Entropy Generation during Turbulent Flow of Zirconia-water and Other Nanofluids in a Square Cross Section Tube with a Constant Heat Flux

Abstract

The entropy generation based on the second law of thermodynamics is investigated for turbulent forced convection flow of ZrO₂-water nanofluid through a square pipe with constant wall heat flux. Effects of different particle concentrations, inlet conditions and particle sizes on entropy generation of ZrO₂-water nanofluid are studied. Contributions from frictional and thermal entropy generations are investigated, and the optimal working condition is analyzed. The results show that the optimal volume concentration of nanoparticles to minimize the entropy generation increases when the Reynolds number decreases. It was also found that the thermal entropy generation increases with the increase of nanoparticle size whereas the frictional entropy generation decreases. Finally, the entropy generation of ZrO₂-water was compared with that from other nanofluids (including Al₂O₃, SiO₂ and CuO nanoparticles in water). The results showed that the SiO₂ provided the highest entropy generation.

1. Introduction

The idea of using nanoparticles in fluids is not quite new. In fact, about one hundred years ago Maxwell [1] found that the thermal conductivity of a suspension of nanoparticles increased compared to that of the base fluid. However, the terminology of “nanofluids” was first introduced in 1995 by Choi [2] for a suspension of nanoparticles in base fluids. Nanoparticles could be silica, polymers, metal dioxides, metals and carbon nanotubes. Oil, water and ethylene glycol are the common base fluids. In many cases the nanofluids have higher thermal conductivity compared to that of the base fluid; consequently, the nanofluids have more desirable heat transfer properties. Nowadays many researchers are studying the potential use of nanofluids in a variety of engineering equipment to improve the energy efficiency and to enhance the system’s thermal performance. Solar collectors [3], car radiators, chillers, boilers, cooling and heating systems in buildings, and micro-channel and heat pipes are just a few examples [4–15].

Kulkarni et al. [16] described the use of nanofluids to retard the decrease of cogeneration efficiency. The thermal conductivity of Cu-water nanofluids produced by a chemical reduction process was assessed by Liu et al. [17]. Haghshenas et al. [18] studied numerically the efficiency of laminar convective heat transfer to nanofluids. Mahmoudi et al. [19] simulated a nanofluid cooling system of natural convection process. Yoo et al. [20] and Xie et al. [21] outlined the methods of preparation and stability and the applications of nanofluids in the fields of energy conversion, mechanical engineering, and biomedical engineering. Maïga et al. [22] studied the hydrodynamic and thermal behavior of turbulent flows in a tube using an Al₂O₃ nanoparticle suspension at various concentrations and constant heat flux boundary conditions, and they reported enhanced thermal conductivity of the Al₂O₃ nanofluids. Vajjha et al. [23] investigated experimentally the forced convective heat transfer of nanofluids comprised of aluminum oxide, copper oxide and silicon dioxide dispersed in ethylene glycol and water in the fully developed turbulent regime, and they suggested an experimental correlation for the corresponding Nusselt number for this class of nanofluids. Recently, using numerical methods Yarmand et al. [24] studied the heat transfer to four different nanofluids in a rectangular heated pipe under a turbulent flow regime subject to constant heat flux boundary conditions. They found that the effect of Reynolds number on heat transfer to a nanofluid is more important than the effect of the concentration of nanoparticles.

In the recent years, entropy analyses of nanofluids for finding the optimal working condition of engineering systems have been studied by several researchers [25,26]. The ratio of irreversibility of a thermal system was estimated by the corresponding entropy generation, and ways to decrease it to improve the working condition was analyzed. Oztop and Al-Salem [27] wrote a review article on entropy generation in mixed and natural convection heat transfer for thermal systems. Sahin [28] examined the entropy generation for different cross-section geometries of ducts at a constant wall temperature. The entropy generation in nanofluid flow between co-rotating cylinders and the changes in the entropy generated in such systems (due to physical properties) have been studied by some researchers [29,30]. Ko et al. [31] found the ideal Reynolds number for laminar flows with different wall heat fluxes in a double sine duct. Leong et al. [32] analyzed entropy generation of an alumina-water nanofluid under turbulent flow regime in a circular pipe under constant wall temperature. They reported that the flow rate and the length of pipe significantly affect the total dimensionless entropy generation. The effect of dynamic viscosity on the entropy generation of a turbulent flow was reported by Bianco et al. [33]. Moghaddamai et al. [34,35] performed entropy analyses on laminar and turbulent flows in a circular tube under constant heat flux. They found that, for the turbulent flow of the Al₂O₃-water nanofluid, the addition of nanoparticles enhances heat transfer when the Reynolds number is less than 40,000. The difference between this work [34] and the results of [35] is the difference in methods for obtaining entropy generation results. Moghaddamai et al. [35] first measured the velocity and temperature fields, and then the entropy generation distribution was evaluated locally [3]. Bianco et al. [36] studied the entropy generation and heat transfer enhancement of the forced convection turbulent flow of Al₂O₃ in a square cross section pipe and found the optimal Reynolds number.

