An experimental investigation of turbulent thermal convection in water ...

PHYSICS OF FLUIDS 23, 022005 共2011兲

An experimental investigation of turbulent thermal convection in water-based alumina nanofluid Rui Ni (倪睿兲,1 Sheng-Qi Zhou (周生启兲,1,2,a兲 and Ke-Qing Xia (夏克青兲1,a兲 1

Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, China Laboratory of Tropical Marine Environmental Dynamics, South China Sea Institute of Oceanology, Chinese Academy of Sciences, Guangzhou 510301, China

2

共Received 23 August 2010; accepted 4 January 2011; published online 11 February 2011兲 We report heat transfer and flow dynamics measurements of alumina nanofluid in turbulent convective flow. Under the condition of fixed temperature at the top plate and fixed input heat flux at the bottom plate, it has been found that the convective heat transfer coefficient, h, Nusselt number, Nu, and Rayleigh number, Ra, all decrease with the increasing volume fraction ␾ of the nanoparticle. In contrast, the velocity of the convective flow showed no significant change within experimental uncertainty and over the range of nanoparticle concentration of the measurement 共from 0% to 1.08%兲. Under the condition of constant nanoparticle concentration 共␾ = 1.08%兲, a second set of measurements of the heat transport and flow properties have been made over a broad range of Ra 共from 2.6⫻ 108 to 7.7⫻ 109兲. For heat transport, a transition near Rac ⯝ 2.5⫻ 109 has been found. For Ra ⬎ Rac, the measured Nu of the nanofluid is roughly the same as that of water in terms of both its magnitude and its scaling relation with Ra, which suggests that the nanofluid can be treated as a single phase fluid in this parameter range. For Ra ⬍ Rac, Nu becomes smaller than that of the water and the deviation becomes larger with decreasing Ra. In the parameter range of Ra ⬍ Rac, the measured instantaneous Nu共t兲 shows strong and quasiperiodic fluctuations, which is absent when Ra ⬎ Rac. This suggests that the significant decrease of the nanofluid Nu comparing to that of water may be caused by the mass diffusion of nanoparticles. Furthermore, measurements of the flow velocity of the bulk nanofluid showed no significant difference from that of water for Ra either above or below Rac. From estimated thermal boundary layer thickness, we found that the deviations of the nanofluid Nu from that of water for Ra ⬍ Rac corresponds to the thickening of the thermal boundary layer at both the top and bottom plates. This thickening of the boundary layer at low input heat flux 共or low driving strength of the convective flow兲 cannot be attributed to possible sedimentation of the nanoparticles. © 2011 American Institute of Physics. 关doi:10.1063/1.3553281兴 I. INTRODUCTION

Nanofluids are liquids in which the nanosized metallic, metal-oxidic, or nonmetallic particles are suspended in the base fluids. Because of their significantly higher thermal conductivity, nanofluids have attracted increasing attention since an original study was carried out in 1995.1–3 It is not surprising that the thermal conductivity of nanofluids is higher than that of their base fluids when the thermal conductivity of these conventional nanoparticles is orders of magnitude larger than that of base fluids. But the enhancement of the thermal conductivity of nanofluids is conspicuously larger than that predicted from the mean-field conduction-based models, 共e.g., the Maxwell–Garnett theory4兲. For example, a small amount 共volume fraction ␾ ⬍ 1%兲 of copper nanoparticles or carbon nanotubes dispersed in the fluids 共ethylene glycol or oil兲 have been reported to increase the thermal conductivity of the liquid by 40% and 161%, respectively.5–7 According to the conventional knowledge of particle-liquid suspensions, high concentrations 共␾ ⬎ 12%兲 of particles are needed to achieve such an enhancement. In addition, coma兲

Authors to whom correspondence should be addressed. Electronic mail: [email protected] 共S.Q.Z.兲; [email protected] 共K.Q.X.兲.

1070-6631/2011/23共2兲/022005/12/$30.00

pared to the thermal application of fluids with micrometersized particles, such nanofluids are very stable 共within one month8兲 and almost free from problems of sedimentation, clogging, abrasion, pressure drop, and erosion. With all these novel features, nanofluids have the potential to meet the increasing demand for high thermal conductive working fluids, which is one of the top challenges in the thermal management of many applications. Thus, nanofluids are expected to have a significant impact on energy conservation and pollution reduction if they can be applied successfully. In addition to their thermal applications, Nanofluids also have great potential in biological 共e.g., polymerase chain reaction of DNA9兲 and medical 共e.g., cancer therapy10兲 applications. In spite of their unique features and potentials, these special fluids are still in the early stages of development. So far, thermal conductivity studies have been the main focus of nanofluid research 共see the review articles1–3,11兲, but ultimately, their flow and heat transfer characteristics in practical applications will reveal their usefulness as advanced thermal fluids. In most thermal applications under varying conditions, the buoyancy-driven convective flow will be involved in the heat transport. Not only their thermal conductivity but the convective heat transfer performance will be of great importance. One central question is whether the con-

23, 022005-1

© 2011 American Institute of Physics

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp

022005-2

Ni, Zhou, and Xia

vective heat transport in nanofluids will be significantly enhanced as that observed in their thermal conductivity enhancement at static condition.1–3,5–7 Such an issue must be solved before these nanofluids can be put into practical use. However, the convective heat transfer of nanofluids has received less attention in the literature, especially in the buoyancy-driven natural convection. In fact, the natural convection of fluids can be used in many applications in the thermal management of electronic devices, engine cooling systems, and in solar collectors. In the theoretical and numerical studies of the natural convection of nanofluids, most effort has been done with the finite-volume approach.12–16 Khanafer et al. solved the twodimensional transport equations and found that the heat transfer rate could be substantially increased by the suspension of nanoparticles and it is increased with an increase of nanoparticle volume fraction.12 Jou and Tzeng found that the heat transfer has been increased by the addition of nanoparticle in the Rayleigh number 共Ra兲 range 共103 – 106兲;13 while Abu-Nada found that the enhancement can be found only for low Ra 共⬃103兲.14 Das et al. also found that the nanoparticles are capable of increasing heat transfer performance in nanofluid and the effect is more pronounced when more nanoparticles are immersed.15 Although Ghasemi and Aminossadati found that adding nanoparticles into pure fluid improves its heat transfer performance, there is an optimum volume fraction that maximizes the heat transfer rate and this optimum volume fraction increases with the increasing Ra, e.g., it is around 0.02 and 0.04 for Ra = 104 and 106 respectively.16 Based on the physical properties deduced from the mixing theory and other models, Jang and Choi estimated that the convective heat transfer coefficient can be enhanced for copper-water nanofluid or diamond-water nanofluid.17 Ogut solved the dynamic and transport equations with the polynomial differential quadrature method and the results showed that the average heat transfer rate increases significantly as particle volume fraction increases.18 With the convective instability analysis, Kim et al. found that the convective motion sets in easily when nanoparticles are suspended and the combination effects of different physical properties of nanoparticles contribute to the enhancement of the heat transfer coefficient of nanofluid.19 By considering the Brownian motion and thermophoresis of nanoparticles, Tzou found that the critical Ra for nanofluids was lower by one or two orders of magnitude than that of regular fluids, which subsequently results in the high heat transfer performance.20 Based on these numerical simulations of natural convection, it may be concluded that in general, the convective heat transfer in nanofluids is enhanced in comparison to that of the base fluids. On the other hand, from the relatively few experimental studies, it appears that the addition of nanoparticles plays a negative role in the convective heat transfer of the base fluid.21–25 Putra et al. found that the heat transfer for alumina nanofluid 共␾ = 4%兲 is only about one half of that of water at low Ra of about 106.21 Wen and Ding found that the heat transfer coefficient can be decreased by 30% in titanium dioxide nanofluid.22,23 Nanna reported that convective heat transfer declines for nanofluid of larger volume fraction

