The effect of particle size on the effective thermal conductivity of Al2O3water nanofluids Calvin H. Li and G. P. Peterson Citation: J. Appl. Phys. 101, 044312 (2007); doi: 10.1063/1.2436472 View online: http://dx.doi.org/10.1063/1.2436472 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v101/i4 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 20 Apr 2013 to 141.161.91.14. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions JOURNAL OF APPLIED PHYSICS 101, 044312 共2007兲 The effect of particle size on the effective thermal conductivity of Al2O3-water nanofluids Calvin H. Li Department of Mechanical, Aeronautical, and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180 G. P. Petersona兲 Department of Mechanical Engineering, University of Colorado at Boulder, Boulder, Colorado 80309-0427 共Received 21 September 2006; accepted 13 December 2006; published online 28 February 2007兲 A steady-state method was used to evaluate the effective thermal conductivity of Al2O3/distilled water nanofluids with nanoparticle diameters of 36 and 47 nm. Tests were conducted over a temperature range of 27– 37 ° C for volume fractions ranging from 0.5% to 6.0%. The thermal conductivity enhancement of the two nanofluids demonstrated a nonlinear relationship with respect to temperature, volume fraction, and nanoparticle size, with increases in the volume fraction, temperature, and particle size all resulting in an increase in the measured enhancement. The most significant finding was the effect that variations in particle size had on the effective thermal conductivity of the Al2O3/distilled water nanofluids. The largest enhancement difference observed occurred at a temperature of approximately 32 ° C and at a volume fraction of between 2% and 4%. The experimental results exhibited a peak in the enhancement factor in this range of volume fractions for the temperature range evaluated, which implies that an optimal size exists for different nanoparticle and base fluid combinations. This phenomenon can be neither predicted nor explained using the theoretical models currently available in the literature. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2436472兴 INTRODUCTION A wide variety of nanofluids have been evaluated over the last decade to determine the variations in thermophysical properties, with particular emphasis on the measurement of the effective thermal conductivity and viscosity. While unusually high thermal conductivities have been measured by a number of investigators,1–21 some of these investigations focused primarily on the effect of temperature on the effective thermal conductivity of these nanofluids, providing very little information on the size effect.10–12,20,21 With the exception of a comparison of the results from two different experiments, obtained for two different sizes of Al2O3/distilled water nanofluids at room temperature3 and one, more recent report on Al2O3/distilled water nanofluids with different sizes of particles,22 none of these previous investigations have attempted to determine if and what the optimal volume fraction for the Al2O3 nanoparticle and distilled water nanofluids might be in order to achieve the highest effective thermal conductivity, based on the temperature and nanoparticle size. Given several of the proposed parameters, which include Brownian motion, volume fraction, particle size, and temperature, for the unusually large increase in the effective thermal conductivity of nanofluids, it is clear that an understanding of both the individual and combined effects of these parameters is necessary in order to develop a more complete understanding of the effective thermal conductivity of these a兲 Author to whom correspondence should be addressed; electronic mail: Bud.Peterson@colorado.edu 0021-8979/2007/101共4兲/044312/5/$23.00 nanofluids and hence more accurate and representative theoretical models, capable of predicting these values. There have been a number of theoretical models developed, capable of predicting the effective thermal conductivity for the mixtures.18–30 The original model developed by Maxwell23 and the Hamilton/Crosser24 共HC兲 model both consider the volume fraction and the thermal conductivity ratio between the particle material and base fluid, and the particle shape in the prediction of the effective thermal conductivity. The modified Maxwell-Garnet 共and the Hasselman and Johnson兲 model consider an additional factor, particle size.25,26 Koo and Kleinstreuer developed a model that considers the impact of the Brownian motion transfer of energy by nanoparticles,27 and more recently, Prasher et al.28 and Li and Peterson18 proposed two equations, which incorporated a somewhat different explanation and representation of the