Modeling of convective turbulent heat transfer of water-based ${\text{Al}}_2{\text{O}}_3$ nanofluids in an uniformly heated pipe

Highlights

• Forced convective turbulent nanofluid flows in an uniformly heated straight pipe.

• The single-phase model can be used confidently up to $\phi$ = 0.8%.

• Benchmark of eight turbulence modelings in their low-Reynolds number formulation.

• The low-Reynolds number k-ω SST model performs the best.

• At $\phi$ = 0.8%, the use of Al₂O₃/water nanofluids is beneficial to the system.

Abstract

Turbulent convective heat transfers of ${\text{Al}}_2{\text{O}}_3$-water nanofluid flowing in a circular tube subjected to an uniform wall heat flux are numerically investigated using different turbulence models. Four nanoparticle volume concentrations $\phi$ up to $2\%$ are considered for bulk Reynolds numbers within the range $3000⩽\mathit{Re}⩽20\text{,}000$. The effects of the nanoparticle concentration and the Reynolds number on the Nusselt number and friction factor are reported. Two different numerical approaches including the single-phase and the mixture two-phase models with variable thermophysical properties are favorably compared to experimental results obtained from the literature for low nanoparticle concentrations ($\phi ⩽0.5\%$). The results at a higher volume fraction $\phi =2\%$ show the necessity to use a mixture model. Eight turbulence models in their low-Reynolds number formulation are also compared to assess their ability to predict the effect of turbulence on the convective heat transfer. The SST k-$\omega$ model was found to perform the best with errors in terms of the average Nusselt number and friction coefficient of 0.44$\%$ and 1.82$\%$ respectively. On the contrary, the linear pressure-strain Reynolds Stress Model completely failed to provide the good values with discrepancies of 41.91$\%$ and 133.54$\%$, respectively. Finally, the benefit of using this nanofluid is discussed regarding four merit criteria.

Introduction

Convective heat transfer plays an important role in various industrial sectors such as air-conditioning, transportation, chemical production, microelectronics and power generation. The conventional heat transfer fluids such as water, ethylene glycol or oil exhibit relatively limited heat transfer properties, which hinders the efficiency of the thermal systems. The recent advance in the field of nanotechnology gave rise to a new type of nanometeric metallic or non-metallic particles characterized by their substantially higher thermal conductivities. These particles, referred as nanoparticles, are dispersed into a conventional fluid, creating a new class of heat transfer fluids named nanofluids. Since the pioneering work of Choi and Eastman (1995), the particularly increased thermal efficiency of nanofluids, compared to conventional fluids, has attracted the attention of researchers and engineers. Alive researches are still done to model appropriately natural convection in enclosures using ionic nanofluids (Minea and El-Maghlany, 2017) or including a porous medium (Toosi and Siavashi, 2017). The literature being too abundant, one will focus in the following on the turbulent convective heat transfer in pipes.