The presented literature survey shows that a second law analysis of the turbulent forced convection flow of ZrO₂-water nanofluids in a square cross sectional duct at constant heat flux has not yet been addressed. In the present work, the entropy generation of ZrO₂, CuO, SiO₂ and Al₂O₃ nanofluids in a square duct was studied using the using the approach of Bianco et al. [37]. The nanofluid flows were in a turbulent forced convection regime with a constant wall heat flux. The effects of different parameters including particle size, volume concentration and Reynolds number on the entropy generation of ZrO₂-water nanofluid were studied. Both the frictional and the thermal entropy generations were included in the analysis. In particular, the concentration of nanoparticles for the optimal working performance of ZrO₂-water nanofluids in terms of minimizing entropy generation under turbulent forced convection flow with a constant heat flux in a square cross section duct was evaluated and discussed. Finally the entropy generation of the ZrO₂-water mixture was compared with the other nanofluids.

2. Methodology

2.1. Thermophysical Properties of Nanofluid

In order to conduct a numerical analysis of a nanofluid flow and heat transfer, its effective thermophysical properties must be determined first. Generally the relevant properties are effective thermal conductivity (k_eff), effective dynamic viscosity, effective mass density (ρ_eff) and effective specific heat (C_peff). These effective properties are typically evaluated using the mixing theory. According to Das [38] the variation of viscosity and thermal conductivity of nanofluids with the temperature is negligible. Therefore, the thermophysical properties of nanofluids are evaluated at the average temperature for different working conditions.

The density of a nanofluid, ρ_nf, is given as [39]:

$${\mathrm{\rho }}_{nf}=\left(1-\mathrm{\phi }\right){\mathrm{\rho }}_f+\mathrm{\phi }{\mathrm{\rho }}_{np}$$

(1)

where ρ_f and ρ_np are, respectively, the mass densities of the base fluid and the solid nanoparticles.

The effective heat capacity of nanofluid at constant pressure (ρ C_p)_nf is given as:

$${\left(\mathrm{\rho }{C}_p\right)}_{nf}=\left(1-\mathrm{\phi }\right){(\mathrm{\rho }{C}_p)}_f+\mathrm{\phi }{(\mathrm{\rho }{C}_p)}_{np}$$

(2)

where (ρ C_p)_f and (ρ C_p)_np are, respectively, the heat capacities of the base fluid and the solid nanoparticles.

The effective viscosity (μ_eff) is obtained using the empirical correlation suggested by Corcione [40]. That is:

$${\mathrm{\mu }}_{\text{eff}}={\mathrm{\mu }}_{\mathrm{f}}\frac{1}{(1-34.87{(d_p/d_f)}^{-0.3}\times {\mathrm{\phi }}^{1.03})}$$

(3)

where:

$$d_f=\left[\frac{6M}{N\pi {\mathrm{\rho }}_{fo}}\right]$$

(4)

Here M is the molecular weight of the base fluid; N is the Avogadro number = 6.022 × 10²³ mol⁻¹; and ρ_fo is the mass density of the base fluid at temperature T₀ = 293 K.

By considering the Brownian motion of nanoparticles, the effective thermal conductivity can be obtained by using the empirical correlation provided in [3]. That is:

$${\mathrm{K}}_{\text{eff}}={\mathrm{k}}_{\text{static}}+{\mathrm{k}}_{\text{Brownian}}$$

(5)

where:

$$k_{static}=k_f\left[\frac{(k_{np}+2k_f)-2\varnothing (k_f-k_{np})}{(k_{np}+2k_f)+\varnothing (k_f+k_{np})}\right]$$

(6)

$${\mathrm{k}}_{\text{brownian}}=5\times {10}^4\mathrm{\beta }\varnothing {\mathrm{\rho }}_f{C}_{p,f}\sqrt{\frac{kT}{2{\mathrm{\rho }}_{np}\cdot R_{np}}}f(T,\varnothing )$$

(7)

Here, R_np is the radius of the nanoparticle, and k = 1.3807 × 10⁻²³ is the Boltzmann constant.