effect of the Brownian motion into the original model of Maxwell.18,28 Jang and Choi have incorporated the impact of the interfacial effects and the effect of Brownian motion.29 While the values predicted by many of these models agree with some of the available experimental data, they are in stark contrast to other experimental data that differs significantly and no single model has been widely accepted. For this reason, it is necessary to develop experimental data that can help to identify the individual and combined effects of as many factors as possible. EXPERIMENTS The current investigation focused on the effect of the nanoparticle size on the effective thermal conductivity of 101, 044312-1 © 2007 American Institute of Physics Downloaded 20 Apr 2013 to 141.161.91.14. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions 044312-2 C. H. Li and G. P. Peterson J. Appl. Phys. 101, 044312 共2007兲 FIG. 1. The experimental data of DI water 共the square is the first test data, the diamond is the second test data, and the line is the table value兲. FIG. 2. The thermal conductivity enhancement of 36 nm diameter Al2O3 / DI nanofluids. nanofluids over a wide range of temperatures and volume fractions. The measurement of the effective thermal conductivity was conducted using a steady-state “cut-bar” apparatus.20 Preliminary tests indicated that the samples were stable for several days after the initial preparation and the tests often required less than 24 h. Throughout all of the tests, care was taken to ensure that there was minimal sedimentation and/or aggregation during the course of the experimental test program. The nanofluid samples were prepared by dispersing Al2O3 spherical gamma crystal nanoparticle powder into de-ionized 共DI兲 water using a twostep procedure. The nanoparticle powder was first evenly blended with DI water and then was vibrated ultrasonically for 90 min until it was well dispersed. The steady-state method used in this investigation has been described previously20 and in order to ensure the validity of the experimental technique, the thermal conductivity of the DI water was measured over a similar temperature range before the test of each size of nanoparticle, and compared with tabular values available in the literature.31 The experimental test facility has a relatively consistent experimental variation that falls well within a range of ±2% as indicated in Fig. 1. To verify the accuracy of this technique, the measured effective thermal conductivity of 47 nm diameter Al2O3 / DI water samples at several different volume fractions obtained using this steady-state cut-bar method were compared with the results obtained on identical samples by other investigators using a transient hot-wire method. The results of this comparison have been presented previously and were in very good agreement for all samples tested.32 Following these calibration tests, the effective thermal conductivity of the nanofluids were evaluated at different temperatures and volume fractions and then normalized using the thermal conductivity of the DI water at the different temperatures, to determine the enhancement factor, defined as the ratio of the effective thermal conductivity of the sample to the effective thermal conductivity of DI water at the same temperature. Figure 2 illustrates the experimental results for the measured thermal conductivity values for the 36 nm diameter Al2O3 / DI nanofluid. As indicated, the volume fractions evaluated were 0.5%, 2%, 4%, and 6% and the temperature range was from 25 to 40 ° C. As shown, for each volume fraction, the thermal conductivity enhancement increases with increases in temperature and the slope increases with increases in the volume fraction. As illustrated, the 0.5% volume fraction nanofluid has a thermal conductivity enhancement of 3.0%; the thermal conductivity enhancement of the 2% volume fraction nanofluid increases from 7.7% at 27.9 ° C to 18.1% at 35.0 ° C; the thermal conductivity enhancement of the 4% volume fraction nanofluid increases from 9.3% at 27.6 ° C to 25.2% at 31.5 ° C; and the thermal conductivity enhancement of the 6% volume fraction nanofluid increases from 11.0% at 27.5 ° C to 28.0% at 35.8 ° C. The 47 nm diameter Al2O3 / DI nanofluids show the same trend on the thermal conductivity enhancement as shown in Fig. 3; with the values obtained for a bulk temperature of approximately 28.0 ° C, thermal conductivity enhancements of 2.9%, 3.5%, 9.7%, and 10.9% were measured for 0.5%, 2%, 4%, and 6% volume fractions, respectively, while at a bulk temperature of approximately 35.0 ° C, the thermal conductivity enhancements were 3.4%, 9.3%, 21.2%, and 26.0%, respectively. This temperature dependence agrees with the report by Patel et al. both qualitatively and quantitatively.11 The measured values at room temperature also fall within the previously reported magnitude range.20 FIG. 3. The thermal conductivity enhancement of 47 nm diameter Al2O3 / DI nanofluids. Downloaded 20 Apr 2013 to 141.161.91.14. