Second Law Analysis of Nanofluid Flow Within a Circular Minichannel

entropy

Article Second Law Analysis of Nanoﬂuid Flow within a Circular Minichannel Considering Nanoparticle Migration

Mehdi Bahiraei * and Navid Cheraghi Kazerooni Mechanical Engineering Department, School of Energy, Kermanshah University of Technology, Kermanshah 6715685420, Iran; [email protected] * Correspondence: [email protected]; Tel.: +98-833-7259-980

Academic Editor: Eliodoro Chiavazzo Received: 19 August 2016; Accepted: 18 October 2016; Published: 21 October 2016 Abstract: In the current research, entropy generation for the water–alumina nanofluid flow is studied in a circular minichannel for the laminar regime under constant wall heat flux in order to evaluate irreversibilities arising from friction and heat transfer. To this end, simulations are carried out considering the particle migration effects. Due to particle migration, the nanoparticles incorporate non-uniform distribution at the cross-section of the pipe, such that the concentration is larger at central areas. The concentration non-uniformity increases by augmenting the mean concentration, particle size, and Reynolds number. The rates of entropy generation are evaluated both locally and globally (integrated). The obtained results show that particle migration changes the thermal and frictional entropy generation rates significantly, particularly at high Reynolds numbers, large concentrations, and coarser particles. Hence, this phenomenon should be considered in examinations related to energy in the field of nanofluids.

Keywords: nanoﬂuid; second law of thermodynamics; entropy generation; particle migration; concentration distribution

1. Introduction Nanofluids—i.e., suspensions of nanometer-sized particles—are the novel generation of heat transfer fluids for different industrial applications, due to their excellent thermal efficiency. Some instances of the applications of nanofluids include various types of heat exchangers [1,2], thermosyphons and heat pipes [3,4], car radiators [5], cooling of electronic devices, chillers, cooling and heating in buildings, medical applications [6], microchannels [7], and solar collectors [8]. A great deal of research work has been performed on nanofluids [9–11]. The initial studies were mainly confined to measurement and modeling of thermal conductivity. To our knowledge, the first research on these suspensions was implemented in 1993 by Masuda et al. [12]. They added Al2O3 and TiO2 nanoparticles at a concentration of 4.3% in base fluid, and illustrated that the thermal conductivity increment values are 32% and 11%, respectively. In comparison with base fluids, Choi and Eastman [13] showed that the thermal conductivities of Cu–water and carbon nanotube (CNT)–water nanofluids are higher. Eastman et al. [14] indicated that Cu–ethylene glycol nanofluid presents a 40% increment in thermal conductivity at 0.3% concentration. In spite of the initial studies (which were often done on the thermal conductivity), the researchers have recently paid more attention to the convective heat transfer, since the nanofluids can have many potential applications in the processes associated with convective heat transfer [15–17]. Sundar and Sharma [18] assessed the influence of the water–Al2O3 concentration on the Nusselt number and friction factor in a circular pipe having a twisted tape for turbulent condition. Their obtained results showed that the heat transfer augments by concentration and Reynolds number increment.

Entropy 2016, 18, 378; doi:10.3390/e18100378 www.mdpi.com/journal/entropy Entropy 2016, 18, 378 2 of 27

Mahdavi et al. [19] evaluated flow and heat transfer characteristics of the laminar flow of a nanofluid inside a straight pipe using the Eulerian–Lagrangian method. In this study, some conventional kinds of nanofluids, including nanoparticles of alumina, zirconia, and silica were examined, and the findings were compared with experimental data. Pressure drop predicted by Eulerian–Lagrangian technique was found to be reliable for concentrations less than 3%. In the research contributions conducted so far, a nanofluid has frequently been presumed to be a uniform suspension which has uniform properties in all positions of the suspension. These assumptions cannot be true, and may lead to mistakes in phenomena associated with nanofluids. In fact, the nanoparticles can have Brownian diffusion resulting from their low mass and size, even if they are in a motionless medium. Therefore, the nanoparticle movement examination is significant to evaluate nanofluids as heat transfer fluids. Additionally, the presented findings on flow and heat transfer features of nanofluids are remarkably inconsistent. These inconsistencies (which are seen to a large extent in the literature) prove that the interactions of base fluid and nanoparticles may considerably affect the amount of heat transfer, and are currently unclear. Although many mechanisms, such as liquid layering, Brownian diffusion, ballistic conduction, etc. have been introduced, there is not any overall approach to demonstrating nanofluid behavior. A significant parameter that may be operative for the proper characterization of nanofluid behavior is particle migration. It is indicated that if the particle migration impacts are properly considered, more truthful findings that are much closer to the physics of the problem can be achieved. A non-uniform concentration distribution is obtained, caused by the particle migration, which can change the distributions of thermo-physical properties—especially the viscosity and thermal conductivity, as these properties are significantly functions of particle distribution. Very few research studies have been carried out addressing the impacts of particle migration on the characteristics of nanofluids [20–22]. Ding and Wen [23] examined nanoparticle motion in the flow of nanofluids for the laminar regime. It was found that the particle concentration near the pipe wall is lower in comparison to central areas. Bahiraei [24] examined the hydrothermal properties of the nanofluids containing magnetite nanoparticles for the turbulent regime. The impacts of non-uniform shear rate, Brownian motion, and viscosity gradient were taken into account. By considering the influences of particle migration, the near-wall concentration was smaller than in central areas. The particle concentration non-uniformity was more obvious for the larger nanoparticles, and was augmented by increment of Reynolds number and mean concentration. Malvandi et al. [25] studied heat transfer properties hydromagnetic alumina/water nanofluid through a micro-annulus, taking various mechanisms of particle migration into account. The mode utilized was capable of regarding particle migration due to Brownian diffusion and thermophoresis. The results proved that augmenting the slip velocity and magnetic field improves the heat transfer efficacy, while raising concentration, ratio of internal wall to outer wall radius, and heat flux ratio, reducing thermal efficiency. Most of the contributions on nanofluids (some of them were mentioned above) have been carried out using the first law of thermodynamics, while it is not properly able to analyze energy saving or wasting alone. Generally, since the 1970s, the application of the second law of thermodynamics in the thermal design of equipment has attracted substantial attention. The second law of thermodynamics is related to energy availability. The most efficient use of available energy can be attained by applying optimization of thermal exchange equipment through the second law instead of the first law. Employing the second law of thermodynamics has influenced the design approaches of various thermal devices to optimize the rates of entropy production, and therefore to maximize achievable work. The use of one of two terms has been considered by many researchers: irreversibility (entropy generation) and exergy (available energy). Lower entropy production in a piece of thermal equipment means lower energy dissipation. Bejan [26] carried out one of the initial research studies on entropy production for convection heat exchange in a number of important applications. Bejan [27] is the most Entropy 2016, 18, 378 3 of 27 famous investigator in this field, and has concentrated on the various factors that are effective on the production of entropy. Although the key goal in heat exchange systems is the heat exchange rate, the entropy production can be great, which may cause a small efficacy of the second law. Therefore, entropy production analysis could be an appropriate approach to investigating a heat transfer device from a second law viewpoint. Several researchers have used the second law of thermodynamics in their investigations for the analysis of nanofluid performance. Moghaddami et al. [28] investigated entropy production for two Al2O3–water and Al2O3–ethylene glycol nanofluids inside a tube under uniform heat flux and for both laminar and turbulent regimes. The authors concluded that the addition of solid nanoparticles decreases entropy production for the laminar regime, whereas there was an optimum Reynolds number in which entropy generation was minimized in turbulent conditions. Sheremet et al. [29] investigated the influences of a temperature-constant panel insertion for a cavity filled with nanofluid that is chilled using a temperature-constant cooler. The model developed as dimensionless parameters was solved through finite volume approach. The research was carried out for various geometrical ratios of block inserted and temperature-constant cooler, Rayleigh number, and particle concentration. The results showed that Bejan number, Nusselt number, and total entropy generation augment by concentration increment. Mahian et al. [30] assessed the entropy production between two cylinders, applying TiO2–water nanofluid under the influence of Magneto-Hydrodynamic (MHD) flow. In addition to the thermal and frictional impacts, the effect of the employed magnetic field on the entropy production was also taken into account. The authors proposed utilizing nanofluids under MHD just for small Brinkman numbers. Boghrati et al. [31] evaluated entropy production due to the flow of water-based carbon nanotubes (CNTs) and Al2O3 nanoparticles inside parallel plates. Between the two plates, a rectangular barrier was situated such that the nanofluid can flow around the barrier. The results revealed that entropy production augments through the addition of solid nanoparticles. Furthermore, it was concluded that the entropy production caused by CNTs is higher than that resulting from Al2O3 nanoparticles. Mahmoudi et al. [32] investigated the effect of MHD on the production of entropy, applying Cu–water nanofluids inside a trapezoidal cavity. It was concluded that the production of entropy decreases as concentration increases, whereas it increases with the magnetic field. Frictional, thermal, and total entropy generation rates for the water–Al2O3 nanofluid flow inside a circular minichannel are examined in the present work, with respect to the particle migration effects. The impacts of parameters such as particle size, Reynolds number, and concentration on entropy production are studied. To the best knowledge of the authors, the current research is the first investigation in which the second law analysis is employed to assess irreversibility in nanofluids with respect to the impacts of particle migration.

2. Particle Migration

This study is conducted on the flow of water–Al2O3 nanofluid within a circular minichannel in order to evaluate the irreversibilities caused by friction and heat transfer with respect to the migration of nanoparticles. Three main factors cause the migration of particles in non-uniform shear flows:

• Particle movement from greater shear rate areas to smaller shear rate ones, caused by shear-induced mechanism; • Particle movement from greater viscosity areas to smaller viscosity ones, due to viscosity gradient; • Particle movement from higher concentration areas to smaller concentration ones, due to Brownian motion [33–35].

These factors affect particle migration differently. For instance, shear-induced particle migration leads to a concentration of the particles in areas with low shear, whereas Brownian diffusion moves the particles from areas of higher concentration to those with smaller concentration. Entropy 2016, 18, 378 4 of 27

Considering mass balance for the solid phase in a nanoﬂuid ﬂow having conditions of steady state and fully developed inside a pipe will present [23]:

dJ J + r = 0 (1) dr where r represents the radial coordinate and J denotes the total particle ﬂux in the r direction. As mentioned above, the total particle migration ﬂux includes three mechanisms:

J = Jµ + Jc + Jb (2) where Jµ, Jc, and Jb stand for the particle flux resulting from viscosity gradient, particle flux due to non-uniform shear rate, and particle flux caused by Brownian diffusion, respectively. Phillips et al. [34] presented these equations: 2 ! . dp dµ J = −K γϕ2 ∇ϕ (3) µ µ µ dϕ

2 2 . . Jc = −Kcdp(ϕ ∇γ + ϕγ∇ϕ) (4)

Jb = −Db∇ϕ (5) . where Kµ and Kc denote phenomenological constants, γ represents the shear rate, ϕ is the concentration, dp is the particle size, µ is the dynamic viscosity, and Db represents the Brownian diffusion coefﬁcient, calculated by: kBT Db = (6) 3πµdp where kB represents Boltzmann’s constant and T is the temperature. By integrating Equation (1) and applying boundary condition at r = 0, Equation (7) is obtained. In fact, the boundary condition of symmetry at the center of the pipe is employed.

