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Applied Sciences
, Volume 6 (11) – Nov 2, 2016

/lp/multidisciplinary-digital-publishing-institute/further-investigation-on-laminar-forced-convection-of-nanofluid-flows-eVMJSWEHp0

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- Multidisciplinary Digital Publishing Institute
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- © 1996-2019 MDPI (Basel, Switzerland) unless otherwise stated
- ISSN
- 2076-3417
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- 10.3390/app6110332
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applied sciences Article Further Investigation on Laminar Forced Convection of Nanoﬂuid Flows in a Uniformly Heated Pipe Using Direct Numerical Simulations † ,† Ghofrane Sekrani and Sébastien Poncet * Faculté de génie, Département de génie mécanique 2500 Boulevard de l’Université, Université de Sherbrooke, Sherbrooke, QC J1K 2R1, Canada; Ghofrane.Sekrani@USherbrooke.ca * Correspondence: Sebastien.Poncet@usherbrooke.ca; Tel.: +1-819-821-8000 (ext. 62150) † These authors contributed equally to this work. Academic Editor: Xianchang Li Received: 22 September 2016; Accepted: 26 October 2016; Published: 2 November 2016 Abstract: In the present paper, laminar forced convection nanoﬂuid ﬂows in a uniformly heated horizontal tube were revisited by direct numerical simulations. Single and two-phase models were employed with constant and temperature-dependent properties. Comparisons with experimental data showed that the mixture model performs better than the single-phase model in the all cases studied. Temperature-dependent ﬂuid properties also resulted in a better prediction of the thermal ﬁeld. Particular attention was paid to the grid arrangement. The two-phase model was used then conﬁdently to investigate the inﬂuence of the nanoparticle size on the heat and ﬂuid ﬂow with a particular emphasis on the sedimentation process. Four nanoparticle diameters were considered: 10, 42, 100 and 200 nm for both copper-water and alumina/water nanoﬂuids. For the largest diameter d = 200 nm, the Cu nanoparticles were more sedimented by around 80%, while the Al O n p 2 3 nanoparticles sedimented only by 2.5%. Besides, it was found that increasing the Reynolds number improved the heat transfer rate, while it decreased the friction factor allowing the nanoparticles to stay more dispersed in the base ﬂuid. The effect of nanoparticle type on the heat transfer coefﬁcient was also investigated for six different water-based nanoﬂuids. Results showed that the Cu-water nanoﬂuid achieved the highest heat transfer coefﬁcient, followed by C, Al O , CuO, TiO , and SiO , 2 3 2 2 respectively. All results were presented and discussed for four different values of the concentration in nanoparticles, namely ' = 0, 0.6%, 1% and 1.6%. Empirical correlations for the friction coefﬁcient and the average Nusselt number were also provided summarizing all the presented results. Keywords: nanoﬂuid; numerical simulation; heat transfer; sedimentation 1. Introduction Heat transfer is one of the most important processes in many industrial and heating-cooling applications, such as microelectronics, transportation, manufacturing, metrology, defense, and energy supply industries [1,2]. However, the inherent low thermal conductivity of conventional ﬂuids, such as water, oils, and ethylene glycol, is a primary limitation in developing efﬁcient heat transfer systems. The Maxwell’s theory [3] showed that an enhancement of the thermal conductivity may be achieved by dispersing millimeter or micrometer-sized solid particles into a base ﬂuid. However one major drawback associated with the use of such large size particles is their rapid settling, which may result into a complete separation of the two phases along with the clogging of heat exchangers due to the sedimentation of the solid aggregates formed by the large size particles. This type of solid-ﬂuid suspensions requires also the addition of a large number of particles resulting in signiﬁcantly greater pressure drop, hence increased pumping power, corrosion of the walls and a noticeable increase in the Appl. Sci. 2016, 6, 332; doi:10.3390/app6110332 www.mdpi.com/journal/applsci Appl. Sci. 2016, 6, 332 2 of 24 wall shear stress. Thus, Choi and Eastman [4] suggested a novel approach to enhance heat transfer processes in industrial applications by exploiting the properties of nanoparticles and their dispersion in a host ﬂuid. These metallic or non-metallic nanoparticles have an equivalent diameter d lower n p than 100 nm. As opposed to milli- or microsized suspensions, very stable suspensions may be achieved by introducing nanoparticles. Moreover, nanoparticles beneﬁt from a 10 times larger surface/volume ratio than that of microparticles and exhibit much higher thermal conductivity than that of base ﬂuids. For examples, the thermal conductivities of copper or alumina at room temperature are about 670 and 70 times greater than that of water, respectively [5]. On the contrary, it leads most of the time to a decrease in the heat capacity [6,7] and an increase in the dynamic viscosity of the mixture [6]. A compromise must be then found between the increase in thermal conductivity without loosing too much heat storage capacity and consuming too much power for pumping. If well stabilized, nanoﬂuids represent nowadays a major technological and economical challenge and should offer very interesting perspectives for any heat transfer process. The exponential increase in the number of publications about nanoﬂuids [8] prevents from making an exhaustive state-of-the-art review on the topic. Many authors concentrated on measuring the thermophysical properties of various nanoﬂuids showing that their properties depend on a large number of parameters such as the type of nanoparticle, their size, their mass or volume fraction, the type and the concentration of the surfactant, the pH of the mixture, the Brownian motion and the thickness of the interfacial nanolayer among other parameters (see in [6–10]). Others developed experimental set-ups to measure the convective heat transfer and temperature proﬁles in pipes [11,12], coaxial [13] or plate [14] heat exchangers among other geometries. Most authors focused on measuring global thermal quantities due to the difﬁculty to measure velocity and temperature proﬁles in such insulated systems. It has relatively slowed down the development of accurate models dedicated to nanoﬂuid ﬂows, especially regarding the agglomeration and sedimentation processes. Only a few in-house solvers have been developed to investigate convective nanoﬂuid ﬂows. Most of them assume the ﬂow as being a single-phase ﬂow with constant or variable nanoﬂuid properties in canonical conﬁgurations. For example, Mehrez