Laminar forced convection of a nanofluid in a microchannel: Effect of flow inertia and external forces on heat transfer and fluid flow characteristics

Highlights

• Flow and external forces acting on nanofluid phases in forced convection are computed.

• Effect of thermophoresis in nanofluid was studied and was found negligible.

• At low Re numbers (Re < 10), Brownian force makes the flow in-homogenous.

• Application of nanofluid is only feasible for low Re numbers (Re < 10).

Abstract

The multi-phase Lattice Boltzmann Method (LBM) is used to explore some unprecedented aspects of laminar forced convection in a bottom heated rectangular microchannel. Important physical parameters, such as forces exerted on fluid parcels as well as on the dispersed nanoparticle phase are studied, in an attempt to elucidate the mechanism that results in establishment of a relative velocity between nanoparticles and the continuous fluid phase (slip velocity). The significance of the external forces, such as the gravitational, thermophoresis and Brownian forces is investigated. A recently established expression for the estimation of thermophoresis force in nanofluids is employed to study the true effect of thermophoresis, as other studies either neglect this effect, or are parametric or employ expressions that overestimate this effect. The results indicate that in laminar forced convection, the Brownian force has a significant effect on flow and heat transfer characteristics for low Re number flows (Re∼1–10), but thermophoresis may be safely neglected for all flow conditions. At low Re number flows, the nanofluid flow is heterogeneous, and heat transfer characteristics of nanofluid compared to the base fluid, such as Nu and convection heat transfer coefficient, significantly increase, while at higher Re numbers, such as Re = 100, flow behaves homogeneously and therefore the application of a nanofluid may not be justified.

Introduction

Microchannels have emerging microfluidic applications in thermal management, cooling of electronic devices, biology and lab-on-a-chip devices, etc. A microchannel heat sink has the capability to dissipate a large amount of heat from a small area with a high heat transfer rate and less fluid inventory [1]. Lack of effective heat dissipation routes in electronic boards and digital computing is a burden for developing faster computers and electronic devices. In the development of microchannels, the main issues are the heat transfer rate, microchannel allowable temperature, and the pumping power. In recent years, application of nanofluids in heated microchannels has been considered as a method for heat transfer augmentation.

The term nanofluid denotes engineered colloids comprised of conductive particles, such as metal and metal oxide nanoparticles, dispersed in a base fluid, usually to improve the fluid thermal characteristics. Owing to small sizes of nanoparticles, in most cases, a stable suspension forms without particle settlement. Most recent and reliable studies indicate an increase in fluid thermal conductivity and heat transfer coefficient, when a nanofluid is used in lieu of pure base fluid. Heat transfer augmentation in natural, mixed and forced convection of nanofluids is believed to be due an enhancement in thermal conductivity, as well as, development of a relative velocity between the main fluid flow and the suspended nanoparticles (slip/drift velocity), which enhances flow mixing and therefore heat transfer rate. Any significant external force acting on nanoparticles may be a source of slip velocity or drift. Due to the presence of slip velocity, homogenous single phase models and even dispersion models may provide only a rough approximation of heat transfer rate in the system and may not be suitable for accurate and fundamental studies on nanofluids. Thus, at least for certain flow conditions, only models that consider the nanofluid as a multiphase flow system are reliable. The proper and general modeling approach is to use a two phase model considering particles as a discrete phase, given that the nanoparticle concentration is low, and the base fluid is a continuum phase. Instead of using the conventional two phase flow modeling approach based on the Navier–Stokes equations, energy and continuity equations, and the force balance on the particles, the two-phase lattice Boltzmann method (LBM) is employed in this work. This is because from a microscopic point of view, the LBM can better reveal the inherent nature of the flow and energy transport processes inside the nanofluid and can better take into account the effect of interactions between the molecules and particles of the mixture. Although the LBM is a multiphase flow simulation tool, the external forces that may be present in the flow need to be defined and included in the model, separately. Below, these external forces are outlined, followed by a review of the pertinent works.