J = Jµ + Jb + Jc = 0 (7)

It should be noted that Equation (7) is applicable for all radial locations under the fully developed and steady state condition (not only at the tube center). Equation (8) is achieved by substituting Equations (3)–(5) into Equation (7):

2 . . dp dµ dγ . dϕ dϕ K γϕ2 + K d2 ϕ2 + K d2 ϕγ + D = 0 (8) µ µ dr c p dr c p dr b dr

For a Newtonian ﬂuid: . 1 dP γ = r (9) 2µ dz

. dv where γ = dr (v denotes the velocity) and P represents the pressure. The variable z is the longitudinal component. By supposing the nanoﬂuid is a Newtonian ﬂuid, Equation (8) is achieved as below:

r 1 dµ K µ d µ K 1 dϕ 1 1 dϕ + ( c ) + ( c ) − = 2 0 (10) µ dr Kµ r dr Kµ ϕ dr ϕ r KµPe dr where 3 r 3πdp (−dP/dz) R r = , Pe = (11) R 2kBT The parameter Pe is Peclet number. This dimensionless number presents the ratio of particle migration due to convection to that of Brownian motion. The parameter R denotes the pipe radius. Entropy 2016, 18, 378 5 of 27

The following equation [36] was used for the nanoﬂuid effective viscosity in Equation (10).

2 µ = µ f (123ϕ + 7.3ϕ + 1) (12) where subscript f refers to the base ﬂuid. A boundary condition is needed to solve Equation (10), which is obtained from Equation (13):

R ϕ(r)dA ϕm = (13) R dA where ϕm is the mean concentration and A denotes the area. Nanoparticle concentration distribution is achieved by solving Equation (10) under various conditions.

3. Governing Equations and Boundary Conditions The continuity, momentum, and energy equations have been presented in the following, which should be solved by applying effective properties. Conservation of mass: 1 ∂(ρrv ) ∂(ρv ) r + x = 0 (14) r ∂r ∂x Conservation of momentum:

∂ρv ∂ρv ∂P ∂ µ ∂(rv ) ∂ ∂v v r + v r = − + r + µ r (15) r ∂r x ∂x ∂r ∂r r ∂r ∂x ∂x

∂ρv ∂ρv ∂P 1 ∂ ∂v ∂ ∂v v x + v x = − + µr x + µ x (16) r ∂r x ∂x ∂x r ∂r ∂r ∂x ∂x Conservation of energy:

1 ∂ ∂ 1 ∂ ∂T ∂ ∂T rρc Tu + ρc Tu = kr + k (17) r ∂r p r ∂x p x r ∂r ∂r ∂x ∂x where ρ represents density, cp denotes speciﬁc heat, and k is thermal conductivity.

3.1. Nanofluid Properties In the research contributions, in which a single-phase method has been applied to simulate nanofluids, uniform properties have been employed. The effective properties, however, are utilized in the current survey as location-dependent after determining the distribution of concentration. The following models were employed to calculate specific heat and density, whereas Equation (12) was applied to evaluate the viscosity.

ρ = ϕρp + (1 − ϕ)ρ f (18)

( − )( ) + ( ) 1 ϕ ρcp f ϕ ρcp p c = (19) p ρ where subscript p refers to the nanoparticles. Furthermore, the model presented by Maiga et al. [36] was employed to evaluate the thermal conductivity: 2 k = k f (4.97ϕ + 2.72ϕ + 1) (20) It is noteworthy that the location-dependent concentration is applied in above equations. Entropy 2016, 18, 378 6 of 27

3.2. Boundary Conditions At the pipe inlet, developed velocity and uniform temperature are utilized. No-slip condition and constant heat ﬂux are employed on the wall. Moreover, atmospheric static pressure is applied at the pipe outlet. The boundary conditions are presented mathematically as follows:

• At the inlet of the minichannel (x = 0):

u = udeveloped, T = T0 (21)

• At the wall: ∂T u = 0, −k = q00 (22) ∂r • At the outlet of the minichannel: P = P0 (23)

4. Entropy Generation Two factors cause local entropy production rate; one of them originates from heat transfer, and other is due to friction. Equation (24) presents the local entropy production rate per unit volume:

. 000 . 000 . 000 St = Sh + S f (24)

. 000 . 000 . 000 where St , Sh and S f represent total, thermal and frictional entropy generation rates, respectively.

" 2 2# . 000 k ∂T ∂T S = + (25) h T2 ∂x ∂r

( " 2 2# 2) . 000 µ ∂v ∂v ∂v ∂v S = 2 x + r + x + r (26) f T ∂x ∂r ∂r ∂x

By integrating from local entropy production rate as per Equation (27), entropy production rate for total volume of the nanoﬂuid is calculated. Equation (27) is usable for total, frictional, and thermal entropy generation. . Z . 000 S = S dV (27)

To measure the contributions of two factors which are effective in entropy production, a dimensionless number called the Bejan number is introduced according to Equation (28), which gives the ratio of thermal entropy production rate to the total one. It should be noted that the Bejan number is examined locally as well as globally. . 000 Sh Be = . 000 (28) St

5. Numerical Method and Validation The investigation was performed through a finite volume approach. To solve the equations of momentum and energy, a second order upwind model was used. Moreover, the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) approach was adopted for coupling the velocity and pressure. The simulation is an axisymmetric problem, and therefore onlyt half of the field was considered two-dimensionally. The diameter of the minichannel was considered to be 1 mm. To finalize the numerical executions, convergence criterion was assumed 10−6 for the variables. Besides, to ensure grid independency, several grids were evaluated, and the one having 20 × 2500 nodes was chosen as the best meshing. Indeed, no considerable change was observed in the findings of smaller meshes. Entropy 2016, 18, 378 7 of 27

The investigation was performed through a finite volume approach. To solve the equations of momentum and energy, a second order upwind model was used. Moreover, the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) approach was adopted for coupling the velocity and pressure. The simulation is an axisymmetric problem, and therefore onlyt half of the field was considered two-dimensionally. The diameter of the minichannel was considered to be 1 mm. To finalize the numerical executions, convergence criterion was assumed 10−6 for the variables. Besides, to ensure grid independency, several grids were evaluated, and the one having 20 × 2500 nodes was chosen as the best meshing. Indeed, no considerable change was observed in the findings of smaller meshes. The findings of grid independency check have been presented in Table 1 for Nusselt number at a dimensionless length of the channel (i.e., x/D = 200).

Table 1. Grid independency study for Nusselt (Nu) number at x/D = 200.

Grid Nu (Numerical) Nu (Equation (29)) 15 × 800 5.295 5.452 Entropy 2016, 18,15 378 × 1000 5.412 5.452 7 of 27 18 × 2000 5.414 5.452 The findings20 of grid× 2500 independency check 5.445 have been presented in Table 5.4521 for Nusselt number at a dimensionless length of the channel (i.e., x/D = 200). 28 × 2900 5.443 5.452 Table 1. Grid independency study for Nusselt (Nu) number at x/D = 200. For validation, the results of the present numerical solution were compared with reliable data Grid Nu (Numerical) Nu (Equation (29)) for pure water as well as nanofluid.15 × 800Hence, a vali 5.295d equation for 5.452 water, and an experimental work for the water–Al2O3 nanofluid were15 applied× 1000 for 5.412investigation of 5.452 the validity of the method. Equation 18 × 2000 5.414 5.452 (29) was utilized for water [37]: 20 × 2500 5.445 5.452 28 × 2900 5.443 5.452 −13/ −≤ 3., 302xx++ 1 0. 00005  For validation, the results=− of the present−13/ numerical solution ≤≤ were compared with reliable data for Nu1.., 302 x++ 0 5 0.. 00005 x 0 0015 (29) pure water as well as nanofluid. Hence, a valid equation for water, and an experimental work for the  3− 0 506 −41x+ 4 364+≥ 8 68 10. 0 0015 water–Al2O3 nanofluid were applied..() for investigationxex++ of the validity , of the. method. Equation (29) was utilized for water [37]:

= x  −1/3 where x+ in which Pr denotes 3.302x+ the− Prandtl1, x+ ≤ 0.00005 number, and D represents the diameter of the  −1/3 DRePr Nu = 1.302x+ − 0.5, 0.00005 ≤ x+ ≤ 0.0015 (29)  3 −0.506 −41x pipe.  4.364 + 8.68(10 x+) e + , x+ ≥ 0.0015

Figure 1 presents ax comparison between the results of the present study and those of Equation where x+ = DRePr in which Pr denotes the Prandtl number, and D represents the diameter of the pipe. (29) for water. ItFigure is observed1 presents athat comparison there is between a very the good results agreement of the present studybetween and those the ofresults. Equation In (29) addition, the convective forheat water. transfer It is observed coefficients that there achieved is a very from good agreementthe current between study the results. were Incompared addition, with the the convective heat transfer coefﬁcients achieved from the current study were compared with the experimental data [38] for water–Al2O3 nanofluid within a circular tube in Table 2. In this table, a experimental data [38] for water–Al2O3 nanoﬂuid within a circular tube in Table2. In this table, proper consistencya proper consistency is noticed is noticedbetween between the data, the data, proving proving thatthat the the solution solution is valid. is valid.

Figure 1. NusseltFigure 1.numberNusselt (Nu) number obtained (Nu) obtained from from the thepresent present work work comparedcompared to validto valid data data [37] for [37] for pure pure water. water.