et al. [15] numerically investigated the entropy generation and the mixed convection heat transfer of copper/water-based nanoﬂuids in an inclined open cavity with uniform heat ﬂux at the wall. During the last decade, many other authors compared the performance of the different single and two-phase models with constant or temperature-dependent properties in the context of nanoﬂuid ﬂows [16–21]. A detailed state-of-the-art review has been besides recently proposed by Kakaç and Pramuanjaroenkij [10]. Bianco et al. [17] compared the predictions of single and two-phase models (discrete phase model) with constant or temperature-dependent properties for a laminar forced convection ﬂow of Al O /water-based 2 3 nanoﬂuids. They concluded that models with temperature-dependent properties lead to higher values of the heat transfer coefﬁcient and Nusselt number, while decreasing the wall shear stress. With variable properties and for a volume fraction of Al O nanoparticles equal to 4%, similar results 2 3 have been found using single- and two-phase models with a maximum difference of 11%. On the contrary, Lotﬁ et al. [18] showed that the mixture model performs better than the single-phase model and the Eulerian one. Akbari et al. [19] compared three different two-phase models and a single-phase model to the experiments of Wen and Ding [11] for Al O /water-based nanoﬂuids. The mixture, 2 3 Volume of Fluid (VOF) and Eulerian models provided very similar results for the thermal ﬁeld, while the single-phase model strongly underestimated the heat transfer coefﬁcient. No clear consensus arises then from these former studies on the choice of the appropriate single- or two-phase ﬂow models. Some attempts have also been achieved to investigate the inﬂuence of constant or variable thermophysical properties on the performances of single-phase ﬂow models. In that, Labonté et al. [16] showed that the model with constant properties tends to underestimate the wall shear stress and overestimate the heat transfer coefﬁcient. Azari et al. [20] found that the single-phase model with constant physical properties provides an acceptable agreement with the experimental data and the temperature-dependent model improves the predictions of the discrete two-phase ﬂow model for Appl. Sci. 2016, 6, 332 3 of 24 low volume fractions in nanoparticles, typically ' = 0.03%. On the contrary, at higher particle concentrations ' = 3.5%, the two-phase ﬂow performs best. Numerical modeling of laminar convective nanoﬂuid ﬂows even in relatively simple geometries remains very challenging, since the choice of the single- or two-phase ﬂow models appears to be very case dependent. Analytical models have also been developed to investigate the entropy generation in similar conﬁgurations. For example, one could cite the recent work of Bianco et al. [22], who investigated the entropy generation of Al O -water nanoﬂuid turbulent forced convection in a pipe with constant 2 3 wall temperature by means of a second law analysis. They showed in particular that the type of inlet conditions greatly inﬂuences the mechanisms responsible for entropy generation. Such analysis could be then very helpful to optimize nanoﬂuid ﬂows from an exergetic point of view. The present paper focuses on the convective heat transfer in a cylindrical pipe for laminar ﬂows of Al O /water-based nanoﬂuids. This choice is justiﬁed by the large number of former works using 2 3 this nanoﬂuid in a similar ﬂow conﬁguration (laminar or developing ﬂows in a pipe with constant heat ﬂux) [23,24]. Moreover, such nanoﬂuid is of a particular interest due to its non-corrosive properties and its good thermal conductivity enhancement using very low volume fractions in nanoparticles. For examples, Wang and Li [25] obtained an enhancement of 13% using only a volume fraction equal to 0.4% and Liu et al. [26] measured an increase of 34% for a nanoparticle diameter equal to 33 nm and a volume fraction of 3%. The reader can refer to the reviews by Kakaç and Pramuanjaroenkij [10,27] for more details about the thermal enhancement using nanoﬂuids. The objective of the present paper is four-fold: (1) to properly revisit the laminar forced convection ﬂows of Al O /water-based nanoﬂuids using direct numerical simulations; (2) to extend the results to 2 3 a wider range of Reynolds numbers as proposed by [19]; (3) to quantify the inﬂuence of the nanoparticle diameter and the type of nanoparticle on the hydrodynamic and thermal ﬁelds with an emphasis on the sedimentation process; (4) to provide useful empirical correlations for the friction coefﬁcient and average Nusselt number. The experimental set-up developed by Wen and Ding [11] and the former numerical simulations of Akbari et al. [19] using the same model have been chosen for comparisons in the case of Al O /water-based nanoﬂuids with the present simulations. The paper is then organized 2 3 as follows: the numerical modeling and its validation are presented in Sections 2 and 3 respectively. The inﬂuence of the Reynolds number, the concentration in nanoparticles, their diameter and the type of nanoparticles on the heat transfer process and the hydrodynamic ﬁeld are then discussed in details in Section 4, before some concluding remarks in Section 5. 2. Numerical Approach Three-dimensional calculations are carried out in the case of forced convection nanoﬂuid ﬂows in a heated pipe. Single- and two-phase ﬂow models are both considered with constant or temperature-dependent properties using a ﬁnite-volume solver. 2.1. Geometrical Modeling The problem under consideration involves nanoparticles of diameter d perfectly monodispersed n p in pure liquid water. The geometry corresponds to the experimental set-up developed by Wen and Ding [11]. It consists of a horizontal cylindrical pipe of a length L = 0.97 m and a diameter D = 2R = 0.0045 m, heated with a uniform heat ﬂux q = 21898 Wm along the wall (Figure 1). w Appl. Sci. 2016, 6, 332 4 of 24 Figure 1. Schematic view of the computational domain with the boundary conditions. 2.2. Numerical Method The governing equations for the conservation of mass, momentum and energy are solved using a finite volume solver in a Cartesian frame. These equations are discretized in space by a second-order upwind scheme. The pressure-velocity coupling is achieved using the SIMPLEC algorithm. All calculations are performed in steady-state. It has been carefully checked that unsteady calculations led to similar results. 