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
]. 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 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
lower than 100 nm. As opposed to milli- or microsized suspensions, very stable suspensions may be achieved by introducing nanoparticles. Moreover, nanoparticles benefit from a
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
] 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
]). Others developed experimental set-ups to measure the convective heat transfer and temperature profiles in pipes [11
], 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
]. 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
/water-based 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
nanoparticles equal to
, similar results have been found using single- and two-phase models with a maximum difference of
. 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
/water-based nanofluids. The mixture, 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 low volume fractions in nanoparticles, typically
. On the contrary, at higher particle concentrations
, 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
-water nanofluid turbulent forced convection in a pipe with constant 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
/water-based nanofluids. This choice is justified by the large number of former works using this nanofluid in a similar flow configuration (laminar or developing flows in a pipe with constant heat flux) [23
]. 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
using only a volume fraction equal to
and Liu et al. [26
] measured an increase of
for a nanoparticle diameter equal to 33 nm and a volume fraction of
. The reader can refer to the reviews by Kakaç and Pramuanjaroenkij [10
] 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
/water-based nanofluids using direct numerical simulations; (2) to extend the results to 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
/water-based nanofluids with the present simulations. The paper is then organized as follows: the numerical modeling and its validation are presented in Section 2
and Section 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
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 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 (), the concentration in nanoparticles () and their diameter ( nm) on the hydrodynamic and thermal fields. /water based nanofluids have been considered first before evaluating the thermal performances of other nanoparticles such as: , C, , , and .
For /water based nanofluids, the average heat transfer coefficient increased linearly with the nanoparticle concentration for all Reynolds numbers. At , the local heat transfer coefficient increased in average by , and for , and , 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 () planes, only weakly influenced by . The maximum value of the axial velocity component observed at (, ) 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 for ).
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.