Phys. Fluids 23, 022005 共2011兲

nanoparticles 共␾ ⬎ 2%兲, but he also claimed that enhancement can be obtained for nanofluid of smaller nanoparticle volume fractions; although within the experimental uncertainty, one may argue that there is no enhancement.24 Recently, an increase of the onset Ra for convection has been observed,25 which disagrees with results from numerical predictions.19,20 This result suggests that the convective heat transfer has been suppressed rather than enhanced by the suspension of nanoparticles.25 What is the cause for the seemingly opposite results from the numerical and experimental studies? We note that the experimental measurements were done at low Rayleigh numbers, all below Ra = 5 ⫻ 108.21–25 It is known that for this parameter range the convective flow is not in a fully turbulent state.26 Thus, one may wonder whether this factor limits the influence of nanoparticles. Another question is: If the nanoparticles have certain influence on the convection of base fluid, in what way the convective flow structures are modified? In this paper, we report turbulent natural convection experiments in the classical Rayleigh–Bénard configuration where a confined enclosure with conductive top and bottom plates and insulating sidewalls is heated from below and cooled from above.26–28 The system can be described with three control parameters, namely the Rayleigh number 共Ra兲, Prandtl number 共Pr兲, and the aspect ratio 共⌫兲. The Rayleigh number is defined as Ra = ␣gL3⌬T / ␯␬, with g being the gravitational acceleration and ␣, ␯, and ␬ being, respectively, the thermal expansion coefficient, the kinematic viscosity, and the thermal diffusivity of working fluid. Moreover, ␬ = k / ␳c p and ␯ = ␮ / ␳ are often deduced from the more basic physical parameters of fluid, where ␳, c p, k, and ␮ stand for the density, specific heat, thermal conductivity, and dynamic viscosity, respectively. The Prandtl number is defined as Pr = ␯ / ␬, and the aspect ratio ⌫ is the ratio of the lateral dimension of the system to its height. In the experiment, Ra was varied from 2.6⫻ 108 to 7.7⫻ 109 so that the global heat transport and velocity field measurements of pure water and water-based alumina nanofluid can be made over a broad range of flow states. The remaining sections of the paper are organized as follows. In Sec. II, we describe the experimental apparatus, the temperature measurement, and the nanofluid preparation. In the construction of convection cell, much attention has been paid to prevent heat exchange with the surrounding environment. In Sec. III, we present and discuss the experimental results of nanofluid with the heat and flow properties of pure water as a benchmark. The heat and flow properties include the convective coefficient 共h兲, the Nusselt number 共Nu兲, and the Reynolds number 共Re兲 in water-based alumina nanofluid of different volume fractions ␾. Then we discuss these quantities 共h, Nu, and Re兲 as a function of Ra. Lastly, a possible explanation has been provided to these experimental results. In Sec. IV, we present the summary and conclusions. In the Appendix, we present the determination of the physical properties of nanofluid.


022005-3

An experimental investigation of turbulent thermal

K J G I H G F E D C B A

FIG. 1. Schematic drawing of the convection cell and related accessories 共drawing not to scale兲. The marked components are explained in the text.

II. EXPERIMENTAL APPARATUS AND PROCEDURE A. Experimental setup

The measurement of nanofluid convective flow was taken inside the Rayleigh–Bénard system where a cylindrical convection cell was employed. Some aspects of the apparatus used here have been described in Ref. 29. As some new features were introduced to improve the precision of the heat transfer and flow structure measurements, we describe the apparatus in detail here. A schematic diagram of the apparatus is shown in Fig. 1. From bottom to top, we find first a rubber heater A of thickness 2 mm and diameter 200 cm 共Model SRFR-8/10, Omega, Inc.兲. The heater provided heat power to a copper pan B above it so that the temperature of the pan was regulated to be close to that of the lower surface of the bottom plate D 共⫾0.1 ° C兲. Such a configuration minimizes the heat exchange between the bottom plate and the surrounding. The bottom base of the cell was made of two copper plates; the bottom one D provides support, while the upper one F was finely machined. The bottom base sat on top of an adiabatic shield of three 23 cm diameter plywood plates C 共total thickness 5.4 cm兲 inside the pan. Plate F was made of pure copper and had a thickness of 3.0 cm and diameter of 23.0 cm. Its surface was coated with nickel to prevent oxidization and provide a mirror-finish surface. The central area 共diameter 19.6 cm兲 of the bottom side of F was carved with uniformspaced parallel straight grooves 0.5 cm in depth and 0.30 cm in width. These grooves had a separation of 0.5 cm and were interconnected by semicircles at ends. Distributed inside the grooves was a heater E, which was made of NiChrome wire 共26 Gauge, Aerocon Systems兲 surrounded by fiberglass sleeving and Teflon tape. This heater had a total length of 3.76 m and a resistance of 52.29 ⍀. The center section of the top surface of F had an extruded step of a diameter of 19.3 cm and height 2.00 cm that was a close fit to the plexiglas sidewall cylinder I. Inside the bottom base, five small holes G were drilled from below to the positions within 0.15 cm distance beneath the top surface of plate F; one hole was in the center and the other four holes


were evenly distributed along the circle of 9.8 cm diameter. Thermistors were inserted into these holes to monitor the temperature distribution of the bottom base. The rest empty spaces between the bottom base and the copper pan were filled with styrofoam fragments for further enhanced thermal insulation. The plexiglas sidewall I had an inner diameter of 19.3 cm and a wall thickness of 0.5 cm. The height was uniform to ⫾0.01 cm around its circumference. As shown in Fig. 1, it extended 2.00 cm below the top surface of plate F. A rubber O-ring was sealed between the plate F and the plexiglas sidewall I. A similar construction was used to terminate the cell wall at the top base. Resembling the bottom base, the top one was made up of two copper top plates, J and K, with the same dimensions. A chamber was constructed between these two plates so that the top base was cooled by flowing water from a refrigerated circulator 共Model 6206, Polyscience, Inc.兲. The chamber had a diameter and height of 19.5⫻ 1.5 cm2 and was connected with the circulator by four nozzles. Water flowed into the chamber with two nozzles from the opposite direction and left the other two nozzles perpendicular to the inlets. Similar to the plate F at the bottom base, the bottom plate J at the top base was nickel-coated too. It had a thickness of 3.0 cm and its bottom part of thickness 2.0 cm was inserted into the plexiglas sidewall. Four equally-spaced small holes were drilled through plate J from the top with a distance 0.2 cm above the fluid-contacting surface, where the calibrated thermistors G were mounted inside to monitor the temperature of the top base. Each of the thermistors was protected from the circulating water in the chamber by an additional small plate with O-ring on the top of the hole. Six 0.8 cm diameter rods 共not shown in Fig. 1兲 were used to assemble the bottom base, plexiglas sidewall, and top base into the convection cell. To minimize the heat exchange through the rods, the middle 1/3 part of the rod was made of Teflon while the two ends were made of stainless steel. When the cell was assembled, the distance between the top and bottom bases was measured to be 19.24 cm, corresponding to ⌫ = 1. Inside the convection cell, three small thermistors were placed at the midheight positions H that are 2 cm away from the sidewall. At one side, the separation between two thermistors was 5.2 mm, which was much larger than their own size. These two thermistors would be used to detect the local velocity near the sidewall. The third thermistor was placed at an axis-symmetric position on the other side. The bulk temperature can be obtained from the weighted average of these three thermistors. Outside the convection cell, the top plate and sidewall were surrounded by two layers of nitrile rubber sheets and one layer of styrofoam for thermal insulation. Finally, the whole apparatus was put inside a thermostat where the temperature was regulated to be close to the mean temperature of the convection system. B. Temperature measurement

