Introduction

Nowadays, “Nanofluid” has converted to a well-known term in the dictionary of thermal engineering. Many studies have been carried out to determine the thermophysical properties of various nanofluids (for example, see Refs. [1–8]) since the design accuracy of thermal systems depends strongly on the properties of working fluid. In parallel, researchers have developed the exploitation of nanofluids in various engineering systems such as automotive sector [9], micro-/minichannels [10], heat exchangers [11–14], and solar energy devices [15, 16].

One of the applications of nanofluids is in flat-plate solar collectors as the most common solar systems. The literature includes a considerable number of studies on the thermal efficiency of solar collectors where a nanofluid is working fluid. However, fewer studies are performed on the exergy efficiency and entropy generation of solar collectors using nanofluids, whereas exergy analysis of a thermal system is vital to recognize the energy loss factors. Here, a brief review is presented on the previous studies on the exergy and entropy generation analysis of flat-plate solar collectors. Alim et al. [17] investigated the exergy efficiency of a flat-plate solar collector using four different nanofluids including ZnO/water, CeO2/water, NiO, and CoO/water nanofluids theoretically. Their results showed a linear increase in exergy efficiency with the volume fraction of nanoparticles and the mass flow rate of nanofluids. They indicated that CeO2 and ZnO nanoparticles show the highest and smallest exergy efficiencies, respectively. Faizal et al. [18] investigated the exergy efficiency of a flat-plate solar collector using SiO2/water nanofluids with volume fractions of 0.2 and 0.4 % where the size of nanoparticles was 15 nm. They found that exergy efficiency increased with increasing the volume fraction, whereas the entropy generation decreased by particle loading. Said et al. [19] examined the exergy efficiency and entropy generation for a solar collector with four different nanoparticles including single-wall carbon nanotubes (SWCNTs), TiO2, SiO2, and Al2O3 nanoparticles dispersed in water as the base fluid analytically. They found that SWCNTs nanofluids provided the minimum entropy generation in the system.

Mahian et al. [20] theoretically evaluated the effects of uncertainties in thermophysical properties on entropy generation of Al2O3/water nanofluids in a solar collector (flat-plate type) where the flow was turbulent. The results were presented in terms of constant mass flow rates and for different values of nanoparticle size (ranging from 25 to 100 nm) by considering the tube roughness. They found that different models that were exploited to calculate the properties had no a significant effect on the entropy generation, independent of nanofluid concentration, and mass flow rate. The tube roughness effects on the entropy generation were pronounced at high mass flow rates. It was also elucidated that the reduction in entropy generation in the solar collector occurs by nanoparticle loading. Later, Mahian et al. [21] assessed the entropy generation due to the flow of boehmite alumina/EG–water nanofluids with volume concentrations up to 4 % in a flat-plate solar collector where nanoparticles had four different shapes including platelets, cylinders, blades, and bricks. Moreover, the entropy generation was examined for collector tubes made of two different materials, i.e., copper and steel. The results unveiled that when the collector tubes are fabricated of copper tubes, using nanofluids containing brick-shaped nanoparticles with concentrations of 2 % concentration leads to entropy generation minimization in the system. However, for steel-based solar collector, 4 % blade-shaped nanoparticles are needed to minimize the entropy generation. It was also found that the entropy generation for copper-based collector is, on average, 11–18 % (depends on the mass flow rate) less than steel-based collector. Mahian et al., in separate studies, investigated the effects of nanofluid pH [22]using four different nanofluids including Cu/water, Al2O3/water, TiO2/water, and SiO2/water [23] on the entropy generation of a solar collector. Said et al. [24] reported the exergy efficiency of a flat-plate solar collector using Al2O3/water nanofluids with volume fractions of 0.1 and 0.3 % and mass flow rates ranging from 0.5 to 1.5 kg min−1 where the nanoparticles size was 13 nm. It was found that the exergy efficiency ameliorated by about 20 % when the mass flow rate was 1 kg min−1 and the volume fraction was 0.1 %. In a similar study, Said et al. [25] examined exergy efficiency of a flat-plate solar collector using SWCNTs/water nanofluids where the properties of nanofluid samples were measured experimentally. Shojaeizadeh et al. [26] analyzed the exergy efficiency of a flat-plate solar collector using Al2O3/water nanofluids. In the study, they found the optimum conditions in which the exergy efficiency was maximized. Shojaeizadeh and Veysi [27] considered the flow of Al2O3/water nanofluids in a flat-plate solar collector and developed a correlation for the optimum exergy efficiency. They indicated that the values of mass flow rate, nanoparticle volume fraction, and inlet temperature in which the exergy efficiency was optimized will be reduced with the environmental parameter that was defined as the ratio of ambient temperature to solar radiation.