2.3. Fluid Properties and Two-Phase Modeling 2.3.1. Water Properties The physical properties of water are considered to be temperature-dependent while those of the solid nanoparticles are kept constant (see Table 1). The following equations are used to evaluate the properties of pure liquid water (henceforth subscripted by b f for base ﬂuid) as a function of temperature T: • Density [28]: 2 4 3 7 4 10 5 = 2446 20.674T + 0.11576T 3.12895 10 T + 4.0505 10 T 2.0546 10 T (1) b f • Viscosity [29]: ( ) Tc = A 10 (2) b f where, A = 2.414 10 , B = 247.8 and C = 140. • Speciﬁc heat [30]: 8.29041 0.012557T C p = exp (3) b f 1 1.52373 10 T • Conductivity [31]: 3 5 2 k = 0.76761 + 7.535211 10 T 0.98249 10 T (4) b f Note that the above equations are similar to those used in the former numerical simulations of Akbari et al. [19] to ensure direct comparisons. Appl. Sci. 2016, 6, 332 5 of 24 Table 1. Thermophysical properties of different types of nanoparticles. 3 1 1 2 1 (kgm ) C (Jkg K ) k (Wm K ) C 220 710 129 Cu 8933 385 401 CuO 6510 540 18 Al O 3880 729 42.3 2 3 TiO 4175 692 8.4 SiO 2220 745 1.4 2.3.2. Single-Phase Model The single-phase model assumes that the phases are in thermal equilibrium and the relative velocity between the base ﬂuid and the nanoparticles is null. It treats then the nanoﬂuid as a homogeneous ﬂuid with effective thermophysical properties evaluated by theoretical models or empirical correlations. All the nanoﬂuid properties are function of the base ﬂuid (b f ) and nanoparticles (n p) properties as well as the volume fraction ' of the nanoparticles. It is recalled that all properties of the base ﬂuid are temperature-dependent and evaluated using Equations (1)–(4). Plenty of correlations are available in the literature [8,10] and it appears crucial to use the most appropriate ones for the effective nanoﬂuid properties to produce accurate results with the single-phase model. The present correlations are chosen to enable direct comparisons with Akbari et al. [19]. Two correlations for the thermal conductivity k and for the dynamic viscosity are considered here. The equations used to evaluate n f n f the nanoﬂuid properties (density [11], heat capacity [32,33], viscosity [34,35]) are as follows: = (1 ') + ' (5) n p n f b f (1 ')(C p) + '(C p) b f n p C p = (6) n f n f = (1 + 0.025' + 0.015' ) (7) n f b f = (1 + 7.3' + 123' ) (8) n f b f Note that the relations for the dynamic viscosity do not take into account the hysteresis cycle observed by Hachey et al. [36] for commercial and highly concentrated solutions of Al O /water-based 2 3 nanoﬂuids. The thermal conductivity k is evaluated using two different correlations suggested n f by [5,37] respectively: " # k (1 + 2) + 2k 2'(k k (1 )) n p b f b f n p k = k (9) n f b f k (1 + 2) + 2k + '(k k (1 )) n p n p b f b f b f k = k (1 ') + k ' + C k Re Pr' (10) n f b f n p d b f n p n p 8 2 where = 2R k =d is the particle Biot number, R = 0.77 10 Km /W is the interfacial thermal b b f n p b resistance, = 0.01 is a constant taking into account the Kapitza resistance per unit area, C = 18 10 and Re the particle Reynolds number deﬁned as: n p C d n p R M Re = (11) n p b f Appl. Sci. 2016, 6, 332 6 of 24 In the present case, the random motion velocity C is ﬁxed to 0.1 m=s as recommended by [5]. R M The general forms of the governing differential equations (conservation of mass, momentum and energy) for the single-phase model are: r (V) = 0 (12) ~ ~ ~ V rV = rP +r (rV) + ~ g (13) ~ ~ r (V H) = r q t : rV (14) 2.3.3. Mixture Model Several approaches exist to model two-phase ﬂows, such as the volume of ﬂuid (VOF) method, the mixture model, the Eulerian model or the discrete phase model (DPM) among other models. Akbari et al. [19] already demonstrated the superiority of two-phase models over the single-phase one. The Eulerian, VOF and mixture models giving very similar results in their case, only the mixture model will be considered here due to its simplicity, stability and lowest computational costs required. The mixture model treats the nanoﬂuid as a single ﬂuid consisting of two strongly coupled phases. It deﬁnes the concept of phase volume fractions, which are continuous functions and their sum equals one. Each phase has its own velocity. The governing equations of the two-phase model are: • Conservation of mass: r ( V ) = 0 (15) m m • Conservation of momentum: ~ ~ ~ ~ ~ V rV = rP +r ( rV ) + g +r ( ' V V ) (16) m m m m m m m å k k dr,k dr,k k=1 where the mixture velocity, density and viscosity are respectively: å ' V k k k k=1 V = (17) = ' (18) m å k k k=1 = ' (19) m å k k k=1 • The drift velocity of the kth phase writes: ~ ~ ~ V = V V (20) dr,k k m • Conservation of energy: ~ ~ r ( ' V H ) = r q t : rV (21) m m m å k k k k k=1 • Conservation of the volume fraction in nanoparticles: ~ ~ r (' V ) = r (' V ) (22) n p n p m n p n p dr,n p Appl. Sci. 2016, 6, 332 7 of 24 • The slip velocity is deﬁned as the velocity of a second phase (np: nanoparticles) relative to the primary phase (bf: base ﬂuid): ~ ~ ~ V = V V (23) n p p f b f • The drift velocity is related to the relative velocity by: k k ~ ~ ~ V = V V (24) dr,n p p f å f k e f f k=1 • The relative velocity is evaluated through the following equation proposed by Manninen et al. [38]: t d ( ) p n p n p e f f V = ~ a (25) p f 18 f b f drag n p where f is the drag function calculated from Schiller and Naumann [39]: drag 0.687 1 + 0.15Re Re 1000 n p n p f = (26) drag 0.0183Re Re > 1000 n p n p with Re = (V d )/ n and~ a = ~ g (~ v r)~ v . n p m n p e f f m m 2.3.4. Boundary Conditions and Grid Resolution The governing equations for the two models are solved with the following boundary conditions: • At the inlet (z = 0): w = w , u = v = 0, T = T = 293K (27) in in • On the pipe wall (r = R = D=2): ¶T u = v = w = 0,k j = q (28) r=R w e f f ¶r • At the pipe outlet, the gauge pressure is set equal to zero and all the normal diffusion ﬂuxes and the mass balance correction are applied. Several different grid distributions were tested to ensure the independence of the numerical results to the grid size. A structured mesh is used throughout the domain, with 140 nodes in the circumferential direction, 220 in the radial direction and 800 in the axial direction. A grid reﬁnement close to the wall and in the pipe entrance is deemed necessary, where the highest velocity and temperature gradients occur (see Figure 2). This mesh grid provides grid-independent solutions for all cases studied. (a) (b) Figure 2. Schematic view of the mesh grid: (a) in a given cross-section; and (b) along the axial direction. Appl. Sci. 2016, 6, 332 8 of 24 The calculations are performed in parallel using Mammouth Parallel 2 of the Calcul Québec cluster with 2 nodes having each 8 processors. The convergence is typically reached after 4000 iterations corresponding to a CPU time of about 6 h. 3. Validation of the Numerical Model In this section, the results are discussed in terms of the inlet Reynolds number Re = w D=n . in n f The maximum value of the Richardson number reached here is Ri = Gr=Re = 0.0115, which max ensures