One of the necessary conditions for heat transfer measurement is to accurately determine the temperature of the system. Here, two types of thermistors were used. One is a


022005-4

Ni, Zhou, and Xia

round metallic probe encapsulated with epoxy 共model 44031, Omega Engineering, Inc.兲, which has an overall diameter of 2.4 mm. It was mounted in the positions G in the top and bottom bases, as shown in Fig. 1. The temperature range for best stability of the thermistor is from 0 to 75 ° C, which covers the temperature range of both top and bottom bases in the present study. The other type is a high-sensitive metal oxides NTC thermistor 共Model AB6E3-B05103J, Thermometrics, Inc.兲. It is welded and hermetically sealed in the front of the glass-coated thermobead and has a very small size 共diameter⬃ 0.3 mm兲. This kind of thermistor was placed inside the flow at position H, as shown in Fig. 1. Based on the estimated Reynolds number of the thermistor, its disturbance to the flow is negligible. An ac Wheatstone bridge was used for measuring local temperature fluctuations inside the convection cell. The temperature measurement apparatus was similar to that described in Ref. 29. The thermistor served as one arm of an ac Wheatstone bridge driven sinusoidally at 1 kHz in frequency and 0.5 V in amplitude. The output of the bridge was first fed to a lock-in amplifier and then digitized by a Labview program with a sampling rate of 128 Hz. The recording time was 4.5 to 9 h to ensure good data statistics. Before the temperature measurement, each thermistor had been individually calibrated with a standard platinum thermal probe. To show the temperature stability of the convection system, we plot in Fig. 2 examples of temperature trace measured in the top and bottom plates and inside the cell. For these data, the temperature difference across the cell was ⌬T = 39.70 K and the volume fraction of alumina nanoparticles was ␾ = 0.38%. Figure 2共a兲 shows the recording of one of the probes in the top plate. For the measurement duration of about 6.5 h, it had a mean value of 17.07 ° C with a standard deviation of 0.11 ° C. The other three probes on the top plate showed comparable temporal fluctuations and the variations among the four probes in the plate were within 0.15 ° C. Figure 2共b兲 shows the time trace of one of the three small thermistors placed inside the cell 共the left one shown in Fig. 1兲. It had an average of 39.71⫾ 0.52 ° C for the duration of the measurement. This time series shows large and asymmetric temperature fluctuations, indicating that the hot plumes are coming up along the left side of the cell. Figure 2共c兲 plots the temperature signal of one of the five probes imbedded in the bottom base, with a time average of 56.76⫾ 0.12 ° C. The variation among the five probes were within 0.15 ° C. From Fig. 2, we see that temperature fluctuations inside the convection cell are much larger than those in the top and bottom plates, which are caused by the presence of thermal plumes convected by the turbulent flow. C. Preparation of nanofluid

The nanofluid samples were supplied by Nanophase Technologies, Inc., where the water-based alumina suspension had the initial volume fraction 共␾兲 of 21.7%. The average nanoparticle diameter was listed as 45 nm. It was claimed that a very small amount of dispersant had been added in the suspension to minimize nanoparticle agglomeration.30 Note that this suspension was also used in


FIG. 2. Normalized temperature traces measured 共a兲 in the top base, 共b兲 in the cell, and 共c兲 in the bottom base. The temperature difference across the cell was ⌬T = 39.70 K and the volume fraction of alumina nanoparticles was ␾ = 0.38%.

Refs. 21, 24, and 31–34. Before each experiment, the initial suspension would be sonicated for 12–24 h using an ultrasonic processor 共Elma, Inc.兲 to break and deagglomerate clustered nanoparticles. To avoid artificial pollution, the ultrosoniced initial suspension was injected into the convection cell directly where the water flow was highly turbulent. The suspension of different volumes in the cell would result in nanofluid samples of different volume fractions 共␾兲.

III. RESULTS AND DISCUSSION

In the investigation of the convective heat transfer of nanofluids, we follow two schemes to check the influences of nanoparticles in turbulent natural convection. In the first scheme, the top plate was kept at constant temperature 共23.17⫾ 0.08 ° C兲 and the bottom one was kept at constant heat flux 共330 W兲, then the heat transfer and flow dynamics of nanofluid were examined as functions of the volume fraction ␾ of nanoparticles. In the second one, the volume fraction of nanoparticles was kept constant at ␾ = 1.08%, the heat transfer and flow dynamics of nanofluid were examined as a function of Ra 共Pr was kept almost constant at 4.43⫾ 0.05兲.


022005-5



ratio of temperature drop ⌬t across the top boundary layer to that of the bottom one, ⌬b,

␹ = ⌬t/⌬b =

FIG. 3. 共a兲 Convective heat transfer coefficient of the nanofluid normalized by that of the pure fluid 共water兲, hnf / h f , and 共b兲 boundary layer temperature drop ratio, ␹, vs nanoparticle volume fraction ␾.

A. Volume fraction dependency

In this section, we present heat transport and flow property measurements of nanofluids with different nanoparticle volume fractions. The experiment was conducted under the condition of constant temperature 共Tt = 23.17⫾ 0.08 ° C兲 at the top plate and constant heat flux 共330 W兲 at the bottom plate while varying the volume fraction ␾ of nanoparticles from 0% to 1.08%. For the above boundary condition, the measured Ra = 7.63⫻ 109, Pr = 4.5, and Nu = 122.4 for the case of pure deionized water 共␾ = 0兲. The convective heat transfer coefficient h is a basic parameter to characterize the heat transfer performance, which is defined as h = Q/⌬T,

共1兲

where Q is the heat flux per unit area. A smaller value of h means heat is transported less efficiently than larger ones. Note that the definition of h is similar to the Nusselt number Nu, the heat transfer coefficient is a quantity that does not have explicit dependence on fluid parameters and depends only on two directly measured quantities of the experiment. As shown in Fig. 3共a兲, within the measured parameter range, h appears to decrease monotonically with increasing volume fraction ␾, hnf measured for the nanofluid has been normalized by that of pure fluid h f . Such a result implies that the addition of alumina nanoparticles deteriorates the convective heat transfer of the fluid. Because the temperature of the top plate is constant, the decreasing h corresponds to the increasing bottom temperature, which may affect the Oberbeck– Boussinesq condition in the convection system.35 The nonOberbeck–Boussinesq effect may be characterized36 by the

