Eulerian–Eulerian two-phase numerical simulation of nanofluid laminar forced convection in a microchannel

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Abstract

In this paper, laminar forced convection heat transfer of a copper–water nanofluid inside an isothermally heated microchannel is studied numerically. An Eulerian two-fluid model is considered to simulate the nanofluid flow inside the microchannel and the governing mass, momentum and energy equations for both phases are solved using the finite volume method. For the first time, the detailed study of the relative velocity and temperature of the phases are presented and it has been observed that the relative velocity and temperature between the phases is very small and negligible and the nanoparticle concentration distribution is uniform. However, the two-phase modeling results show higher heat transfer enhancement in comparison to the homogeneous single-phase model. Also, the heat transfer enhancement increases with increase in Reynolds number and nanoparticle volume concentration as well as with decrease in the nanoparticle diameter, while the pressure drop increases only slightly.

Introduction

Emerging developments in MEMS (Micro-Electro-Mechanical-Systems) make it possible to fabricate very small scale devices. On the other hand, these small scale devices can generate a high amount of heat flux that should be taken away by a cooling system to guarantee their appropriate performance. One possible way to cool these devises can be the use of so-called nanofluids. A nanofluid is a suspension of nano-sized (10–100 nm) metallic and non-metallic solid particles in a conventional cooling liquid such as water, ethylene glycol, or oil. The term nanofluid for the first time was used by Choi (1995) for such a suspension. After that, many researchers focused on studying the thermophysical and also heat and fluid flow properties of nanofluids. Most of the studies concentrated on the modeling of the effective thermal conductivity of nanofluids (e.g. Xuan et al., 2003, Koo and Kleinstreuer, 2004, Feng et al., 2007). Recently, researchers have focused on the nanofluid heat and fluid flow behavior.

There are many experimental studies for nanofluids on macro and micro-scales (e.g. Wen and Ding, 2004, Heris et al., 2006, Jung et al., 2009, Wu et al., 2009). Wen and Ding (2004) investigated the heat transfer of an Al2O3–water nanofluid in the entrance region of a 4.5 mm diameter copper tube under the constant heat flux conditions. Their measurements showed enhancement in heat transfer especially in the entrance region of the tube. They described this behavior as the particle migration effect (non-uniform nanoparticle volume concentration) that reduces the thermal boundary layer thickness. For an annular tube with a 6 mm inner diameter copper tube and a 32 mm outer diameter stainless steel tube, Heris et al. (2006) studied CuO and alumina nanoparticles in water. They compared the experimental results with homogeneous model results (single-phase correlations with nanofluid effective properties) and reported that the homogeneous modeling under-estimates the heat transfer enhancement, especially in higher volume concentrations.

Jung et al. (2009) did experiments for Al2O3–water nanofluids in rectangular microchannels. The particle diameter in their experiments was 170 nm. With only 1.8% of volume concentration they reported a 32% increase of the heat transfer coefficient in comparison to single distilled water. Also, experiments on nanofluid heat transfer in trapezoidal silicon microchannels have been performed by Wu et al. (2009). For channels with a hydraulic diameter of 194.5 μm and an Al2O3–water nanofluid, they reported an increase in the Nusselt number with increasing particle concentration, Reynolds and Prandtl numbers, while the pressure drop increased slightly when compared to pure water.

For the theoretical study of the pressure-driven nanofluid heat and fluid flow commonly homogenous (single-phase) and two-phase models are used. In homogeneous modeling it is assumed that the particles and the base fluid have the same temperature and velocity and thus, the single-phase equations along with the appropriate effective thermophysical properties (thermal conductivity, viscosity, specific heat and density) for the nanofluid are solved. In this method, the accuracy of the models used as effective thermophysical properties is very important. Most of the theoretical studies in this field are based on the homogeneous approach (e.g. Koo and Kleinstreuer, 2005, Li and Kleinstreuer, 2008, Santra et al., 2009). In addition to the pressure-driven nanofluid flows, it is possible to use electroosmotic transport for nanofluids especially in the micro-scale. In this case, the thickness of the electrical double layer and effective electrical conductivity of the nanofluid can affect the nanofluid heat transfer behavior (Chakraborty and Padhy, 2008, Chakraborty and Roy, 2008).