2.2. Governing Equations

Entropy generation for a nanofluid flowing through a square cross section is a combination of frictional and thermal entropy generation [26,41]. Accordingly:

$${\mathrm{S}}_{\text{gen},\mathrm{T}}={\mathrm{S}}_{\text{gen},\mathrm{t}}+{\mathrm{S}}_{\text{gen},\mathrm{f}}$$

(8)

The thermal and frictional entropy generations for the unit length of the duct are given as (Bejan [26]):

$${\mathrm{S}}_{\text{gen},\mathrm{T}}(\mathrm{X})=\frac{\dot{q}.\Delta T(x)}{T^2}+\frac{m}{\rho T}(-\frac{dp}{dx})$$

(9)

The first term in Equation (9) is the thermal entropy generation, and the second term is the frictional entropy generation. By integrating Equation (9), Bejan [26] obtained the following formula for a circular duct:

$${\mathrm{S}}_{\text{gen},\mathrm{T}}=\frac{q2.P.D.L}{Nu.k.T_{ave}}+\frac{2.m^3.f.L}{\rho .T_{ave}.D.A_c^2}$$

(10)

For a square cross sectional pipe, Bianco et al. [37] found:

$${\mathrm{S}}_{\text{gen},\mathrm{T}}=\frac{q2.\pi .D_h^2.L}{Nu.k.T_{ave}}+\frac{32.m^3.f.L}{{\rho }^2.T_{ave}.D_h^5.\pi }$$

(11)

where D_h and T_ave are given as:

$$D_h=\frac{4A_c}{P}$$

(12)

$${\mathrm{T}}_{\text{ave}}=\frac{T_{\mathit{in}}-T_{out}}{\text{ln}(\frac{T_{\mathit{in}}}{T_{out}})}$$

(13)

where A_c is the cross-sectional area and P is the perimeter.

Using an energy balance (first law) for the tube with a constant heat flux, T_out can be found as:

$${\mathrm{T}}_{\text{out}}={\mathrm{T}}_{\text{in}}+\frac{q\cdot A_s}{m\cdot C_p}$$

(14)

The corresponding thermal and frictional entropy generations are then, respectively, given as:

$${\mathrm{S}}_{\text{gen},\mathrm{t}}=\frac{q2.\pi .D_h^2.L}{Nu.k.T_{ave}}$$

(15)

$${\mathrm{S}}_{\text{gen},\mathrm{f}}=\frac{32.m^3.f.L}{{\rho }^2.T_{ave}.D_h^5.\pi }$$

(16)

where f is the friction factor and is given by [28,32]:

$$f={\left[0.79\text{ln}\left(\text{Re}\right)-1.64\right]}^{-2}$$

(17)

The Nusselt number for turbulent forced convection of nanofluids may be obtained from the experimental study of Dittus and Boelter [42]. That is:

$$\text{Nu}=\mathrm{0.023.}{\text{Re}}^{0.8}.{\text{Pr}}^{0.4}$$

(18)

where the Reynolds number is defined as:

$$\text{Re}=\frac{{\rho }_{nf}.v.D_h}{{\mu }_{nf}}$$

(19)

To find the relative influences of thermal and frictional entropy generations on the total entropy generation, the non-dimensional Bejan number (Be) is used. The range of Be is from 0 to 1, where Be = 0 implies that all the entropy generation is due to friction and Be = 1 corresponds to the case where all the entropy generation is thermal. The Bejan number is defined as:

$$\text{Be}=\frac{S_{\mathit{gen},t}}{S_{\mathit{gen},T}}$$

(20)

2.3. Description of the Problem

The second law analysis of a ZrO₂-water nanofluid in a square cross sectional pipe is investigated in this section. Schematic of the duct is shown in Figure 1. The length of the duct is 1 meter, and its hydraulic diameter is 0.01 m. The duct wall is at a constant heat flux of 50,000 W/m², and the inlet temperature is 300 K.