Tc − Tt , Tb − Tc

共2兲

where the subscripts t, b, and c refer to the top plate, bottom plate, and the bulk of the system. When the Oberbeck– Boussinesq assumption is satisfied, ␹ = 1, implying topbottom symmetry. Figure 3共b兲 plots ␹ as a function of ␾. It shows that at ␾ = 0, ␹ = 1.183, suggesting that there exists a non-Oberbeck–Boussinesq effect at this temperature difference before any nanoparticles are added. This result is consistent with previous experimental measurements.37,38 However, with the increasing ␾, the non-Oberbeck–Boussinesq effect appears to become smaller. This result is somewhat different from a previous study in a sidewall-heated configuration,24 which found that ␹ is about 1 in pure water and the deviations from the Oberbeck–Boussinesq approximation increases with increasing nanoparticles concentration. Besides the difference in the experimental configurations, the convective flow in Ref. 24 was less turbulent 共Ra ⬃ 107兲 than ours 共see the following section兲, which may be a reason for the discrepancy between the two studies. We now examine how the dimensionless parameters Ra, Pr, Nu, and Re of the system change in response to the addition of nanoparticles. In general, the changes in these quantities come from two sources. One comes from the change in flow states and the other from the changes in fluid physical properties, such as its thermal conductivity, viscosity, and heat capacity, and density 共see the Appendix兲. For the Prandtl number Pr, we find that it varies less than 1% for all nanoparticle volume fractions used in the present study, which is within the experimental uncertainty of our measurement. From its definition, the near constancy of Pr suggests that when combined, the effects of k, ␮, and c p cancel out, although individually each of these fluid parameters change significantly when nanoparticles are added. On the other hand, Ra and Nu show strong volume fraction dependencies. As can be seen in Fig. 4共a兲, the Rayleigh number Ranf of the nanofluid normalized by that of the pure fluid Ra f decreases monotonically by as much as 15% when the volume fraction ␾ is increased from 0% to 1.08%. A similar trend has also been found in previous investigations but with less magnitude of decrease.24 This difference is probably due to the fact that the convective flow in the present study is more turbulent than that in the previous work.24 As shown in Fig. 4共b兲, the normalized Nusselt number Nu also deceases with the increase of volume fraction. Previous numerical simulations of nanofluid natural convection in the present volume fraction range but at much lower Ra 12,15,16,39 have found an increase rather than a decrease in Nu as compared to the pure fluid case. At larger volume fractions, these studies also found that Nu decreases with ␾. Experimentally, a study using a side-heated enclosure showed no significant change of Nu within the experimental uncertainty for 0 ⱕ ␾ ⱕ 2% 共although the authors stated a Nu increase in this range兲 and decreases for larger ␾.24 We note that in these earlier numeri-


022005-6

Ni, Zhou, and Xia

FIG. 4. 共a兲 Rayleigh number of the nanofluid normalized by that of the pure fluid 共water兲, Ranf / Ra f , and 共b兲 normalized Nusselt number, Nunf / Nu f , vs nanoparticle volume fraction, ␾. The dashed line in 共a兲 indicates the result in Ref. 24.

cal and experimental studies, the flows were not as turbulent as in the present work 共Ra ⬍ 106 in Refs. 12, 15, and 24 and Re ⬍ 1500 in Ref. 39兲. To find out how the addition of nanoparticles changes the flow dynamics of the system, we measured the convective flow velocity. As the nanofluid is optically nontransparent at the particle concentrations of the present study, optical techniques such as particle image velocimetry 共PIV兲 and laser Doppler velocimetry 共LDV兲, are not applicable here. Two indirect methods were thus used. It is well-known that, for turbulent thermal convection in pure fluid, there exists a coherent low-frequency oscillation in the convective flow, which can be detected from both temperature and velocity measurements,40 and that the oscillation period, tT, is associated with the typical velocity of convective flow by the following expression, Vosc = 4L / tT. The second method is based on the simultaneous measurement of temperature from two neighboring probes 共see Fig. 1兲 with a known separation,41 here d = 0.0052 m. By making short-time correlation measurements between temperature signals from the two probes for various time delays and by identifying the time delay, ␶, that yields the largest correlation, one infers a velocity, Vcor = d / ␶ for the mean convection flow at local position. Both Vosc and Vcor as functions of the volume fraction ␾ are shown in Fig. 5共a兲. In a way, Vosc represents the velocity of global convective flow while Vcor is the local mean velocity. Therefore the two velocities are not expected to be the same in terms of their absolute values. Note that their relative magnitudes also depend on the definitions, i.e., the numerical factor 4 in Vosc, which is somewhat arbitrary 共For a detailed discussion on this, please see Ref. 42.兲. Within the experimental uncertainty, it is seen that Vosc and Vcor remain un-


FIG. 5. 共a兲 Velocity, Vosc 共solid circles兲, and Vcor, 共open circles兲, and 共b兲 Reynolds number, Reosc 共solid circles兲, and Recor, 共open circles兲, as a function of nanoparticle volume fraction, ␾.

changed in the volume fraction range of the experiment. This result suggests that the addition of nanoparticles does not significantly perturb the flow dynamics of the convecting fluid. Such a result suggests that fluid phase and nanoparticles are in a thermal equilibrium state and they move at the same velocity as that of pure fluid, which is the main assumption of a one-phase model in numerical simulation.12 It contradicts to the findings in the two-phase simulation where the increases in the particle concentration increased the drag force between the phases, causing a slower motion of both the fluid and particle phases.43 On the other hand, as shown in Fig. 5共b兲, the Reynolds numbers, Reosc共cor兲 = Vosc共cor兲L / ␯, based on either Vosc or Vcor, exhibit a gradual decrease with the increasing volume fraction ␾. This may be explained by the fact that the fluid simply becomes more viscous with the increasing nanoparticle concentration 共see the Appendix兲. A larger viscosity would suppress the turbulence level of the flow. B. Rayleigh number dependency

In this section, we present and discuss results from convective heat transport and flow property measurements of nanofluid with ␾ = 1.08% over a broad range of experimental conditions. Results from pure water will also be presented as a benchmark. All the physical quantities of both pure water and the nanofluid are calculated based on their respective fluid properties. Unless ambiguities arise, the subscripts nf and f are omitted for ease of presentation. We started the measurement by varying Ra from high to low values in the first two weeks and repeated the whole


022005-7



FIG. 6. The convective heat transfer coefficient, h, as a function of the input heat flux density, Q, for water 共solid circles兲 and for the nanofluids measured from the first run 共open circles兲 and from the second run 共open squares兲. Error bar is smaller than the symbol.

process 共with different values of Ra兲 in the next two weeks to check the repeatability of the experiment and to determine whether sedimentation of nanoparticles has occurred. During the measurement, the bulk temperature was held almost constant 共at Tc = 39.6⫾ 0.3 ° C兲 for all Ra so as to avoid the influence of fluid property variations. The convective heat transfer coefficient h from both measurements is shown in Fig. 6. It can be seen that data from the two runs are consistent with each other. Generally speaking, at the same heat current density, Q, the convective heat transfer coefficient h of the nanofluid is smaller than it is in pure water and this trend becomes more apparent for smaller heat current density. This tendency is also clearly demonstrated in Fig. 7共a兲, where the Ra dependency of Nu has been plotted. In pure fluid turbulent thermal convection, Nu has been found to have a relationship of Nu ⬃ Ra0.306 for the present range of Ra, which is consistent with many previous studies 共see for example, Refs. 44 and 45兲. However, the measured Nu for the nanofluid cannot be described by a single power law scaling and it shows a transition over the Ra range of the present work. Figure 7共b兲 is a compensated plot of the Nusselt number, i.e., Nu multiplied by Ra−0.3, which enabled us to examine the data with higher sensitivity.44,45 It is seen that the transition appears at a threshold value Rac ⬃ 2.5⫻ 109, suggesting that below a certain value of Ra, the nanofluid’s ability of transporting heat deteriorates. For Ra greater than the threshold value, the relationship between Nu and Ra of the nanofluid appears to follow that of the pure water. Here, the nanofluid can be treated as one-phase fluid. The slightly low Nu may be caused by the Pr difference between pure water and the nanofluid. In order to understand this behavior of the nanofluid, we examined the instantaneous Nu共t兲 at different values of Ra for both water and nanofluid. Two examples are shown in Fig. 8. At higher Ra 共above Rac兲, as shown in Fig. 8共a兲, the instantaneous Nu共t兲 of the nanofluid varies little in the duration of the measurements 共⬃9 h兲 and, except having a smaller magnitude, it is no different from that of the pure water. At an Ra 共=1.6⫻ 109兲 lower than Rac, as shown in Fig. 8共b兲, the instantaneous Nu共t兲 of nanofluid exhibits much larger fluctuations than that of pure water at a

FIG. 7. 共a兲 Nusselt number, Nu, and 共b兲 the compensated Nusselt number, NuRa−0.3, as a function of Rayleigh number, Ra, for water 共solid circles兲 and the nanofluids 共open circles兲. The solid line in 共a兲 is the best power-law fit: Nu = 0.115Ra0.306.