that a forced convection regime is achieved (Gr the Grashof number based on D, the nanoﬂuid properties and the temperature difference T T ) for all the cases studied. r=R,max in 3.1. Performances of the Mixture Model In order to ﬁrst validate the selected numerical model, the local heat transfer coefﬁcient h(z) is evaluated along the pipe length. The mixture model is used together with temperature-dependent properties for water and Equations (7) and (10) for the nanoparticle properties. Due to the lack of precise information on the temperature measurement procedure in the experiments of Wen and Ding [11], four different averaging methods are used to evaluate the wall temperature of the simulated cases. Comparisons with the measurements of Wen and Ding [11] and the numerical simulations of Akbari et al. [19], for Re = 1600 and ' = 0.6%, are performed using the mixture model. Figure 3a illustrates that using an average over an upper arc of 45 leads to closer results to the experiments with an average error between 0.37% and 13.84% along the pipe. The temperature was probably measured experimentally near the top of the tube where the ﬂuid is warmer due to buoyancy forces. This averaging method will be used adopted for the remainder of the study. Wen and Ding (2004) Circumferential average Upper-half average Upper 45 max Akbari et al. (2011) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 z(m) (a) (b) Figure 3. Axial variations of the local heat transfer coefﬁcient h(z) predicted by the mixture model for Al O water-based nanoﬂuids: (a) different methods for the evaluation of the wall temperature T for 2 3 ' = 0.006; (b) three nanoparticle concentrations (' = 0.006, 0.01 and 0.016). Comparisons for Re = 1600 and d = 42 nm with the experimental data of Wen and Ding [11] and the numerical simulations of n p Akbari et al. [19]. Figure 3b compares the values of the convective heat transfer coefﬁcient h(z) of the present simulations with the numerical results of Akbari et al. [19] and the experimental data of Wen and Ding [11] for Re = 1600 and three volume fractions of Al O 2 3 nanoparticles ' = 0.6%, 1% and 1.6%, respectively. The present numerical results agree fairly well with the experiments [11] with an exponential decrease of the heat transfer coefﬁcient along the tube as expected from energy balance equation. The main discrepancy is observed close to the inlet due to the choice of the boundary conditions in the present calculations. It is noteworthy that the present h(W/m K) Appl. Sci. 2016, 6, 332 9 of 24 simulations improve the predictions of Akbari et al. [19] using exactly the same solver and methods. It points out in particular the necessity to use an appropriate mesh grid in the near-wall regions and globally a more dense grid. Compared to [19], the number of mesh points is multiplied by a factor 86. The mesh grid sensitivity studies are generally performed using small increments of values such that no noticeable effect is observed. Figure 4 provides further comparisons in terms of the average heat transfer coefﬁcient h for the av same three cases. The present results obtained using the mixture model show an acceptable agreement with the experimental ones. The numerical data exhibit an average enhancement of about 36% with increasing ' from 0.6% to 1.6%, which is to be compared to the value 24.3% in the experiments. Once again, the present simulations improve the previous ones of Akbari et al. [19] pointing out the inﬂuence of the mesh grid. This improvement may be attributed to the increase of the thermal conductivity and some authors [11,12,40] proposed that it is also associated to the decrease of the thermal boundary layer thickness. Wen and Ding (2004) Present simulation Akbari et al. (2011) 0.0 0.5 1.0 1.5 2.0 (%) Figure 4. Inﬂuence of the concentration of Al O nanoparticles ' on the average heat transfer coefﬁcient 2 3 for Re = 1600 and d = 42 nm. Comparisons with the experimental data of Wen and Ding [11] and the n p numerical simulations of Akbari et al. [19]. 3.2. Comparative Analysis of Single-Phase and Mixture Models Many authors [18,19,30,41] have already shown that the mixture model performs better than single-phase or other two-phase models like the Eulerian or volume of ﬂuid models. Nevertheless, it is crucial to use appropriate correlations for the effective nanoﬂuid properties to obtain accurate results with single-phase models. First, the single-phase (using Equations (7) and (10) for and k respectively) and the mixture n f n f models with temperature-dependent are compared in Figure 5 for Re = 1600 and four concentrations of nanoparticles. The single-phase model fails to predict the right axial distributions of the local heat transfer coefﬁcient h(z). An exponential decrease of h(z) with the axial distance z is observed however with a noticeable underestimation for all cases with ' 6= 0. As expected, for ' = 0, both models predict the same proﬁle. Though the nanoﬂuid properties take into account the volume fraction of nanoparticles ', the single-phase model appears insensitive to ' as the same heat transfer coefﬁcient distribution is obtained whatever the value of ' 0.016. This conﬁrms the previous results of Akbari et al. [19] and Bianco et al. [17]. On the other hand, the heat transfer coefﬁcient predicted by the h (W/m K) av Appl. Sci. 2016, 6, 332 10 of 24 mixture model clearly increases with increased nanoparticle concentration in agreement with previous observations [18,19,30,41]. Figure 5. Local heat transfer coefﬁcient obtained for Re = 1600, d = 42 nm and n p Al O -water based nanoﬂuids: comparison between the single-phase and mixture models with 2 3 temperature-dependent properties. The superiority of the mixture model may be easily explained. In fact, the latter ensures a more accurate treatment of the two-phase mixture, compared to the single-phase model, which does not take into account either the spatial variations of the distribution in nanoparticles in the base ﬂuid, nor the relative velocity of each phase. The mixture model seems a better model to describe the nanoﬂuid ﬂow. In fact, the slip velocity between the ﬂuid and the nanoparticles is not zero due to several factors such as the Brownian motion or gravity, which induces for example the sedimentation of the solid particles. One could argue that the single-phase model does not perform well due to inappropriate correlations for the thermal conductivity and the dynamic viscosity of the nanoﬂuid. Thus, several correlations have been tested and the results are summarized in Table 2 for Re = 1600, d = 42 nm n p and Al O -water based nanoﬂuids. Two volume fractions ' = 0.006 and 0.016 have been considered. 