The thermal conductivity of aqueous nanofluids containing ceria nanoparticles

JOURNAL OF APPLIED PHYSICS 107, 066101 共2010兲

The thermal conductivity of aqueous nanofluids containing ceria nanoparticles Michael P. Beck, Yanhui Yuan, Pramod Warrier, and Amyn S. Tejaa兲 School of Chemical and Biomolecular Engineering, Georgia Institute of Technology, Atlanta, Georgia 303320100, USA

共Received 22 June 2009; accepted 30 January 2010; published online 16 March 2010兲 We present data for the thermal conductivity of nanofluids consisting of two sizes of ceria nanoparticles dispersed in water at 298 K. We also demonstrate that the effective thermal conductivity of these heterogeneous nanofluids can be described by a simple “mixing” rule that incorporates the size dependence of the thermal conductivity of the nanoparticles themselves. Our data follow the same trends with particle size as shown for alumina nanofluids in our previous work, and provide additional validation for that data. © 2010 American Institute of Physics. 关doi:10.1063/1.3330506兴 I. INTRODUCTION

Nanofluids have attracted attention because of the large enhancements in the thermal conductivity that have been reported when small amounts of solid nanoparticles are dispersed in common heat transfer fluids.1–3 The reported thermal conductivity enhancements are greater than those predicted by classical theories such as those of Maxwell,4 and others.5 There is, however, some disagreement between different data sets with respect to the magnitude of the enhancement and its dependence on particle size.5 In the present work, we report measurements of the thermal conductivity of ceria nanofluids in water. We have used these data to show that mixing rules for the effective thermal conductivity of composites work well for ceria nanofluids, provided that the size dependence of the thermal conductivity of ceria nanoparticles is taken into account in these mixing rules.

ing heterogeneous system to ultrasonic treatment to obtain uniform dispersions. The pH of each sample was adjusted with HCl to a value of 4.0⫾ 0.2. This pH is much less than the isoelectric point of ceria 共6.7–8.7兲 and minimizes particle aggregation during the experiments. The thermal conductivity of each nanofluid was measured using a liquid metal transient hot-wire device described in detail in our earlier work.7 We have used this device to measure the thermal conductivity of electrically conducting liquids7 and nanofluids.6,8 Our transient hot-wire consists of a mercury-filled glass capillary that is suspended vertically in the dispersion. The glass capillary insulates the mercury “wire” from the electrically conducting dispersion and prevents current leakage when a voltage is applied to the

II. EXPERIMENTS

Two sizes of ceria nanoparticles were purchased from Nanostructured and Amorphous Materials, Inc. 共Houston, TX兲 with average particle diameters of 15–30 nm and 70– 100 nm as reported by the manufacturer. We characterized these samples by employing BET 共Brunauer–Emmett–Teller兲 surface area measurements using nitrogen adsorption to show that the average diameter of the smaller particles was 12 nm, and that of the larger particles was 74 nm. We assumed that the particles were spherical in order to convert BET surface area measurements to particle diameters. 关Although the particles were aggregated 共Fig. 1兲, we have shown elsewhere6 that average particle sizes determined via BET measurements work well in nanofluid thermal conductivity correlations. Polydispersity in particle sizes generally leads to larger error bars being associated with the measurements and hence poorer fits of the data.兴 Nanofluids were prepared by dispersing preweighed quantities of the nanoparticle samples into deionized water, and subjecting the resulta兲

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0021-8979/2010/107共6兲/066101/3/$30.00

FIG. 1. SEM micrographs of 共a兲 15–30 and 共b兲 70–100 nm ceria particles. Scale bars ⫽ 200 nm. 107, 066101-1

© 2010 American Institute of Physics

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TABLE I. Experimental and calculated thermal conductivities of ceria nanofluids at 298 K.

d / nm

␸ 共%兲

Experiment keff / W m−1 K−1

Equations 共3兲 and 共4兲a keff / W m−1 K−1

Equation 共1兲 keff / W m−1 K−1

12 12 12 74 74 74

2.00 3.00 4.00 2.00 3.00 4.00

0.630⫾ 0.006 0.649⫾ 0.010 0.671⫾ 0.013 0.659⫾ 0.010 0.693⫾ 0.016 0.730⫾ 0.020

0.637 0.653 0.669 0.664 0.695 0.724

0.617 0.623 0.628 0.634 0.650 0.663

a

with n = 0.31 and A = 0.025 nm−1.

“wire.” The resistance of the mercury wire as it is heated is measured using a Wheatstone bridge circuit. The change in resistance with time is used to compute the temperature of the wire, and hence the thermal conductivity of the dispersion by solving Fourier’s equation for a linear heat source of infinite length 共the “wire”兲 in an infinite medium 共the dispersion兲. A linear relationship between the temperature change of the wire and the natural log of time is used to confirm that the primary mode of heat transfer during the measurement is conduction. Corrections to the equation are included for the insulating capillary around the wire, the finite dimensions of the wire, the finite volume of the fluid, and heat loss due to radiation. The transient hot-wire apparatus was calibrated with reference fluids to obtain an effective wire length that allows for nonuniform capillary thickness and end effects. In the present study, water9 and dimethyl phthalate10 were used as the reference fluids. Additional details of the apparatus and method are available elsewhere.7 The experiment was performed five times per sample, and each data point reported in this work thus represents an average of five measurements with an estimated error in the effective thermal conductivity of ⫾2%. III. RESULTS AND DISCUSSION

Table I lists our measured values of the effective thermal conductivity of aqueous ceria nanofluids at 298 K together with the associated standard deviations. The results are plotted in Fig. 2 and show that the effective thermal conductivity

