Nanofluid convective heat transfer in a parallel-disk system

Abstract

Inherently low thermal conductivities of basic fluids form a primary limitation in high-performance cooling which is an essential requirement for numerous thermal systems and micro-devices. Nanofluids, i.e., dilute suspensions of, say, metal-oxide nanoparticles in a liquid, are a new type of coolants with better heat transfer performances than their pure base fluids alone. Using a new, experimentally validated model for the thermal conductivity of nanofluids, numerical simulations have been executed for alumina-water nanofluid flow with heat transfer between parallel disks. The results indicate that, indeed, nanofluids are promising new coolants when compared to pure water. Specifically, smoother mixture flow fields and temperature distributions can be achieved. More importantly, given a realistic thermal load, the Nusselt number increases with higher nanoparticle volume fraction, smaller nanoparticle diameter, reduced disk-spacing, and, of course, larger inlet Reynolds number, expressed in a novel form as Nu = Nu(Re and Br). Fully-developed flow can be assumed after a critical radial distance, expressed in a correlation R_crit = fct(Re), has been reached and hence analytic solutions provide good approximations. Nanofluids reduce the system’s total entropy generation rate while hardly increasing the required pumping power for any given Re_in. Specifically, minimization of total entropy generation allows for operational and geometric system-optimization in terms of S_gen = fct (Re and δ).

Introduction

For pure fluids, inherently low thermal conductivities form a primary limitation in high-performance cooling which is the essential requirement for numerous thermal systems and micro-devices. Nanofluids, i.e., dilute suspensions of, say, metal-oxide nanoparticles in a liquid, are a new type of coolants with better heat transfer performances than their pure base fluids alone. Specifically, numerous experiments with nanofluids have demonstrated thermal conductivity enhancement, indicating promising applications to micro-scale cooling. Compared to pure liquids, experimental evidence showed that the thermal conductivity of nanofluids, k_nf, significantly increases comparing at small nanoparticle volume fractions, with decreasing nanoparticle diameter and elevated mixture temperature. For example, Lee and Choi [1] investigated CuO-water/ethylene glycol nanofluids with particle diameters 18.6 and 23.6 nm as well as Al₂O₃-water/ethylene glycol nanofluids with particle diameter 24.4 and 38.4 nm and discovered a 20% thermal conductivity increase at a volume fraction of just 4%. Wang et al. [2] determined experimentally a 12% increase in k_nf with 28 nm-diameter Al₂O₃-water and 23 nm-diameter CuO-water nanofluids at 3% volume fraction. Eastman et al. [3] reported a 40% thermal conductivity increase for 10 nm-diameter Cu-water nanofluids with a volume fraction of only 0.3%. Li and Peterson [4] provided the thermal conductivity expression in terms of temperature (T) and volume fraction (φ) by using curve-fitting for CuO-water and Al₂O₃-water nanofluids. Additionally, Xie et al. [5] investigated SiC-water nanofluids and Hong et al. [6] focused on Fe-water nanofluids. Recently, Chopkar et al. [7] investigated Ag₂Al-water nanofluids and Al₂Cu-water nanofluids and reported a 130% increase in thermal conductivity with a volume fraction less than 1%. Considering particle diameters of 47 and 36 nm, Mintsa et al. [8] provided new thermal conductivity expressions for Al₂O₃-water and CuO-water nanofluids. Murshed et al. [9] reported a 27% increase in 4% TiO₂-water nanofluids with a particle diameter of 15 nm and a 20% increase for Al₂O₃-water nanofluids. However, Duangthongsuk and Wongwaises [10] observed a more moderate increase for TiO₂-water nanofluids. Das et al. [11] commented systematically on the relationship between k_nf and temperature demonstrating that the thermal conductivity of nanofluids will significantly increase with higher temperatures. Patel et al. [12] confirmed the temperature effect obtained by Das et al. [11] as well as the findings of Lee and Choi [1]. They also showed the inverse dependence of particle size on thermal conductivity enhancement, considering three sizes of alumina nanoparticles suspended in water.

Still, controversies for k_nf (φ, T) persist [13], [14], [15], [16], [17], [18]. More recently, scientists used optical measurement methods [14], [15], [16], [17], [18] to obtain the effective thermal conductivities of nanofluids and found no anomalous increase, which brings the transient hot-wire (THW) method into question. For example, Ju et al. [19] analyzed and commented on the error possibilities of the THW method and investigated 20, 30 and 45 nm Al₂O₃ nanoparticle-water suspensions up to a volume fraction of 10%. They did not discover any strong relationship between effective thermal conductivity enhancement and temperature increase. There are other parameters which may influence k_nf, e.g., pH value [20] and particle shape/clustering [9], [21].

