Effective thermal conductivity of nanofluids – A new model taking into consideration Brownian motion

Highlights

• New analytical model for the effective thermal conductivity of nanofluids developed.

• Significant influence of convective resistance due to Brownian motion identified.

• Prediction considers temperature and particle volume fraction, diameter, and shape.

• Good agreement with Hamilton-Crosser model for spherical and non-spherical particles.

• Increasing volume-specific surface area is of advantage for non-spherical particles.

Abstract

In this study, a new analytical model for the effective thermal conductivity of liquids containing dispersed spherical and non-spherical nanometer particles was developed. In addition to heat conduction in the base fluid and the nanoparticles, we also consider convective heat transfer caused by the Brownian motion of the particles. For nanoparticle suspensions, the latter mechanism has significant influence on the effective thermal conductivity, which is reduced compared to a system in which only conduction is considered. The simple model developed allows for the prediction of the effective thermal conductivity of nanofluids as a function of volume fraction, diameter, and shape of the nanoparticles as well as temperature. Due to the inconsistency of experimental data in the literature, the model has been compared with the established Hamilton–Crosser model and other empirical models for the systems aluminum oxide (Al₂O₃) and titanium dioxide (TiO₂) suspended in water and ethylene glycol. The theoretical estimates show no anomalous enhancement of the effective thermal conductivity and agree very well with the Hamilton–Crosser model within relative deviations of less than 8% for volume fractions of spherical particles up to 0.25. In accordance with the Hamilton–Crosser model for non-spherical particles, our model reveals that a more distinct increase in the enhancement of the effective thermal conductivity can be achieved using non-spherical nanoparticles having a larger volume-specific surface area.

Introduction

Over the past decade, the dispersions of nanometer-sized particles with typical diameters ranging from 1 to 100 nm in a liquid medium, usually called nanofluids, have been reported to possess substantially higher thermal conductivities than anticipated from Maxwell’s classical theory [1]. As shown in the review by Tertsinidou et al. [2], a large number of experimental results have reported an anomalous increase in the thermal conductivity of nanoparticle suspensions. This would make them very attractive as potential heat transfer fluids for many applications. However, results from other experiments have not shown any anomalous increase in thermal conductivity [2], [3], [4], [5]. This has triggered controversy regarding the actual value of the thermal conductivity of nanofluids and the reliability of the experimental methods. To solve this problem, a fundamental understanding of the heat transfer mechanisms present in suspensions of nanoparticles is necessary, which may lead to a physical model for the description of their effective thermal conductivity.

Debate continues regarding the potential causes of abnormal behavior ranging, e.g., from the effect of nanoparticle clustering [6] over the layering of the liquid at the liquid-particle interface [7] to the role of the Brownian motion of nanoparticles [8]. Regarding the impact of Brownian motion on the effective thermal conductivity, one opinion in the literature [8], [9], [10] is that this effect is the main reason for the high thermal conductivity of nanofluids. Yet, the molecular dynamics simulations performed by Evans et al. [11] reveal that the enhancement effect due to Brownian motion of the nanoparticles is rather insignificant. Therefore, disagreement also exists among researchers on the role of Brownian motion of nanoparticles.

Maxwell was one of the first to use the effective medium theory to study the properties of a solid bulk material consisting of one material distributed as spherical inclusions within a continuous material [1]. His static model for the effective electrical conductivity of solid-based systems was used by Hamilton and Crosser [12] to determine the effective thermal conductivity of two-phase, two-component solids because of the similar mathematical formulations of the two transport phenomena. Their model is also applicable for systems of liquids containing solid particles with different shapes and predicts their effective thermal conductivity k_eff by

$$\frac{k_{\text{eff}}}{k_{\text{bf}}}=\frac{k_{\text{p}}+(n-1)k_{\text{bf}}-(n-1)\phi (k_{\text{bf}}-k_{\text{p}})}{k_{\text{p}}+(n-1)k_{\text{bf}}+\phi (k_{\text{bf}}-k_{\text{p}})}\text{.}$$