As a series of simulations are performed, where the flow Reynolds number is varied in the range of 50,000 to 70,000, the nanoparticle concentrations are varied in the range of 0% to 7%, and the nanoparticle diameters are varied in the range of 10 nm to 50 nm. While the ZrO₂-water nanofluids are of main concern, the effects of using different nanoparticle-water mixtures on the entropy generation are also investigated.

2.4. Validation

For validation the present model predictions are compared with the experimental data of Williams et al. [43] in Figure 2. Comparisons are done for three concentrations of 0.2%, 0.5% and 0.9% for ZrO₂-water nanofluids. This figure shows that there is a good agreement between the entropy productions estimated by the present model and the experimentally measured data. The average difference for the three concentrations shown is about 0.85%. This slight difference may be attributed to experimental error or the inaccuracies in the formula used for the effective thermal conductivity and/or effective viscosity of the ZrO₂-water mixtures.

3. Results and Discussion

In this section the effects of nanoparticle concentration, Reynolds number, nanoparticle diameter and type of nanoparticle material on entropy generation are investigated. The ZrO₂-water nanofluid is used for studying the effect of variations in nanoparticle concentration, Reynolds number, and nanoparticle diameter. For the case of studying the effect of nanoparticle material, the entropy generation for suspensions of Al₂O₃, ZrO₂, SiO₂ and CuO in water are evaluated and compared with each other.

3.1. Effects of Nanoparticle Concentration

Here the variations of thermal and frictional entropy generations with nanoparticle concentration are investigated. For the turbulent flow of ZrO₂-water nanofluids at Re = 60,000 in the square cross section duct with a wall heat flux of 50,000 w/m², Figure 3 shows the predicted variations of total entropy generation, as well as the thermal and frictional entropy generations with particle volume concentrations in the range of 0 to 7%. Here the diameter of the ZrO₂ particles is d_p = 20 nm. It is seen that the thermal entropy generation is dominant compared with the frictional entropy generation in this case. Furthermore, the thermal and frictional entropy generations have opposing trends of variation with increasing concentration of nanoparticles. That is, with the increase of nanoparticle concentration, the thermal entropy generation decreases while the frictional entropy generation increases. The decreasing trend of entropy generation may be justified by examining Equation (15), where the increase of the effective thermal conductivity and Nusselt number with the increase of particle concentration leads to a decrease in the thermal entropy generation. The frictional entropy generation, however, increases with particle concentration since the effective viscosity increases. Although with the increase of nanoparticle concentration both the viscosity and density of nanofluid increase, the increase of viscosity is far more than the increase of density which leads to an increase of the shear stress [36]. Thus, to maintain a constant Reynolds number, the velocity has to be increased. The increase of velocity (mass flow rate) has a direct effect on the increasing trend of frictional entropy generation at a constant Reynolds number.

Due to the opposing trends of the thermal and frictional entropy generations with particle volume fraction, the total entropy generation gets a minimum value that can be seen in Figure 3. This figure suggests that the optimal working condition with minimum net entropy generation for the ZrO₂-water nanofluid at a constant Reynolds number flow in a square duct with a constant heat flux is at a nanoparticle concentration of about 5% to 6%.

3.2. Effects of Reynolds Number

In this section, the effects of the Reynolds number on the thermal and frictional entropy generations are studied. Figure 4 shows variations of thermal, frictional and total entropy generations, as well as the Bejan (Be) number versus nanoparticle concentrations at different Reynolds numbers (Re = 50,000, 60,000 and 70,000) at d_p = 20 nm.

Figure 4a shows that the thermal entropy generation decreases with the increase of the Reynolds number. In contrast, Figure 4b shows that the frictional entropy generation increases with the increase of the Reynolds number. This trend is due to the fact that the Nusselt number increases with the increase of Reynolds number, as shown by Equation (18). The increase of Nusselt number implies that the thermal entropy generation as given by Equation (15) decreases. The increase of frictional entropy generation is due to the increase of the friction factor, as given by Equation (17), which increases with the increase in Reynolds number.

According to Figure 4b the frictional entropy generations for the three Reynolds numbers studied are quite low and roughly the same order for low solid volume concentrations (≤3%). As the concentration of the nanoparticle increases, however, the differences become noticeable, and the flow with higher Reynolds number produces higher frictional entropy. Furthermore, the rate of change of the frictional entropy generation versus the nanoparticle concentration has a higher slope at the higher Reynolds number.