2 3 Simulation 1 combines Equations (7) and (10), whereas simulation 2 uses Equations (8) and (9). It is clear that the two sets of correlations provide rather the same results in terms of the average value of the heat transfer coefﬁcient with differences of about 17% and 33% for ' = 0.006 and 0.016, respectively, compared to the experiments. It shows in particular that for these sets of parameters, the correlations do not considerably inﬂuence the accuracy of the single-phase model. As shown previously, the results are not inﬂuenced by the particle volume fraction. Though the effect of Brownian motion is accounted for in Equation (10), it has no noticeable inﬂuence in laminar ﬂows. This conﬁrms the previous work of Keblinski et al. [42], who suggested that the motion of nanoparticles due to the Brownian motion is too slow to transport a signiﬁcant amount of heat through a nanoﬂuid. They ignored the effect of Brownian motion in the enhancement of thermal conductivity of nanoﬂuids. Appl. Sci. 2016, 6, 332 11 of 24 2 1 Table 2. Inﬂuence of the different correlations on the average heat transfer coefﬁcient h (Wm K ) av for Re = 1600, d = 42 nm and Al O -water based nanoﬂuids. The relative error is given in brackets. n p 2 3 Simulation 1 combines Equations (7) and (10), whereas simulation 2 uses Equations (8) and (9). Results obtained using the single-phase model. ' Experiments [11] Simulation 1 Simulation 2 1089.66 1086.37 0.006 1313.54 (17.04%) (17.29%) 1085.69 1075.72 0.016 1626.12 (33.23%) (33.85%) For the sake of simplicity when developing new numerical models, it may be interesting to consider the mixture model with constant properties if satisfactory results may be obtained. Figure 6 displays the axial distributions of the local heat transfer coefﬁcient for Re = 1600, d = 42 nm and n p Al O -water based nanoﬂuids with four concentrations of nanoparticles. The results are obtained 2 3 using Equations (7) and (10) either with constant (CP) or variable (VP) properties for the base ﬂuid. In all cases, the local heat transfer coefﬁcient h decreases exponentially with the axial distance z. Using temperature-dependent properties (VP) leads to very satisfactory results as already shown in Figure 3a. On the contrary, using constant properties (CP) leads to a strong overestimation of the heat transfer coefﬁcient, with more pronounced differences towards the thermally fully developed ﬂow region. In fact, the thermal conductivity of the nanoﬂuid increases drastically with decreasing values of both temperature and density. Using VP, the circumferential wall temperature appears to be non-uniformly distributed in the tangential direction, whereas the CP model exhibits a more uniform and axisymmetric behavior. The VP model takes then into account buoyancy effects, which result in a noticeable increase of the ﬂuid temperature in the upper half of the pipe. This conﬁrms previous results such as those suggested by [17] except from the previous work of Labonte et al. [16], who showed that CP lead to an underestimation of the heat transfer coefﬁcient. This difference may be attributed to the different multiphase models used. CP VP CP VP CP VP CP VP 0.0 0.2 0.4 0.6 0.8 1.0 z(m) Figure 6. Local heat transfer coefﬁcient obtained for Re = 1600, d = 42 nm and Al O -water based n p 2 3 nanoﬂuids: Inﬂuence of temperature-dependent properties on the performances of the mixture model. h(W/m K) Appl. Sci. 2016, 6, 332 12 of 24 It is important to note that taking into account temperature-dependent properties does not increase the computational cost. Using the same mesh grid, both calculations take about 4 h using 8 processors on Mammouth Parallel 2. Calculations using VP lead to a rather faster convergence as compared to the CP case. 4. Results and Discussion All the results presented in the following have been obtained using the mixture model with temperature-dependent properties and Equations (7) and (10) to model the nanoﬂuid properties. The inﬂuence of the Reynolds number Re, the volume fraction ', the diameter d of the nanoparticles n p and the type of nanoparticles on the hydrodynamic and the thermal ﬁelds are successively discussed in details in the following sections. 4.1. Inﬂuence of the Volume Fraction of Nanoparticles and Reynolds Number for Al O /Water-Based 2 3 Nanoﬂuids The combined effects of the Reynolds number and the volume fraction of nanoparticles on the average heat transfer coefﬁcient are plotted in Figure 7. It can be clearly observed that an average enhancement of the convective average heat transfer coefﬁcient of about 40% is achieved when the Reynolds number increases from Re = 600 to 1600 for all nanoparticle concentrations '. A linear dependency of h is obtained against to ' for the three Re numbers. av Re = 600 Re = 1000 Re = 1600 0.0 0.4 0.8 1.2 1.6 (%) Figure 7. Effects of the Al O nanoparticle concentration ' and Reynolds number Re on the average 2 3 heat transfer coefﬁcient h for d = 42 nm. av n p Figure 8 shows the axial distribution of the heat transfer coefﬁcient ratio h =h along the pipe n f b f for d = 42 nm, Re = 1600 and three volume fractions of Al O nanoparticles. It clearly indicates n p 2 3 an average thermal enhancement of 28%, 48% and 75.6% for ' = 0.006, 0.01 and 0.016, respectively. This ratio h =h is rather constant in the axial direction z=D with a local maximum around z=D ' 85 n f b f whatever ' and a second local maximum at z=D ' 175 for ' = 0.01. h (W/m K) av Appl. Sci. 2016, 6, 332 13 of 24 1.9 1.8 1.7 = 0.006 = 0.01 1.6 = 0.016 1.5 1.4 1.3 1.2 20 40 60 80 100 120 140 160 180 200 z/D Figure 8. Axial variations of the heat transfer coefﬁcient ratio h =h for d = 42 nm, Re = 1600 and n p n f b f three volume fractions in Al O nanoparticles: ' = 0.006, 0.01 and 0.016. 2 3 Figure 9 illustrates the temperature contours at four axial positions z = 0.2 m (z=D = 44.4), 0.4 m (z=D = 88.9), 0.6 m (z=D = 133.3) and 0.8 m (z=D = 177.8) for d = 42 nm, Re = 1600 n p and four concentrations '. The temperature contours change from a circular form at z = 0.2 m, to an elliptical one at z = 0.4 m then to a kidney shape from z = 0.6 m to the tube outlet for all concentrations of nanoparticles. The circumferential wall temperature appears to be non-uniformly distributed in the tangential direction, especially at z = 0.8 m, with maximum values at the top of the tube. It can be simply explained by density variations, since the warm nanoﬂuid has a lowest density and can rise due to the buoyancy force to the upper half of the tube inducing a stratiﬁcation of the ﬂuid temperature. This suggests the necessity to consider temperature-dependent properties for the nanoﬂuid in order to predict this effect. The hot temperature region located at the top of tube for ' = 0 shown in Figure 9d progressively disappears when increasing the nanoparticle volume fraction. For