FIG. 8. 共Color online兲 Measured instantaneous Nu共t兲 for values of Ra both larger and smaller than Rac 共see text兲. 共a兲 Ra ⬎ Rac; black line: Nanofluid at Ra = 4.1⫻ 109; gray 共red兲 line: Water at Ra = 5.0⫻ 109. 共b兲 Ra ⬍ Rac; black line: Nanofluid at Ra = 1.6⫻ 109; gray 共red兲 line: Water at Ra = 7.4⫻ 108.


022005-8


Ni, Zhou, and Xia

FIG. 9. 共a兲 The velocity, Vcor, and 共b兲 Reynolds number, Recor, as a function of Rayleigh number, Ra, for water 共solid circles兲 and the nanofluids 共open circles兲. The solid lines in 共a兲 and 共b兲 are the respective best fits: Vcor = 1.289⫻ 10−7Ra0.525 and Recor = 0.043Ra0.518 to the water data.

value of Ra 共=7.4⫻ 108兲 also below Rac. It appears that Nu共t兲 oscillates with an approximate period of 2 h. At other lower values of Rac that we studied, similar low-frequency oscillations were also observed in the nanofluid. It is clear that below and above the transitional Rac, the convective flow is in two different dynamic states. This change in the dynamic behavior reflects not only in the properties of the instantaneous Nu共t兲 and the time-averaged Nu, but also in the thermal boundary layer thickness ␦ 共see Fig. 10 below兲 as Nu and ␦ are inversely related to each other. It is not clear what is the cause for this oscillatory behavior. One possible reason could be that it is produced by the mass diffusion of nanoparticles. In a previous natural convection study of water-fine particle suspensions 共particles size 5 ␮m兲,46 a saw-toothed oscillation had been found in the instantaneous heat transfer coefficient h and it is attributed to the double diffusive convection based on the temperature and concentration gradients of the suspension. Next we examine the Ra-dependency of flow velocity and the Reynolds number. Figure 9共a兲 plots the velocity Vcor versus Ra measured in both pure water and the nanofluid. Within the experimental resolution, the two sets of data showed no difference. For the measured Vosc, a similar behavior was also observed. Figure 9共b兲 plots the Reynolds number, Recor, versus Ra for both the nanofluid and water, which shows that the magnitude of Re for nanofluid is smaller than that of water. This could be understood by the fact that the nanofluid has a larger viscosity, which leads to a smaller Reynolds number for the same flow speed. We note that the volume fraction dependency of Re 共Fig. 5兲 can be similarly explained, i.e. a larger ␾ leads to a larger viscosity

FIG. 10. Thermal boundary layer thickness determined indirectly at 共a兲 the top plate 共␦t兲 and 共b兲 the bottom one 共␦b兲 as functions of the input heat flux density 共Q兲 for both water 共solid circles兲 and the nanofluid 共␾ = 1.08%兲共open circles兲.

and hence lower Re. Otherwise, the behavior is similar to the measured velocity. Combining the results of both ␾ and Ra dependencies of the measured velocity/Reynolds number, we can conclude that the flow dynamics in the bulk of the convecting fluid is not significantly altered by the dispersion of nanoparticles. On the other hand, it should be noted that the measured Re does not show a clear transition at Rac, which is the case for Nu. This is an important difference between the two quantities. As the bulk fluid shows no significant changes in terms of flow dynamics, we look at the thermal boundary layer. In the absence of direct boundary layer 共BL兲 measurement, we estimate thermal BL thickness ␦t and ␦b at the top and bottom plates from the measured temperature drops ⌬t = Tc − Tt and ⌬b = Tb − Tb across the top and bottom boundary layers, respectively. Assuming that within the thermal BL heat is mainly transported by conduction, then the heat flux passing through the top BL is Qt = knf

⌬t

␦t

共3兲

,

and that across the bottom BL is Qb = knf

⌬b

␦b

,

共4兲

where knf is the bulk-temperature-based thermal conductivity of the nanofluid as determined in the Appendix. In the Appendix, it is also shown that the knf has a relatively weak dependence on temperature 关Fig. 11共b兲 shows that the varia-


022005-9



convective flow of the bulk fluid. Finally, we remark that as the density of alumina nanoparticles is larger than that of pure water, one may suspect that sedimentation has occurred at lower flow strength of the convecting fluid 共which corresponds to lower heat flux density兲. However, this argument is not supported by the fact that the top and bottom boundary layers both become thicker. Further work is certainly needed to understand this phenomenon.

IV. CONCLUSION

FIG. 11. 共a兲 Thermal conductivity, knf / k f , as a function of volume fraction, ␾, at different temperatures in alumina nanofluid 共from Ref. 51兲: T = 21 ° C 共open circles兲, 36 ° C 共solid circles兲, and 51 ° C 共open squares兲. The fitting formula Eq. 共A5兲共in the Appendix兲 is plotted as solid lines for the three temperatures, respectively. 共b兲 Circles: Experiment data for a nanofluid with ␾ = 1% and a particle size of 47 nm from Ref. 48. The solid line is according to Eq. 共A5兲 with ␾ = 1%.

tion of knf over that of the base fluid is less than 10% over a temperature range of 50 °C兴. When the sidewall heat leakage is negligible, Qt = Qb, and in the present case, we take this to be the input heat flux, i.e., Qt = Qb = Q, as the heat leakage of the whole system is negligibly small. Figures 10共a兲 and 10共b兲 show the thermal BL thickness ␦t and ␦b as functions of the input heat flux Q. It should be mentioned that the thermal conductivity used here for the estimation of the thermal boundary layer is based on the mean temperature of the bulk fluid, which may be different from that in the thermal boundary layer where a steep temperature gradient exists. Thus, the estimations made here should not be used to substitute the real thermal boundary layer measurement, which should certainly be the subject of future studies. First, we look at the top thermal boundary. As shown in Fig. 10共a兲, ␦t of the nanofluid is almost the same as 共or slightly thicker than兲 that of the pure water at larger heat flux density, Q ⲏ 2000 W / m2. But for smaller values of Q, which would correspond to those values of Ra smaller than Rac ␦t of nanofluid departs from that of pure water and increases much faster than that of water. At Q ⬇ 400 W / m2, ␦t is more than two times thicker than that of pure water. A similar trend is also seen at the bottom BL. A notable difference is the datum point corresponding to Q ⬇ 400 W / m2, where ␦b is only about 1.5 times thicker than that of pure water. The behavior of the thermal BL is consistent with the measured properties of the Nusselt number and shows that the decrease in the Nu of the nanofluid is associated with the thickening of the thermal BL rather than a change 共or slow-down兲 of