Basic theoretical models for the thermal conductivity of dilute spherical particle suspensions relied on the static model of Maxwell [22]. Hamilton and Crosser [23] extended Maxwell’s result to non-spherical particles. For other classical models please refer to Refs. [24], [25], [26]. The classical models, based on continuum-mechanics formulations, typically involve only the particle size/shape and volume fraction and assume diffusive heat transfer in both fluid and solid phases. While they provide good predictions for micrometer or larger-size particle suspensions, they usually underestimate any enhanced thermal conductivity increase with volume fraction, nanoparticle diameter, and mixture temperature.

In contrast to the classical models which treat particles stationary to the base fluids, dynamical models take the effect of nanoparticles’ random motion into account. The basic mechanisms of anomalous thermal conductivity enhancement of nanofluids are as follows:

• Brownian motion of nanoparticles.

• Liquid molecule-layering on the nanoparticle surface.

• Enhanced heat conduction in the nanoparticles.

• Effect of nanoparticle clustering.

Other mechanisms may be categorized into conduction, nano-scale convection, near-field radiation [27], and thermal wave propagation [28]. The main underlying mechanism of the present F–K model is based on the micro-mixing effect due to Brownian motion. Although Wang et al. [2] and Keblinski et al. [29] concluded that Brownian motion is not a significant mechanism accounting for the anomalous enhancement of the thermal conductivity of nanofluids, they failed to consider the surrounding fluid motion induced by the random movements of the nanoparticles. Based on in-house research [30], Brownian motion effect is a significant mechanism for the enhancement of the thermal conductivity of nanofluid.

Several theoretical k_nf-models have been published based on the Brownian motion effect. For example, Jang and Choi [31] focusing on the heat transfer between nanoparticles and carrier fluid, proposed four modes of energy transport and introduced the idea that a Brownian nanoparticle produces a convection-like effect at the nano-scale. Kumar et al. [32] suggested two models: a stationary particle model and a moving particle model, building a relationship between the effective thermal conductivity and the average particle velocity which is determined by the temperature T. Koo and Kleinstreuer [33] considered the effective thermal conductivity to be composed of two parts: k_static is the static thermal conductivity due to the higher thermal conductivity of nanoparticles, following Maxwell’s theory, while k_Brownian is the enhanced thermal conductivity part generated by the additional convective heat transfer of nanoparticles’ Brownian motion and related ambient fluid induced motion. Li [34] extended the model of Koo and Kleinstreuer [33]. Bao [35] also considered the effective thermal conductivity, consisting of a static part and Brownian motion part. Different from the KKL model [34], he only focused on one time interval of Brownian motion, which implies that the velocity of the particle is constant, and treated the ambient fluid around nanoparticle as steady flow. Most recently, a new thermal conductivity theory for nanofluids has been developed, labeled the F–K model; being based on first principles, it does not require any matching functions and predicts benchmark experimental data sets very well [30], [36].

One effective cooling device applicable to needs in high-tech industries is the impinging-jet parallel-disk system using nanofluids. However, only a few publications have focused on nanofluid convective heat transfer between two parallel disks [5], [6]. Original work on radial flow between rotating (or stationary) disks goes back to McGinn [39], with a most recent contribution by Achintya [40]. Roy and his research group [5], [6] found that for Al₂O₃-water and Al₂O₃-ethylene glycol nanofluids the Nusselt number rapidly increases with elevated alumina volume fractions, say, above 2%. For their numerical calculations they relied on the Maxwell correlation for k_nf (see Eq. (7)) which may not be suitable for this case [30], [36], [37].

Extending a basic mixture-flow analysis which demonstrated the usefulness of the new k_nf-model [41], for this paper thermal nanofluid flow between parallel disks has been simulated to present temperature fields and discuss Nusselt numbers as compared to pure water for a realistic heat load. In addition, operational system data as well as entropy generation were analyzed for system-optimization considerations. Based on the extensive literature review, the present computer simulation of radial thermal nanofluid flow between parallel disks and entropy-minimization analysis best device design are novel contributions.