example, the wall temperature T decreases noticeably when ' increases: at z = 0.8 m, the maximum wall temperature decreases from 332 K to 293 K for ' = 0 (Figure 9d) and ' = 0.016 (Figure 9p), respectively. The maximum temperature difference is inversely proportional to the nanoparticle volume fraction. It is reduced almost by a factor 2 between ' = 0.016 and ' = 0. More generally, the introduction of even higher volume fractions (for this range of parameters) tends to homogeneize the temperature distribution at a given cross-section. It may be explained by considering nano-convection effect, which is linearly related to ' [43]. This effect is induced by the Brownian motion of the nanoparticles. Brownian motion caused by the thermal interaction between the nanoparticles and the base ﬂuid is stronger within regions of higher ﬂuid temperature that is why the upper half of the tube is more affected. The inﬂuence of Brownian force on the thermal conductivity enhancement is strongly debated in the scientiﬁc community, some authors assume that it plays a key role [43,44], while others ignore its effect [42]. Note that the same phenomena are observed for the two other values of the Reynolds number. h /h nf bf Appl. Sci. 2016, 6, 332 14 of 24 z=0.2m z=0.4m z=0.6m z=0.8m ' = 0 (a) (b) (c) (d) ' = 0.006 (e) (f) (g) (h) ' = 0.01 (i) (j) (k) (l) ' = 0.016 (m) (n) (o) (p) (q) Figure 9. Maps of temperature T at four axial positions: z = 0.2 m (a,e,i,m); z = 0.4 m (b,f,j,n); z = 0.6 m (c,g,k,o); and z = 0.8 m (d,h,l,p). Results obtained for d = 42 nm, Re = 1600 and four n p volume fractions of Al O nanoparticles: ' = 0 (a–d); 0.006 (e–h); 0.01 (i–l); and 0.016 (m–p). 2 3 Figure 10 displays the corresponding streamlines colored by the axial velocity component w at four cross-sections along the pipe. Due to the increased temperature at the wall, a secondary ﬂow is observed. It consists of a pair of symmetrical counter-rotating vortices with respect to the tube axis. These vortices are induced by buoyancy forces: an upward ﬂow restricted in a thin layer along the wall rises up a warm ﬂuid and a downward ﬂow along the tube axis drops a cool ﬂuid [45,46]. For pure water (' = 0), Figure 10a shows clearly that the buoyancy force already appears at z = 0.2 m and induces the secondary ﬂow. The contours of the axial velocity component are axisymmetric at this axial position for all nanoparticle concentrations and the recirculation cells are symmetric. This indicates that the velocity is not yet affected by the buoyancy force, which is due to the fact that, at this location, the circumferential temperature gradients are very small. Appl. Sci. 2016, 6, 332 15 of 24 z=0.2m z=0.4m z=0.6m z=0.8m ' = 0 (a) (b) (c) (d) ' = 0.006 (e) (f) (g) (h) ' = 0.01 (i) (j) (k) (l) ' = 0.016 (m) (n) (o) (p) (q) Figure 10. Streamline patterns colored by the axial velocity component w at four axial positions: z = 0.2 m (a,e,i,m); z = 0.4 m (b,f,j,n); z = 0.6 m (c,g,k,o); and z = 0.8 m (d,h,l,p). Results obtained for d = 42 nm, Re = 1600 and four concentrations of Al O nanoparticles: ' = 0 (a–d); 0.006 (e–h); n p 2 3 0.01 (i–l); and 0.016 (m–p). When the ﬂuid moves further downstream, the recirculations are slightly shifted and moves across the median plane, above the plane at z = 0.4 m and just below the plane at z = 0.8 m, for all values of '. The maximum value of w is also slightly shifted downward below the horizontal tube axis when moving to the pipe outlet. This shift results from the important increase in the intensity of the secondary ﬂow. This loss of axisymmetry is due to both the boundary layer development and the increasing inﬂuence of the buoyancy force, which becomes more pronounced along the tube, increasing then the strength of the secondary ﬂow. At z = 0.8 m, the axial velocity contours exhibit an ellipsoid-shaped form for ' = 0. At the same time, the circular streamlines in the lower half of the tube indicate a weaker secondary ﬂow, while the curved ones in the upper half indicate that a hot ﬂuid Appl. Sci. 2016, 6, 332 16 of 24 is conﬁned and accompanied by a more intense secondary ﬂow. At z = 0.8 m, the form of the axial velocity contours change from an ellipsoid-shaped pattern for ' = 0 to a rather more circular form for ' = 0.016 as illustrated in Figure 10d,p, respectively. For ' = 0.016, the shape of the velocity contours remains practically unchanged indicating a fully developed region from z = 0.4 m. This result agrees well with the previous observations conﬁrming that the nanoparticles suppress the buoyancy force induced secondary ﬂow, stabilizes the ﬂow with a strong homogeneity of the ﬂuid temperature within the tube. The same behavior is observed for all volume fractions of nanoparticles indicating that the latter has no remarkable inﬂuence on the hydrodynamic ﬁeld. Nevertheless, the intensity of the secondary ﬂow decreases when increasing ' for all axial positions especially at z = 0.8 m as shown in Figure 11. This is consistent with previous observations [47–49]. The wall layer vorticity also decreases with increased values of '. The weak inﬂuence of the nanoparticle concentration on the velocity and temperature ﬁelds results from the ability of nanoparticles to homogenize the ﬂuid temperature and therefore impeding buoyancy forces. Only few studies considered the inﬂuence of the nanoparticles on the development of the secondary ﬂow and the homogenization of the temperature ﬁeld. When ' is increased, the molecular diffusion increases accompanied with an increase in the thermal conductivity and a reduction in the speciﬁc heat capacity, as proposed by [48]. Colla et al. [49] invoke the role of the Brownian diffusion. (a) (b) (c) (d) Figure 11. Contours of the streamwise vorticity at z = 0.8 m: (a) ' = 0; (b) ' = 0.006; (c) ' = 0.01; and (d) ' = 0.016. Results obtained for d = 42 nm, Re = 1600 and Al O nanoparticles. n p 2 3 4.2. Inﬂuence of the Nanoparticle Diameter for Al O and Cu/Water-Based Nanoﬂuids 2 3 In real thermal engineering applications, increasing the nanoparticle diameter leads to higher agglomeration effects resulting in the sedimentation of the agglomerates. Such particle-particle interactions are not taken into account in the present model. Only the Brownian motion and the ratio between gravity and buoyancy forces are modeled here. The nanoparticle diameter d has then no n p remarkable effect on the heat transfer coefﬁcients as it will be shown in the following. More interestingly however, the inﬂuence of d on the axial distributions of the nanoparticle concentration ' is n p quite remarkable. Figure 12 displays the axial proﬁles of ' at three radial locations r=R = 0.98 (top wall), r=R = 0 (axis) and r=R = 0.98 (bottom wall) for two types of nanoﬂuid (copper and alumina water-based), three