In this work, precision heat transfer and flow dynamics measurements of the nanofluid 共alumina particles in water兲 have been performed in turbulent natural convection in a Rayleigh–Bénard configuration. The effects of the alumina nanoparticles have been investigated under the conditions of varying volume fraction of nanoparticles at fixed input heat flux 共fixed driving strength兲 and of varying input heat flux at fixed nanoparticle volume fraction. With the fixed temperature 共Tt = 23.17 ° C兲 at the top plate and fixed heating power 共330 W兲 through the bottom plate, it has been found that the convective heat transfer coefficient, h, Nusselt number, Nu, and Rayleigh number, Ra, all decrease with the increasing volume fraction ␾ from 0% to 1.08%. These results suggest that the convective heat transfer of the fluid is deteriorated when nanoparticles are suspended in the fluid. On the other hand, the convective flow of the fluid is not slowed down by the addition of nanoparticles. But as the viscosity of nanofluid increases with the large volume fraction ␾, the Reynolds number, Re, becomes smaller. At a constant nanofluid volume fraction 共␾ = 1.08%兲, the heat transport and flow properties have been measured over a broad range of Ra. For heat transport, a transition near Rac ⯝ 2.5⫻ 109 has been found. At Ra ⬎ Rac, the measured Nu of the nanofluid is roughly the same as that of water both in terms of its magnitude and its scaling relation with Ra, which suggests that the nanofluid can be treated as single phase fluid in this parameter range. For Ra ⬍ Rac, Nu becomes smaller than that of pure water and the deviation becomes larger with decreasing Ra. In the parameter range of Ra ⬍ Rac, the measured instantaneous Nu共t兲 shows strong and quasiperiodic fluctuations, which is absent when Ra ⬎ Rac. This suggests that the significant decrease of the nanofluid Nu comparing to that of water may be caused by the mass diffusion of nanoparticles. Furthermore, measurements of the flow velocity of the bulk nanofluid showed no significant difference from that of water for Ra either above or below Rac. From the measured temperature drop across the top and bottom plates, we determined the thermal boundary layer thickness indirectly. The results showed that the deviations of the nanofluid Nu from that of water for Ra ⬍ Rac corresponds to the thickening of the thermal boundary layer at both the top and bottom plates. We emphasize that in addition to the thickening of the thermal boundary layer, the behavior of the measured Nusselt number of the nanofluid can be caused by a number of effects, such as the flow structures and dynamics, plumes, and boundary


022005-10


Ni, Zhou, and Xia

layer structures. The present work is only a first step toward the understanding of the convective flow properties of nanofluid in highly turbulent state. ACKNOWLEDGMENTS

We thank X. Z. Zhao for assistance in the experiment and helpful discussions. This work was supported by a GRF Grant No. CUHK404808 from the Hong Kong Research Grants Council, a China NSF Grant No. 11072253 and Guang Dong NSF Grant No. 10251030101000000. APPENDIX: PHYSICAL PROPERTIES OF THE NANOFLUID

According to the definitions of Rayleigh number and Prandtl number, the convective heat transport of a certain fluid not only depends on its thermal conductivity but also on its heat capacity, density, and dynamic viscosity. The combined effect of these properties as well as the flow dynamics will determine the overall thermal performance of the convecting fluid. Thus, these physical properties should be determined before the measurement. Fortunately, the waterbased alumina suspension is one of the most used nanofluids in experiments and numerical simulations,12,21,24,31–34,47–49 which provide benchmarking references for our study. It is widely accepted that the density of a mixture of nanoparticles and base fluid can be expressed as

␳nf = 共1 − ␾兲␳ f + ␾␳n .

共A1兲

Here, ␳ is the density, ␾ is the volume fraction of nanoparticles in the suspension, and the subscripts nf, f, and n refer to nanofluid, base fluid, and nanoparticles, respectively. This formula has been confirmed with experiment.50 Specific heat is another important parameter. Generally, two formulas have been applied in the experimental and numerical nanofluid investigations.12,19,50–63 They are c p,nf = ␾c p,n + 共1 − ␾兲c p,f ,

共A2兲

and c p,nf =

␾共␳c p兲n + 共1 − ␾兲共␳c p兲f . ␾␳n + 共1 − ␾兲␳ f 64

共A3兲

The recent work by two of the present authors shows that the specific heat of a nanofluid can well be described by Eq. 共A3兲. The prediction of thermal conductivity k of a nanofluid has been reported in many sources 共see, for example, Table V of Ref. 65兲. Nevertheless, the discrepancy among the various results reported in the literature was so great that it limited their general applications. In the present study, we determine the thermal conductivity of our nanofluid based on data from an experimental study of water-based alumina nanofluid with a particle size nominally close to ours,51 where the nanofluid was 38.4 nm sized alumina nanoparticles dispersed in water. However, Ref. 51 does not have the thermal conductivity data for all the temperatures and concentrations used in our study. To our knowledge, there exists no theoretical or empirical formulas for the temperature de-

FIG. 12. Normalized nanofluid viscosity, ␮nf / ␮ f , as a function of the nanoparticle volume fraction. Data were obtained in alumina nanofluid from various experimental sources. Solid diamonds: Present study 共T = 25 ° C兲. Open squares: Ref. 66 共temperature unknown兲. Solid squares: Ref. 67 共temperature unknown兲. Solid circles: Ref. 21, T = 20 ° C. Open circles: Ref. 21, T = 40 ° C. Solid line is according to Eq. 共A6兲.

pendency of the nanofluid thermal conductivity. For the volume fraction dependency of k, there is an empirical formula, knf = k f 关1 + A␾ + B␾2兴,

共A4兲

in Ref. 54, A = 2.72 and B = 4.97. To find an empirical relationship between knf and ␾ and the temperature T, here we assume that the coefficients A and B both have a linear dependence on the temperature, i,e, A = c1 + c2T and B = c3 + c4T. To determine the coefficients ci, we substitute A共T兲 and B共T兲 into Eq. 共A4兲 and fitted the equation to three sets of data for different temperatures, each set for various values of ␾, from Ref. 51. The results are c1 = −49.796, c2 = 0.178, c3 = 535.576, and c4 = −1.840. 共knf / k f = 1 at ␾ = 0% has been used in the fitting.兲 As a result, the thermal conductivity of the present nanofluid can be written as knf = k f 关1 + 共− 49.796 + 0.178T兲␾ + 共535.576 − 1.840T兲␾2兴,

共A5兲

where T is in Kelvin. Shown in Fig. 11共a兲 are the three sets of data from Ref. 51 together with the fitting curves according to Eq. 共A5兲. To see whether the above result is reasonable, we compare Eq. 共A5兲 with results from other experiments of alumina-water nanofluids over comparable ranges of temperature and concentration but different particle size. In Fig. 11共b兲, we plot the experimental data 共circles兲 from Ref. 48 in which thermal conductivity was measured for various temperatures in an alumina-water nanofluid of ␾ = 1% with nanoparticle size of 47 nm. The solid line is according to Eq. 共A5兲 for ␾ = 1%. It is seen that there is an excellent agreement between the empirical result and this independent set of data. We also checked the applicability of Eq. 共A5兲 against other independent data sets,34,47 and found good agreement. Finally, we note that at room temperature and for sufficiently small ␾, Eq. 共A5兲 can be approximately