The present paper aims to analyze the second law of thermodynamics for a flat-plate solar collector using SiO2 nanoparticles with a diameter of 40 nm suspended in a mixture of EG and water (50:50 vol%). Experimental data reported in our previous work [28] are used in this study. For the analysis, the effects of concentration and mass flow rate of nanofluids on the exergy efficiency and a dimensionless entropy generation number (N s), which is defined as a function of dimensional entropy generation, ambient temperature, and solar radiation, are evaluated. It should be noted that based on the best knowledge of the authors, it is the first time that the dimensionless entropy generation number is exploited to investigate the nanofluid-based solar collectors.

Experimental

As the details of experiments are available in Ref. [28], here just a brief description of nanofluid preparation and experimental setup is presented.

Nanofluid preparation

Nanofluids have been prepared in three different volume fractions including 0.5, 0.75, and 1 %. For this purpose, specified amounts of SiO2 nanoparticles having a diameter of 40 nm were dispersed in the mixture of EG and water (50:50 vol%) by the aid of a mixer and an ultrasonic processor. Specified amount of nanoparticles (depends on the volume fraction) was gradually added to the base fluid, and at the same time, the suspension was stirred for about 30 min. After mixing, the suspension is sonicated with an ultrasonic processor for about 2 h. No surfactant was used in the preparation process. It was observed that the nanofluids have high stability even after months, and no sedimentation was seen with naked eyes. Figure 1 displays a photograph of nanofluid samples.

Fig. 1
figure 1

Prepared samples of SiO2/EG–water nanofluids

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Experiments on solar collector

Figure 2 depicts a schematic of the experimental setup, and Fig. 3 illustrates a photograph of the flat-plate solar collector. Also, the specifications of the solar collector are summarized in Table 1. In each cycle of nanofluid circulation, the fluid is heated up in the solar collector and returns to the tank for the cooling process. Nanofluids flow in the collector with three different mass flow rates including 1, 1.75, and 2.5 L min−1. Experiments have been performed from 10:30 a.m. to 2:30 p.m. The accuracy of thermocouples, pyranometer, timer, and the vessel that is used for measuring the mass flow rate is 0.1 °C, 0.1 W m−2, 0.01 s, and 1 mL, respectively.

Fig. 2
figure 2

A schematic of the experimental setup

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Fig. 3
figure 3

A photograph of the solar collector

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Table 1 Specifications of the solar collector
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Data analysis

The entropy generation rate in the solar collector is obtained by the following relation [20–23]:

$$\begin{aligned} \dot{S}_{\text{gen}} & = \underbrace {{\eta_{\text{o}} G_{\text{t}} A_{\text{c}} \left( {\frac{1}{{T_{\text{p}} }} - \frac{1}{{T_{\text{S}} }}} \right) + \dot{m}\,C_{{{\text{P}},{\text{nf}}}} \left( {\ln \left( {\frac{{T_{\text{out}} }}{{T_{\text{in}} }}} \right) - \frac{{\left( {T_{\text{out}} - T_{\text{in}} } \right)}}{{T_{\text{P}} }}} \right) + U_{\text{L}} A_{\text{c}} \left( {1 - \frac{{T_{\text{a}} }}{{T_{\text{p}} }}} \right)\left( {\frac{{T_{\text{p}} }}{{T_{\text{a}} }} - 1} \right)}}_{{\dot{S}_{\text{gen}} )_{\text{H}} }} \\ & \quad + \underbrace {{\frac{{\dot{m}\Delta P}}{{\rho_{\text{nf}} }}\frac{{\,\ln \left( {\frac{{T_{\text{out}} }}{{T_{\text{a}} }}} \right)}}{{(T_{\text{out}} - T_{\text{in}} )}}}}_{{\dot{S}_{\text{gen}} )_{\text{F}} }} \\ \end{aligned}$$
(1)

where η o is the optical efficiency and is equal to 0.84, G t is the solar radiation, A c is the solar collector surface area, \(\dot{m}\) is the mass flow rate, C p is the heat capacity, ρ nf is the density of nanofluid, and ΔPis the pressure drop. Also, T S is the apparent sun temperature and approximately equals to 4350 K, i.e., 75 % of blackbody temperature of the sun [29, 30], T p is the absorber plate temperature, and it is calculated by:

$$T_{\text{p}} = T_{\text{in}} + \frac{{\dot{m}C_{\text{p,nf}} (T_{\text{out}} - T_{\text{in}} )}}{{A_{\text{c}} F_{\text{R}} U_{\text{L}} }}(1 - F_{\text{R}} )$$
(2)

in which inlet and outlet temperatures are replaced by measured data. Two parameters including F R (heat removal factor) and F R U L (removed energy parameter) are obtained based on the experiments in our former work [28].

