Technical Note
Empirical correlating equations for predicting the effective thermal conductivity and dynamic viscosity of nanofluids

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Abstract

In this paper, two empirical correlations for predicting the effective thermal conductivity and dynamic viscosity of nanofluids, based on a high number of experimental data available in the literature, are proposed and discussed. It is found that, given the nanoparticle material and the base fluid, the ratio between the thermal conductivities of the nanofluid and the pure base liquid increases as the nanoparticle volume fraction and the temperature are increased, and the nanoparticle diameter is decreased. Additionally, also the ratio between the dynamic viscosities of the nanofluid and the pure base liquid increases as the nanoparticle volume fraction is increased, and the nanoparticle diameter is decreased, being practically independent of temperature. The ease of application of the equations proposed, and their wide regions of validity (the ranges of the nanoparticle diameter, volume fraction and temperature are 10–150 nm, 0.002–0.09 and 294–324 K for the thermal conductivity data, and 25–200 nm, 0.0001–0.071 and 293–323 K for the dynamic viscosity data), make such equations useful by the engineering point of view, for both numerical simulation purposes and thermal design tasks.

Introduction

Nanofluids are a new type of heat transfer fluids obtained by suspending nano-sized particles into a base liquid. The term nanofluid was coined in 1995 by Choi [1], who showed that the uniform dispersion of low concentrations of nanoparticles into a traditional liquid such as water, oil, and ethylene glycol, could noticeably improve its thermal performance. Since then, owing to the potential of their impact upon several industrial sectors, nanofluids have attracted the interest of an increasing number of scientists, as clearly reflected by the significant research effort dedicated to this topic, whose main discoveries are summarized in the recent review-papers written by Wang and Mujumdar [2], Trisaksri and Wongwises [3], Daungthongsuk and Wongwises [4], and Murshed et al. [5].

One of the major outcomes emerging from a thorough analysis of the available literature is that in most cases the models originally developed for composites and mixtures with micro-sized and milli-sized inclusions – namely, the models developed by Maxwell [6], Hamilton and Crosser [7], and Davis [8] – fail dramatically in predicting the anomalously increased thermal conductivity of nanoparticle suspensions (at least when the nanofluid temperature is one or some tens degrees higher than “room” temperature), likely because such traditional models include only the effect of the nanoparticle concentration. This has motivated the development of a number of new theoretical models for the evaluation of the effective thermal conductivity of nanofluids, that basically account for the effects of the phenomena occurring at the solid/liquid interface and/or the micro-mixing convection caused by the Brownian motion of the nanoparticles, which is the case of the models proposed by Yu and Choi [9], Xue [10], Kumar et al. [11], Koo and Kleinstreuer [12], Jang and Choi [13], [14], Xie et al. [15], Patel et al. [16], Ren et al. [17], Prasher et al. [18], [19], Leong et al. [20], Xuan et al. [21], Prakash and Giannelis [22], and Murshed et al. [23]. However, these models show large discrepancies among each other, which clearly represents a restriction to their applicability. Moreover, most of these models include empirical constants of proportionality whose values were often determined on the basis of a limited number of experimental data, or were not clearly defined.

Indeed, the thermal conductivity enhancement is not the only noteworthy effect originating from the suspension of nanoparticles into a base fluid. In fact, a contemporary growth in dynamic viscosity occurs, which could be a serious limitation, either in terms of an exaggerated pressure drop increase in forced convection applications, or in terms of a drastic fluid motion decrease in natural convection situations. Accordingly, the possibility of calculating the effective dynamic viscosity of a nanoparticle suspension seems crucial to establish if its use is actually advantageous with respect to the pure base liquid. In spite of this, leaving aside the theories developed time ago for traditional colloid dispersions by Einstein [24], [25], Brinkman [26], Lundgren [27], and Batchelor [28], whose predictions typically under-estimate the dynamic viscosity of nanoparticle suspensions, only few models have recently been proposed for describing the rheological behaviour of nanofluids, such as those developed by Koo [29], and Masoumi et al. [30]. Actually, as these models contain empirical correction factors based on an extremely small number of experimental data, their region of validity is someway limited.

Framed in this general background, the aim of the present paper is to introduce and discuss two easy-to-apply empirical correlating equations for predicting the effective thermal conductivity and dynamic viscosity of nanofluids, that, matching sufficiently well a high number of experimental data available in the literature, may be usefully employed for numerical simulation purposes and thermal engineering design tasks.

Section snippets

Correlation for the effective thermal conductivity

The equation proposed for the nanofluid effective thermal conductivity, keff, normalized by the thermal conductivity of the base fluid, kf, is derived from a wide variety of experimental data relative to nanofluids consisting of alumina, copper oxide, titania and copper nanoparticles with a diameter in the range between 10 nm and 150 nm, suspended in water or ethylene glycol (EG). These data are extracted from the following sources: Masuda et al. [31] for TiO2(27 nm) + H2O; Pak and Cho [32] for TiO2

Correlation for the effective dynamic viscosity

As for the thermal conductivity, also the equation proposed for the nanofluid effective dynamic viscosity, μeff, normalized by the dynamic viscosity of the base liquid, μf, is derived from a wide selection of experimental data available in the literature. These data, relative to nanofluids consisting of alumina, titania, silica and copper nanoparticles with a diameter ranging between 25 nm and 200 nm, suspended in water, ethylene glycol (EG), propylene glycol (PG) or ethanol (Eth), are taken out

Conclusions

In sum, in this paper two empirical equations for predicting the effective thermal conductivity and dynamic viscosity of nanofluids, based on a high number of experimental data available in the literature, have been proposed and discussed. According to the best-fit of the literature data, once the nanoparticle material and the base fluid are assigned, the ratio between the thermal conductivities of the nanofluid and the pure base liquid increases as the nanoparticle volume fraction and the

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