In despite of the homogeneous modeling, in the two-phase modeling, the nanoparticle and the base fluid are considered as two different phases with different velocities and temperatures. In this method, the interactions between the phases are taken into account in the governing equations. There are a few studies that used two-phase approach to study nanofluids. Behzadmehr et al. (2007) used a two-phase mixture model to study the turbulent nanofluid convection inside a circular tube. Comparing with an experimental study they reported that the two-phase results are more precise than the homogeneous modeling results. However, they considered thermal equilibrium conditions (the same temperature) for the phases. Mirmasoumi and Behzadmehr (2008a) used the same method as in Behzadmehr et al. (2007) to study the mixed convection of the nanofluid in a tube. Also, Mirmasoumi and Behzadmehr, 2008b, Akbarinia and Laur, 2009 investigated the nanoparticles size effect on the mixed convective heat transfer of a nanofluid using the two-phase mixture method. In both studies an increase in heat transfer with a decrease in the nanoparticles size was reported. Bianco et al. (2009) modeled the nanofluid flow and heat transfer inside a tube. They used both single-phase and two-phase methods. For the two-phase method, they implemented Lagrangian approach to model the particle motion. They reported a maximum difference of 11% between the single and two-phase results. Kurowski et al. (2009) used three different homogeneous, Eulerian–Lagrangian and mixture methods to simulate nanofluid flow inside a minichannel. Their results showed almost the same behavior for all the methods. Fard et al. (2010) studied the nanofluid heat transfer inside a tube considering both the single and two-phase methods. For a 0.2% copper–water nanofluid, they reported that the average relative error between the experimental data and single-phase model was 16% while for the two-phase method it was 8%. On the other hand, Lotfi et al. (2010) used homogeneous, two-phase Eulerian and mixture models for nanofluid flow inside a tube. They reported that among these methods, the two-phase mixture method is more precise than the others.

According to the literature, there is a non-uniform nanoparticle volume concentration distribution in the entrance region (Wen and Ding, 2004) and the homogeneous model under-estimates the observed heat transfer enhancement in the experiments (Heris et al., 2006, Behzadmehr et al., 2007, Bianco et al., 2009, Fard et al., 2010, Lotfi et al., 2010). Thus, the two-phase modeling can be an alternative method. On the other hand, the existing studies for the two-phase method do not consider the temperature difference between the phases (Behzadmehr et al., 2007, Mirmasoumi and Behzadmehr, 2008a, Mirmasoumi and Behzadmehr, 2008b, Akbarinia and Laur, 2009) or do not present detailed results on the relative velocity and temperature between the phases and the volume concentration distribution (Bianco et al., 2009, Fard et al., 2010). The amount of the relative velocity and temperature between the phases along with the nanoparticle concentration distribution can provide an estimation of the accuracy of the assuming nanofluid as a homogeneous solution. On the other hand, all the above mentioned two-phase studies are for macro-sized circular tubes and to the best of the knowledge of the authors there is no such study for microchannels. So, this paper aims to study the nanofluid laminar forced convection in a deep rectangular (parallel plate) microchannel with isothermally heated walls, using the Eulerian–Eulerian two-phase model. To do this, mass, momentum and energy conservation equations for both phases are solved with the iterative numerical methods. The two-phase results are compared with the single-phase results from the literature and then the nanoparticle size, nanoparticle concentration and Reynolds number effects on the nanofluid heat transfer behavior are studied. Also, the relative velocity and temperature between the phases and the particle volume concentration distribution in the field are investigated. To the best knowledge of the authors, this is the first paper reporting the detailed two-phase nanofluid modeling in a microchannel that considers different velocity and temperatures for the phases.