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Table 2. Convective heat transfer coefficient obtained from the present study compared to the Entropyexperimental2016, 18, 378data [38] for ϕ = 1% at different Reynolds numbers. 8 of 27

Present Study (with Present Study (without Re Experimental Results [38] Table 2. Convective heat transfer coefﬁcientParticle obtained Migration) from the present studyParticle compared Migration) to the experimental data [38] for ϕ = 1% at different Reynolds numbers. 300 2555 2604 2637

900 2761 Present 2691 Study (with Present Study (without 2680 Re Experimental Results [38] 1500 2889 Particle 2823 Migration) Particle Migration) 2792 2000 3003107 2555 3148 2604 2637 3174 900 2761 2691 2680 6. Results and1500 Discussion 2889 2823 2792 2000 3107 3148 3174 Numerical research is conducted in order to assess the influence of particle migration on entropy6. Results production and Discussion for the water–Al2O3 nanofluid flow within a circular minichannel under a constant Numericalwall heat flux research (i.e., is 5000 conducted W/m2). in The order numerical to assess the runs influence are carrie of particled out for migration mean concentrations on entropy of 1%,production 3% and 5%, for the the water–Al Reynolds2O 3numbersnanofluid of flow 200, within 1000, aand circular 2000, minichannel and particle under sizes a constantof 10, 50, wall and 90 nm. Forheat this flux goal (i.e., 5000at first, W/m the2). distribution The numerical of runs the areconcentration carried out for at meanthe cross-section concentrations of of the 1%, pipe 3% and should be determined5%, the Reynolds by solving numbers Equation of 200, 1000,(10) for and different 2000, and states. particle sizes of 10, 50, and 90 nm. For this goal atFigure first, the2 depicts distribution the concentration of the concentration distribution at the cross-section for various ofmean the pipe concentrations should be determined at Re = 2000 by and solving Equation (10) for different states. dp = 90 nm at a tube cross-section. As can be observed, non-uniformity of the concentration distributionFigure intensifies2 depicts theby concentrationmean concentration distribution increment, for various so mean that concentrations the nanofluid at Reattains= 2000 a and higher d = 90 nm at a tube cross-section. As can be observed, non-uniformity of the concentration distribution concentrationp in the central areas of the tube. This observation is attributed to 1/ϕ2 in the latest term intensifies by mean concentration increment, so that the nanofluid attains a higher concentration in of Equation (10), which considerably reduces with the increase of concentration, due to the power of the central areas of the tube. This observation is attributed to 1/ϕ2 in the latest term of Equation (10), secondwhich order considerably in the denominator. reduces with Indeed, the increase the ofimpa concentration,ct of Brownian due to motion—which the power of second tends order to inmake the the particledenominator. distribution Indeed, more the uniform—reduces impact of Brownian motion—whichby the concentration tends to makeincrement the particle as compared distribution to the effectmore of the uniform—reduces other factors. by the concentration increment as compared to the effect of the other factors.

FigureFigure 2. Concentration 2. Concentration distribution distribution at at a cross-section of of the the tube tube for for different different mean mean concentrations concentrations at at ReRe = 2000= 2000 and and ddp p== 90 90 nm. nm.

The concentration distribution is presented in Figure 3 for different Reynolds numbers at ϕm = The concentration distribution is presented in Figure3 for different Reynolds numbers at ϕm = 5% 5% andand ddpp == 90 90 nm. nm. It It can can be be noticed noticed thatthat augmentingaugmenting the the Reynolds Reynolds number number makes makes the the concentration concentration distributiondistribution rather rather non-uniform non-uniform atat the the tube tube cross-section. cross-section. This is This attributed is attributed to an increment to an in increment diffusion in diffusiondue to due non-uniform to non-uniform shear induced shear atinduced the higher at Reynoldsthe higher numbers, Reynolds which numbers, can increase which gathering can increase of gatheringthe nanoparticles of the nanoparticles in the central in areas.the central areas.

Entropy 2016, 18, 378 9 of 27 Entropy 2016,, 18,, 378378 9 of 27

Figure 3. Concentration distribution for different Reynolds numbers at ϕm = 5% and dp = 90 nm. FigureFigure 3.3. ConcentrationConcentration distributiondistribution forfor differentdifferent ReynoldsReynolds numbersnumbers atat ϕϕmm = 5% and dp == 90 90 nm. nm.

The concentration distribution at the tube cross-section is illustrated in Figure 4 for different The concentrationconcentration distributiondistribution at at the the tube tube cross-section cross-section is illustratedis illustrated in Figure in Figure4 for 4 different for different sizes m sizes of the nanoparticles at Re = 2000 and ϕm = 5%. As can be observed, the concentration of the nanoparticles at Re = 2000 and ϕm = 5%. As can be observed, the concentration distribution distribution becomes more non-uniform by particles’ enlargement. The non-uniform shear rate leads distributionbecomes more becomes non-uniform more non-uniform by particles’ by enlargement.particles’ enlargement. The non-uniform The non-uniform shear rate shear leads rate to leads the to the migration of nanoparticles to the central areas of the tube, while the Brownian diffusion tomigration the migration of nanoparticles of nanoparticles to the centralto the central areasof areas the tube,of the while tube, thewhile Brownian the Brownian diffusion diffusion guides guides the nanoparticles opposite to the concentration gradient. Thus, these two mechanisms work guidesthe nanoparticles the nanoparticles opposite opposite to the to concentration the concentration gradient. gradient. Thus, Thus, these these two two mechanisms mechanisms work work in in two opposite directions. By increasing the particle size, the impact of the Brownian diffusion intwo two opposite opposite directions. directions. By increasingBy increasing the particle the particle size, thesize, impact the impact of the Brownianof the Brownian diffusion diffusion reduces, reduces, while the influence of the non-uniform shear rate augments. Thus, at a given mean whilereduces, the while inﬂuence the ofinfluence the non-uniform of the non-unifor shear ratem augments. shear rate Thus, augments. at a given Thus, mean at a concentration, given mean concentration, a higher agglomeration of the nanoparticles accumulates in the central areas for concentration,a higher agglomeration a higher of agglomeration the nanoparticles of accumulatesthe nanoparticles in the accumulates central areas forin the coarser central nanoparticles, areas for coarser nanoparticles, such that the concentration value increases just 0.36% from the wall to the coarsersuch that nanoparticles, the concentration such value that the increases concentration just 0.36% value from increases the wall tojust the 0.36% tube centerfrom forthe thewall particles to the tube center for the particles of 10 nm size, while it increases by almost four times from the wall to the tubeof 10 center nm size, for whilethe particles it increases of 10 nm by almostsize, while four it times increases from by the almost wall tofour the times tube from center the for wall particles to the tube center for particles of 90 nm size. tubeof 90 center nm size. for particles of 90 nm size.

Figure 4. Concentration distribution at the tube cross-section for different sizes of nanoparticles at Figure 4. Concentration distribution at at the tube cross-section for different sizes of nanoparticles at Re = 2000 and ϕm = 5%. Re = 2000 and ϕm = 5%. Re = 2000 and ϕm = 5%.

The effects of various parameters on entropy production rates will be discussed in the The effects of various parameters on entropy production rates will be discussed in the following, following, taking into consideration the particle migration impacts. Local investigations are carried taking into consideration the particle migration impacts. Local investigations are carried out on out on a cross-section of 0.9 m distance from the tube inlet, unless otherwise mentioned for the a cross-section of 0.9 m distance from the tube inlet, unless otherwise mentioned for the relevant relevant cross-section location. cross-section location. p Figure 5 illustrates the thermal entropy generation rate at Re = 2000 and dp = 90 nm for various Figure5 illustrates the thermal entropy generation rate at Re = 2000 and d = 90 nm for various concentrations. As seen from this figure, increasing the concentration leads top a reduction in the concentrations. As seen from this ﬁgure, increasing the concentration leads to a reduction in the thermal thermal entropy production. The reason for this is that the increased concentration improves the entropy production. The reason for this is that the increased concentration improves the nanoﬂuid nanofluid thermal conductivity, which results in a reduction of the temperature gradient (see Figures 6 and 7). On the other hand, the greater concentration increases the nanofluid viscosity, and because the investigations are done at a constant Reynolds number, the nanofluid velocity shall be

Entropy 2016, 18, 378 10 of 27 thermal conductivity, which results in a reduction of the temperature gradient (see Figures6 and7 ). On theEntropy other 2016, hand,18, 378 the greater concentration increases the nanofluid viscosity, and because10 of 27 the investigations are done at a constant Reynolds number, the nanofluid velocity shall be augmented augmented (since the viscosity is in the Reynolds number denominator), which causes a nanofluid (sinceEntropy the viscosity 2016, 18, 378 is in the Reynolds number denominator), which causes a nanofluid temperature10 of 27 temperature decrement. The lower temperature enhances the thermal entropy generation rate decrement. The lower temperature enhances the thermal entropy generation rate because the augmentedbecause the (sincetemperature the viscosity is located is in in the the Reynolds denomina numbertor of the denominator), equation, which which describes causes athe nanofluid thermal temperaturetemperatureentropy generation is located decrement. (Equation in the The denominator 25).lower However, temperature ofsince the thee equation,nhances temperature the which thermal gradient describes entropy decrease thegeneration dominates thermal rate the entropy generationbecausetemperature (Equationthe temperature decrease, (25)). isthe located However, thermal in the sinceentropy denomina the getor temperatureneration of the equation, rate gradientreduces which decreasedescribeswith concentration the dominates thermal the temperatureentropyaugmentation. generation decrease, the(Equation thermal 25). entropy However, generation since the temperature rate reduces gradient with concentration decrease dominates augmentation. the temperature decrease, the thermal entropy generation rate reduces with concentration augmentation.

p FigureFigure 5. Thermal 5. Thermal entropy entropy generation generation rate rate for for differentdifferent concentrations concentrations at atReRe = 2000= 2000 and and d = d90 nm.= 90 nm. p

Figure 5. Thermal entropy generation rate for different concentrations at Re = 2000 and dp = 90 nm.

Figure 6. Thermal conductivity at a tube cross-section for different concentrations.

FigureFigure 6. Thermal 6. Thermal conductivity conductivity atat aa tube cross-section cross-section for for different different concentrations. concentrations.

Figure 7. Temperature gradient at a tube cross-section for different concentrations.

FigureFigure 7. Temperature 7. Temperature gradient gradient atat aa tubetube cross-section for for different different concentrations. concentrations.

Entropy 2016, 18, 378 11 of 27

Figure8 shows the distributions of thermal entropy generation rate for two different cross-sections—one just close to the inlet (i.e., at x = 0.05 m), and the other close to the outlet (i.e., at x = 0.9 m)—at Re = 2000, dp = 90 nm, and ϕm = 5%. It is clearly seen that at the cross-section near the inlet, the thermal entropy generation rate is insigniﬁcant in a wide area of this cross-section. The reason for this is that the temperature gradient is negligible in the central areas because the thermal boundary layer has not developed much, and has not yet reached central regions as a consequence. However, at the cross-section near the outlet, this rate has become signiﬁcant in a wider area of the cross-section due to more development of the thermal boundary layer. In addition, it is noticed that the thermal entropy production rate near the wall for the cross-section close to the outlet is smaller than that for the one close to the inlet. The reason is that on one hand, the temperature gradient on the wall is the same for both of the cross-sections, in accordance with the Fourier law (Equation (30)), and on the other hand, for the cross-section close to the outlet, the temperature is higher, and consequently, the thermal entropy generation near the wall for this cross-section will be smaller (as per Equation (25)).

∂T q00 = k (30) ∂r

Figure 8. Proﬁles of thermal entropy generation rate at two different cross-sections for Re = 2000, dp = 90 nm, and ϕm = 5%.

