Effective thermal conductivity of nanofluids – A new model taking into consideration Brownian motion
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 keff by
The terms kbf and kp 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
The sphericity of a particle is defined as the ratio of the surface area of a sphere, Ap,sph, having the same volume as the particle, Vp, to the surface area of the particle, Ap, according to
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 equationfor 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 kp ≫ kbf and small φ, Eq. (1) reduces to
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,
For spherical isotropic inclusions, the constants a and b are given as a = 2.25 and b = 2.27 for kp/kbf = 10 as well as a = 3.00 and b = 4.51 for kp/kbf → ∞. 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.
Section snippets
Description of the model
To develop an analytical model for the thermal conductivity of liquids with suspended nanoparticles, the thermal resistances of the base fluid and the nanoparticles as well as of convection induced in the fluid due to Brownian motion of the nanoparticles are determined. Other possible heat transfer effects in form of thermal radiation or thermal diffusion of the nanoparticles due to a temperature gradient were found to be negligible.
In the following, the model is derived on basis of a nanofluid
Results and discussion
To analyze the quality of our model for the description of the effective thermal conductivity of nanofluid systems by Eq. (24), we selected the four simple systems Al2O3/H2O, TiO2/H2O, Al2O3/EG, and TiO2/EG. These systems are common, inexpensive, and have been widely tested experimentally [2]. The origin of the thermophysical properties of the nanoparticles and base fluids at atmospheric pressure required for all subsequent calculations is summarized in the following.
The thermal conductivity kp
Conclusions
A new analytical model to determine the effective thermal conductivity in fluids containing well-dispersed spherical and non-spherical nanoparticles was presented. The model takes thermal resistances in connection with the base fluid, the nanoparticles, and the micro-convection between the nanoparticles and the fluid due to Brownian motion of the particles into account. Furthermore, the model is based on well-defined properties and does not include any empirical constants. It has revealed the
Acknowledgements
This work was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) by funding the Erlangen Graduate School in Advanced Optical Technologies (SAOT) within the German Excellence Initiative. K.N. Shukla acknowledges support from the Alexander von Humboldt Foundation in sponsoring a renewed research stay at the Friedrich-Alexander-University Erlangen-Nuremberg (FAU).
References (24)
- et al.
Nanofluids for thermal transport
Mater. Today
(2005) - et al.
Effective thermal conductivity and thermal diffusivity of nanofluids containing spherical and cylindrical nanoparticles
Exp. Therm. Fluid Sci.
(2007) - et al.
Thermal conductivity measurement and sedimentation detection of aluminum oxide nanofluids by using the 3ω method
Int. J. Heat Fluid Flow
(2008) A Treatise on Electricity and Magnetism
(1892)- et al.
The apparent thermal conductivity of liquids containing solid particles of nanometer dimensions: a critique
Int. J. Thermophys.
(2015) - et al.
Experimental study on the effective thermal conductivity and thermal diffusivity of nanofluids
Int. J. Thermophys.
(2006) - et al.
Experimental investigation of heat conduction mechanisms in nanofluids. Clue on clustering
Nano Lett.
(2009) - et al.
The role of interfacial layers in the enhanced thermal conductivity of nanofluids: A renovated Hamilton–Crosser model
J. Nanopart. Res.
(2004) - et al.
Role of Brownian motion in the enhanced thermal conductivity of nanofluids
Appl. Phys. Lett.
(2004) - et al.
Effect of aggregation kinetics on the thermal conductivity of nanoscale colloidal solutions (nanofluid)
Nano Lett.
(2004)
A new thermal conductivity model for nanofluids
J. Nanopart. Res.
Role of Brownian motion hydrodynamics on nanofluid thermal conductivity
Appl. Phys. Lett.
Cited by (55)
Guarded Parallel-Plate Instrument for the Determination of the Thermal Conductivity of Gases, Liquids, Solids, and Heterogeneous Systems
2023, International Journal of Heat and Mass TransferEffective thermal conductivity of microemulsions consisting of water micelles in n‐decane
2023, International Journal of Heat and Mass TransferCitation Excerpt :For this, we limit the range of modeling approaches within the present work to two models, which both have a physical basis and could be applied successfully for the prediction of λeff of nanofluids [20,21]. Next to the commonly employed effective medium theory reflected by the Hamilton-Crosser (HC) model [19], a new analytical resistance model developed [23] and further improved [21] at AOT-TP is considered. Details about the two models can be found in the given references.
Significant enhancement of thermal conductivity in toluene-based nanofluids employing highly dispersed and concentrated ZrO<inf>2</inf> nanodots
2022, International Journal of Thermal SciencesHeat transfer enhancement with nanofluids in automotive
2022, Advances in Nanofluid Heat TransferNumerical modeling of nanofluids’ flow and heat transfer
2022, Advances in Nanofluid Heat Transfer