The pressure drop in the system is estimated by the following relation [20–23]:

$$\Delta P = P_{1} - P_{2} = \rho_{\text{nf}} \,g\,\left( {L_{\text{r}} \text{Sin} \beta + h_{\text{L}} } \right)$$
(3)

where β is the collector slope and L r is length of riser. The total head loss, h L, is obtained by:

$$h_{L} = \frac{{8\dot{m}_{\text{r}}^{2} }}{{(\rho_{\text{nf}} )^{2} g\pi^{2} D_{\text{i}}^{4} }}\left( {f\frac{{L_{\text{r}} }}{{D_{\text{i}} }} + \sum {K_{\text{L}} } } \right)$$
(4)

where \(\dot{m}_{\text{r}}\) is the corresponding mass flow rate of one riser, D i is the inner diameter of riser, K L is loss coefficient(for laminar flow when flow enters the pipe is 0.5 and when the flow exits the pipe is 2), and friction factor (f) equals \(\frac{64}{\text{Re}}\) since the flow regime is laminar in the present work. The density and specific heat capacity of nanofluids are calculated by [31]:

Density:

$$\rho_{\text{nf}} = \rho_{\text{f}} \left( {1 - \phi } \right) + \rho_{\text{p}} \phi$$
(5)

Specific heat capacity:

$$C_{\text{p,nf}} = \frac{{\rho_{\text{f}} C_{\text{p,f}} (1 - \phi ) + \rho_{\text{p}} C_{\text{p,p}} \phi }}{{\rho_{\text{nf}} }}$$
(6)

The subscripts of f, p, and nf stand for base fluid, particle, and nanofluid, respectively. The variations of heat capacity and density for volume fractions of 0–1 % are shown in Fig. 4. As seen, with increasing the nanofluid concentration, the density increases linearly, while the heat capacity decreases nonlinearly.

Fig. 4
figure 4

Variations of heat capacity and density with nanoparticle volume fraction

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Since the experiments for different volume fractions and mass flow rates have been done in different days (different weather conditions), so it is not reasonable to compare the entropy generation rate in different weather conditions with each other. To resolve this problem, a dimensionless entropy generation number (N s) could be introduced as follows:

$$N_{\text{S}} = \frac{{\dot{S}_{\text{gen}} T_{\text{a}} }}{{G_{\text{t}} A_{\text{c}} }}$$
(7)

The Bejan number that shows the ratio of irreversibility due to heat transfer to the total irreversibility is written as:

$${\text{Be}} = \frac{{\dot{S}_{\text{gen}} )_{\text{H}} }}{{\dot{S}_{\text{gen}} }}$$
(8)

in the above, \(\dot{S}_{\text{gen}} )_{\text{H}}\) includes three terms that is indicated in Eq. (1).

Finally, the energy efficiency of the solar collector is obtained by:

$$\eta_{\text{II}} = \frac{{\dot{m}\left[ {C_{\text{p,nf}} \left( {T_{\text{out}} - T_{\text{in}} - T_{\text{a}} \ln \left( {\frac{{T_{\text{out}} }}{{T_{\text{in}} }}} \right)} \right) - \frac{\Delta P}{{\rho_{\text{nf}} }}} \right]}}{{G_{\text{t}} A_{\text{c}} \left( {1 - \frac{{T_{\text{a}} }}{{T_{\text{s}} }}} \right)}}$$
(9)

To use the above equations, semi-steady-state conditions have been considered.

Results and discussion

The results of this study are presented in terms of exergy efficiency, dimensionless entropy generation number, and Bejan number for different volume fraction and mass flow rates. Figure 5 illustrates the variations of exergy efficiency with volume fraction for different mass flow rates. It is found that with rising nanofluid concentration from 0 to 1 %, exergy efficiency enhances by 62.7 % for a mass flow rate of 1 L min−1, whereas the corresponding increases in mass flow rates of 1.75 and 2.5 L min−1 are 45.2 and 39.7 %, respectively. In general, it is seen that the exergy efficiency is highest for the maximum mass flow rate, i.e., 2.5 L min−1. For volume fractions greater than 0.5 %, it is observed that the exergy efficiency associated with the mass flow rate of 1 L min−1 is higher than a mass flow rate of 1.75 L min−1. Referring Eq. (9), it is found that exergy efficiency is a function of mass flow rate, pressure drop, heat capacity, and density. The interaction of these parameters determines the exergy efficiency.