022005-11

reduced to the theoretical result from the effective medium theory, knf / k f = 1 + 3␾.34 Experimental data of the nanofluid viscosity ␮nf are relatively scarce. Maiga et al.54 obtained an empirical formula,

␮nf = ␮ f 共1 + 7.2␾ + 123␾2兲,

共A6兲

based on the experimental data of Ref. 66, where the measurement was made at room temperature using nanofluid of alumina particles with an average diameter of 20 nm. Both the data from Ref. 66 and the Eq. 共A6兲 are shown in Fig. 12. We also made our own viscosity measurements for two samples of respective volume fraction ␾ = 1.25% and 5.0% 共at T = 25 ° C兲 with a stress-controlled rheometer 共Model ARG2, TA Instruments, Delaware, USA兲. As shown in Fig. 12, the measured viscosity data are in excellent agreement with the empirical formula Eq. 共A6兲. In this figure, we also plot experimental data from two other sources,21,67 where the particle size was 131.2 nm. 共In the figure caption where we labeled data with “temperature unknown,” the measurements were probably made at room temperature.兲 From the figure, we can see that the measured viscosity of the nanofluid does not seem to have a strong temperature dependency and, given the data scatter, the empirical formula can give a reasonably good representation of most of the data point. 1



S. U. S. Choi, “Enhancing thermal conductivity of fluids with nanoparticles,” in Developments and Applications of Non-Newtonian Flows, edited by D. A. Siginer and H. P. Wang 共ASME, New York, 1995兲, Vol. 66, pp. 99–105. 2 J. A. Eastman, S. R. Phillpot, S. U. S. Choi, and P. Keblinski, “Thermal transport in nanofluids,” Annu. Rev. Mater. Res. 34, 219 共2004兲. 3 P. Keblinski, J. Eastman, and D. Cahill, “Nanofluids for thermal transport,” Mater. Today 8, 36 共2005兲. 4 J. C. Maxwell, A Treatise on Electricity and Magnetism 共Clarendon, Oxford, 1881兲, 2nd ed., Vol. 1, p. 435. 5 J. A. Eastman, S. U. S. Choi, S. Li, W. Yu, and L. J. Thompson, “Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles,” Appl. Phys. Lett. 78, 718 共2001兲. 6 S. U. S. Choi, Z. G. Zhang, W. Yu, F. E. Lockwood, and E. A. Grulke, “Anomalous thermal conductivity enhancement in nanotube suspensions,” Appl. Phys. Lett. 79, 2252 共2001兲. 7 S. Shaikh, K. Lafdi, and R. Ponnappon, “Thermal conductivity improvement in carbon nanoparticle doped PAO oil: An experimental study,” J. Appl. Phys. 101, 064302 共2007兲. 8 V. Trisaksri and S. Wongwises, “Critical review of heat transfer characteristics of nanofluids,” Renewable Sustainable Energy Rev. 11, 512 共2007兲. 9 M. Li, Y.-C. Lin, C.-C. Wu, and H.-S. Liu, “Enhancing the efficiency of a PCR using gold nanoparticles,” Nucleic Acids Res. 33, e184 共2005兲. 10 D. P. ÓNeal, L. R. Hirsch, N. J. Halas, J. D. Payne, and J. L. West, “Photo-thermal tumor ablation in mice using near infrared-absorbing nanoparticles,” Cancer Lett. 209, 171 共2004兲. 11 S. K. Das, S. U. S. Choi, and H. E. Patel, “Heat transfer in nanofluids—A review,” Heat Transfer Eng. 27, 3 共2006兲. 12 K. Khanafer, K. Vafai, and M. Lightstone, “Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids,” Int. J. Heat Mass Transfer 46, 3639 共2003兲. 13 R. Y. Jou and S. C. Tzeng, “Numerical research of nature convective heat transfer enhancement filled with nanofluids in rectangular enclosures,” Int. Commun. Heat Mass Transfer 33, 727 共2006兲. 14 E. Abu-Nada, “Effects of variable viscosity and thermal conductivity of Al2O3-water nanofluid on heat transfer enhancement in natural convection,” Int. J. Heat Mass Transfer 30, 679 共2009兲. 15 M. K. Das and P. S. Ohal, “Natural convection heat transfer augmentation in a partially heated and partially cooled square cavity utilizing nanofluids,” Int. J. Numer. Methods Heat Fluid Flow 19, 411 共2009兲. 16 B. Ghasemi and S. M. Aminossadati, “Natural convection heat transfer in an inclined enclosure filled with a water-CuO nanofluid,” Numer. Heat

Transfer, Part A 55, 807 共2009兲. S. P. Jang and S. U. Choi, “Free convection in a rectangular cavity 共Bénard convection兲 with nanofluids,” Proceedings of the 2004 ASME International Mechanical Engineering Congress and Exposition, Anaheim, California, 2004, Vol. 196, p. 147. 18 E. B. Öğüt, “Natural convection of water-based nanofluids in an inclined enclosure with a heat source,” Int. J. Therm. Sci. 48, 2063 共2009兲. 19 J. Kim, Y. T. Kang, and C. Choi, “Analysis of convective instability and heat transfer characteristics of nanofluids,” Phys. Fluids 16, 2395 共2004兲. 20 D. Y. Tzou, “Thermal instability of nanofluids in natural convection,” Int. J. Heat Mass Transfer 51, 2967 共2008兲. 21 N. Putra, W. Roetzel, and S. K. Das, “Natural convection of nano-fluids,” Heat Mass Transfer 39, 775 共2003兲. 22 D. S. Wen and Y. L. Ding, “Formulation of nanofluids for natural convective heat transfer applications,” Int. J. Heat Fluid Flow 26, 855 共2005兲. 23 D. S. Wen and Y. L. Ding, “Natural convective heat transfer of suspensions of titanium dioxide nanoparticles 共nanofluids兲,” IEEE Trans. Nanotechnol. 5, 220 共2006兲. 24 A. G. Agwu Nnanna, “Experimental model of temperature-driven nanofluid,” J. Heat Transfer 129, 697 共2007兲. 25 A. V. G. Donzelli and R. Cerbino, “Bistable heat transfer in a nanofluid,” Phys. Rev. Lett. 102, 104503 共2009兲. 26 E. D. Siggia, “High Rayleigh number convection,” Annu. Rev. Fluid Mech. 26, 137 共1994兲. 27 W. V. R. Malkus, “The heat transport and spectrum of thermal turbulence,” Proc. R. Soc. London, Ser. A 225, 196 共1954兲. 28 L. N. Howard, “Convection at high Rayleigh number,” Proceedings of the 11th Congress on Applied Mechanics, edited by H. Grtler, 1964, p. 1109. 29 S.-Q. Zhou and K.-Q. Xia, “Spatially correlated temperature fluctuations in turbulent convection,” Phys. Rev. E 63, 046308 共2001兲. 30 Private communication with the Nanophase Technologies, Inc. 31 C. H. Li and G. P. Petersona, “Experimental investigation of temperature and volume fraction variations on the effective thermal conductivity of nanoparticle suspensions 共nanoflu-ids兲,” J. Appl. Phys. 99, 084314 共2006兲. 32 D. S. Wen and Y. L. Ding, “Experimental investigation into convective heat transfer of nanofluids at the entrance region under laminar flow conditions,” Int. J. Heat Mass Transfer 47, 5181 共2004兲. 33 C. T. Nguyen, G. Roy, C. Gauthier, and N. Galanis, “Heat transfer enhancement using Al2O3-water nanofluid for an electronic liquid cooling system,” Appl. Therm. Eng. 27, 1501 共2007兲. 34 E. V. Timofeeva, A. N. Gavrilov, J. M. McCloskey, Y. V. Tolmachev, S. Sprunt, L. M. Lopatina, and J. V. Selinger, “Thermal conductivity and particle agglomeration in alumina nanofluids: Experiment and theory,” Phys. Rev. E 76, 061203 共2007兲. 35 J. Boussinesq, Theorie Analytique de la Chaleur 共Gauthier-Villars, Paris, 1903兲, Vol. 2. 36 X.-Z. Wu and A. Libchaber, “Scaling relations in thermal turbulence: The aspect-ratio dependence,” Phys. Rev. A 45, 842 共1992兲. 37 C. Sun, L.-Y. Ren, H. Song, and K.-Q. Xia, “Heat transport by turbulent Rayleigh-Bénard convection in 1 m diameter cylindrical cells of widely varying aspect ratio,” J. Fluid Mech. 542, 165 共2005兲. 38 A. Nikolaenko, E. Brown, D. Funfschilling, and G. Ahlers, “Heat transport by turbulent Rayleigh-Bénard convection in cylindrical cells with aspect ratio one and less,” J. Fluid Mech. 523, 251 共2005兲. 39 A. K. Santra, S. Sen, and N. Chakraborty, “Study of heat transfer due to laminar flow of copper-water nanofluid through two isothermally heated parallel plates,” Int. J. Therm. Sci. 48, 391 共2009兲. 40 S.-Q. Zhou, C. Sun, and K.-Q. Xia, “Measured oscillations of the velocity and temperature fields in turbulent Rayleigh-Bénard convection in a rectangular cell,” Phys. Rev. E 76, 036301 共2007兲. 41 M. Sano, X.-Z. Wu, and A. Libchaber, “Turbulence in helium-gas freeconvection,” Phys. Rev. A 40, 6421 共1989兲. 42 C. Sun and K.-Q. Xia, “Scaling of the Reynolds number in turbulent thermal convection,” Phys. Rev. E 72, 067302 共2005兲. 43 M. A. Al-Subaie and A. J. Chamkha, “Steady natural convection flow of a particulate suspension through a circular pipe,” Heat Mass Transfer 40, 673 共2004兲. 44 G. Ahlers, E. Brown, F. F. Araujo, D. Funfschilling, S. Grossmann, and D. Lohse, “Non-Oberbeck-Boussinesq effects in strongly turbulent RayleighBénard convection,” J. Fluid Mech. 569, 409 共2006兲. 45 K.-Q. Xia, S. Lam, and S.-Q. Zhou, “Heat-flux measurements in highPrandtl-number Rayleigh-Bénard convection,” Phys. Rev. Lett. 88, 064501 共2002兲. 17


