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
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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
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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.
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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.
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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
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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.
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