Figure9 illustrates the frictional entropy generation rate at Re = 2000 and dp = 90 nm for various concentrations. Increasing the concentration intensifies the frictional entropy production rate. One reason is that the nanofluid viscosity increases at higher concentrations (see Figure 10a), which can increase the frictional entropy generation rate (as per Equation (26)). On the other hand, since the examination is done at a constant Reynolds number, and the viscosity—which is in denominator of the Reynolds number calculation—increases by increasing the concentration, greater velocities are adopted at higher concentrations (see Figure 10b), which can augment the frictional entropy generation due to the velocity gradient increment. Another reason for the increased frictional entropy generation rate at higher concentrations is that the nanofluid temperature decreases by concentration augmentation (see Figure 10c). The reason for the temperature reduction is that although the nanofluid specific heat decreases at greater concentrations (and the specific heat decrement can increase the nanofluid temperature), the increased velocity compensates for this specific heat reduction, such that the temperature eventually decreases. Thus, since the temperature is in the denominator of Equation (26), the temperature reduction can increase the frictional entropy generation rate. Additionally, due to the more intensive particle migration to the central areas at higher Entropy 2016, 18, 378 12 of 27 concentrations (and thus, more non-uniform distribution of the concentration at the cross-section of the pipe), the nanofluid viscosity reduces considerably near the wall, which can intensify the velocity gradient there (see Figure 11). As shown in Figure9, this causes significant discrepancy in the frictional entropy generation rates adjacent to the wall for different concentrations.

Figure 9. Frictional entropy generation rate for different concentrations at Re = 2000 and dp = 90 nm.

(a) (b)

(c)

Figure 10. Effect of concentration change at Re = 2000 and dp = 90 nm on: (a) viscosity; (b) velocity; (c) temperature. Entropy 2016, 18, 378 13 of 27 EntropyEntropy 2016 2016, 18, ,18 378, 378 13 of13 27of 27

Figure 11. Velocity gradient at tube cross-section for different concentrations. Figure 11. Velocity gradient at tube cross-section forfor differentdifferent concentrations.concentrations. The total entropy generation rate versus the concentration is illustrated in Figure 12 for Re = The total entropy generation rate versus the concentration is illustrated in Figure 12 for Re = 2000The and total dp entropy = 90 nm. generation It is noticed rate versus that variations the concentration of the total is illustrated entropy ingeneration Figure 12 ratefor Reat =the 2000 and2000concentrationsd pand= 90 d nm.p = 90 Itof is nm.3% noticed and It is5% that noticed have variations a similarthat ofvariations trend the total to those of entropy the of thetotal generation frictional entropy entropy rate generation at the generation concentrations rate rate.at the ofconcentrations 3%This and is because 5% have of the3% a contribution similarand 5% trend have of to athe similar those frictional of trend the en frictional tropyto those generation of entropy the frictional is generation much greaterentropy rate. than generation This that is becauseof the rate. theThisthermal contribution is because entropy the of thecontributiongeneration frictional at entropyof these the frictionalconcentratio generation entropyns, is muchas generation is greaterevident thanisfrom much that a greatercomparison of the thermal than betweenthat entropy of the generationthermalFigures entropy 5 at and these 9. generationHowever, concentrations, at at the these concentration as is evidentconcentratio from of 1%,ns, a comparisonthe as frictionalis evident between and from thermal Figures a comparisonentropy5 and 9generation. However, between atFiguresrates the concentration have5 and close 9. However, values; of 1%, therefore, at the the frictional concentration the trend and of thermalvariationsof 1%, the entropy in frictional the total generation andentropy thermal ratesgeneration entropy have closerate generation at values; this therefore,ratesconcentration have the close trend isvalues; affected of variations therefore, by both in thermal the totaltrend and entropy of frictional variations generation entropy in the rate generation total at thisentropy concentrationrates. generation is affectedrate at this by bothconcentration thermal and is affected frictional byentropy both thermal generation and frictional rates. entropy generation rates.

p Figure 12. Total entropy generation rate in terms of the concentration at Re = 2000 and d = 90 nm.

Figure 12. Total entropy generation rate in terms of the concentration at Re = 2000 and dp = 90 nm.m FigureFigure 12. 13Total depicts entropy the thermal generation entropy rate ingenerati terms ofon the rate concentration for different at ReynoldsRe = 2000 numbers and dp = 90at ϕ nm. = 5% and dp = 90 nm. It is seen that the thermal entropy generation rate near the wall is greater at the larger ReynoldsFigure 13numbers. depicts Thisthe thermal is because entropy at greater generati Reynoldson rate numbersfor different (as Reynoldsdiscussed numbersbefore, Figure at ϕm 3),= 5% Figure 13 depicts the thermal entropy generation rate for different Reynolds numbers at ϕm = 5% andmigration dp = 90 nm. of theIt is particles seen that to the the th centralermal areasentropy occurs generation with more rate intensity, near the wallso that is greaterthe concentration at the larger and d = 90 nm. It is seen that the thermal entropy generation rate near the wall is greater at the Reynoldsdecreasesp numbers. more significantly This is because near the atwall. greater This causReynoldses a decrease numbers of the (as thermal discussed conductivity before, adjacentFigure 3), larger Reynolds numbers. This is because at greater Reynolds numbers (as discussed before, Figure3), migrationto the wall of atthe the particles higher Reynolds to the central numbers. areas Therefore, occurs withwith respect more intensity,to the constant so that amount the concentrationof heat flux migration of the particles to the central areas occurs with more intensity, so that the concentration decreaseson wall, morethe temperature significantly gradient near the at thewall. vicinity This causof thees wall a decrease will be ofgreater the thermal at larger conductivity Reynolds numbers adjacent decreases more signiﬁcantly near the wall. This causes a decrease of the thermal conductivity adjacent to the(see wall Figure at 14).the higherApproaching Reynolds the centralnumbers. regions, Therefore, the op withposite respect trend is to noted. the constant In other amountwords, for of higherheat flux to the wall at the higher Reynolds numbers. Therefore, with respect to the constant amount of heat onReynolds wall, the temperaturenumbers at gradientthe central at theregions, vicinity the of th theermal wall conductivity will be greater increases; at larger consequently, Reynolds numbers the ﬂux on wall, the temperature gradient at the vicinity of the wall will be greater at larger Reynolds (seetemperature Figure 14). gradientApproaching decreases the central in comparison regions, withthe op lowerposite Reynolds trend is noted.numbers. In otherThis willwords, reduce for higher the numbers (see Figure 14). Approaching the central regions, the opposite trend is noted. In other words, Reynoldsthermal entropynumbers production at the central rate in theregions, central the areas th ermalat higher conductivity Reynolds numbers. increases; consequently, the for higher Reynolds numbers at the central regions, the thermal conductivity increases; consequently, temperature gradient decreases in comparison with lower Reynolds numbers. This will reduce the the temperature gradient decreases in comparison with lower Reynolds numbers. This will reduce the thermal entropy production rate in the central areas at higher Reynolds numbers. thermal entropy production rate in the central areas at higher Reynolds numbers.

Entropy 2016, 18, 378 14 of 27

Figure 13. Thermal entropy generation rate for different Reynolds numbers at ϕm = 5% and dp = 90 nm. Figure 13. Thermal entropy generation rate for different Reynolds numbers at ϕm = 5% and dp = 90 nm.

Figure 13. Thermal entropy generation rate for different Reynolds numbers at ϕm = 5% and dp = 90 nm.

Figure 14. Temperature gradient for different Reynolds numbers at ϕm = 5% and dp = 90 nm. Figure 14. Temperature gradient for different Reynolds numbers at ϕm = 5% and dp = 90 nm. Frictional entropy generation rate is demonstrated in Figure 15 for different Reynolds numbers at Frictionalϕm = 5% and entropy dp = 90 nm. generation As is clearly rate seen is from demonstrated this figure, the in frictional Figure entropy15 for differentgeneration Reynoldsrate increases numbers at the higherFigure Reynolds 14. Temperature numbers. gradient The difference for different betw Reynoldseen the frictional numbers entropyat ϕm = 5% generation and dp = 90 rates nm. becomes at ϕm = 5% and dp = 90 nm. As is clearly seen from this figure, the frictional entropy generation rate more considerable near the wall. One can notice from Figure 16 that increasing the Reynolds number has increases at the higher Reynolds numbers. The difference between the frictional entropy generation a significantFrictional effect entropy on thegeneration velocity rategradient is demonstrated intensification. in Figure Here, 15in foraddition different to Reynoldsthe effect numbersof velocity at rates becomesincreaseϕm = 5% andon more velocity dp = considerable90 nm. gradient As is clearlyincr nearement, seen the thefrom wall. concentration this figure, One canthe distribution frictional notice fromentropy is more Figure generation non-uniform 16 thatrate at increases increasinggreater the ReynoldsReynoldsat the number higher numbers Reynolds has a(Figure significant numbers. 3). This The effectwill difference reduce on the the betw velocityvisceenosity the nearfrictional gradient the wall, entropy intensification. and generationtherefore the rates Here,gradient becomes in of addition to the effectthemore velocity considerable of velocity will be near increasemore the wall.signific onOneant velocity can there; notice finally, gradient from Figureit causes increment, 16 thata significant increasing the concentration discrepancythe Reynolds between number distribution hasthe is more non-uniformfrictionala significant entropy effect at generation greateron the velocity Reynoldsrates near gradient the numbers wall intensific for different (Figureation. ReynoldsHere,3). in This numbers.addition will to reduce the effect the of viscosity velocity near increase on velocity gradient increment, the concentration distribution is more non-uniform at greater the wall, and therefore the gradient of the velocity will be more significant there; finally, it causes Reynolds numbers (Figure 3). This will reduce the viscosity near the wall, and therefore the gradient of a significantthe velocity discrepancy will be more between signific theant frictionalthere; finally, entropy it causes generation a significant rates discrepancy near the between wall for the different Reynoldsfrictional numbers. entropy generation rates near the wall for different Reynolds numbers.

Figure 15. Frictional entropy generation rate for different Reynolds numbers at ϕm = 5% and dp = 90 nm.

Figure 15. Frictional entropy generation rate for different Reynolds numbers at ϕm = 5% and Figure 15. Frictional entropy generation rate for different Reynolds numbers at ϕm = 5% and dp = 90 nm. dp = 90 nm.

Entropy 2016, 18, 378 15 of 27 Entropy 2016, 18, 378 15 of 27

Figure 16. Velocity gradient for different Reynolds numbers at ϕm = 5% and dp = 90 nm. FigureFigure 16. 16.Velocity Velocity gradient gradient for for different different Reynolds Reynolds numbersnumbers at ϕm == 5% 5% and and dd =p 90=90 nm. nm.