Figure 4c illustrates the variation of total entropy generation versus the nanoparticle concentration for different Reynolds numbers. As noted before, due to the decreasing and increasing trends of thermal and frictional entropy generations, the total entropy generation exhibits an optimum condition with minimum entropy generation. It is seen that the minimum entropy generation shifts to the lower concentration as Re increases. That is, the minimum entropy production occurs at about 6% to 7% for Re = 50,000, but shifts to about 5% to 6% for Re = 60,000. For Re = 70,000, the minimum total entropy production occurs at about 4% to 5% nanoparticle concentration. When Re increases, the frictional entropy generation becomes more significant and the total entropy generation increases; thus, the value of concentration which minimizes the total entropy generation reduces.

The variations of the Bejan number with particle concentration for different Re are shown in Figure 4d. This figure shows that the thermal entropy generation is dominant at lower particle concentrations, but the effect of frictional entropy generation increases at higher particle concentrations, particularly, at higher Reynolds numbers. At low Reynolds numbers, the entropy generation is dominated by the irreversibility of heat transfer, whereas with increasing values of Re and particle concentration, the entropy generation, due to friction losses, becomes more important.

3.3. Effect of Nanoparticle Diameter

In this section the effect of nanoparticle diameter on the thermal and frictional entropy generations is investigated. Figure 5 shows variations of thermal, frictional and total entropy generations, as well as, Bejan number versus particle concentration in the range of φ = 0% to 7% for different nanoparticle diameters (d_p = 10, 20, and 50 nm) at Reynold number = 60,000. Figure 5a shows that the thermal entropy generation increases as nanoparticle diameter increases. Teng et al. [44] and Kleinstreuer and Feng [45] among others have reported that the thermal conductivity of nanofluids decreases with an increase in nanoparticle size. Equation (15) for the thermal entropy production shows that the thermal entropy generation is inversely proportional to the thermal conductivity. As a result, an increase in particle size leads to an increase in thermal entropy generation. This fact is due to the enhancement of heat transfer due to the increase of thermal conductivity of nanofluids, which is inversely proportional to the nanoparticle size at the same volume concentration. This is because smaller particles have higher surface area for interaction with the liquid phase.

Figure 5b indicates that at a constant volume concentration and constant Reynolds number, the frictional entropy generation decreases with the increase of nanoparticle size. The decrease of frictional entropy generation is due to decreasing of the frictional losses which is associated with the decrease in mass flow rate. Equation (3) shows that the effective viscosity decreases as nanofluid particle size increases; therefore at a constant Re, the mass flow rate decreases and Equation (16) shows that the frictional entropy generation which is proportional to cube of mass flow rate decreases.

Figure 5c shows the variation of the total entropy generation versus the nanoparticle concentration for different nanoparticle diameters. Since the thermal and frictional entropy generations follow opposite trends with increase of concentration and nanoparticle size, there is a critical concentration for the minimum total entropy production. Figure 5c shows the concentration for the minimum entropy generation varies significantly with the nanoparticle diameter. For 10 nm particles, there is a clear minimum value at the concentration of about 4%, and critical concentration shifts to about 5% to 6% for dp = 20 nm. For 50 nm particles, the total entropy production continues to decrease with the increase of concentration, indicating that the minimum entropy occurs at concentrations larger than 6%. Figure 5c also shows that the total entropy production for different size particles is only slightly different for concentrations less than 4% or 5% and follows the trend of thermal entropy production. For higher concentrations, when the frictional contribution becomes important, the differences become more noticeable.

Variation of the Bejan number with concentrations for different nanoparticle sizes is shown in Figure 5d. It is seen that Be is about 1 and the effect of particle size is negligible for concentrations less than 3%, indicating that the entropy generation is mainly thermal. For larger concentrations, the effect of frictional entropy generation on Bejan number becomes more noticeable particularly for 10 nm particles.

In summary, Figures 5a–d reveal that the total entropy production of ZrO₂-water nanofluids is dominated by the thermal generation at all concentrations, particularly for larger diameter nanoparticles. It can be seen that when d_p is smaller and concentration is higher, the frictional entropy generation starts to become more significant.

3.4. Effect of Different Nanoparticles

In this section, entropy generations for water-based nanofluids with different nanoparticle materials are studied. Figure 6 compares the variations of entropy generation of different nanofluids versus solid volume concentration. Nanofluids with four different nanoparticle materials (Al₂O₃, ZrO₂, SiO₂ and CuO) are studied. For all nanofluids, the same inlet condition with Re = 60,000 and the same nanoparticle size dp = 20 nm are used in the analysis.