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions 044312-3 J. Appl. Phys. 101, 044312 共2007兲 C. H. Li and G. P. Peterson FIG. 4. The thermal conductivity enhancement comparison between 36 and 47 nm diameter Al2O3 / DI nanofluids vs temperature at different volume fraction. FIG. 6. The thermal conductivity enhancement difference between 36 nm diameter and 47 nm diameter nanofluids vs volume fraction at different temperature. It is also interesting to note in Figs. 4 and 5 that the 36 nm diameter Al2O3 / DI nanofluids have a higher thermal conductivity enhancement than the 47 nm diameter Al2O3 / DI nanofluids at each and every volume fraction and temperature, with the only exception being the enhancement at 28 ° C and a volume fraction of 4%. Figure 4 indicates that at a temperature of approximately 28.0 ° C, the enhancement for each of the nanoparticle sizes evaluated is relatively small for all volume fractions, while for increases in the bulk temperature to 30.5 or 35.5 ° C, the 36 nm diameter Al2O3 / DI nanofluids show increasingly higher thermal conductivity enhancements than the 47 nm diameter Al2O3 / DI nanofluids for all volume fractions. Figure 5 shows that at 0.5% volume fraction, the differences between the thermal conductivity enhancement for the two sizes of nanofluids are very small at all temperatures. With changes in the volume fraction, the enhancement changes as a function of both the temperature and the particle size. The relative increase resulting from the variation of these two parameters is more clearly apparent in Fig. 6 where the thermal conductivity enhancement differences between the 36 and 47 nm Al2O3 / DI nanofluids are compared directly. In Fig. 6, it is clearly apparent that the enhancement differences at each temperature are small at a volume fraction of 0.5% and reach a peak value at a volume fraction of approximately 2%. The relative value of this enhancement difference reaches a maximum value at 2% and decreases for both increased and decreased volume fractions. In addition, the higher the temperature, the higher the value ultimately reached. Figure 7 illustrates this even more clearly as the enhancement difference at a volume fraction of 2% is consistently the highest among the volume fractions evaluated at each temperature, with the enhancement difference decreasing towards both ends of the volume fraction spectrum, regardless of whether the volume fraction increases or decreases. FIG. 5. The thermal conductivity enhancement comparison between 36 and 47 nm diameter Al2O3 / DI nanofluids vs volume fraction at different temperature. DISCUSSION The recent models of Jang and Choi and Prasher et al.,28,29 are shown as Eqs. 共1兲 and 共2兲 below, keff = kBF共1 − f兲 + knano f + 3C1 dBF 2 kBF Rednano Pr f , dnano 共1兲 FIG. 7. The thermal conductivity enhancement difference between 36 nm diameter and 47 nm diameter nanofluids vs temperature at different volume fraction. Downloaded 20 Apr 2013 to 141.161.91.14. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions 044312-4 J. Appl. Phys. 101, 044312 共2007兲 C. H. Li and G. P. Peterson 冋 册 共1 + 2␣兲 + 2共1 − ␣兲 k . = 共1 + A Rem Pr0.333兲 kf 共1 + 2␣兲 − 共1 − ␣兲 共2兲 As illustrated, both included a term to compensate for the contribution of the Brownian motion; however, both models assume that the only impact of the Brownian motion is the convective heat transfer between the base fluid and nanoparticles. In reality, the Brownian motion of nanoparticles in suspension is a slow dynamic phenomena and the time scale for the Brownian motion of nanoparticles is much larger than = 1 / 3共c pa / h兲, which is the time scale needed for the nanoparticle to obtain the equilibrium temperature of the surrounding base fluid molecules, and has the order of 10−12 s 共1 ps兲. Here, a and c p are the radius and thermal capacity of Al2O3 nanoparticle, respectively, and h is the heat transfer coefficient of the Al2O3 nanoparticle and water in the Stokes flow region, and is calculated with the equation h = 共k f / a兲关1 + 共1 / 4兲Re Pr兴.33 Also, the Biot number for Al2O3 nanoparticles in water is of the order of 10−2, much smaller than 0.1, which means that the center temperature of an Al2O3 nanoparticle will be in equilibrium to the surrounding base fluid temperature in a time period of the order of 10−12 s. Hence, an Al2O3 nanoparticle could not bring heat energy from the hot zone to the cold zone through its Brownian motion translation. However, the