Fig. 5
figure 5

Variations of exergy efficiency with volume fraction for different mass flow rates

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Figure 6 presents the variations of entropy generation number with volume fraction for different mass flow rates and various times of day that correspond with different solar radiation intensities. The figure unveils that in general the entropy generation number is lowest for the mass flow rate of 2.5 L min−1 and is highest for the mass flow rate of 1 L min−1. Also, the results elucidate that with increasing the volume fraction of nanoparticles, the entropy generation number reduces continuously. The calculations show that the pressure drop has the minimum contribution to the entropy generation in Eq. (1). The length of risers is not so long (just about 1.7 m), and on the other hand, the mass flow rates are low so that the flow in the risers is laminar. Therefore, the pressure drop magnitude that mainly depends on mass flow rate and tube dimensions will be small, and hence, its effect on entropy generation is negligible. The first term on the right-hand side of Eq. (1) has the maximum contribution (more than 90 %) to entropy generation, and also, T S is considered to be 4350 K; therefore, the term “\(\frac{1}{{T_{\text{S}} }}\)” is negligible; so Eq. (1) can be reduced as:

$$\dot{S}_{\text{gen}} \approx \frac{{\eta_{\text{o}} G_{\text{t}} A_{\text{c}} }}{{T_{\text{p}} }}$$
(10)
Fig. 6
figure 6

Variations of entropy generation number with volume fraction for different mass flow rates and times of day

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Moreover, based on the definition of entropy generation number (N S) in Eq. (7) and considering Eq. (2), one can reach the following statement:

$$N_{\text{S}} = {\text{function}}{\kern 1pt} \;(T_{\text{in}} ,\dot{m}{\kern 1pt} {\kern 1pt} ,\;C_{{{\text{p}},{\text{nf}}}} ,T_{\text{out}} ,F_{\text{R}} U_{\text{L}} )$$
(11)

As seen, many parameters affect the value of N S, and the interaction of these parameters determines the final value of entropy generation number.

Figure 6 also reveals that the maximum reduction in entropy generation number occurs at 12 p.m. and for 2.5 L min−1. When the volume fraction increases from 0 to 1 %, the entropy generation number decreases about 1.2 %. In the previous work [28], it was concluded that nanofluids with volume fractions of 0.75 and 1 % provide nearly the same enhancements in the energy efficiency of the solar collector, but by considering the preparation cost, it was suggested to use a volume concentration of 0.75 %. In the present investigation, since the values of N s at volume fractions of 0.75 and 1 % are very close (with <1 % difference), the volume fraction of 0.75 % may be the optimum concentration for using in the solar collector if the price of nanoparticles is involved. It should be noted a more comprehensive study is needed to select the exact optimum concentration where energy efficiency, entropy generation magnitude, and preparation costs are involved in the analysis.

Figure 7 displays the variations of Bejan number with volume fraction for different mass flow rates and times of the day. It is seen that in a given mass flow rate, the Bejan number augments with increasing the volume fraction; this implies that by particle loading, the contribution of heat transfer to total entropy generation increases. The increment of Bejan number with particle loading is more noticeable for higher mass flow rates. At a specific nanofluid concentration, it is also found that the Bejan number diminishes with increasing the mass flow rate; this means that the contribution of heat transfer to total irreversibility decreases with increasing the velocity of nanofluids in the solar collector.

Fig. 7
figure 7

Variations of Bejan number with volume fraction for different mass flow rates and times of day

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Figure 8 is plotted to show the variations of Bejan number with time for different volume fractions and mass flow rates. It is observed that as time passes, the Bejan number reduces, and the decline is more visible at high mass flow rates. With passing the time, especially after 12 p.m., the inlet and absorber plate temperatures increase, and on the other hand, the solar irradiation decreases; therefore, the entropy generation due to heat transfer and consequently the Bejan number fall.

Fig. 8
figure 8

Variations of Bejan number with time for different volume fractions and mass flow rates

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Conclusions

A theoretical study based on experimental data was performed to evaluate the second law of thermodynamics for a flat-plate solar collector where SiO2/EG–water nanofluids with volume fractions up to 1 % were exploited. A new dimensionless entropy generation number was introduced to assess the irreversibility magnitude in the solar collector, i.e., a function of solar radiation and ambient temperature. The analysis was done for three different mass flow rates and different times of the day. The main findings of the study are summarized as follows:

  • With nanoparticle loading, the exergy efficiency of solar collector enhances. The amount of increase in the exergy efficiency depends on mass flow rate. A lower mass flow rate results in a higher augmentation in exergy efficiency.

  • When nanofluid concentration changes from 0 to 1 %, exergy efficiency enhances by 62.7 % for a mass flow rate of 1 L min−1, while for a mass flow rate of 2.5 L min−1, the exergy efficiency ameliorates about 40 %.

  • The entropy generation number decreases with increasing the nanofluid concentration.

  • The entropy generation number is lowest for the mass flow rate of 2.5 L min−1.

  • Bejan number rises with adding nanoparticles to the base liquid. The increase highlights high mass flow rates.

  • Bejan number reduces when time passes, especially after 12 p.m. when the solar radiation intensity reduces.

  • The volume concentration of 0.75 % may be the optimum concentration where energy efficiency, entropy generation, and cost of nanoparticles have been considered.