In a fluid flow, fluid parcels/molecules move as a result of gravity, shear and pressure forces. When a second phase, such as nanoparticles, is dispersed in the continuous phase, some fluid parcels are displaced by the newly embedded particles. This rearrangement may result in realization and creation of additional forces. Some of these forces include but are not limited to Brownian (random motion), thermophoresis (thermal diffusion as a result of a temperature gradient), lift, Magnus (particle rotation), diffusiophoresis (diffusion as a result of a concentration gradient), fluid drainage, and so on. As a result of the presence of these external forces, particles may attain velocities different from the velocity of the displaced fluid parcels (drift or slip velocity). The relative magnitude of external forces exerted on particles compared to the magnitude of the forces that would otherwise exert on the displaced fluid parcels depends on several factors, such as the nature of flow (laminar vs. turbulent), geometry and boundary conditions, particle size and shape, etc. The effect of various forces that may cause slip velocity in various flow conditions has been studied by several researchers, usually based on an approximate and parametric time scale analysis, e.g. Ref. [2]. In laminar flows, turbulent eddies, which induce an abrupt change in the flow direction, are absent. Also, forces such as diffusiophoresis, Magnus effect, fluid drainage, are insignificant in nanofluids and will not be considered here. In-line with this argument, in our previous works on natural convection in a cavity [3], [4], it was observed that, in the order of importance, the Brownian, thermophoresis, and gravitational forces are responsible for slip velocity in, particularly at moderate Ra numbers (Ra ∼ 10⁶). Development of a slip velocity is equivalent to having a heterogeneous flow. Below is a review of recent pertinent works, including numerical and experimental investigations. Most numerical works use the conventional numerical schemes, while few use the LBM. In most cases, the thermophoresis and/or Brownian forces are neglected or modeled inadequately.

Experimental data on forced convection in channels and microchannels are quite abundant, e.g., [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. While in natural convection, particle agglomeration is a potential source of error and discrepancy in the experimental data, in forced convection particle agglomeration and clustering is minimized due to the fluid flow. Nevertheless, particle precipitation on channel inner surfaces has been reported as a source of disagreement between experimental data and numerical simulations. Overall, the experimental studies predict an increase in heat transfer rate and pumping power with an increase in nanoparticle volume concentration and Re number. Chein and Huang [5] analyzed the performance of nanofluids in a silicon microchannel where it was found that Nu number increases significantly with an increase in Re and particle loading. Up to 15% reduction in thermal resistance was observed with no significant increase in pressure drop. Jang and Choi [6] and Chein and Chuang [7] performed experimental analyses in microchannel heat sinks, where it was concluded that utilizing nanofluids at low flow rates reduces the thermal resistance of the heat sink and enhances the cooling performance. However, at high flow rates the advantage of using nanofluids, as far as heat transfer is concerned, is negated. This conclusion is in-line with the finding of the present work. Singh et al. [8] adopted a conventional Eulerian–Lagrangian approach with Brownian and thermophoresis forces considered although, the expression used for thermophoresis force is only valid for gases, and therefore overestimates the effect of thermophoresis up to several orders of magnitude. Nevertheless, their numerical results match with their experimental data where an increase in Nu number is observed with an increase in Re number and particle volume concentration. Asirvatha et al. [9] performed an experimental study in a small channel considering laminar, transition and turbulent flows. A 69% increase in convective heat transfer coefficient was achieved at particle volume concentration of 0.9%. Their results show that the Dittus–Boelter correlation used with the nanofluid physical properties highly underestimates the Nu number. In another experimental work, the performance of different loadings of water-based alumina nanofluids was investigated in a commercially available cooling system of a computational processing unit [10]. An enhancement up to 18% in the convective heat transfer coefficient was reported at particle volume fraction of up to 1.5%. Ho et al. [11] conducted experiments to investigate forced convective cooling performance of a copper microchannel heat sink with a nanofluid as the coolant. The nanofluid, compared to the base fluid, had a significantly higher average heat transfer coefficient and lower thermal resistance and wall temperature, in the expense of slightly increased pumping power. In another experimental and numerical work, Kalteh followed the conventional two-phase flow Eulerian–Lagrangian approach; however Brownian and thermophoresis forces were neglected [12]. Numerical predictions were in qualitative agreement with their experimental data. In another experimental study, both an increase and decrease in the heat transfer rate were observed by Anoop et al. [13]. Heat transfer deterioration was attributed to nanoparticle precipitation (fouling) on microchannel surfaces. Refs. [14], [16] predict an increase in heat transfer rate when a nanofluid is utilized, whereas in Ref. [17] for Re numbers in the range of 500–2500, an increase in the heat transfer rate was observed for particle volume fractions of 0.24% and 1.03%, but a decrease for particle volume fraction of 4.5%. The pumping power in all cases increases significantly, in such a way that the application of a nanofluid was found infeasible. In contrast to Ref. [17] which predicts only a slight increase in heat transfer rate, in Ref. [18] a significant increase in the heat transfer rate up to 250% is reported when a nanofluid is used to cool a microchannel in the range of Re = 50 to 800. Murshed et al. [19] also reported a significant increase in heat transfer rate when TiO₂ nanofluid was used within the range of Re = 900 to 1700 in a circular channel.