Figure 17 shows the total entropy generation rate at ϕm = 5% and dp = 90 nm for various Figure 17 shows the total entropy generation rate at ϕ = 5% and d = 90 nm for various Reynolds Reynolds numbers. As can be noticed, the diagram of the mtotal entropy pgeneration rate at Re = 2000 is numbers.very like As that can of be the noticed, frictional the entropy diagram generation of the total rate. entropy This is because generation at this rate Reynolds at Re = 2000 number is very (as is like thatobvious of the frictionalin Figures entropy 13 and generation 15), the rate.frictional This isentropy because generation at this Reynolds has a number significantly (as is obviousgreater in Figurescontribution 13 and 15compared), the frictional to the entropy generation due has ato signiﬁcantly heat transfer. greater At Re contribution = 1000, wherein compared the to thecontributions entropy generation of the friction due and to heatheat transfer.transfer in At entropyRe = 1000, generation wherein are the almost contributions equal, the ofdiagram the friction of andthe heat total transfer entropy in generation entropy generation rate has been are almost affected equal, by both the diagramfactors equally. of the totalHowever, entropy thegeneration thermal rateentropy has been generation affected completely by both factors dominates equally. the However,frictional entropy the thermal generation entropy at generationRe = 200, and completely that is dominateswhy the diagram the frictional of the entropytotal entropy generation generation at Re rate= 200,in Figure and that17 is is to why a large the extent diagram similar of theto the total entropythermal generation entropy production rate in Figure rate. 17 is to a large extent similar to the thermal entropy production rate.

Figure 17. Total entropy generation rate for different Reynolds numbers at ϕm = 5% and dp = 90 nm. Figure 17. Total entropy generation rate for different Reynolds numbers at ϕm = 5% and dp = 90 nm.

Figure 18 depicts the impact of particle size on thermal entropy generation rate at ϕm = 5% and ReFigure = 2000. 18 It depictsis observed the that impact the ofthermal particle entropy size on generation thermal entropyrate near generationthe wall intensifies rate at ϕ bym =an 5% andincreaseRe = 2000 of the. It nanoparticle is observed size. that theGoing thermal far from entropy the wall generation and approaching rate near to the the wall tube intensiﬁes center, the by anopposite increase trend of the is nanoparticle observed, where size. an Going increase far fromin the the nanoparticle wall and approachingsize corresponds to the to tubea smaller center, thethermal opposite entropy trend isgeneration observed, rate where in anthe increase central inre thegions. nanoparticle This is because, size corresponds in accordance to a smallerwith thermalEquation entropy (11), generation as the nanoparticle rate in the central size increase regions.s, This the is Peclet because, number in accordance will be withgreater, Equation which (11), as theintensifies nanoparticle migration sizeincreases, of the nanoparticles the Peclet number to the will central be greater, regions. which As intensiﬁes a consequence, migration ofthe the nanoparticlesconcentration to distribution the central regions.will be more As anon-uniform consequence, at the tube concentration cross-section distribution (Figure 4). will For belarger more non-uniformparticles, the at concentration the tube cross-section will thus (Figurebe lower4). adjacent For larger to the particles, wall, and the higher concentration in the central will thusareas. be lowerThis adjacent can reduce to the the wall, thermal and conductivity higher in the in central regions areas. adjacent This canto the reduce wall, the and thermal enhance conductivity it in the incentral regions areas. adjacent Thus, to thefor wall,larger and nanoparticles, enhance it inthe the temperature central areas. gradient Thus, foraugments larger nanoparticles,at the wall thevicinity temperature and reduces gradient in the augments central atareas. the wall vicinity and reduces in the central areas.

Entropy 2016, 18, 378 16 of 27 Entropy 2016, 18, 378 16 of 27

Figure 18. Effect of nanoparticle size on thermal entropy generation rate at ϕm = 5% and Re = 2000. Figure 18. Effect of nanoparticle size on thermal entropy generation rate at ϕm = 5% and Re = 2000.

FigureFigure 1918. displaysEffect of nanoparticlethe frictional size entropy on thermal generation entropy generationrate at ϕm rate = 5% at ϕandm = 5%Re and= 2000 Re = for2000. various Figure 18. Effect of nanoparticle size on thermal entropy generation rate at ϕm = 5% and Re = 2000. Figureparticle 19diameters. displays It theis observed frictional that entropy the friction generational entropy rate generation at ϕm = rate 5% reduces and Re with= 2000 increasing for various particlenanoparticle diameters.Figure 19 size. displays It isFor observed larger the frictional nanopa thatrticles, the entropy frictional although generation entropy the velocityrate generation at ϕ gradm = 5%ient rateand is greater Re reduces = 2000 adjacent withfor various increasingto the Figure 19 displays the frictional entropy generation rate at ϕm = 5% and Re = 2000 for various particle diameters. It is observed that the frictional entropy generation rate reduces with increasing nanoparticleparticlewall (Figure diameters. size. 20), For the largerIt isfrictional observed nanoparticles, entropy that the generation friction althoughal rateentropy the is velocitylower. generation The gradient reason rate reduces for is greater this with is that adjacentincreasing for the to the nanoparticle size. For larger nanoparticles, although the velocity gradient is greater adjacent to the wall (Figurenanoparticlelarger nanoparticles, 20), size. the frictionalFor the larger viscosity nanopa entropy becomesrticles, generation althoughsmaller ratenear the isthevelocity lower. wall graddue The toient reasonmore is greater considerable for thisadjacent is particle that to the for the wall (Figure 20), the frictional entropy generation rate is lower. The reason for this is that for the largerwallmigration nanoparticles, (Figure toward 20), the the frictionalcentral viscosity areas entropy becomes (see Figuregeneration smaller 21). Meanwhile, rate near is thelower. wall the The velocity due reason to gradient more for this considerable decreases is that for in the particle largercentral nanoparticles, regions due to the the viscosity velocity profilebecomes flattening smaller fornear the the greater wall duenanoparticles, to more considerable due to the viscosity particle migrationlarger towardnanoparticles, the central the viscosity areas (see becomes Figure smaller 21). Meanwhile, near the wall the due velocity to more gradient considerable decreases particle in the migration toward the central areas (see Figure 21). Meanwhile, the velocity gradient decreases in the migrationincrement. toward This reduction the central in the areas velocity (see Figure gradient 21). wi Meanwhile,ll decrease thethe velocityfrictional gradient entropy decreases generation in rate the centralcentral regions regions due due to theto the velocity velocity profile profile flattening flattening for the the greater greater nanoparticles, nanoparticles, due due to the to viscosity the viscosity centralthere when regions compared due to theagainst velocity the caseprofile using flattening finer particles. for the greater nanoparticles, due to the viscosity increment.increment. This This reduction reduction in in the the velocity velocity gradient gradient willwill decrease decrease the the frictional frictional entropy entropy generation generation rate rate increment. This reduction in the velocity gradient will decrease the frictional entropy generation rate therethere when when compared compared against against the the case case using using finer finer particles.particles. there when compared against the case using finer particles.

Figure 19. Frictional entropy generation rate at ϕm = 5% and Re = 2000 for different particle sizes.

Figure 19. Frictional entropy generation rate at ϕm = 5% and Re = 2000 for different particle sizes. FigureFigure 19. Frictional 19. Frictional entropy entropy generation generation rate rate at at ϕϕmm = 5% 5% and and ReRe = 2000= 2000 forfor different different particle particle sizes. sizes.

Figure 20. Velocity gradient at different particle sizes.

Figure 20. Velocity gradient at different particle sizes. FigureFigure 20. 20.Velocity Velocity gradient gradient at different particle particle sizes. sizes.

Entropy 2016, 18, 378 17 of 27 Entropy 2016, 18, 378 17 of 27 Entropy 2016, 18, 378 17 of 27

Figure 21. Nanofluid viscosity at different particle sizes. Figure 21. NanofluidNanoﬂuid viscosity at different particle sizes.

Figure 22 illustrates the total entropy production rate for various nanoparticle sizes at ϕm = 5% Figure 22 illustrates the total entropy productionproduction rate forfor variousvarious nanoparticlenanoparticle sizes atat ϕ m = 5% and Re = 2000. It is noticed that in all particle sizes, the total entropy generation rate changesm with a and Re = 2000. 2000. It It is is noticed noticed that that in in all all particle particle sizes, sizes, the the total total entropy entropy generation generation rate rate changes changes with with a trend that is similar to that of the frictional entropy production rate. This is attributed to the greater trenda trend that that is issimilar similar to to that that of of the the frictional frictional entrop entropyy production production rate. rate. This This is is attributed attributed to to the the greater contribution of the frictional entropy production compared to that of the thermal entropy generation contribution of the frictional entr entropyopy production compared to that of the thermal entropy generation at Re = 2000 and ϕm = 5% (see Figures 18 and 19). Furthermore, it is noticed that an increase of the at Re = 2000 and ϕm = 5% 5% (see (see Figures Figures 1818 andand 19).19). Furthermore, Furthermore, it it is is noticed noticed that that an an increase of the nanoparticle size decreasesm the total entropy production rate, such that the total entropy production nanoparticle size decreases the total entropy produc productiontion rate, such that the total entropy production rate decreases by about 20% on average by increasing the particle size from 10 to 90 nm. rate decreases by about 20% on average by increasingincreasing thethe particleparticle sizesize fromfrom 1010 toto 9090 nm.nm.

Figure 22. Total entropy generation rate for different nanoparticle sizes at ϕm = 5% and Re = 2000. m Figure 22. Total entropyentropy generationgeneration raterate forfor differentdifferent nanoparticlenanoparticle sizessizes atat ϕϕm = 5% and Re = 2000. The Bejan number is employed to determine the contribution of two factors in entropy The Bejan number is employed to determine the contribution of two factors in entropy productionThe Bejan (i.e., number friction is and employed heat transfer). to determine Figure the 23 contribution depicts the oflocal two Bejan factors number in entropy at ϕ productionm = 5% and production (i.e., friction and heat transfer). Figure 23 depicts the local Bejan number at ϕm = 5% and (i.e.,Re = friction2000 for and various heat transfer).particle sizes. Figure As 23 can depicts be noti theced, local the Bejan Bejan number number at isϕ higher= 5% than and Re0.5= in 2000 the Re = 2000 for various particle sizes. As can be noticed, the Bejan number is higherm than 0.5 in the forcentral various regions, particle and consequently, sizes. As can the be contribution noticed, the of Bejan the thermal number entropy is higher generation than 0.5 is in greater the central than central regions, and consequently, the contribution of the thermal entropy generation is greater than regions,that of the and frictional consequently, one. Nevertheless, the contribution near of thethe thermalwall, the entropy trend generationis opposite, is and greater since than the that Bejan of that of the frictional one. Nevertheless, near the wall, the trend is opposite, and since the Bejan thenumber frictional is smaller one. Nevertheless,than 0.5, the contribution near the wall, of thefrictional trend isentropy opposite, production and since is the higher Bejan than number that of is number is smaller than 0.5, the contribution of frictional entropy production is higher than that of smallerthe thermal than entropy 0.5, the contributiongeneration. Furthe of frictionalrmore, entropylooking productionat this figure is higherwill reveal than that that as of the the particles thermal the thermal entropy generation. Furthermore, looking at this figure will reveal that as the particles entropyincreasegeneration. in size, the Furthermore, Bejan number looking changes at this from ﬁgure values will revealover 0.5 that to as values the particles below increase 0.5, occurring in size, increase in size, the Bejan number changes from values over 0.5 to values below 0.5, occurring thefurther Bejan away number from changes the centerline. from values The over reason 0.5 to for values this below is attributed 0.5, occurring to the further rather away non-uniform from the further away from the centerline. The reason for this is attributed to the rather non-uniform centerline.concentration The distribution reason for this for is the attributed larger nanoparticles. to the rather non-uniform concentration distribution for the concentration distribution for the larger nanoparticles. larger nanoparticles.