Section snippets

Governing equations

The geometry of the present problem is shown in Fig. 1. It consists of a parallel plate microchannel with height 200 μm and the length L is 100 times larger than the height (L/H = 100). The origin of the Cartesian coordinate system is considered to be at the plate symmetry axis and only the top half of the channel is used for numerical simulation. In this problem laminar nanofluid flow that is a mixture of water and copper nanoparticles enters the channel with a uniform velocity and temperature

Numerical method

The non-dimensional form of the mass, momentum and energy conservation equations for liquid and particle phases along with the interphase correlations and boundary conditions are discretized using the finite volume method on the upper half of the channel (Patankar, 1980, Versteeg and Malalasekera, 1995). A non-uniform grid is employed in the computational domain. The grids are finer close to the wall and also in the channel entrance region using the cosine weighting function for control volume

Grid-independence study

To check for the independency of the results from the number of grid points used, a grid independency study is done by considering the amount of the calculated average Nusselt number. To do this, different numbers of grid points are used in the x- and y-directions. The results are shown in Table 1, where the flow Reynolds number is 1500. According to this study, the number of the grid points in x- and y-directions are considered 500 and 30 respectively in the present study.

Code validation

Due to the lack of experimental data for nanofluid flow in a parallel plate microchannel, the calculated average Nusselt numbers for the special case of pure water flow (φp = 0.0) at different Reynolds numbers are compared to corresponding available data in the literature in order to check the accuracy of the written computer code. For a single-phase fluid flow in an isothermally-heated parallel plate channel, the average Nusselt number is calculated as (Ebadian and Dong, 1998)Nu¯=7.55+0.024x-

Results and discussion

Since in the present study the particle phase is considered as a continuum, its viscosity μp has to be obtained. In fact, due to the lack of experimental data, the solid viscosity for a liquid–solid two-phase mixture is not available. So, for the first degree of approximation the following method is adopted in the present study: The corresponding pressure drop and the average Nusselt number of a highly dilute nanofluid with volumetric concentration of 0.00001 (which is quite close to pure

Conclusions

Pressure drop and heat transfer due to copper–water nanofluid flow inside an isothermally-heated parallel plate microchannel is studied numerically for a wide range of Reynolds numbers, nanoparticle volume concentrations and nanoparticle diameters. To do this, the nanofluid flow is modeled using the Eulerian two-fluid model. In this method, the difference between the velocity and temperature for liquid and nanoparticle phases are considered and the governing equations for both phases are solved

Acknowledgements

The first author would like to acknowledge the Ministry of Science, Research and Technology of the Islamic Republic of Iran for financial support to perform the research at the Eindhoven University of Technology. The authors also like to thank Anton Darhuber for fruitful discussions.

References (36)

  • R. Lotfi et al.

    Numerical study of forced convective heat transfer of nanofluids: comparison of different approaches

    Int. Commun. Heat Mass Transfer

    (2010)
  • S. Mirmasoumi et al.

    Numerical study of laminar mixed convection of a nanofluid in a horizontal tube using two-phase mixture model

    Appl. Thermal Eng.

    (2008)
  • S. Mirmasoumi et al.

    Effect of nanoparticles mean diameter on mixed convection heat transfer of a nanofluid in a horizontal tube

    Int. J. Heat Fluid Flow

    (2008)
  • G.L. Morini

    Viscous heating in liquid flows in micro-channel

    Int. J. Heat Mass Transfer

    (2005)
  • A.K. Santra et al.

    Study of heat transfer due to laminar flow of copper–water nanofluid through two isothermally heated parallel plates

    Int. J. Thermal Sci.

    (2009)
  • D. Wen et al.

    Experimental investigation into convective heat transfer of nanofluids at the entrance region under laminar flow conditions

    Int. J. Heat Mass Transfer

    (2004)
  • J.X. Bouillard et al.

    Porosity distributions in a fluidized bed with an immersed obstacle

    AIChE J.

    (1989)
  • S. Chakraborty et al.

    Anomalous electrical conductivity of nanoscale colloidal suspensions

    ACS NANO

    (2008)
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