Koo and Kleinstreuer [20] performed a numerical simulation where Brownian motion was considered, as an external force. In another study, heat transfer enhancement in the cooling system of microprocessors and electronic components was studied, where a considerable enhancement in convective heat transfer coefficient around 40% was reported for 6.8% particle volume fraction [21]. An analytical and numerical investigation was performed considering the mixed convection of nanofluids in a vertical channel [22], where Brownian and thermophoresis effects were considered as two variable parameters. A significant change in the flow and heat transfer characteristics was observed with a change in Brownian and thermophoresis forces in the form of non-dimensional parameters. However, it is noted that real strength of both of these forces can be estimated using available theories and changing these parameters within an unrealistic range may be misleading. Fan et al. [23] performed a similar parametric study in a horizontal channel. In another study that considers thermophoresis and Brownian forces, Nield and Kuznetsov [24] investigated forced convection in a parallel plate channel filled with a nanofluid with or without a porous medium. Surprisingly, their results show that the combined effect of thermophoresis and Brownian diffusion reduces the Nu number. Authors of Ref. [24] were consulted regarding their results; they recently published an erratum to address some issues in their original work and made some corrections and clarifications and limitations to the range of applicability of their results [25]. Akbarinia et al. [26] performed a numerical investigation on forced convection in a microchannel where slip and non-slip boundary conditions (non-zero Kn number) and Brownian motion were considered. It was concluded that with an increase in nanoparticle concentration, nanofluid viscosity increases and therefore the channel inlet velocity should be increased in order to keep the Re number constant. Thus it was concluded that the increase in the Nu number or heat transfer rate at constant Re is due to an increase in the flow velocity and not the presence of nanoparticles. In other words, it was argued that at constant inlet velocity, an increase in particle volume concentration has no significant effect on heat transfer rate; but it is noted that in this case the flow Re number substantially decreases. There are many other papers that have considered various aspects of nanofluids forced convection in a microchannel, using conventional numerical schemes, either single phase or multiphase, e.g. Refs. [27], [28], [29], [30], [31]. In recent years, however, the multiphase Lattice Boltzmann Method (LBM) has proved to be an effective tool to simulate multiphase flows, problems with complex boundaries, and problems subjected to various external forces [32]. Below is a brief review of works that use the LBM to simulate nanofluid laminar flow in a microchannel.

Using LBM, simulations were conducted by Yang and Lai [33], [34] at low Re numbers in a microchannel, where it was found that the average Nu number increases with an increase in Re number and particle volume concentration. In another work, Hung et al. [35] predicted nanoparticle optimum concentration for obtaining maximum heat transfer coefficient. Sidik et al. [36] studied nanofluid laminar flow in a bottom heated finned channel, where the presence of fins was found to enhance heat transfer rate. Their LBM simulation takes into account the Brownian motion of nanoparticles. Importance of slip velocity and temperature at the boundaries in laminar nanofluid flow was studied by Karimipour et al. [37], where significant temperature jump was observed at the entrance, which resulted in a decrease in the Nu number. Brownian motion was taken into account in their LBM simulation. A comprehensive review of the LBM simulation approach as well as a review of recent works on natural and forced convection in cavities and microchannels is given in Ref. [38]. A list of forced convection correlations and more discussion on various aspects of forced convection in channels and microchannels may be found in Refs. [39], [40], [41].

The main findings of the existing literature may be summarized as follows: (1) Most experimental studies predict an increase in heat transfer rate (or Nu number) with inclusion of nanoparticles, with some differences in details of enhancement; (2) most numerical studies have considered a single phase fluid approach, which may oversimplify the problem and obscure the interaction between the base fluid and nanoparticles, at least for a range of Re number; (3) most studies have neglected the effect of thermophoresis and Brownian forces that may be considerable in laminar flows. Very few studies have considered thermophoresis effect, while inaccurate or unsuitable expressions have been used to investigate the effect of thermophoresis. Based on the forgoing summary, the objective of this work is to explore the effect of thermophoresis and Brownian motion in fluid and heat transfer characteristics of laminar forced convection flow in a microchannel. To this end, the two phase lattice Boltzmann method (LBM) is employed along with the most accurate expression for thermophoresis in nanofluids. The analysis is novel and different from existing works, in that the fluid flow and external forces are calculated and compared to gain insight into the physics of the problem and to address and answer some fundamental questions about the significance of external forces and validity and the range of applicability of single phase modeling approach.