heat decreased when using the correlation prescribed while thermal conductivity and viscosity increased.
e increase in viscosity is due to the fact that nanoparticles are at low concentrations (0.15%). e increase in viscosity has exceeded the improvement of the thermal conductivity which is manifested in the figure by the remarkable effect of viscosity compared to other thermophysical parameters effects. is also explains some inconsistencies among experimental evaluations of the thermal performance of nanofluids in thermal systems [31,32]. Especially when the uncertainty in the heat specific capacity of the nanofluid increases hugely, the nanofluid improves the outlet temperature without increasing the efficiency. Figure 14 shows the variation of the exergy efficiency versus the inlet temperature with E-G and E-G + MCNT nanofluid at 0.15v% using thermophysical properties found by Kumaresan and Velraj [24] and by using Maxwell's model, Brinkman's model, and Xuan and Roetzel's correlation. By using prescribed theoretical correlations, the exergy efficiency takes values close to those of the collector in the absence of nanoparticles. By using values obtained experimentally by Kumaresan and Velraj [24], the exergy increases substantially. e maximum enhancement in exergy efficiency is obtained for T in = 75.76°C and is equal to 1.18%. is result reflects the fact that theoretical correlations of thermophysical properties have a great influence on heat transfer characteristics and hence on the energy and exergy efficiency of the collector.

Conclusions
In this work, a 1D mathematical model was developed and implemented in order to assess the sensitivity of energy and exergy performances of a Flat Plate Solar Collector (FPSC) according to EN 12975 procedure. e proposed method allows quantifying effects of uncertainties of both operating conditions and thermophysical properties of working fluid on the performance estimation of the collector. e main results of numerical and experimental analysis gave the following conclusions: (1) Results of the computational program are in good agreement with experimental measurements and benchmark solutions available in literature in terms of collector efficiency curve and outlet temperature, respectively. e proposed program was shown to be a reliable method for understanding major measurement parameter sensitivities in efficiencies and their uncertainties. (2) During the test procedure, the ambient temperature is found to be the most influencing parameter on the performance estimation and a deviation about ±1°C may affect the energy efficiency by 13.7% and the exergy efficiency by 3.9%. (3) When 0.15v% MWCNT-E-G (30 : 70) nanofluid is used as working fluid, uncertainties in its thermophysical properties lead to important deviations in both energy and exergy efficiencies, and results show that common correlations are not able to predict the collector's performance neither in terms of energy nor in exergy. Especially, the deviation in exergy efficiency reaches 1.18%.

Data Availability
e data used to support the findings of this study are included within the article.