Entropy 2016, 18, 378 18 of 27 Entropy 2016, 18, 378 18 of 27

Entropy 2016, 18, 378 18 of 27

Figure 23. Local Bejan number for different particle sizes at ϕm = 5% and Re = 2000. Figure 23. Local Bejan number for different particle sizes at ϕm = 5% and Re = 2000. Up to here, theFigure effects 23. Local of particleBejan number size, for Reynolds different particle number, sizes and at ϕ mparticle = 5% and concentration Re = 2000. on the rate Up to here, the effects of particle size, Reynolds number, and particle concentration on the rate of of entropy production were assessed considering particle migration. In most of the papers available entropy productionUp to here, werethe effects assessed of particle considering size, Reynolds particle number, migration. and In particle most ofconcentration the papers availableon the rate in the in the related literature, nanofluids have been considered as homogeneous fluids with uniform relatedof literature,entropy production nanofluids were have assessed been consideredconsidering asparticle homogeneous migration. fluidsIn most with of the uniform papers concentrationavailable concentration distribution. The effect of regarding the phenomenon of nanoparticle migration is distribution.in the related The effectliterature, ofregarding nanofluids thehave phenomenon been considered of nanoparticle as homogeneous migration fluids with is studied uniform in the studiedconcentration in the following distribution. on the The generation effect of regardin of entropyg the in phenomenon the nanofluid of nanoparticleas compared migration to the state is in following on the generation of entropy in the nanofluid as compared to the state in which the migration whichstudied the migration in the following of nanoparticles on the generation is not taken of entropy into accountin the nanofluid at all. In as the compared following to theinvestigations, state in of nanoparticles is not taken into account at all. In the following investigations, the non-uniform model the non-uniformwhich the migration model of nanoparticlesmeans regarding is not taken the into impact account of at all.particle In the followingmigration, investigations, which causes means regarding the impact of particle migration, which causes concentration gradients, and thus, concentrationthe non-uniform gradients, model and thus,means the regarding non-uniform the impactdistribution of particle of thermophys migration,ical which properties causes at the the non-uniform distribution of thermophysical properties at the cross-section of the tube. In addition, cross-sectionconcentration of the gradients, tube. In andaddition, thus, the the non-uniform uniform model distribution means of ignoring thermophys theical particle properties migration, at the the the uniformcross-section model of the means tube. ignoring In addition, the the particle uniform migration, model means the ignoring result of the which particle would migration, be a uniformthe result of which would be a uniform concentration distribution, and therefore, uniform concentrationresult of distribution,which would and be therefore, a uniform uniform concentr thermophysicalation distribution, properties and throughouttherefore, uniform the domain. thermophysical properties throughout the domain. Figurethermophysical 24 shows properties the rate ofthroughout thermal entropythe domain. generation for two uniform and non-uniform models Figure 24 shows the rate of thermal entropy generation for two uniform and non-uniform at ϕm = 5%,FigureRe =24 2000, shows and thed ratep = 90of nm.thermal As entropy can be observed,generation thefor two thermal uniform entropy and non-uniform production rate models at ϕm = 5%, Re = 2000, and dp = 90 nm. As can be observed, the thermal entropy production for themodels non-uniform at ϕm = 5%, model Re = 2000, is greater and dp = than 90 nm. that As for can the be observed, uniform onethe thermal near the entropy wall. production This is due to rate forrate the for non-uniform the non-uniform model model is isgreater greater than than thatthat for thethe uniform uniform one one near near the the wall. wall. This This is due is todue to migration of the particles, which reduces the concentration there. Therefore, the thermal conductivity migrationmigration of theof theparticles, particles, which which reducesreduces thethe concentration there. there. Therefore, Therefore, the thethermal thermal will also be smaller in this region, and the temperature gradient increases due to the wall heat flux conductivityconductivity will will also also be besmaller smaller in in this this region region,, andand thethe temperaturetemperature gradient gradient increases increases due dueto the to the being constant (see Equation (30)). It is also noticed that the effect of considering the particle migration wall heatwall heatflux fluxbeing being constant constant (see (see Equation Equation (30)). (30)). It is alsoalso noticed noticed that that the the effect effect of considering of considering the the near the wall is higher than that in the central regions. Furthermore, in the central areas, the thermal particleparticle migration migration near near the the wall wall is ishigher higher thanthan that inin thethe central central regions. regions. Furthermore, Furthermore, in the in the entropycentral production areas, the rate thermal for the entropy case of consideringproduction rate the particlefor the case migration of considering is smaller the in comparisonparticle central areas, the thermal entropy production rate for the case of considering the particle with themigration result obtainedis smaller from in comparison the uniform with model, the result since theobtained concentration from the isuniform greater model, for the since non-uniform the migration is smaller in comparison with the result obtained from the uniform model, since the modelconcentration there. is greater for the non-uniform model there. concentration is greater for the non-uniform model there.

Figure 24. Thermal entropy generation rate for two uniform and non-uniform models at ϕm = 5%, Re Figure 24. Thermal entropy generation rate for two uniform and non-uniform models at ϕm = 5%, = 2000, and dp = 90 nm. Re = 2000, and dp = 90 nm. Figure 24. Thermal entropy generation rate for two uniform and non-uniform models at ϕm = 5%, Re

= 2000, and dp = 90 nm.

Entropy 2016, 18, 378 19 of 27

EntropyEntropy 20162016,, 1818,, 378378 1919 ofof 2727 Figure 25 demonstrates the frictional entropy generation rate for two models (namely, uniform and FigureFigure 2525 demonstratesdemonstrates thethe fricfrictionaltional entropyentropy generationgeneration raterate forfor twotwo modelsmodels (namely,(namely, uniformuniform non-uniform) at ϕm = 5%, Re = 2000, and dp = 90 nm. It is seen that taking particle migration into and non-uniform) at ϕm = 5%, Re = 2000, and dp = 90 nm. It is seen that taking particle migration into and non-uniform) at ϕm = 5%, Re = 2000, and dp = 90 nm. It is seen that taking particle migration into consideration signiﬁcantly reduces the frictional entropy generation. Figure 26 indicates that, due to considerationconsideration significantlysignificantly reducesreduces thethe frictionalfrictional enentropytropy generation.generation. FigureFigure 2626 indicatesindicates that,that, duedue toto migration of the nanoparticles to the central areas and the viscosity increase there, the velocity migrationmigration ofof thethe nanoparticlesnanoparticles toto thethe centralcentral areasareas andand thethe viscosityviscosity increaseincrease there,there, thethe velocityvelocity distribution of the non-uniform model is ﬂatter than that of the uniform model. This leads to the distributiondistribution ofof thethe non-uniformnon-uniform modelmodel isis flatterflatter ththanan thatthat ofof thethe uniformuniform model.model. ThisThis leadsleads toto thethe velocity gradient related to the former to be below that of the later near the tube centerline, and thus velocityvelocity gradientgradient relatedrelated toto thethe formerformer toto bebe belowbelow thatthat ofof thethe laterlater nearnear thethe tubetube centerline,centerline, andand thusthus inin thesein thesethese regions, regions,regions, the thethe frictional frictionalfrictional entropy entropyentropy productionproductionproduction raterate forfor for thethe the non-uniformnon-uniform non-uniform modelmodel model willwill will bebe belower.lower. lower. AdjacentAdjacentAdjacent to toto the thethe wall, wall,wall, in inin spite spitespite of ofof the thethe higher higherhigher velocity velocityvelocity gradientgradient inin thethe non-uniformnon-uniform non-uniform model,model, model, thethe the frictionalfrictional frictional entropyentropyentropy generation generationgeneration rate raterate is isis lower lowerlower thanthan the the oneone one obtainedobtained obtained fromfrom from thethe the uniformuniform uniform model.model. model. ThisThisThis isis becausebecause is because ofof ofthe the smaller smaller viscosityviscosity forfor for thethe the non-uniformnon-uniform non-uniform modelmodel model nearnear near thethe the wallwall wall asas as aa aconsequenceconsequence consequence ofof of thethe the particleparticle particle migrationmigrationmigration toward towardtoward the thethe central centralcentral regions. regions.regions. The TheThe discrepancy discrediscrepancypancy between betweenbetween the thethe frictional frictionalfrictional entropy entropyentropy production productionproduction rates adjacentratesrates adjacentadjacent to the wall toto forthethe twowallwall approaches forfor twotwo approachesapproaches is approximately isis approximatelyapproximately 14%, which 14%,14%, gradually whichwhich decreasesgraduallygradually approaching decreasesdecreases theapproachingapproaching centerline. thethe centerline.centerline.

FigureFigureFigure 25. 25.25.Frictional FrictionalFrictional entropy entropyentropy generation generationgeneration rate raterate forforfor twotwo models models (namely,(namely, uniformuniform uniform andand and non-uniform)non-uniform) non-uniform) atat at ϕm = 5%, Re = 2000, and dp = 90 nm. ϕmϕ=m 5%,= 5%,Re Re= = 2000, 2000, and andd dpp = 90 nm. nm.

Figure 26. Velocity profiles for uniform and non-uniform models. FigureFigure 26. 26.Velocity Velocity proﬁlesprofiles forfor uniformuniform and non-uniform non-uniform models. models.