nanoparticle diameters d = 42, 100 and 200 nm and for Re = 1600 and ' = 1.6%. Firstly, the n p concentration is rather constant along the center line of the tube (r=R = 0) for the two nanoﬂuids and all the nanoparticle diameters, remaining between 1.6% and 1.596%. For d = 42 nm, along n p the top wall of the tube, the nanoparticle concentration decreases by 1.5% and 3% for Al O and 2 3 Cu nanoparticles, respectively (Figure 12a). As the nanoparticle diameter grows, the concentration decreases along the top wall due to gravity effects. For example, ' is reduced by 22.5% and 43.75% for Al O and Cu nanoparticles, respectively, for d = 200 nm. By conservation of the average value 2 3 n p of ' at a given cross-section, the concentration of nanoparticles along the bottom wall increases. Because of their higher density, this effect is more noticeable for copper/water-based nanoﬂuids as illustrated in Figure 12c,f,i. The latter shows that ' increases by 0.75%, 17.5% and 87.5% for d = 42, n p Appl. Sci. 2016, 6, 332 17 of 24 100 and 200 nm, respectively. The concentration of Al O appears quite constant, with an increase of 2 3 only 2.5% for d = 200 nm. This behavior can be easily explained since the density of copper is twice n p the alumina one. r=R = 0.98 r=R = 0 r=R = 0.98 (a) (b) (c) (d) (e) (f) (g) (h) (i) Figure 12. Axial variations of the volume fraction ' at three radial locations: r=R = 0.98 (a,d,g); r=R = 0 (b,e,h); and r=R = 0.98 (c,f,i). Results obtained for Cu and Al O nanoparticles of three 2 3 different diameters d = 42 nm (a–c); 100 nm (d–f); and 200 nm (g–i) with ' = 0.016 and Re = 1600. n p The inﬂuence of the mean diameter of the Al O nanoparticles on the heat transfer is not shown 2 3 here, but it can be noticed that increasing d does not affect the average heat transfer coefﬁcient. n p 2 2 For ' = 0.016 and Re = 1600, h = 1796.7 W/m K and 1795.5 W/m K for d = 10 and 200 nm, av n p respectively. However, the average heat transfer coefﬁcient may be strongly modiﬁed by the size of the nanoparticles for copper-water nanoﬂuids, which sediment more with only 12% of the particles still in suspension in the base ﬂuid as shown previously for d = 200 nm (Figure 12g,i). n p Figure 13 illustrates the effect of the global Reynolds number Re on the distributions of the concentration in copper nanoparticles along the tube for d = 200 nm, ' = 0.016 and two Reynolds n p numbers Re = 600 and 1600 at three radial locations r=R = 0.98 (near the top wall), r=R = 0 (pipe axis) and r=R = 0.98 (near the bottom wall). Even for this large nanoparticles, the concentration of nanoparticles along the tube axis remains almost constant. Throughout the pipe length, the variation of ' is 0.13% for Re = 1600 and 0.75% for Re = 1600. At r=R = 0.98, ' decreases by a factor 2 then 4 for Re = 1600 and Re = 600, respectively. It results in a huge increase in nanoparticle concentration at the Appl. Sci. 2016, 6, 332 18 of 24 bottom of the pipe (Figure 13c): ' increases by 87.5% and 137.5% for Re = 1600 and 600, respectively. Almost all the copper nanoparticles are sedimented and there are only few copper nanoparticles suspended in pure water at the top of the tube for Re = 600. The Reynolds number plays an important role to keep the nanoparticles well dispersed in the base ﬂuid and reduce the sedimentation process, inducing a better stability of the nanoﬂuid and higher resulting heat transfer. r=R = 0.98 r=R = 0 r=R = 0.98 Re = 600 Re = 600 0.016 0.040 Re = 600 Re = 1600 0.01600 Re = 1600 Re = 1600 0.014 0.01598 0.035 0.012 0.01596 0.030 0.010 0.01594 0.025 0.008 0.01592 0.020 0.006 0.01590 0.015 0.004 0.01588 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 z (m) z(m) z(m) (a) (b) (c) Figure 13. Axial variations of the volume fraction ' in Cu nanoparticles at three radial locations: (a) r=R = 0.98; (b) r=R = 0; and (c) r=R = 0.98. Results obtained for Cu nanoparticles of diameter d = 200 nm with ' = 0.016 and two Reynolds numbers Re = 600 and 1600. n p 4.3. Inﬂuence of the Type of Nanoparticles for Water-Based Nanoﬂuids The axial variations of the heat transfer coefﬁcient of six types of nanoparticles for ' = 1.6%, d = 42 nm and Re = 1600 are shown in Figure 14. This later ﬁgure illustrates that the local heat transfer n p coefﬁcients for all water-based nanoﬂuids exhibit the same behavior, with an exponential decrease with increased distance along the pipe. The nanoﬂuid with copper nanoparticles achieves the highest heat transfer coefﬁcient, followed by C, Al O , CuO, TiO , and SiO , respectively. This behavior was 2 3 2 2 expected since the copper has the highest thermal conductivity amongst the other nanoﬂuids as shown in Table 1. Cu Al2O3 CuO TiO2 SiO2 0.0 0.2 0.4 0.6 0.8 1.0 z(m) Figure 14. Axial variations of the local heat transfer coefﬁcient h for six types of nanoparticles with d = 42 nm, ' = 0.016 and Re = 1600. n p In the other hand, a major hindrance associated with the use of copper nanoparticles is the sedimentation phenomenon, as illustrated earlier for ' = 1.6% and d = 200 nm. They sediment 34 n p h(W/m K) Appl. Sci. 2016, 6, 332 19 of 24 times higher than the Al O nanoparticles under the same operating conditions. For this reason, the 2 3 choice of the appropriate nanoﬂuid does not depend only on its thermal conductivity, but should also take into account their stability and the suspension of the nanoparticles in the base ﬂuid for a long term use. 4.4. Summary For engineering applications, empirical correlations are of primary importance to predict the average heat transfer coefﬁcient as a function of the ﬂow and geometrical parameters for an effective design of thermal systems such as heat exchangers. This is particularly challenging for convective nanoﬂuid ﬂows as opposed to single-phase ﬂows, because of the inﬂuence of various parameters due to the presence of the solid nanoparticles. An attempt has been done in the following. First, the well-known correlation proposed by Shah [50] for laminar ﬂows under a constant heat ﬂux boundary condition and used by Wen And Ding [11] is considered: 1=3 D D Nu = 1.953 RePr 33.3 RePr (29) Shah x x A second correlation was proposed by [50] for RePr 33.3 but it was carefully checked here that it led to very similar results. For simplicity, only Equation (29) will be used in the following. All the results obtained in this paper using direct numerical simulations may be expressed in terms of Nusselt number using the following correlation: 1=3 Re D Nu = Nu 1 + 1.7 ' 6.6 RePr 46.5 Pr 1 (30) D NS Shah Pr x It is noteworthy that the validity range of Equation (30) has been extended to 6.6 RePr 46.5 compared to Equation (29). The present correlation is valid for all values tested in the present work except for the simulations involving copper nanoparticles for which Pr 1. Both geometrical (through x and D) and ﬂow (Re and Pr) parameters are considered in Equation (30). The inﬂuence of the solid nanoparticles is taken into account through ' but also through their thermophysical properties used to deﬁne Re and Pr. Figure 15 conﬁrms that Equation (30) ﬁts particularly well with all simulations for Pr 1. 