The variations of thermal entropy production of different nanofluids are shown in Figure 6a. It is seen that the thermal entropy productions for different nanofluids decrease with increase of concentration. ZrO₂ nanofluid typically leads to the highest thermal entropy generation, and Al₂O₃ nanofluids generate the lowest thermal entropy among the nanofluids studied. This is perhaps due to the fact that Al₂O₃ has the highest thermal conductivity compared to the other nanoparticles studied. Equation (15) shows that the thermal entropy generation is inversely proportional to the effective thermal conductivity. In other words, the Al₂O₃ nanofluids have better heat transfer characteristics compared to the other nanoparticles used in the present study and thus lead to lower thermal entropy generation.

Figure 6b compares the frictional entropy generation for different nanofluids. It is seen that the frictional entropy productions of different nanofluids increase with concentration but are roughly the same at the low solid volume fractions. For concentrations higher than 4%, the trend of increase with volume fraction becomes sharper, and the differences in frictional entropy productions become more significant. Figure 6b also shows that the SiO₂-water nanofluid has the highest S_gen,f among the nanofluids considered at the same Reynolds number and concentration. This is because SiO₂ has the highest density, and Equation (16) shows that the frictional entropy generation is roughly linearly proportional to the effective density of the nanofluid. (Equation (16) shows that S_gen,f is proportional to the cube of mass flow rate divided by density square). In summary, the nanofluid with higher density generates more friction loses which leads to higher friction entropy generation.

Variations of the total entropy production with solid volume fraction for different nanofluids are shown in Figure 6c. Since concentration range is extended to 7%, the v-shape variation of total entropy production can be clearly seen in the figure. This figure shows that all nanofluids studied have an optimum concentration for minimum entropy production at the concentration range of 5% to 6%. As mentioned before, for ZrO₂-water nanofluid, the optimal working condition is at a solid volume fraction of about 5% to 6%. The SiO₂-water nanofluid generates the highest total entropy production, with its minimum value at a concentration of 5%. The optimal working condition for the Al₂O₃-water nanofluid, which has the lowest total entropy generation, is also at a concentration of 5%. The minimum entropy production of CuO-water nanofluid occurs at about 6% concentration. Figure 6c shows that the effect of frictional entropy generation becomes markedly significant at higher concentrations.

Figure 6d compares the variation of Bejan number as a function of volume concentration for different nanofluids. It is seen that the trend of variation of Bejan number is roughly the same for all the nanofluids studied. Only at concentrations larger than 5%, do the differences become noticeable. It is seen that the ZrO₂-water nanofluid leads to the highest value of Bejan number, while SiO₂-water nanofluid leads to the lowest values.

This study includes an analysis of the entropy generation of Al₂O₃-water and a comparison with nanofluids with other nanoparticles (ZrO₂, CuO, SiO₂). The present results for ZrO₂-water nanofluid were verified through a comparison with the results of Bianco et al. [36,37]. Figures 6a–d show that zirconia-water nanofluid used in the present study and Al₂O₃-water nanofluid used in references [36,37] exhibit a trend of variation similar to the variation trend of the volume fraction of nanoparticles as well as Reynolds number and diameter of the particles. However, as discussed earlier the entropy generations for ZrO₂-water were generally more than those for Al₂O₃-water nanofluid.

4. Conclusions

In this paper the second law analysis of turbulent forced convection of ZrO₂-water nanofluid flow at constant heat flux is performed. Based on the results presented, the following conclusions are drawn:

• At a constant Reynolds number and for a given nanoparticle size, the thermal entropy generation decreases with the increase of concentration, while the frictional entropy generation has an opposite trend. The total entropy generation decreases to a minimum at 6% solid volume fraction and then increases with further increase of volume concentration.
• With an increase in Reynolds number, the total and the thermal entropy generation decrease, while the frictional entropy generation increases. The optimal volume concentration for minimum entropy generation decreases with the increase in Reynolds number.
• As nanoparticle size increases, the thermal and the total entropy generation increase, but the frictional entropy production decreases. The optimal concentration for minimum total entropy generation becomes higher for larger nanoparticle diameters.
• Al₂O₃-water nanofluid has the lowest thermal entropy generation, and the ZrO₂-water nanofluid has the highest. In terms of total entropy generation, the SiO₂-water nanofluid generated the highest entropy generation among the tested nanofluids.