FigureFigure 2727 comparescompares thethe totaltotal entropyentropy producproductiontion ratesrates achievedachieved fromfrom thethe uniformuniform andand non-uniformFigure 27 compares models at the ϕ totalm = 5%, entropy Re = production 2000, and ratesdp = 90 achieved nm. Under from thethese uniform conditions, and non-uniform since the non-uniform models at ϕm = 5%, Re = 2000, and dp = 90 nm. Under these conditions, since the modelsfrictionalfrictional at ϕ entropymentropy= 5%, generationgenerationRe = 2000, dominatesdominates and dp = 90 thethe nm. thermatherma Underll entropyentropy these conditions,production,production, sinceandand thethe the frictionalfrictional frictional entropyentropy entropy generationproductionproduction dominates ofof thethe uniformuniform the thermal modelmodel isis entropy greatergreater production, thanthan ththatat ofof thethe and non-uniformnon-uniform the frictional modelmodel entropy (Figure(Figure production 25),25), thethe totaltotal of the uniformentropyentropy model productionproduction is greater forfor the thanthe uniformuniform that of model themodel non-uniform isis greatergreater thanthan model thatthat (Figure ofof thethe non-uniformnon-uniform25), the total model.model. entropy production for theFigure uniformFigure 2828 model illustratesillustrates is greater thethe therther thanmalmal that entropyentropy of the non-uniformproductionproduction raterate model. forfor thethe uniformuniform andand non-uniformnon-uniform models at two different Reynolds numbers for ϕm = 5% and dp = 90 nm. It is clear that the discrepancy modelsFigure at 28 two illustrates different theReynolds thermal numbers entropy for production ϕm = 5% and rate dp for = 90 the nm. uniform It is clear and that non-uniform the discrepancy models between the amounts of thermal entropy production rate obtained from these two models at the atbetween two different the amounts Reynolds of thermal numbers entropy for ϕm production= 5% and ratedp = obtained 90 nm. from It is clearthese thattwo themodels discrepancy at the

Entropy 2016, 18, 378 20 of 27

Entropy 2016, 18, 378 20 of 27 betweenEntropy the 2016 amounts, 18, 378 of thermal entropy production rate obtained from these two models at20 the of 27 higher Entropy 2016, 18, 378 20 of 27 Reynoldshigher number Reynolds is number greater is in greater comparison in comparison with the with lower the lower Reynolds Reynolds number, number, which which is due is due to to higher higher Reynolds number is greater in comparison with the lower Reynolds number, which is due to non-uniformityhigher non-uniformity of concentration of concentrat at theion greater at the Reynoldsgreater Reynolds number. number. higher non-uniformityReynolds number of isconcentrat greater inion comparison at the greater with Reynolds the lower number. Reynolds number, which is due to SimilarSimilar to the to the thermal thermal entropy entropy production, production, thethe difference in in the the frictional frictional entropy entropy production production higherSimilar non-uniformity to the thermal of concentrat entropy ionproduction, at the greater the difference Reynolds innumber. the frictional entropy production obtainedobtained from from these these two two models models increases increases by by increasing increasing the Reynolds Reynolds number number (Figure (Figure 29). 29 ). obtainedSimilar from to these the thermal two models entropy increases production, by increasing the difference the Reynolds in the numberfrictional (Figure entropy 29). production obtained from these two models increases by increasing the Reynolds number (Figure 29).

Figure 27. Total entropy generation rates obtained from the uniform and non-uniform models at ϕm = FigureFigure 27. Total27. Total entropy entropy generation generation rates obtained obtained from from the theuniform uniform and non-uniform and non-uniform models at models ϕm = at 5%, Re = 2000, and dp = 90 nm. m ϕm =5%,Figure ReRe =27.= 2000, 2000,Total and entropy and dp =d p90 =generation nm. 90 nm. rates obtained from the uniform and non-uniform models at ϕ = 5%, Re = 2000, and dp = 90 nm.

(a) (b) (a) (b) Figure 28. Thermal entropy(a) generation rate obtained from uniform and non-uniform(b) models for ϕm = Figure 28. Thermal entropy generation rate obtained from uniform and non-uniform models for ϕm = Figure5% 28. andThermal dp = 90 nm entropy at: (a) Re generation= 200; (b) Re rate= 1000. obtained from uniform and non-uniform models for m 5%Figure and 28. dp =Thermal 90 nm at: entropy (a) Re =generation 200; (b) Re rate = 1000. obtained from uniform and non-uniform models for ϕ = ϕ = 5% and d = 90 nm at: (a) Re = 200; (b) Re = 1000. m 5% and dp p= 90 nm at: (a) Re = 200; (b) Re = 1000.

(a) (b) (a) (b) Figure 29. Frictional entropy(a) generation rate obtained from uniform and(b) non-uniform models for Figure 29. Frictional entropy generation rate obtained from uniform and non-uniform models for ϕFigurem = 5% 29. and Frictional dp = 90 nm entropy at: (a) Regeneration = 200; (b) rateRe = obtained1000. from uniform and non-uniform models for Figureϕm 29. = 5%Frictional and dp = 90 entropy nm at: (a generation) Re = 200; (b rate) Re = obtained 1000. from uniform and non-uniform models for ϕm = 5% and dp = 90 nm at: (a) Re = 200; (b) Re = 1000. ϕ m = 5% and dp = 90 nm at: (a) Re = 200; (b) Re = 1000.

Entropy 2016, 18, 378 21 of 27

Entropy 2016, 18, 378 21 of 27 Figure 30 provides both thermal and frictional entropy generation rates obtained from the uniform Figure 30 provides both thermal and frictional entropy generation rates obtained from the model and non-uniform one for the particle size of 10 nm at ϕm = 5% and Re = 2000. As observed in uniform model and non-uniform one for the particle size of 10 nm at ϕm = 5% and Re = 2000. As this figure, in contrast with Figures 24 and 25 (which were for the particles of 90 nm), since the particle observed in this figure, in contrast with Figures 24 and 25 (which were for the particles of 90 nm), distribution is very uniform for dp = 10 nm, no significant difference is noticed between the entropy since the particle distribution is very uniform for dp = 10 nm, no significant difference is noticed generation rates of the uniform and non-uniform models. Moreover, a comparison between this figure between the entropy generation rates of the uniform and non-uniform models. Moreover, a with Figures 24 and 25 clarifies that the effect of changing the particle size on the difference between comparison between this figure with Figures 24 and 25 clarifies that the effect of changing the theparticle uniform size and on the non-uniform difference between models forthe uniform frictional and entropy non-uniform production models is for greater frictional in comparison entropy withproduction thermal entropyis greater production. in comparison Based with on thermal Figures entropy 24 and production. 25 (which Based are depicted on Figures for d24p =and 90 25 nm), the(which difference are depicted between for the d ratesp = 90 ofnm), thermal the difference entropy generationbetween the obtained rates of thermal from the entropy two models generation near the wallobtained is about from 5%, the and two the models difference near between the wall the is ratesabout of5%, frictional and the entropy difference generation between obtainedthe rates fromof thefrictional two models entropy near generation the wall is aboutobtained 14.5%, from while the two these models differences near arethe negligiblewall is about for 1014.5%, nm particles,while asthese shown differences in Figure are30. negligible for 10 nm particles, as shown in Figure 30.

(a) (b)

Figure 30. Entropy generation rates obtained from the uniform and non-uniform models for particle Figure 30. Entropy generation rates obtained from the uniform and non-uniform models for particle size of 10 nm at ϕm = 5% and Re = 2000: (a) thermal; (b) frictional. size of 10 nm at ϕm = 5% and Re = 2000: (a) thermal; (b) frictional. Figure 31 demonstrates the variations of the thermal, frictional, and total entropy production ratesFigure of water 31 demonstrates upon adding particles the variations of 90 nm of size the with thermal, the concentration frictional, and of 5% total at Re entropy = 2000. productionIt is seen ratesthat of the water thermal upon entropy adding production particles of rate 90 nmof the size base with fluid the concentrationdecreases, and of its 5% frictional at Re = entropy 2000. It is seenproduction that the thermalrate increases; entropy however, production due rateto the of thedominance base fluid of decreases,the latter to and the its former, frictional the entropytotal productionentropy generation rate increases; increases. however, due to the dominance of the latter to the former, the total entropy generationFigure increases. 32 demonstrates the variations in the thermal, frictional, and total entropy production ratesFigure by adding 32 demonstrates nanoparticles the of variations90 nm size inwith the the thermal, concentration frictional, of 5% and at totalRe = entropy200. As seen production from ratesFigure by adding32c, in contrast nanoparticles to the case of 90of nmRe = size2000, with addition the concentration of the nanoparticles of 5% reduces at Re = the 200. total As entropy seen from Figureproduction 32c, in rate contrast by about to the 5.5% case for of theRe base= 2000, fluid. addition Therefore, of the from nanoparticles the second law reduces standpoint, the total in entropythese productionconditions, rate the by nanofluid about 5.5% will forbe of the lower base irreversibility fluid. Therefore, in comparison from the second with the law base standpoint, fluid; thus, in it these is conditions,more optimal the nanofluidthan the base will fluid be of for lower effectual irreversibility use of the in available comparison energy. with Th theis baseis because fluid; thus,at this it is moreReynolds optimal number, than the heat base transfer fluid forhas effectual a greater use contribution of the available to the energy. generation This isof because entropy at in this comparison with the friction, and (as shown in Figure 32a) the rate of thermal entropy production Reynolds number, heat transfer has a greater contribution to the generation of entropy in comparison related to the nanofluid is smaller than the water. The addition of nanoparticles to the water will on with the friction, and (as shown in Figure 32a) the rate of thermal entropy production related to the average multiply the frictional entropy generation rate by approximately four (see Figure 32b), but nanofluid is smaller than the water. The addition of nanoparticles to the water will on average multiply the rate of thermal entropy generation decreases by about 7.5%, and finally, the total entropy the frictional entropy generation rate by approximately four (see Figure 32b), but the rate of thermal production decreases as a result of higher contribution of the thermal entropy production. entropy generation decreases by about 7.5%, and finally, the total entropy production decreases as a result of higher contribution of the thermal entropy production.

Entropy 2016, 18, 378 22 of 27 Entropy 2016, 18, 378 22 of 27 Entropy 2016, 18, 378 22 of 27

(a) (b) (a) (b)

(c) (c) Figure 31. Variations of entropy generation rates of base fluid upon adding particles of 90 nm size Figure 31. Variations of entropy generation rates of base ﬂuid upon adding particles of 90 nm size with Figurewith concentration 31. Variations of 5%of entropy at Re = 2000: generation (a) thermal; rates of(b )base frictional; fluid upon(c) total. adding particles of 90 nm size concentrationwith concentration of 5% at ofRe 5%= 2000:at Re = ( a2000:) thermal; (a) thermal; (b) frictional; (b) frictional; (c) total. (c) total.