022005-12 46

Ni, Zhou, and Xia

M. Okada and T. Suzuki, “Natural convection of water-fine particle suspension in a rectangular cell,” Int. J. Heat Mass Transfer 40, 3201 共1997兲. 47 S. Lee, S. Choi, S. Li, and J. Eastman, “Measuring thermal conductivity of fluids containing oxide nanoparticles,” J. Heat Transfer 121, 280 共1999兲. 48 C. H. Chon, K. D. Kihm, S. P. Lee, and S. U. S. Choi, “Empirical correlation finding the role of temperature and particle size for nanofluid 共Al2O3兲 thermal conductivity enhancement,” Appl. Phys. Lett. 87, 153107 共2005兲. 49 Y.-S. Chang, “An experimental study of natural convection heat transfer of nanofluids in a square cavity,” M.S. thesis, National Cheng Kung University, Taiwan, 2006. 50 B. C. Pak and Y. I. Cho, “Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles,” Exp. Heat Transfer 11, 151 共1998兲. 51 S. K. Das, N. Putra, P. Thiesen, and W. Roetzel, “Temperature dependence of thermal conductivity enhancement for nanofluids,” J. Heat Transfer 125, 567 共2003兲. 52 L. Gosselin and A. K. da Silva, “Combined ‘heat transfer and power dissipation’ optimization of nanofluid flows,” Appl. Phys. Lett. 85, 4160 共2004兲. 53 J. Lee and I. Mudawar, “Assessment of the effectiveness of nanofluids for single-phase and two-phase heat transfer in micro-channels,” Int. J. Heat Mass Transfer 50, 452 共2007兲. 54 S. E. B. Maïga, C. T. Nguyen, N. Galanis, and G. Roy, “Heat transfer behaviours of nanofluids in a uniformly heated tube,” Superlattices Microstruct. 35, 543 共2004兲. 55 S. E. B. Maïga, S. J. Palm, C. T. Nguyen, G. Roy, and N. Galanis, “Heat transfer behaviours of nanofluids in a uniformly heated tube,” Int. J. Heat Fluid Flow 26, 530 共2005兲. 56 S. E. B. Maïga, C. T. Nguyen, N. Galanis, G. Roy, T. Maré, and M. Coqueux, “Heat transfer enhancement in turbulent tube flow using Al2O3

Phys. Fluids 23, 022005 共2011兲 nanoparticle suspension,” Int. J. Numer. Methods Heat Fluid Flow 16, 275 共2006兲. 57 S. J. Palm, G. Roy, and C. T. Nguyen, “Heat transfer enhancement with the use of nanofluids in radial flow cooling systems considering temperature-dependent properties,” Appl. Therm. Eng. 26, 2209 共2006兲. 58 G. Polidori, S. Fohanno, and C. T. Nguyen, “A note on heat transfer modelling of newtonian nanofluids in laminar free convection,” Int. J. Therm. Sci. 46, 739 共2007兲. 59 K. S. Hwang, J. H. Lee, and S. P. Jang, “Buoyancy-driven heat transfer of water-based Al2O3 nanofluids in a rectangular cavity,” Int. J. Heat Mass Transfer 50, 4003 共2007兲. 60 J. Buongiorno, “Convective transport in nanofluids,” J. Heat Transfer 128, 240 共2006兲. 61 J. Avsec and M. Oblak, “The calculation of thermal conductivity, viscosity and thermodynamic properties for nanofluids on the basis of statistical nanomechanics,” Int. J. Heat Mass Transfer 50, 4331 共2007兲. 62 Y. M. Xuan and W. Roetzel, “Conceptions for heat transfer correlation of nanofluids,” Int. J. Heat Mass Transfer 43, 3701 共2000兲. 63 J. Kim, Y. T. Kang, and C. K. Choi, “Soret and dufour effects on convective instabilities in binary nanofluids for absorption application,” Int. J. Refrig. 30, 323 共2007兲. 64 S.-Q. Zhou and R. Ni, “Measurement of the specific heat capacity in water-based Al2O3 nanofluid,” Appl. Phys. Lett. 92, 093123 共2008兲. 65 X.-Q. Wang and A. S. Mujumdar, “Heat transfer characteristics of nanofluids: A review,” Int. J. Therm. Sci. 46, 1 共2007兲. 66 X. Wang, X. Xu, and S. U. S. Choi, “Thermal conductivity of nanoparticle fluid mixture,” J. Thermophys. Heat Transfer 13, 474 共1999兲. 67 J. H. Lee and S. P. Jang, “Fluid flow characteristics of Al2O3 nanoparticles suspended in water,” Transactions of SAREK Winter Annual Conference, 2005, pp. 546–551.