The terms k_bf and k_p denote the thermal conductivities of the base fluid (bf), that is the continuous phase, and the solid particle (p), that is the dispersed phase. φ is the volume fraction of the dispersed particles, while n is an empirical shape factor. The latter is connected with the sphericity ψ via

$$n=\frac{3}{\psi }\text{.}$$

The sphericity of a particle is defined as the ratio of the surface area of a sphere, A_p,sph, having the same volume as the particle, V_p, to the surface area of the particle, A_p, according to

$$\psi =\frac{{\pi }^{1/3}{(6V_{\text{p}})}^{2/3}}{A_{\text{p}}}\text{.}$$

Particles with strong deviations from spherical shape show smaller ψ and thus larger n values. For spherical particles with ψ = 1 and n = 3, Eq. (1) can be simplified to the original Maxwell-like equation

$$\frac{k_{\text{eff}}}{k_{\text{bf}}}=\frac{k_{\text{p}}+2k_{\text{bf}}-2\phi (k_{\text{bf}}-k_{\text{p}})}{k_{\text{p}}+2k_{\text{bf}}+\phi (k_{\text{bf}}-k_{\text{p}})}$$

for predicting the enhancement of the effective thermal conductivity of nanofluids containing spherical solid particles.

The effective model of Hamilton and Crosser [12] can be applied to low particle volume fractions and shows no dependence on the particle diameter and a very weak dependence on the temperature. For k_p ≫ k_bf and small φ, Eq. (1) reduces to

$$\frac{k_{\text{eff}}}{k_{\text{bf}}}=1+n\phi \text{.}$$

Eq. (5) presents a simple linear relation for the effective thermal conductivity of diluted suspensions. Based on the continuum model of Hamilton and Crosser [12], there have been some modifications considering additional effects. For example, Lu and Lin [13] considered near- and far-field pair interactions applicable to spherical and non-spherical inclusions and modified Eq. (5) to a second-order polynomial,

$$\frac{k_{\text{eff}}}{k_{\text{bf}}}=1+a\phi +b{\phi }^2\text{.}$$

For spherical isotropic inclusions, the constants a and b are given as a = 2.25 and b = 2.27 for k_p/k_bf = 10 as well as a = 3.00 and b = 4.51 for k_p/k_bf → ∞. For non-spherical inclusions, anisotropic effects have to be considered which results in the formulation of an effective thermal conductivity tensor for an anisotropic medium. Besides the models described above, there are numerous other predictive methods – most of them empirical or complex – for the effective thermal conductivity that cannot all be mentioned here. A review about theoretical studies on the effective thermal conductivity of nanofluids is given by Kleinstreuer and Feng [14].

Recently, Tertsinidou et al. [2] evaluated extensive data on the effective thermal conductivity of nanoparticle suspensions containing aluminum oxide (Al₂O₃)_, copper (Cu), copper oxide (CuO), and titanium dioxide (TiO₂) suspended in water (H₂O) and ethylene glycol (EG). When results for the same thermodynamic system are obtained using proven experimental techniques, they concluded that the effective thermal conductivity of nanofluids exhibits no inconsistency with the continuum model of Hamilton and Crosser [12]. The broad span of values for the enhancement of effective thermal conductivity in the literature, with relative deviations of several tens of percents for a given system, is rather attributed to poor characterization of the thermodynamic system and/or the application of experimental techniques of unproven validity [2]. A systematic benchmark study [15] on the thermal conductivity of nanofluids, performed over 30 laboratories worldwide and using a variety of experimental techniques, has also shown good agreement between the experimental data and the corresponding data predicted by the Hamilton–Crosser model [12].

Nevertheless, the continuum model does not explicitly account for effects that are induced by nanoparticles with respect to the effective thermal conductivity of nanofluids. Nanoparticles may cause additional energy transport due to Brownian motion because of their small size, large specific surface area, and morphology. The motion of nanoparticles and base fluid molecules is affected by the combined effect of hydrodynamic and Brownian forces that produce micro-convection in the nanofluids. The aim of the present work was to develop a new model for the effective thermal conductivity of macroscopically static nanofluids taking into account the heat transfer mechanisms caused by convection as well as thermal conduction of the particles and the base fluid. A comparison of the model with the established model of Hamilton and Crosser [12] and the predictions of Lu and Lin [13] is drawn for selected nanofluid systems as a function of the parameters volume fraction, particle diameter, particle shape, and temperature.