One of the most common canonical experiments used to study the convective heat transfer performance of nanofluids, is the turbulent flow through a straight uniformly heated pipe as considered for example by Sundar and Sharma (2010). Both the average heat transfer coefficient and the friction factor of an ${\text{Al}}_2{\text{O}}_3$-water based nanofluid were measured in a straight pipe (with and without inserts) subjected to a constant heat flux at the wall for different axial Reynolds numbers $\mathit{Re}=W_{0}D/{\nu }_{\mathit{nf}}$ ($W_0$ being inlet axial velocity, D the pipe diameter and ${\nu }_{\mathit{nf}}$ the kinematic viscosity of the nanofluid) and nanoparticle volume fractions $\phi$. They observed that, for $\phi$ = 0.5$\%$, the heat transfer coefficient increased by $15.62\%$ and $54.54\%$ for Reynolds numbers equal to 3000 and 18,000, respectively, compared to pure water. Their results will serve, in the following, as the experimental database to validate the present simulations. Li and Xuan (2002) measured the heat transfer coefficient and friction factor for Cu/water nanofluid flowing inside a tube in both laminar and turbulent flow regimes. They noted an enhancement up to $60\%$ for $\phi =2\%$ compared to pure water at the same Reynolds number. Turbulent convective heat transfers and pressure drop of $\gamma$-${\text{Al}}_2{\text{O}}_3$-water nanofluid inside a circular tube were investigated experimentally by Fotukian and Esfahany (2010). They affirmed that the addition of small quantity of alumina nanoparticles to pure water increased heat transfer remarkably. For example, for $\mathit{Re}={10}^4$ and $\phi =0.045\%$, the heat transfer coefficient was increased by $48\%$. Heyhat et al. (2012) experimentally studied the turbulent heat transfer behavior of alumina/water nanofluid in a circular pipe under constant wall temperature condition. Their results showed that the heat transfer coefficient of ${\text{Al}}_2{\text{O}}_3$-water nanofluid was increased by $23\%$ for $\phi =2\%$ compared to pure water at Re around 12,000. Noghrehabadi and Pourrajab (2016) investigated experimentally the convective heat transfer of $\gamma$-${\text{Al}}_2{\text{O}}_3$-water nanofluid in a circular tube with constant heat flux at the wall. Their results showed that the average heat transfer coefficient was increased by $16.8\%$ for $\phi =0.9\%$ compared to distilled water at $\mathit{Re}=2070$. They observed that the enhancement was particularly significant in the entrance region and decreased with the axial distance. The thermal enhancement by the use of nanofluids has been then widely demonstrated experimentally mainly by global temperature measurements as shown in the detailed review of Kakaç and Pramuanjaroenkij (2009). Numerical simulations appear then as a powerful tool to get a better insight into the flow dynamics and heat transfer processes associated with nanofluids and explain in detail the main mechanisms responsible for this enhancement.

Due to their excessive computational cost, only limited attention has been paid to use direct numerical simulations (DNS) (Zonta et al., 2008, Juan-Cheng et al., 2015) or even large eddy simulations (LES) to investigate nanofluid turbulent flows in pipes or channels. Hu et al. (2013) investigated by a LES-Lagraneg method the flow characteristics of nanofluids (water-based Cu or ${\text{SiO}}_2$) through a straight circular tube at $\mathit{Re}=25\text{,}000$ and $\phi =1\%$. Their mesh grid is composed of $1.3$ millions of cells and their results sampled over 5 mean flow residence times but nothing is said about the computional resources required. Turbulence intensities are enhanced by the presence of nanoparticles, which may be responsible for the heat transfer enhancement. For any nanofluid, there was no evidence of coherent vortical structures within the flow requiring the use of advanced unsteady 3D calculations. Peng et al. (2014) performed LES of turbulent nanofluid flows inside a cylindrical pipe and compared the predictions of Eulerian-Eulerian, Euler-Lagrangian, and Lagrangian multiphase models, in an attempt to better explain the flow field behavior and the mechanisms responsible for the heat transfer enhancement. The Lagrangian model was found to perform better than the two other models due to its capability to provide a more detailed information about the development and the interaction of the turbulent eddies with the nanoparticles.