(a) (b) (a) (b)

(c) (c) Figure 32. Variations of entropy generation rates of base fluid upon adding particles of 90 nm size Figure 32. Variations of entropy generation rates of base fluid upon adding particles of 90 nm size Figurewith 32. concentrationVariations of of entropy 5% at Re generation = 200: (a) thermal; rates of (b base) frictional; ﬂuid upon (c) total. adding particles of 90 nm size with with concentration of 5% at Re = 200: (a) thermal; (b) frictional; (c) total. concentration of 5% at Re = 200: (a) thermal; (b) frictional; (c) total. Entropy 2016, 18, 378 23 of 27

In the following, the entropy generated in the total volume of the nanofluid will be evaluated and discussed under different conditions. These values are obtained by integrating over the entire domain of the nanofluid,Entropy 2016 according, 18, 378 to Equation (27). 23 of 27 Figure 33 illustrates the rates of thermal, frictional, and total entropy production for different In the following, the entropy generated in the total volume of the nanofluid will be evaluated concentrationsand discussed at Re under= 2000 different and conditions.dp = 90 These nm forvalues two are obtained states withby integrating and without over the entire the migration of nanoparticlesdomain of (i.e., the nanofluid, uniform according and non-uniform to Equation (27). models). It is seen that for the two models, augmenting theFigure concentration 33 illustrates the decreases rates of thermal, the thermal frictional, entropy and total productionentropy production and for raises different the frictional entropy production,concentrations such at Re that = 2000 by and increasing dp = 90 nm the for concentration two states with from and without 1% to 5%the formigration the non-uniform of nanoparticles (i.e., uniform and non-uniform models). It is seen that for the two models, augmenting model, thethe rate concentration of thermal decreases entropy the production thermal entropy reduces production by about and 14%,raises whereasthe frictional the entropy rate of frictional entropy productionproduction, such augments that by increasing by about the 120%. concentrat It causesion from the 1% totalto 5% entropyfor the non-uniform production model, increment the rate to 73%. It israte also of thermal obvious entropy from production Figure 33 reducesthat at by the ab lowout 14%, concentration, whereas the sincerate of the frictional particle entropy distribution is almost uniform,production there augments would by be about no significant120%. It causes difference the total entropy between production the data increment achieved rate to from 73%.the It uniform is also obvious from Figure 33 that at the low concentration, since the particle distribution is almost model anduniform, the non-uniform there would one.be no However,significant differen at thece higher between concentrations, the data achieved the from difference the uniform between the results of thesemodel twoand the models non-uniform increases. one. However, Therefore, at the for higher the concentrations, 5% concentration, the difference the frictional between the and thermal entropy generationresults of these rates two obtained models increases. from the Therefore, non-uniform for the 5% model concentration, will be the around frictional 16% and and thermal 3% lower and higher, respectively,entropy generation than thoserates obtained of the uniformfrom the non- model.uniform Therefore, model will the be resultsaround 16% of the and uniform 3% lower model are and higher, respectively, than those of the uniform model. Therefore, the results of the uniform acceptablemodel only are at acceptable low concentrations. only at low concentrations. Meanwhile, Meanwhile, by considering by considering particle particle migration, migration, a lowera total entropy productionlower total entropy rate will production be obtained rate will as be compared obtained as tocompared the uniform to the uniform model. model.

Figure 33. Entropy generation rates for different concentrations at Re = 2000 and dp = 90 nm for two Figure 33. Entropy generation rates for different concentrations at Re = 2000 and d = 90 nm for cases with and without particle migration (non-uniform and uniform, respectively). p two cases with and without particle migration (non-uniform and uniform, respectively).

Table 3 lists the rates of thermal, frictional, and total entropy production at ϕm = 5% and dp = 90 nm for the uniform and non-uniform models at different Reynolds numbers. It is noticed that Tableaugmenting3 lists the rates the Reynolds of thermal, number frictional, decreases and the totalrate of entropy thermal entropy production production at ϕm and= 5%raises and thed p = 90 nm for the uniformrate of andfrictional non-uniform entropy production models for at both different models. Reynolds Increase of numbers. the frictional It isentropy noticed production that augmenting the Reynoldsrate numberoccurs more decreases considerably the than rate the of decrease thermal of entropythe thermal production entropy production and raises rate. Meanwhile, the rate of frictional entropy productionfor smaller Reynolds for both numbers, models. great Increase discrepancy of the is frictionalnot seen between entropy the productionresults of the rateuniform occurs more model and the non-uniform one, though this difference becomes more considerable at greater considerablyReynolds than numbers. the decrease As discussed of the earlier, thermal it is entropy because productionthe particle migration rate. Meanwhile, happens more for smaller Reynoldssignificantly numbers, greatat the high discrepancyer Reynolds isnumbers. not seen between the results of the uniform model and the non-uniform one, though this difference becomes more considerable at greater Reynolds numbers. Table 3. Entropy generation rates at ϕm = 5% and dp = 90 nm for the uniform and non-uniform models As discussed earlier, it is because the particle migration happens more signiﬁcantly at the higher at various Reynolds numbers. Reynolds numbers. Entropy Generation Rate Entropy Generation Rate Re (Uniform Model) W/K (with Particle Migration) W/K Table 3. EntropyThermal generation ratesFrictional at ϕm = 5% andTotald p = 90 nmThermal for the uniformFrictional and non-uniformTotal models at various200 Reynolds 0.0002655 numbers. 8.43 × 10−6 0.0002740 0.0002656 8.11 × 10−6 0.0002737 1000 0.0002489 0.0002155 0.0004645 0.0002514 0.0001887 0.0004401 2000 0.0002174 0.0008647 0.0010823 0.0002230 0.0007202 0.0009432 Entropy Generation Rate Entropy Generation Rate Re (Uniform Model) W/K (with Particle Migration) W/K Thermal Frictional Total Thermal Frictional Total 200 0.0002655 8.43 × 10−6 0.0002740 0.0002656 8.11 × 10−6 0.0002737 1000 0.0002489 0.0002155 0.0004645 0.0002514 0.0001887 0.0004401 2000 0.0002174 0.0008647 0.0010823 0.0002230 0.0007202 0.0009432 Entropy 2016, 18, 378 24 of 27 Entropy 2016, 18, 378 24 of 27

To evaluateevaluate the contributions of of two two factors factors in in the the total total entropy entropy production, production, the the Bejan Bejan number number is ispresented presented in inTable Table 4 4for for two two models models at atϕm ϕ=m 5%= 5%and and dp =d 90p = nm 90 nmfor different for different Reynolds Reynolds numbers. numbers. It is Itobvious is obvious that thatthe Bejan the Bejan number number reduces reduces for forgreater greater Reynolds Reynolds numbers, numbers, which which indicates indicates a decrease a decrease of ofthe the heat heat transfer transfer contribution contribution in incomparison comparison with with the the friction. friction. At At ReRe == 200, 200, more more than than 90% 90% of the entropy generation is originated from heat transfer, while the heat transfer contribution is about 20% at Re == 20002000.. ForFor thethe casecase ofof consideringconsidering particleparticle migration,migration, a greater Bejan number is obtainedobtained in comparison with the uniform model. Therefore, in in th thisis case, the heat transfertransfer contributioncontribution is more signiﬁcantsignificant in comparison with the uniformuniform model.model. This becomes even more signiﬁcantsignificant at higherhigher Reynolds numbers.

Table 4. Bejan number for two models at ϕ = 5% and d = 90 nm for various Reynolds numbers. Table 4. Bejan number for two models at ϕmm = 5% and dpp = 90 nm for various Reynolds numbers.

Re Re UniformUniform Model WithWith Particle Particle Migration Migration 200 2000.969 0.969 0.9700.970 10001000 0.536 0.536 0.5710.571 20002000 0.201 0.201 0.2360.236

Figure 3434 shows shows the the rates rates of thermal,of thermal, frictional, friction andal, totaland entropytotal entropy production, production, considering considering particle particle migration for various particle sizes at Re = 2000 and ϕm = 5%. It is noticed that the rate of migration for various particle sizes at Re = 2000 and ϕm = 5%. It is noticed that the rate of thermal entropythermal productionentropy production remains almostremains constant almost withconsta changingnt with particlechanging size, partic sincele assize, the since particle as sizethe changes,particle size the localchanges, rate ofthe thermal local entropyrate of thermal generation entropy increases generation in some regionsincreases and in decreases some regions elsewhere, and asdecreases shown inelsewhere, Figure 18. as Therefore, shown in its Figure global value18. Therefor (whiche, is its indicative global value of the thermal(which is entropy indicative generation of the throughoutthermal entropy the nanoﬂuidgeneration domain) throughout remains the nanofl almostuid unchanged. domain) remains Looking almost at Figure unchanged. 34 clariﬁes Looking the factat Figure that the 34 rateclarifies of frictional the fact entropy that the production rate of frictional reduces entropy with particle production enlargement, reduces because with particle of the reductionenlargement, in local because rates of of the frictional reduction entropy in local production rates of frictional (see Figure entropy 19). In prod addition,uction the (see total Figure entropy 19). productionIn addition, rate the decreasestotal entropy with production particle enlargement, rate decreases such with that particle from 10 enlargement, nm to 90 nm, such its value that from reduces 10 bynm about to 90 13%.nm, its value reduces by about 13%.

Figure 34.34. Entropy generation rates considering particle migration at Re == 2000 and ϕmm = 5% for different particle sizes.

Applying the second law of thermodynamics, fl flowow of a nanofluidnanofluid within a minichannelminichannel was evaluated in the present survey. However, sincesince mostmost ofof thethe conductedconducted investigations on nanofluidsnanofluids have followed the the first first law law of of thermodynamics, thermodynamics, and and since since the the great great importance importance of ofthe the second second law law of ofthermodynamics thermodynamics is in is inthe the optimization optimization of ofsyst systemsems and and the the efficient efficient use use of of energy, energy, much more investigations areare neededneeded inin thisthis area.area.

7. Conclusions

Entropy 2016, 18, 378 25 of 27

7. Conclusions A second law analysis was carried out in this research by numerical solution using control volume method for the water–alumina nanofluid flow in the laminar regime through a circular minichannel. For this purpose, the entropy generation rates resulting from both heat transfer and friction were evaluated. In the simulations, particle migration was considered to study its effect on the entropy generation. Particle migration led to non-uniform profiles of the thermophysical properties of the nanofluid due to the development of a non-uniformity in the concentration distribution. Thus, it is effective on both velocity and temperature profiles. The entropy generation rates were assessed locally and globally (integrated). At the higher Reynolds numbers, greater concentrations, and larger particle sizes, the particle migration involved a more considerable influence on the entropy production, since the non-uniformity in the properties’ distribution was intensified. For investigation of the rates of local entropy production, the profiles of thermophysical properties and velocity were analyzed. The results indicated that in some conditions, by adding the nanoparticles to the water, the rate of total entropy production decreases, which is a beneficial occurrence for effectual utilization of energy. For the case considering particle migration, a greater Bejan number was obtained in comparison with the uniform model. Therefore, in this case, the heat transfer contribution is more significant in comparison with the uniform model.

Acknowledgments: The authors gratefully acknowledge the editor and three anonymous reviewers for their helpful comments for the improvement of this paper. Author Contributions: Mehdi Bahiraei carried out the analyses and wrote the manuscript. Navid Cheraghi Kazerooni performed the numerical simulations and computer runs. Both authors have read and approved the final manuscript. Conflicts of Interest: The authors declare no conflict of interest.

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