FIG. 3. 共Color online兲 Thermal conductivity vs particle size for aqueous nanofluids containing ceria at room temperature. The lines represent calculations using Eqs. 共3兲 and 共4兲 with n = 0.31 and A = 0.025 nm−1.

increases linearly with volume fraction of particles at each particle size, in agreement with the literature.5 Also, the effective thermal conductivity decreases with the particle size 共Fig. 3兲, in agreement with our earlier findings for alumina particles.6 The data were correlated as described below. The most widely used equation for correlating the effective thermal conductivity of nanofluids was derived by Maxwell4 who studied a dilute system of spherical particles dispersed in a liquid and obtained keff = kl

冋

共1兲

where keff, k p, and kl are the thermal conductivity of the nanofluid, the 共solid兲 particles, and the liquid, respectively, and ␸ is the volume fraction of the particles. The spherical particles were assumed to be far from each other so that temperature and heat fluxes in the neighborhood of one particle were not affected by other particles. As discussed by Turian et al.,11 the Maxwell model gives good predictions of thermal conductivity enhancements in heterogeneous systems when the ratio k P / kl ⬃ 1, and increasingly poor predictions as this ratio increases. Large values of keff, in apparent conflict with the Maxwell equation have been reported by Eastman et al.12 and Choi et al.13 in the case of copper nanofluids in water, and nanofluids obtained by dispersing carbon nanotubes in oil. Such conflicts have led to many modifications of the Maxwell equation to account for factors such as the nonsphericity of the particles,14 phenomena such as Brownian motion of particles,15 the existence of an ordered layer of fluid at the particle surface,16 and particle aggregation.17 These modifications have generally failed to account for the actual behavior of the thermal conductivity of nanofluids.5 Several models for the effective thermal conductivity of composites have also been proposed11,18 in the form 共keff,m兲n = 共k P兲n␸ + 共kl兲n共1 − ␸兲

FIG. 2. 共Color online兲 Thermal conductivity vs volume fraction for aqueous nanofluids containing ceria at room temperature. The lines represent calculations using Eqs. 共3兲 and 共4兲 with n = 0.31 and A = 0.025 nm−1.

册

k P + 2kl + 2共k P − kl兲␸ , k P + 2kl − 共k P − kl兲␸

− 1 ⬍ n ⬍ 1.

共2兲

For n = 1, this equation reduces to the arithmetic mean of the thermal conductivities of the two materials, which provides a good representation for conduction in materials arranged in parallel. Similarly, when n = −1, Eq. 共2兲 reduces to the har-

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monic mean of the two thermal conductivities, which provides a good representation for conduction in materials arranged in series. Finally, for n approaching zero, Eq. 共2兲 reduces to the geometric mean of the thermal conductivity of the two materials as follows:

冉 冊冉冊 keff kp = kl kl

␸

.

共3兲

Turian et al.11 have shown that Eq. 共3兲 works well for heterogeneous suspensions in which k P / kl ⬎ 4, whereas the Maxwell model provides a lower bound for the thermal conductivity for dilute suspensions or when k P / kl ⬃ 1. Their conclusions agree with those of Prasher et al.17 who also suggest that particle aggregation may lead to thermal conductivity enhancements that are greater than those predicted by the Maxwell equation. It would therefore appear that the value of n in Eq. 共2兲 provides a measure of particle density and/or aggregation in heterogeneous systems. We have shown6 that Eq. 共3兲 works well for thermal conductivity enhancement in alumina nanofluids if we account for the temperature dependence of kl as well as the particle size and temperature dependence of k P. In our previous study, the particle size dependence of the thermal conductivity for alumina nanofluids was expressed as k p = k p,bulk共1 − e−Ad兲,

共4兲

where d is the diameter of the particles in nanometer and k p,bulk is the bulk thermal conductivity of the solid. This expression was based on the assumption that the thermal conductivity of very small semiconducting particles decreases with particle size, as has been shown for silicon nanoparticles below 10 nm using molecular dynamics simulations.19 In our previous study,6 combining Eqs. 共3兲 and 共4兲 allowed us to correlate an extensive data set on the thermal conductivity of alumina nanofluids using one adjustable parameter A. In the present work, we have calculated the effective thermal conductivity of ceria nanofluids using Eqs. 共2兲 and 共4兲. Although only two particle sizes were studied in this work, we have correlated the data assuming that the same particle size dependence 关Eq. 共4兲兴 is valid for both alumina and ceria particles. Also, the same value of A 共 =0.025 nm−1兲 reported in our earlier work on five alumina nanofluids was used to correlate our ceria nanofluid data. Calculated values are shown in Table I together with Maxwell equation values. As can be seen from the table, the particle size and volume fraction dependence of the effective

thermal conductivity of ceria nanofluids can be correlated very well with n = 0.31, which is close to the value of n = 0.333 suggested by Landau and Lifshitz20 for the properties of heterogeneous mixtures. As noted above, the parameter n in Eq. 共2兲 provides a measure of particle density and/or aggregation in heterogeneous systems 共n = 1 implying a fullydeveloped network of ceria particles in the nanofluid兲. Scanning electron microscope pictures 共Fig. 1兲 show that our particles were certainly aggregated. However, the extent of particle aggregation or network formation cannot be calculated from the value of n obtained. The development of a mathematical relationship between the value of n and the extent of aggregation will require further study. ACKNOWLEDGMENTS

P.W. thanks the Office of Naval Research for partial financial support under Award No. N000140811057. 1

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