The use of advanced DNS or LES models in the context of turbulent nanofluid flows in realistic geometries remains then marginal and most authors focused on Reynolds-Averaged Navier-Stokes (RANS) turbulence closures. Though being more simple and requiring less computational resources, they might be able to provide accurate data when coupled to the appropriate single or two-phase model and to the appropriate correlations for the nanofluid properties. The standard k-$\epsilon$ model has been successively used in the past to investigate turbulent nanofluid flows and heat transfers inside a cylindrical pipe (Behzadmehr et al., 2007, Bianco et al., 2011, Davarnejad and Jamshidzadeh, 2015, Behroyan et al., 2015, Salman et al., 2016). For water-based Cu nanofluids, Behzadmehr et al. (2007) reported discrepancies of around $7\%$ in terms of averaged Nusselt number at $\mathit{Re}=15\text{,}000$ and $\phi =1\%$ when using the standard k-$\epsilon$ model coupled to the mixture model. Akbari et al. (2012) used also the Realizable k-$\epsilon$ model to evaluate the turbulent forced convection in a horizontal heated tube filled with water-based ${\text{Al}}_2{\text{O}}_3$ or Cu nanofluids. A relatively good agreement was found compared to the experimental data of Sundar and Sharma (2010) using a single-phase approach. However the rate of increase of the Nusselt number with the Reynolds number was underestimated by the two k-$\epsilon$ models at low nanoparticle concentrations $\phi <1\%$. Roy et al. (2012) considered turbulent convective flows of three water-based nanofluids flowing inside a radial cooling system using a single-phase model. They compared the predictions of four turbulence models using air as the working fluid and claimed that the shear stress transport SST k-$\omega$ model was the appropriate level of closure, compared to the RNG k-$\epsilon$, k-$\omega$ and ${\vartheta }^2$-f models, exhibiting a good agreement in terms of local Nusselt number and wall pressure distribution with published experimental data at $\mathit{Re}=23\text{,}000$. Saha and Paul (2014) considered numerically the heat transport behavior of single-phase water-based alumina and titanium nanofluids in a circular pipe under turbulent flow condition. For pure water at $\mathit{Re}=21\text{,}800$, they compared the predictions of three k-$∊$ models and concluded that the realizable k-$\epsilon$ model was the most appropriate turbulence closure. It has then been extensively used for Re up to ${10}^6$ and $\phi =6\%$ with a close agreement in terms of the averaged Nusselt number compared to the Pak and Cho’s correlation (Pak and Cho, 1998). Recently, Boertz et al. (2016) modeled the flows of ${\text{SiO}}_2$ ethylene–glycol or water-based nanofluids in a tube with constant heat flux at the wall. Their results obtained for $6000⩽\mathit{Re}⩽12\text{,}000$ and $\phi ⩽10\%$ using a single-phase model showed that the SST k-$\omega$ better predicted the Nusselt number compared to the standard k-$\epsilon$ or k-$\omega$ models with a mean deviation of $5\%$ compared to published experimental data. It better predicted also the friction factor but with much larger deviations with the experiments. As a conclusion, there is not a clear consensus about the best turbulence model for investigating turbulent nanofluid flows in a cylindrical pipe and the confidence level depends also strongly on the choice of the single- or two-phase approach and of the modeling of the nanofluid thermophysical properties. An excellent review on different numerical approaches for the simulation of nanofluid flows can be found in the references (Bahiraei, 2014, Kakaç and Pramuanjaroenkij, 2016). One can notice that the standard k-$\epsilon$ coupled to a single-phase modeling is still widely considered today to model turbulent forced convective heat transfer of Cu (Ganesan et al., 2016) or ${\text{TiO}}_2$ (Hussein et al., 2017) water-based nanofluids in a single pipe.

To the best of our knowledge, there is no detailed study evaluating in detail the performance of eight RANS turbulence models on the turbulent flow and forced convective heat transfer of nanofluids in a pipe. This work is then an attempt to fill this gap. A carefull attention should be paid also to the choice of the appropriate single- or two-phase approach. The solver will be validated first against the experimental data of Sundar and Sharma (2010) for low volume fractions $\phi ⩽0.5\%$. Eight turbulence models in their low-Reynolds formulation will be then compared for $\mathit{Re}=13\text{,}380$ and $\phi =0.1\%$. A deep insight into the turbulence modeling enables to explain why the Reynolds Stress Model fails to predict such a flow compared to the SST $k-\omega$ model, which performs the best. The performances of water-based ${\text{Al}}_2{\text{O}}_3$ nanofluids in the forced convective turbulent regime will be finally discussed using four merit criteria. The rigourous built-in of the flow solver, the detailed comparisons of the turbulence models and the evaluation of the nanofluid performances certainly constitute the main novelties of the present paper compared to existing literature.