8.0 7.5 7.0 6.5 6.0 5.5 Nu DNS 5.0 4.5 4.0 3.5 3.0 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 4 1/3 Nu [1+1.7 (Re/Pr ) ] Shah Figure 15. Veriﬁcation of Equation (30). Appl. Sci. 2016, 6, 332 20 of 24 1=3 The second term 1.7 Re=Pr ' in Equation (30) is small compared to 1 in all cases. The results are then well ﬁtted with Equation (29) and so with the correlation provided by Rea et al. [35] for Al O /water-based nanoﬂuids, 431 Re 2000 and ' 0.06. It corresponds to Equation (29) with 2 3 a prefactor equal to 2.0398 instead of 1.953. The second interesting quantity for engineering applications is the variations of the average friction factor f as a function of the Reynolds number Re. The friction factor is calculated as proposed av by Choi and Cho [51]: 8t f = (31) where u is the mean ﬂuid velocity and t is the wall shear stress. It may be convenient to m w ﬁnd correlations under the form: f = ARe . Figure 16 summarizes the results obtained for Al O /water-based nanoﬂuids with alumina nanoparticles of diameter d = 42 nm and for four 2 3 n p volume fractions '. As for pure water ﬂows, the average friction factor f decreases for increased av values of Re. Complementary calculations have been performed for turbulent ﬂows up to Re = 15, 000. 0.756 In the laminar regime, f varies according to the relation f = 11.381Re , whereas, in the av av 0.358 turbulent regime, f follows: f = 1.05Re . A discontinuity is also observed in the transitional av av regime. The presence of nanoparticles in the base ﬂuid affects the variations of the average friction factor compared to pure water ﬂows. However, the friction factor f appears to be insensitive to av the nanoparticle concentration '. The results are to be compared to the classical Darcy relations for single-phase ﬂows in pipes, where: A = 64 and = 1 for laminar ﬂows and A = 0.3164 and = 0.25 for turbulent ﬂows in smooth pipes. 0.10 0.09 0.08 0.756 0.07 f = 11.381/Re av 0.06 0.05 0.358 f = 1.05/Re av 0.04 0.03 0 4000 8000 12000 16000 Re Figure 16. Inﬂuence of the Reynolds number Re on the average friction coefﬁcient f . Results obtained av for all values of ', d and all types of nanoﬂuids. n p In the laminar regime, the results agree particularly well with the correlation of Suresh et al. [52] obtained for Al O -Cu/water-based nanoﬂuid with Re < 2300 and ' 0.1%: A = 26.4 f (') and 2 3 = 0.8737. 5. Conclusions Laminar forced convection ﬂows of water-based nanoﬂuids through a uniformly heated tube were revisited here using direct numerical simulations. The single-phase and mixture models with constant and temperature-dependent properties were compared to the experimental data of Wen and Ding [11] and to the numerical simulations of Akbari et al. [19]. The mixture model with temperature-dependent av Appl. Sci. 2016, 6, 332 21 of 24 properties was shown to perform best with a close agreement to the experimental data. The former simulations of Akbari et al. [19] using the same model were signiﬁcantly improved with the use of an appropriate mesh grid. The numerical model was then used conﬁdently and extensively to investigate the inﬂuences of the Reynolds number (600 Re 1600), the concentration in nanoparticles ( 1.6%) and their diameter (42 d 200 nm) on the hydrodynamic and thermal ﬁelds. Al O /water based nanoﬂuids n p 2 3 have been considered ﬁrst before evaluating the thermal performances of other nanoparticles such as: Cu, C, CuO, TiO , and SiO . 2 2 For Al O /water based nanoﬂuids, the average heat transfer coefﬁcient increased linearly with 2 3 the nanoparticle concentration for all Reynolds numbers. At Re = 1600, the local heat transfer coefﬁcient increased in average by 29%, 46% and 74% for ' = 0.006, 0.01 and 0.016, respectively. Increasing the nanoparticle concentration led to a more homogenous temperature ﬁeld, impeding the hot temperature region observed at the top of the pipe wall for pure water ﬂows. The ﬂow ﬁeld revealed two recirculation regions for all (r,) planes, only weakly inﬂuenced by '. The maximum value of the axial velocity component observed at (r=R ' 0.2, = 90 ) was also weakly affected by '. The volume fraction in nanoparticles affected signiﬁcantly the streamwise vorticity of the two recirculation cells. The ﬂow and temperature ﬁelds exhibited a more homogeneous behavior. A particular attention was also paid to the sedimentation of the nanoparticles, which, as expected, increased for large size or high density nanoparticles. Finally, empirical correlations to predict both the Nusselt number and the average friction coefﬁcient have been provided, summarizing all simulations presented here (in the range of Pr 1 for Nu). Further calculations are now required to extend the present simulations to the turbulent flow regime using large-eddy simulations. Further developments are also planned to improve the numerical model to take into account more complex phenomena like the thermophoresis effect and particle-particle interactions. Acknowledgments: The authors would like to thank the NSERC chair on industrial energy efﬁciency funded by Hydro-Québec, CanmetEnergy and Rio Tinto Alcan established at Université de Sherbrooke for the period 2014-2019. The ﬁnancial support of the CREEPIUS research center and the HPC resources of the Compute Canada network are also gratefully acknowledged. Author Contributions: G.S. performed the numerical simulations; G.S. and S.P. performed the physical analysis and wrote the paper. Conﬂicts of Interest: The authors declare no conﬂict of interest. Abbreviations The following abbreviations are used in this manuscript: 1 1 C Speciﬁc heat, JK kg D Tube diameter, m d Nanoparticle diameter, m n p f Friction factor, - 1 1 h Heat transfer coefﬁcient, Wm K 2 1 k Thermal conductivity, Wm K L Tube length, m q Heat ﬂux, Wm R Tube radius, m r Radial location, m Re Global Reynolds number, - T Temperature, K w Axial velocity component, ms z Axial position, m ' Volume fraction, - Dynamic viscosity, Pas Density, kgm t Wall shear stress, Pa av Average Appl. Sci. 2016, 6, 332 22 of 24 b f Base ﬂuid e f f Effective in Inlet m, mix Mixture n f Nanoﬂuid n p Nanoparticles w Wall References 1. Wong, K.V.; De Leon, O. 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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

Applied Sciences – Multidisciplinary Digital Publishing Institute

**Published: ** Nov 2, 2016

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