Brownian motion of nanoparticles could have another effect, which is the mixing effect. This effect is the result of the nanoparticles pushing and pulling the base fluid molecules and effectively mixing the higher temperature zone base fluid molecules with the lower temperature zone base fluid molecules across the temperature contours throughout the entire three-dimensional space in the nanofluids. This mixing effect can be seen using the following time scale analysis. The pushing and dragging of nanoparticles to base fluid molecules surrounding nanoparticles should have a time scale equal or greater than the sound propagation time c = a / c which is the time required for a sound wave with a speed of C to travel a distance equal to the radius a of the nanoparticle. The time, = a2 p / , required for the base fluid molecules subjected to Brownian motion to acquire a velocity due to the viscosity effect would have the a characteristic time of a viscous shear wave, created by the Brownian motion of the nanoparticle. For a nanoparticle with a size ranging from 1 to 100 nm of any nanofluids, the ratio / c is from 100 to 10 000. This means that in the relaxation time, the base fluid molecules surrounding the Brownian motion nanoparticles will have the same velocity as the nanoparticles.34 This mixing effect was reported and has been shown to be the principal mechanism behind the phenomenon of unusual high thermal conductivity of nanofluids.35,36 There is another empirical equation 关Eq. 共3兲兴 which if analyzed using a regression technique with 95% confidence would yield a very good prediction of the effective thermal conductivity for the experimental data,22 ke/k f = 1 + const共1/d p兲0.369关T1.2321/102.4642B/共T−C兲兴. 共3兲 However, comparing the predicted values of this equa- FIG. 8. The comparison of thermal conductivity enhancement of current 47 nm diameter nanofluids and previous report. tion with the current experimental data, the difference is still large. The explanation for this is that the equation represents an empirical equation regressed with the measured data obtained from a hot-wire method at different temperatures. The hot-wire method and the steady-state method have been shown to yield identical experimental results at room temperature,37 but it has been reported that when testing nanofluids in the high temperature regime using the hot-wire method, some natural convection heat transfer will augment the final results of effective thermal conductivity of the nanofluids.38 The experimental data of the same 47 nm diameter Al2O3 nanofluids from the current experiment were compared with the experimental results on which this empirical equation was developed and are shown below in Fig. 8. As illustrated, it is very clear that the experimental data for this empirical equation are larger than the experimental data from the current test at the same temperatures. CONCLUSION The previous investigation has indicated that when the diameter of the nanoparticles used in nanofluids increases, the relaxation time will also increase; however, the Brownian motion velocity will decrease39 and the organic base fluids will result in longer relaxation times than water.40 For a different pair of nanoparticle and base fluid materials, the interfacial thermal conductivity at the surface of the nanoparticle will also deviate from the predicted value of the theoretical model h = 共k f / a兲兵1 + 共1 / 4兲Re Pr兴.41 The phenomena observed here suggests that the thermal conductivity enhancement of Al2O3 / DI nanofluids has a nonlinear relationship with temperature, volume fraction, and nanoparticle size and that there is an optimal combination of nanoparticle size and volume fraction for each different pair of nanoparticle and base fluid materials at a certain bulk temperature. It is very hard to maintain a certain bulk temperature in the heat transfer process, but the optimal volume fraction for the nanoparticle size and the pair of nanoparticle and base fluid materials should not deviate much from the original value. In order to fully understand the individual and combined impact of variations in the volume fraction and particle size and to verify the existing theoretical models, additional experimental thermal conductivity data Downloaded 20 Apr 2013 to 141.161.91.14. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions 044312-5 for Al2O3 / DI nanofluids with other different diameter size gamma phase spherical nanoparticles and different base fluids are required. The authors acknowledge the support of the Office of Naval Research through Grant No. ONR N000140010454 and the National Science Foundation through Grant No. CTS-0312848. 1 J. Appl. 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